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Index notation for tensors
The cross product can alternatively be defined in terms of the Levi-Civita tensor Eijk and a dot product ηmi, which are useful in converting vector notation for tensor applications:
where the indices correspond to vector components. This characterization of the cross product is often expressed more compactly using the Einstein summation convention as
in which repeated indices are summed over the values 1 to 3.
In a positively-oriented orthonormal basis ηmi = δmi (the Kronecker delta) and (the Levi-Civita symbol). In that case, this representation is another form of the skew-symmetric representation of the cross product:
In classical mechanics: representing the cross product by using the Levi-Civita symbol can cause mechanical symmetries to be obvious when physical systems are isotropic. (An example: consider a particle in a Hooke's Law potential in three-space, free to oscillate in three dimensions; none of these dimensions are "special" in any sense, so symmetries lie in the cross-product-represented angular momentum, which are made clear by the abovementioned Levi-Civita representation).
Mnemonic
The word "xyzzy" can be used to remember the definition of the cross product.
If
where:
then:
The second and third equations can be obtained from the first by simply vertically rotating the subscripts, . The problem, of course, is how to remember the first equation, and two options are available for this purpose: either to remember the relevant two diagonals of Sarrus's scheme (those containing i), or to remember the xyzzy sequence.
Since the first diagonal in Sarrus's scheme is just the main diagonal of the above-mentioned 3×3 matrix, the first three letters of the word xyzzy can be very easily remembered.
Cross visualization
Similarly to the mnemonic device above, a "cross" or X can be visualized between the two vectors in the equation. This may be helpful for remembering the correct cross product formula.
If
then:
If we want to obtain the formula for we simply drop the and from the formula, and take the next two components down: | Cross product | Wikipedia | 454 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
When doing this for the next two elements down should "wrap around" the matrix so that after the z component comes the x component. For clarity, when performing this operation for , the next two components should be z and x (in that order). While for the next two components should be taken as x and y.
For then, if we visualize the cross operator as pointing from an element on the left to an element on the right, we can take the first element on the left and simply multiply by the element that the cross points to in the right-hand matrix. We then subtract the next element down on the left, multiplied by the element that the cross points to here as well. This results in our formula –
We can do this in the same way for and to construct their associated formulas.
Applications
The cross product has applications in various contexts. For example, it is used in computational geometry, physics and engineering.
A non-exhaustive list of examples follows.
Computational geometry
The cross product appears in the calculation of the distance of two skew lines (lines not in the same plane) from each other in three-dimensional space.
The cross product can be used to calculate the normal for a triangle or polygon, an operation frequently performed in computer graphics. For example, the winding of a polygon (clockwise or anticlockwise) about a point within the polygon can be calculated by triangulating the polygon (like spoking a wheel) and summing the angles (between the spokes) using the cross product to keep track of the sign of each angle.
In computational geometry of the plane, the cross product is used to determine the sign of the acute angle defined by three points and . It corresponds to the direction (upward or downward) of the cross product of the two coplanar vectors defined by the two pairs of points and . The sign of the acute angle is the sign of the expression
which is the signed length of the cross product of the two vectors.
To use the cross product, simply extend the 2D vectors to co-planar 3D vectors by setting for each of them.
In the "right-handed" coordinate system, if the result is 0, the points are collinear; if it is positive, the three points constitute a positive angle of rotation around from to , otherwise a negative angle. From another point of view, the sign of tells whether lies to the left or to the right of line | Cross product | Wikipedia | 499 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
The cross product is used in calculating the volume of a polyhedron such as a tetrahedron or parallelepiped.
Angular momentum and torque
The angular momentum of a particle about a given origin is defined as:
where is the position vector of the particle relative to the origin, is the linear momentum of the particle.
In the same way, the moment of a force applied at point B around point A is given as:
In mechanics the moment of a force is also called torque and written as
Since position linear momentum and force are all true vectors, both the angular momentum and the moment of a force are pseudovectors or axial vectors.
Rigid body
The cross product frequently appears in the description of rigid motions. Two points P and Q on a rigid body can be related by:
where is the point's position, is its velocity and is the body's angular velocity.
Since position and velocity are true vectors, the angular velocity is a pseudovector or axial vector.
Lorentz force
The cross product is used to describe the Lorentz force experienced by a moving electric charge
Since velocity force and electric field are all true vectors, the magnetic field is a pseudovector.
Other
In vector calculus, the cross product is used to define the formula for the vector operator curl.
The trick of rewriting a cross product in terms of a matrix multiplication appears frequently in epipolar and multi-view geometry, in particular when deriving matching constraints.
As an external product
The cross product can be defined in terms of the exterior product. It can be generalized to an external product in other than three dimensions. This generalization allows a natural geometric interpretation of the cross product. In exterior algebra the exterior product of two vectors is a bivector. A bivector is an oriented plane element, in much the same way that a vector is an oriented line element. Given two vectors a and b, one can view the bivector as the oriented parallelogram spanned by a and b. The cross product is then obtained by taking the Hodge star of the bivector , mapping 2-vectors to vectors: | Cross product | Wikipedia | 423 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
This can be thought of as the oriented multi-dimensional element "perpendicular" to the bivector. In a d-dimensional space, Hodge star takes a k-vector to a (d–k)-vector; thus only in d = 3 dimensions is the result an element of dimension one (3–2 = 1), i.e. a vector. For example, in d = 4 dimensions, the cross product of two vectors has dimension 4–2 = 2, giving a bivector. Thus, only in three dimensions does cross product define an algebra structure to multiply vectors.
Handedness
Consistency
When physics laws are written as equations, it is possible to make an arbitrary choice of the coordinate system, including handedness. One should be careful to never write down an equation where the two sides do not behave equally under all transformations that need to be considered. For example, if one side of the equation is a cross product of two polar vectors, one must take into account that the result is an axial vector. Therefore, for consistency, the other side must also be an axial vector. More generally, the result of a cross product may be either a polar vector or an axial vector, depending on the type of its operands (polar vectors or axial vectors). Namely, polar vectors and axial vectors are interrelated in the following ways under application of the cross product:
polar vector × polar vector = axial vector
axial vector × axial vector = axial vector
polar vector × axial vector = polar vector
axial vector × polar vector = polar vector
or symbolically
polar × polar = axial
axial × axial = axial
polar × axial = polar
axial × polar = polar
Because the cross product may also be a polar vector, it may not change direction with a mirror image transformation. This happens, according to the above relationships, if one of the operands is a polar vector and the other one is an axial vector (e.g., the cross product of two polar vectors). For instance, a vector triple product involving three polar vectors is a polar vector.
A handedness-free approach is possible using exterior algebra. | Cross product | Wikipedia | 428 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
The paradox of the orthonormal basis
Let (i, j, k) be an orthonormal basis. The vectors i, j and k do not depend on the orientation of the space. They can even be defined in the absence of any orientation. They can not therefore be axial vectors. But if i and j are polar vectors, then k is an axial vector for i × j = k or j × i = k. This is a paradox.
"Axial" and "polar" are physical qualifiers for physical vectors; that is, vectors which represent physical quantities such as the velocity or the magnetic field. The vectors i, j and k are mathematical vectors, neither axial nor polar. In mathematics, the cross-product of two vectors is a vector. There is no contradiction.
Generalizations
There are several ways to generalize the cross product to higher dimensions.
Lie algebra
The cross product can be seen as one of the simplest Lie products, and is thus generalized by Lie algebras, which are axiomatized as binary products satisfying the axioms of multilinearity, skew-symmetry, and the Jacobi identity. Many Lie algebras exist, and their study is a major field of mathematics, called Lie theory.
For example, the Heisenberg algebra gives another Lie algebra structure on In the basis the product is
Quaternions
The cross product can also be described in terms of quaternions.
In general, if a vector is represented as the quaternion , the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. The real part will be the negative of the dot product of the two vectors.
Octonions
A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence of nontrivial vector-valued cross products of two vectors in other dimensions is related to the result from Hurwitz's theorem that the only normed division algebras are the ones with dimension 1, 2, 4, and 8.
Exterior product | Cross product | Wikipedia | 441 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
In general dimension, there is no direct analogue of the binary cross product that yields specifically a vector. There is however the exterior product, which has similar properties, except that the exterior product of two vectors is now a 2-vector instead of an ordinary vector. As mentioned above, the cross product can be interpreted as the exterior product in three dimensions by using the Hodge star operator to map 2-vectors to vectors. The Hodge dual of the exterior product yields an -vector, which is a natural generalization of the cross product in any number of dimensions.
The exterior product and dot product can be combined (through summation) to form the geometric product in geometric algebra.
External product
As mentioned above, the cross product can be interpreted in three dimensions as the Hodge dual of the exterior product. In any finite n dimensions, the Hodge dual of the exterior product of vectors is a vector. So, instead of a binary operation, in arbitrary finite dimensions, the cross product is generalized as the Hodge dual of the exterior product of some given vectors. This generalization is called external product.
Commutator product
Interpreting the three-dimensional vector space of the algebra as the 2-vector (not the 1-vector) subalgebra of the three-dimensional geometric algebra, where , , and , the cross product corresponds exactly to the commutator product in geometric algebra and both use the same symbol . The commutator product is defined for 2-vectors and in geometric algebra as:
where is the geometric product. | Cross product | Wikipedia | 303 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
The commutator product could be generalised to arbitrary multivectors in three dimensions, which results in a multivector consisting of only elements of grades 1 (1-vectors/true vectors) and 2 (2-vectors/pseudovectors). While the commutator product of two 1-vectors is indeed the same as the exterior product and yields a 2-vector, the commutator of a 1-vector and a 2-vector yields a true vector, corresponding instead to the left and right contractions in geometric algebra. The commutator product of two 2-vectors has no corresponding equivalent product, which is why the commutator product is defined in the first place for 2-vectors. Furthermore, the commutator triple product of three 2-vectors is the same as the vector triple product of the same three pseudovectors in vector algebra. However, the commutator triple product of three 1-vectors in geometric algebra is instead the negative of the vector triple product of the same three true vectors in vector algebra.
Generalizations to higher dimensions is provided by the same commutator product of 2-vectors in higher-dimensional geometric algebras, but the 2-vectors are no longer pseudovectors. Just as the commutator product/cross product of 2-vectors in three dimensions correspond to the simplest Lie algebra, the 2-vector subalgebras of higher dimensional geometric algebra equipped with the commutator product also correspond to the Lie algebras. Also as in three dimensions, the commutator product could be further generalised to arbitrary multivectors.
Multilinear algebra
In the context of multilinear algebra, the cross product can be seen as the (1,2)-tensor (a mixed tensor, specifically a bilinear map) obtained from the 3-dimensional volume form, a (0,3)-tensor, by raising an index. | Cross product | Wikipedia | 388 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
In detail, the 3-dimensional volume form defines a product by taking the determinant of the matrix given by these 3 vectors. By duality, this is equivalent to a function (fixing any two inputs gives a function by evaluating on the third input) and in the presence of an inner product (such as the dot product; more generally, a non-degenerate bilinear form), we have an isomorphism and thus this yields a map which is the cross product: a (0,3)-tensor (3 vector inputs, scalar output) has been transformed into a (1,2)-tensor (2 vector inputs, 1 vector output) by "raising an index".
Translating the above algebra into geometry, the function "volume of the parallelepiped defined by " (where the first two vectors are fixed and the last is an input), which defines a function , can be represented uniquely as the dot product with a vector: this vector is the cross product From this perspective, the cross product is defined by the scalar triple product,
In the same way, in higher dimensions one may define generalized cross products by raising indices of the n-dimensional volume form, which is a -tensor.
The most direct generalizations of the cross product are to define either:
a -tensor, which takes as input vectors, and gives as output 1 vector – an -ary vector-valued product, or
a -tensor, which takes as input 2 vectors and gives as output skew-symmetric tensor of rank – a binary product with rank tensor values. One can also define -tensors for other k.
These products are all multilinear and skew-symmetric, and can be defined in terms of the determinant and parity.
The -ary product can be described as follows: given vectors in define their generalized cross product as:
perpendicular to the hyperplane defined by the
magnitude is the volume of the parallelotope defined by the which can be computed as the Gram determinant of the
oriented so that is positively oriented.
This is the unique multilinear, alternating product which evaluates to , and so forth for cyclic permutations of indices.
In coordinates, one can give a formula for this -ary analogue of the cross product in Rn by: | Cross product | Wikipedia | 468 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
This formula is identical in structure to the determinant formula for the normal cross product in R3 except that the row of basis vectors is the last row in the determinant rather than the first. The reason for this is to ensure that the ordered vectors (v1, ..., vn−1, Λvi) have a positive orientation with respect to (e1, ..., en). If n is odd, this modification leaves the value unchanged, so this convention agrees with the normal definition of the binary product. In the case that n is even, however, the distinction must be kept. This -ary form enjoys many of the same properties as the vector cross product: it is alternating and linear in its arguments, it is perpendicular to each argument, and its magnitude gives the hypervolume of the region bounded by the arguments. And just like the vector cross product, it can be defined in a coordinate independent way as the Hodge dual of the wedge product of the arguments. Moreover, the product satisfies the Filippov identity,
and so it endows Rn+1 with a structure of n-Lie algebra (see Proposition 1 of ).
History
In 1773, Joseph-Louis Lagrange used the component form of both the dot and cross products in order to study the tetrahedron in three dimensions.
In 1843, William Rowan Hamilton introduced the quaternion product, and with it the terms vector and scalar. Given two quaternions and , where u and v are vectors in R3, their quaternion product can be summarized as . James Clerk Maxwell used Hamilton's quaternion tools to develop his famous electromagnetism equations, and for this and other reasons quaternions for a time were an essential part of physics education.
In 1844, Hermann Grassmann published a geometric algebra not tied to dimension two or three. Grassmann developed several products, including a cross product represented then by . ( | Cross product | Wikipedia | 402 | 157092 | https://en.wikipedia.org/wiki/Cross%20product | Mathematics | Algebra | null |
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product (or rarely the projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see Inner product space for more).
Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euclidean magnitudes of the two vectors and the cosine of the angle between them. These definitions are equivalent when using Cartesian coordinates. In modern geometry, Euclidean spaces are often defined by using vector spaces. In this case, the dot product is used for defining lengths (the length of a vector is the square root of the dot product of the vector by itself) and angles (the cosine of the angle between two vectors is the quotient of their dot product by the product of their lengths).
The name "dot product" is derived from the dot operator " · " that is often used to designate this operation; the alternative name "scalar product" emphasizes that the result is a scalar, rather than a vector (as with the vector product in three-dimensional space).
Definition
The dot product may be defined algebraically or geometrically. The geometric definition is based on the notions of angle and distance (magnitude) of vectors. The equivalence of these two definitions relies on having a Cartesian coordinate system for Euclidean space.
In modern presentations of Euclidean geometry, the points of space are defined in terms of their Cartesian coordinates, and Euclidean space itself is commonly identified with the real coordinate space . In such a presentation, the notions of length and angle are defined by means of the dot product. The length of a vector is defined as the square root of the dot product of the vector by itself, and the cosine of the (non oriented) angle between two vectors of length one is defined as their dot product. So the equivalence of the two definitions of the dot product is a part of the equivalence of the classical and the modern formulations of Euclidean geometry.
Coordinate definition
The dot product of two vectors and specified with respect to an orthonormal basis, is defined as: | Dot product | Wikipedia | 494 | 157093 | https://en.wikipedia.org/wiki/Dot%20product | Mathematics | Algebra | null |
where denotes summation and is the dimension of the vector space. For instance, in three-dimensional space, the dot product of vectors and is:
Likewise, the dot product of the vector with itself is:
If vectors are identified with column vectors, the dot product can also be written as a matrix product
where denotes the transpose of .
Expressing the above example in this way, a 1 × 3 matrix (row vector) is multiplied by a 3 × 1 matrix (column vector) to get a 1 × 1 matrix that is identified with its unique entry:
Geometric definition
In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction to which the arrow points. The magnitude of a vector is denoted by . The dot product of two Euclidean vectors and is defined by
where is the angle between and .
In particular, if the vectors and are orthogonal (i.e., their angle is or ), then , which implies that
At the other extreme, if they are codirectional, then the angle between them is zero with and
This implies that the dot product of a vector with itself is
which gives
the formula for the Euclidean length of the vector.
Scalar projection and first properties
The scalar projection (or scalar component) of a Euclidean vector in the direction of a Euclidean vector is given by
where is the angle between and .
In terms of the geometric definition of the dot product, this can be rewritten as
where is the unit vector in the direction of .
The dot product is thus characterized geometrically by
The dot product, defined in this manner, is homogeneous under scaling in each variable, meaning that for any scalar ,
It also satisfies the distributive law, meaning that
These properties may be summarized by saying that the dot product is a bilinear form. Moreover, this bilinear form is positive definite, which means that is never negative, and is zero if and only if , the zero vector.
Equivalence of the definitions
If are the standard basis vectors in , then we may write
The vectors are an orthonormal basis, which means that they have unit length and are at right angles to each other. Since these vectors have unit length,
and since they form right angles with each other, if ,
Thus in general, we can say that:
where is the Kronecker delta. | Dot product | Wikipedia | 499 | 157093 | https://en.wikipedia.org/wiki/Dot%20product | Mathematics | Algebra | null |
Also, by the geometric definition, for any vector and a vector , we note that
where is the component of vector in the direction of . The last step in the equality can be seen from the figure.
Now applying the distributivity of the geometric version of the dot product gives
which is precisely the algebraic definition of the dot product. So the geometric dot product equals the algebraic dot product.
Properties
The dot product fulfills the following properties if , , and are real vectors and , , and are scalars.
Commutative which follows from the definition ( is the angle between and ): The commutative property can also be easily proven with the algebraic definition, and in more general spaces (where the notion of angle might not be geometrically intuitive but an analogous product can be defined) the angle between two vectors can be defined as
Bilinear (additive, distributive and scalar-multiplicative in both arguments)
Not associative Because the dot product is not defined between a scalar and a vector associativity is meaningless. However, bilinearity implies This property is sometimes called the "associative law for scalar and dot product", and one may say that "the dot product is associative with respect to scalar multiplication".
Orthogonal Two non-zero vectors and are orthogonal if and only if .
No cancellation
Unlike multiplication of ordinary numbers, where if , then always equals unless is zero, the dot product does not obey the cancellation law: If and , then we can write: by the distributive law; the result above says this just means that is perpendicular to , which still allows , and therefore allows .
Product rule If and are vector-valued differentiable functions, then the derivative (denoted by a prime ) of is given by the rule
Application to the law of cosines
Given two vectors and separated by angle (see the upper image), they form a triangle with a third side . Let , and denote the lengths of , , and , respectively. The dot product of this with itself is:
which is the law of cosines.
Triple product
There are two ternary operations involving dot product and cross product.
The scalar triple product of three vectors is defined as | Dot product | Wikipedia | 462 | 157093 | https://en.wikipedia.org/wiki/Dot%20product | Mathematics | Algebra | null |
Its value is the determinant of the matrix whose columns are the Cartesian coordinates of the three vectors. It is the signed volume of the parallelepiped defined by the three vectors, and is isomorphic to the three-dimensional special case of the exterior product of three vectors.
The vector triple product is defined by
This identity, also known as Lagrange's formula, may be remembered as "ACB minus ABC", keeping in mind which vectors are dotted together. This formula has applications in simplifying vector calculations in physics.
Physics
In physics, the dot product takes two vectors and returns a scalar quantity. It is also known as the "scalar product". The dot product of two vectors can be defined as the product of the magnitudes of the two vectors and the cosine of the angle between the two vectors. Thus, Alternatively, it is defined as the product of the projection of the first vector onto the second vector and the magnitude of the second vector.
For example:
Mechanical work is the dot product of force and displacement vectors,
Power is the dot product of force and velocity.
Generalizations
Complex vectors
For vectors with complex entries, using the given definition of the dot product would lead to quite different properties. For instance, the dot product of a vector with itself could be zero without the vector being the zero vector (e.g. this would happen with the vector This in turn would have consequences for notions like length and angle. Properties such as the positive-definite norm can be salvaged at the cost of giving up the symmetric and bilinear properties of the dot product, through the alternative definition
where is the complex conjugate of . When vectors are represented by column vectors, the dot product can be expressed as a matrix product involving a conjugate transpose, denoted with the superscript H:
In the case of vectors with real components, this definition is the same as in the real case. The dot product of any vector with itself is a non-negative real number, and it is nonzero except for the zero vector. However, the complex dot product is sesquilinear rather than bilinear, as it is conjugate linear and not linear in . The dot product is not symmetric, since
The angle between two complex vectors is then given by
The complex dot product leads to the notions of Hermitian forms and general inner product spaces, which are widely used in mathematics and physics. | Dot product | Wikipedia | 496 | 157093 | https://en.wikipedia.org/wiki/Dot%20product | Mathematics | Algebra | null |
The self dot product of a complex vector , involving the conjugate transpose of a row vector, is also known as the norm squared, , after the Euclidean norm; it is a vector generalization of the absolute square of a complex scalar (see also: Squared Euclidean distance).
Inner product
The inner product generalizes the dot product to abstract vector spaces over a field of scalars, being either the field of real numbers or the field of complex numbers . It is usually denoted using angular brackets by .
The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is sesquilinear instead of bilinear. An inner product space is a normed vector space, and the inner product of a vector with itself is real and positive-definite.
Functions
The dot product is defined for vectors that have a finite number of entries. Thus these vectors can be regarded as discrete functions: a length- vector is, then, a function with domain , and is a notation for the image of by the function/vector .
This notion can be generalized to square-integrable functions: just as the inner product on vectors uses a sum over corresponding components, the inner product on functions is defined as an integral over some measure space :
For example, if and are continuous functions over a compact subset of with the standard Lebesgue measure, the above definition becomes:
Generalized further to complex continuous functions and , by analogy with the complex inner product above, gives:
Weight function
Inner products can have a weight function (i.e., a function which weights each term of the inner product with a value). Explicitly, the inner product of functions and with respect to the weight function is
Dyadics and matrices
A double-dot product for matrices is the Frobenius inner product, which is analogous to the dot product on vectors. It is defined as the sum of the products of the corresponding components of two matrices and of the same size:
And for real matrices,
Writing a matrix as a dyadic, we can define a different double-dot product (see ) however it is not an inner product.
Tensors
The inner product between a tensor of order and a tensor of order is a tensor of order , see Tensor contraction for details.
Computation
Algorithms
The straightforward algorithm for calculating a floating-point dot product of vectors can suffer from catastrophic cancellation. To avoid this, approaches such as the Kahan summation algorithm are used.
Libraries | Dot product | Wikipedia | 500 | 157093 | https://en.wikipedia.org/wiki/Dot%20product | Mathematics | Algebra | null |
A dot product function is included in:
BLAS level 1 real , ; complex , , ,
Fortran as or
Julia as or standard library LinearAlgebra as
R (programming language) as for vectors or, more generally for matrices, as
Matlab as or or or
Python (package NumPy) as or or
GNU Octave as , and similar code as Matlab
Intel oneAPI Math Kernel Library real p?dot ; complex p?dotc | Dot product | Wikipedia | 92 | 157093 | https://en.wikipedia.org/wiki/Dot%20product | Mathematics | Algebra | null |
Cut, copy, and paste are essential commands of modern human–computer interaction and user interface design. They offer an interprocess communication technique for transferring data through a computer's user interface. The cut command removes the selected data from its original position, and the copy command creates a duplicate; in both cases the selected data is kept in temporary storage called the clipboard. Clipboard data is later inserted wherever a paste command is issued. The data remains available to any application supporting the feature, thus allowing easy data transfer between applications.
The command names are an interface metaphor based on the physical procedure used in manuscript print editing to create a page layout, like with paper. The commands were pioneered into computing by Xerox PARC in 1974, popularized by Apple Computer in the 1983 Lisa workstation and the 1984 Macintosh computer, and in a few home computer applications such the 1984 word processor Cut & Paste.
This interaction technique has close associations with related techniques in graphical user interfaces (GUIs) that use pointing devices such as a computer mouse (by drag and drop, for example). Typically, clipboard support is provided by an operating system as part of its GUI and widget toolkit.
The capability to replicate information with ease, changing it between contexts and applications, involves privacy concerns because of the risks of disclosure when handling sensitive information. Terms like cloning, copy forward, carry forward, or re-use refer to the dissemination of such information through documents, and may be subject to regulation by administrative bodies.
History
Origins
The term "cut and paste" comes from the traditional practice in manuscript editing, whereby people cut paragraphs from a page with scissors and paste them onto another page. This practice remained standard into the 1980s. Stationery stores sold "editing scissors" with blades long enough to cut an 8½"-wide page. The advent of photocopiers made the practice easier and more flexible.
The act of copying or transferring text from one part of a computer-based document ("buffer") to a different location within the same or different computer-based document was a part of the earliest on-line computer editors. As soon as computer data entry moved from punch-cards to online files (in the mid/late 1960s) there were "commands" for accomplishing this operation. This mechanism was often used to transfer frequently-used commands or text snippets from additional buffers into the document, as was the case with the QED text editor. | Cut, copy, and paste | Wikipedia | 498 | 157115 | https://en.wikipedia.org/wiki/Cut%2C%20copy%2C%20and%20paste | Technology | User interface | null |
Early methods
The earliest editors (designed for teleprinter terminals) provided keyboard commands to delineate a contiguous region of text, then delete or move it. Since moving a region of text requires first removing it from its initial location and then inserting it into its new location, various schemes had to be invented to allow for this multi-step process to be specified by the user. Often this was done with a "move" command, but some text editors required that the text be first put into some temporary location for later retrieval/placement. In 1983, the Apple Lisa became the first text editing system to call that temporary location "the clipboard".
Earlier control schemes such as NLS used a verb—object command structure, where the command name was provided first and the object to be copied or moved was second. The inversion from verb—object to object—verb on which copy and paste are based, where the user selects the object to be operated before initiating the operation, was an innovation crucial for the success of the desktop metaphor as it allowed copy and move operations based on direct manipulation.
Popularization
Inspired by early line and character editors, such as Pentti Kanerva's TV-Edit, that broke a move or copy operation into two steps—between which the user could invoke a preparatory action such as navigation—Lawrence G. "Larry" Tesler proposed the names "cut" and "copy" for the first step and "paste" for the second step. Beginning in 1974, he and colleagues at Xerox PARC implemented several text editors that used cut/copy-and-paste commands to move and copy text.
Apple Computer popularized this paradigm with its Lisa (1983) and Macintosh (1984) operating systems and applications. The functions were mapped to key combinations using the key as a special modifier, which is held down while also pressing for cut, for copy, or for paste. These few keyboard shortcuts allow the user to perform all the basic editing operations, and the keys are clustered at the left end of the bottom row of the standard QWERTY keyboard.
These are the standard shortcuts:
Control-Z (or ) to undo
Control-X (or ) to cut
Control-C (or ) to copy
Control-V (or ) to paste | Cut, copy, and paste | Wikipedia | 465 | 157115 | https://en.wikipedia.org/wiki/Cut%2C%20copy%2C%20and%20paste | Technology | User interface | null |
The IBM Common User Access (CUA) standard also uses combinations of the Insert, Del, Shift and Control keys. Early versions of Windows used the IBM standard. Microsoft later also adopted the Apple key combinations with the introduction of Windows, using the control key as modifier key. For users migrating to Windows from DOS this was a big change as DOS users used the "COPY" and "MOVE" commands.
Similar patterns of key combinations, later borrowed by others, are widely available in most GUI applications.
The original cut, copy, and paste workflow, as implemented at PARC, utilizes a unique workflow: With two windows on the same screen, the user could use the mouse to pick a point at which to make an insertion in one window (or a segment of text to replace). Then, by holding shift and selecting the copy source elsewhere on the same screen, the copy would be made as soon as the shift was released. Similarly, holding shift and control would copy and cut (delete) the source. This workflow requires many fewer keystrokes/mouse clicks than the current multi-step workflows, and did not require an explicit copy buffer. It was dropped, one presumes, because the original Apple and IBM GUIs were not high enough density to permit multiple windows, as were the PARC machines, and so multiple simultaneous windows were rarely used.
Cut and paste | Cut, copy, and paste | Wikipedia | 282 | 157115 | https://en.wikipedia.org/wiki/Cut%2C%20copy%2C%20and%20paste | Technology | User interface | null |
Computer-based editing can involve very frequent use of cut-and-paste operations. Most software-suppliers provide several methods for performing such tasks, and this can involve (for example) key combinations, pulldown menus, pop-up menus, or toolbar buttons.
The user selects or "highlights" the text or file for moving by some method, typically by dragging over the text or file name with the pointing-device or holding down the Shift key while using the arrow keys to move the text cursor.
The user performs a "cut" operation via key combination ( for Macintosh users), menu, or other means.
Visibly, "cut" text immediately disappears from its location. "Cut" files typically change color to indicate that they will be moved.
Conceptually, the text has now moved to a location often called the clipboard. The clipboard typically remains invisible. On most systems only one clipboard location exists, hence another cut or copy operation overwrites the previously stored information. Many UNIX text-editors provide multiple clipboard entries, as do some Macintosh programs such as Clipboard Master, and Windows clipboard-manager programs such as the one in Microsoft Office.
The user selects a location for insertion by some method, typically by clicking at the desired insertion point.
A paste operation takes place which visibly inserts the clipboard text at the insertion point. (The paste operation does not typically destroy the clipboard text: it remains available in the clipboard and the user can insert additional copies at other points).
Whereas cut-and-paste often takes place with a mouse-equivalent in Windows-like GUI environments, it may also occur entirely from the keyboard, especially in UNIX text editors, such as Pico or vi. Cutting and pasting without a mouse can involve a selection (for which is pressed in most graphical systems) or the entire current line, but it may also involve text after the cursor until the end of the line and other more sophisticated operations.
The clipboard usually stays invisible, because the operations of cutting and pasting, while actually independent, usually take place in quick succession, and the user (usually) needs no assistance in understanding the operation or maintaining mental context. Some application programs provide a means of viewing, or sometimes even editing, the data on the clipboard.
Copy and paste | Cut, copy, and paste | Wikipedia | 473 | 157115 | https://en.wikipedia.org/wiki/Cut%2C%20copy%2C%20and%20paste | Technology | User interface | null |
The term "copy-and-paste" refers to the popular, simple method of reproducing text or other data from a source to a destination. It differs from cut and paste in that the original source text or data does not get deleted or removed. The popularity of this method stems from its simplicity and the ease with which users can move data between various applications visually – without resorting to permanent storage.
Use in healthcare documentation and electronic health records are sensitive, with potential for the introduction of medical errors, information overload, and fraud. | Cut, copy, and paste | Wikipedia | 110 | 157115 | https://en.wikipedia.org/wiki/Cut%2C%20copy%2C%20and%20paste | Technology | User interface | null |
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of the mathematical field of combinatorics that focuses on the appearance of order in a substructure given a structure of a known size. Problems in Ramsey theory typically ask a question of the form: "how big must some structure be to guarantee that a particular property holds?"
Examples
A typical result in Ramsey theory starts with some mathematical structure that
is then cut into pieces. How big must the original structure be in order to ensure that at least one of the pieces has a given interesting property? This idea can be defined as partition regularity.
For example, consider a complete graph of order n; that is, there are n vertices and each vertex is connected to every other vertex by an edge. A complete graph of order 3 is called a triangle. Now colour each edge either red or blue. How large must n be in order to ensure that there is either a blue triangle or a red triangle? It turns out that the answer is 6. See the article on Ramsey's theorem for a rigorous proof.
Another way to express this result is as follows: at any party with at least six people, there are three people who are all either mutual acquaintances (each one knows the other two) or mutual strangers (none of them knows either of the other two). See theorem on friends and strangers.
This also is a special case of Ramsey's theorem, which says that for any given integer c, any given integers n1,...,nc, there is a number, R(n1,...,nc), such that if the edges of a complete graph of order R(n1,...,nc) are coloured with c different colours, then for some i between 1 and c, it must contain a complete subgraph of order ni whose edges are all colour i. The special case above has c = 2 and n1 = n2 = 3.
Results
Two key theorems of Ramsey theory are: | Ramsey theory | Wikipedia | 413 | 157175 | https://en.wikipedia.org/wiki/Ramsey%20theory | Mathematics | Combinatorics | null |
Van der Waerden's theorem: For any given c and n, there is a number V, such that if V consecutive numbers are coloured with c different colours, then it must contain an arithmetic progression of length n whose elements are all the same colour.
Hales–Jewett theorem: For any given n and c, there is a number H such that if the cells of an H-dimensional n×n×n×...×n cube are coloured with c colours, there must be one row, column, etc. of length n all of whose cells are the same colour. That is: a multi-player n-in-a-row tic-tac-toe cannot end in a draw, no matter how large n is, and no matter how many people are playing, if you play on a board with sufficiently many dimensions. The Hales–Jewett theorem implies Van der Waerden's theorem.
A theorem similar to van der Waerden's theorem is Schur's theorem: for any given c there is a number N such that if the numbers 1, 2, ..., N are coloured with c different colours, then there must be a pair of integers x, y such that x, y, and x+y are all the same colour. Many generalizations of this theorem exist, including Rado's theorem, Rado–Folkman–Sanders theorem, Hindman's theorem, and the Milliken–Taylor theorem. A classic reference for these and many other results in Ramsey theory is Graham, Rothschild, Spencer and Solymosi, updated and expanded in 2015 to its first new edition in 25 years. | Ramsey theory | Wikipedia | 345 | 157175 | https://en.wikipedia.org/wiki/Ramsey%20theory | Mathematics | Combinatorics | null |
Results in Ramsey theory typically have two primary characteristics. Firstly, they are unconstructive: they may show that some structure exists, but they give no process for finding this structure (other than brute-force search). For instance, the pigeonhole principle is of this form. Secondly, while Ramsey theory results do say that sufficiently large objects must necessarily contain a given structure, often the proof of these results requires these objects to be enormously large – bounds that grow exponentially, or even as fast as the Ackermann function are not uncommon. In some small niche cases, upper and lower bounds are improved, but not in general. In many cases these bounds are artifacts of the proof, and it is not known whether they can be substantially improved. In other cases it is known that any bound must be extraordinarily large, sometimes even greater than any primitive recursive function; see the Paris–Harrington theorem for an example. Graham's number, one of the largest numbers ever used in serious mathematical proof, is an upper bound for a problem related to Ramsey theory. Another large example is the Boolean Pythagorean triples problem.
Theorems in Ramsey theory are generally one of the following two types. Many such theorems, which are modeled after Ramsey's theorem itself, assert that in every partition of a large structured object, one of the classes necessarily contains its own structured object, but gives no information about which class this is. In other cases, the reason behind a Ramsey-type result is that the largest partition class always contains the desired substructure. The results of this latter kind are called either density results or Turán-type result, after Turán's theorem. Notable examples include Szemerédi's theorem, which is such a strengthening of van der Waerden's theorem, and the density version of the Hales-Jewett theorem. | Ramsey theory | Wikipedia | 388 | 157175 | https://en.wikipedia.org/wiki/Ramsey%20theory | Mathematics | Combinatorics | null |
Glass fiber (or glass fibre) is a material consisting of numerous extremely fine fibers of glass.
Glassmakers throughout history have experimented with glass fibers, but mass manufacture of glass fiber was only made possible with the invention of finer machine tooling. In 1893, Edward Drummond Libbey exhibited a dress at the World's Columbian Exposition incorporating glass fibers with the diameter and texture of silk fibers. Glass fibers can also occur naturally, as Pele's hair.
Glass wool, which is one product called "fiberglass" today, was invented some time between 1932 and 1933 by Games Slayter of Owens-Illinois, as a material to be used as thermal building insulation. It is marketed under the trade name Fiberglas, which has become a genericized trademark. Glass fiber, when used as a thermal insulating material, is specially manufactured with a bonding agent to trap many small air cells, resulting in the characteristically air-filled low-density "glass wool" family of products.
Glass fiber has roughly comparable mechanical properties to other fibers such as polymers and carbon fiber. Although not as rigid as carbon fiber, it is much cheaper and significantly less brittle when used in composites. Glass fiber reinforced composites are used in marine industry and piping industries because of good environmental resistance, better damage tolerance for impact loading, high specific strength and stiffness.
Fiber formation
Glass fiber is formed when thin strands of silica-based or other formulation glass are extruded into many fibers with small diameters suitable for textile processing. The technique of heating and drawing glass into fine fibers has been known for millennia, and was practiced in Egypt and Venice. Before the recent use of these fibers for textile applications, all glass fiber had been manufactured as staple (that is, clusters of short lengths of fiber).
The modern method for producing glass wool is the invention of Games Slayter working at the Owens-Illinois Glass Company (Toledo, Ohio). He first applied for a patent for a new process to make glass wool in 1933. The first commercial production of glass fiber was in 1936. In 1938 Owens-Illinois Glass Company and Corning Glass Works joined to form the Owens-Corning Fiberglas Corporation. When the two companies joined to produce and promote glass fiber, they introduced continuous filament glass fibers. Owens-Corning is still the major glass-fiber producer in the market today. | Glass fiber | Wikipedia | 481 | 157606 | https://en.wikipedia.org/wiki/Glass%20fiber | Technology | Materials | null |
The most common type of glass fiber used in fiberglass is E-glass, which is alumino-borosilicate glass with less than 1% w/w alkali oxides, mainly used for glass-reinforced plastics. Other types of glass used are A-glass (Alkali-lime glass with little or no boron oxide), E-CR-glass (Electrical/Chemical Resistance; alumino-lime silicate with less than 1% w/w alkali oxides, with high acid resistance), C-glass (alkali-lime glass with high boron oxide content, used for glass staple fibers and insulation), D-glass (borosilicate glass, named for its low dielectric constant), R-glass (alumino silicate glass without MgO and CaO with high mechanical requirements as reinforcement), and S-glass (alumino silicate glass without CaO but with high MgO content with high tensile strength). | Glass fiber | Wikipedia | 206 | 157606 | https://en.wikipedia.org/wiki/Glass%20fiber | Technology | Materials | null |
Pure silica (silicon dioxide), when cooled as fused quartz into a glass with no true melting point, can be used as a glass fiber for fiberglass, but has the drawback that it must be worked at very high temperatures. In order to lower the necessary work temperature, other materials are introduced as "fluxing agents" (i.e., components to lower the melting point). Ordinary A-glass ("A" for "alkali-lime") or soda lime glass, crushed and ready to be remelted, as so-called cullet glass, was the first type of glass used for fiberglass. E-glass ("E" because of initial electrical application), is alkali free, and was the first glass formulation used for continuous filament formation. It now makes up most of the fiberglass production in the world, and also is the single largest consumer of boron minerals globally. It is susceptible to chloride ion attack and is a poor choice for marine applications. S-glass ("S" for "Strength") is used when high tensile strength (modulus) is important, and is thus important in composites for building and aircraft construction. The same substance is known as R-glass ("R" for "reinforcement") in Europe. C-glass ("C" for "chemical resistance") and T-glass ("T" is for "thermal insulator" – a North American variant of C-glass) are resistant to chemical attack; both are often found in insulation-grades of blown fiberglass.
Chemistry
The basis of textile-grade glass fibers is silica, SiO2. In its pure form it exists as a polymer, (SiO2)n. It has no true melting point but softens up to 1200 °C, where it starts to degrade. At 1713 °C, most of the molecules can move about freely. If the glass is extruded and cooled quickly at this temperature, it will be unable to form an ordered structure. In the polymer it forms SiO4 groups which are configured as a tetrahedron with the silicon atom at the center, and four oxygen atoms at the corners. These atoms then form a network bonded at the corners by sharing the oxygen atoms. | Glass fiber | Wikipedia | 464 | 157606 | https://en.wikipedia.org/wiki/Glass%20fiber | Technology | Materials | null |
The vitreous and crystalline states of silica (glass and quartz) have similar energy levels on a molecular basis, also implying that the glassy form is extremely stable. In order to induce crystallization, it must be heated to temperatures above 1200 °C for long periods of time.
Although pure silica is a perfectly viable glass and glass fiber, it must be worked with at very high temperatures, which is a drawback unless its specific chemical properties are needed. It is usual to introduce impurities into the glass in the form of other materials to lower its working temperature. These materials also impart various other properties to the glass that may be beneficial in different applications. The first type of glass used for fiber was soda lime glass or A-glass ("A" for the alkali it contains). It is not very resistant to alkali. A newer, alkali-free (<2%) type, E-glass, is an alumino-borosilicate glass. C-glass was developed to resist attack from chemicals, mostly acids that destroy E-glass. T-glass is a North American variant of C-glass. AR-glass is alkali-resistant glass. Most glass fibers have limited solubility in water but are very dependent on pH. Chloride ions will also attack and dissolve E-glass surfaces.
E-glass does not actually melt, but softens instead, the softening point being "the temperature at which a 0.55–0.77 mm diameter fiber 235 mm long, elongates under its own weight at 1 mm/min when suspended vertically and heated at the rate of 5 °C per minute". The strain point is reached when the glass has a viscosity of 1014.5 poise. The annealing point, which is the temperature where the internal stresses are reduced to an acceptable commercial limit in 15 minutes, is marked by a viscosity of 1013 poise.
Properties
Thermal
Fabrics of woven glass fibers are useful thermal insulators because of their high ratio of surface area to weight. However, the increased surface area makes them much more susceptible to chemical attack. By trapping air within them, blocks of glass fiber make good thermal insulation, with a thermal conductivity of the order of 0.05 W/(m·K).
Selected properties | Glass fiber | Wikipedia | 471 | 157606 | https://en.wikipedia.org/wiki/Glass%20fiber | Technology | Materials | null |
Mechanical properties
The strength of glass is usually tested and reported for "virgin" or pristine fibers—those that have just been manufactured. The freshest, thinnest fibers are the strongest because the thinner fibers are more ductile. The more the surface is scratched, the less the resulting tenacity. Because glass has an amorphous structure, its properties are the same along the fiber and across the fiber. Humidity is an important factor in the tensile strength. Moisture is easily adsorbed and can worsen microscopic cracks and surface defects, and lessen tenacity.
In contrast to carbon fiber, glass can undergo more elongation before it breaks. Thinner filaments can bend further before they break. The viscosity of the molten glass is very important for manufacturing success. During drawing, the process where the hot glass is pulled to reduce the diameter of the fiber, the viscosity must be relatively low. If it is too high, the fiber will break during drawing. However, if it is too low, the glass will form droplets instead of being drawn out into a fiber.
Manufacturing processes
Melting
There are two main types of glass fiber manufacture and two main types of glass fiber product. First, fiber is made either from a direct melt process or a marble remelt process. Both start with the raw materials in solid form. The materials are mixed together and melted in a furnace. Then, for the marble process, the molten material is sheared and rolled into marbles which are cooled and packaged. The marbles are taken to the fiber manufacturing facility where they are inserted into a can and remelted. The molten glass is extruded to the bushing to be formed into fiber. In the direct melt process, the molten glass in the furnace goes directly to the bushing for formation. | Glass fiber | Wikipedia | 368 | 157606 | https://en.wikipedia.org/wiki/Glass%20fiber | Technology | Materials | null |
Formation
The bushing plate is the most important part of the machinery for making the fiber. This is a small metal furnace containing nozzles for the fiber to be formed through. It is almost always made of platinum alloyed with rhodium for durability. Platinum is used because the glass melt has a natural affinity for wetting it. When bushings were first used they were pure platinum, and the glass wetted the bushing so easily that it ran under the plate after exiting the nozzle and accumulated on the underside. Also, due to its cost and the tendency to wear, the platinum was alloyed with rhodium. In the direct melt process, the bushing serves as a collector for the molten glass. It is heated slightly to keep the glass at the correct temperature for fiber formation. In the marble melt process, the bushing acts more like a furnace as it melts more of the material.
Bushings are the major expense in fiber glass production. The nozzle design is also critical. The number of nozzles ranges from 200 to 4000 in multiples of 200. The important part of the nozzle in continuous filament manufacture is the thickness of its walls in the exit region. It was found that inserting a counterbore here reduced wetting. Today, the nozzles are designed to have a minimum thickness at the exit. As glass flows through the nozzle, it forms a drop which is suspended from the end. As it falls, it leaves a thread attached by the meniscus to the nozzle as long as the viscosity is in the correct range for fiber formation. The smaller the annular ring of the nozzle and the thinner the wall at exit, the faster the drop will form and fall away, and the lower its tendency to wet the vertical part of the nozzle. The surface tension of the glass is what influences the formation of the meniscus. For E-glass it should be around 400 mN/m.
The attenuation (drawing) speed is important in the nozzle design. Although slowing this speed down can make coarser fiber, it is uneconomic to run at speeds for which the nozzles were not designed. | Glass fiber | Wikipedia | 451 | 157606 | https://en.wikipedia.org/wiki/Glass%20fiber | Technology | Materials | null |
Continuous filament process
In the continuous filament process, after the fiber is drawn, a size is applied. This size helps protect the fiber as it is wound onto a bobbin. The particular size applied relates to end-use. While some sizes are processing aids, others make the fiber have an affinity for a certain resin, if the fiber is to be used in a composite. Size is usually added at 0.5–2.0% by weight. Winding then takes place at around 1 km/min.
Staple fiber process
For staple fiber production, there are a number of ways to manufacture the fiber. The glass can be blown or blasted with heat or steam after exiting the formation machine. Usually these fibers are made into some sort of mat. The most common process used is the rotary process. Here, the glass enters a rotating spinner, and due to centrifugal force is thrown out horizontally. The air jets push it down vertically, and binder is applied. Then the mat is vacuumed to a screen and the binder is cured in the oven.
Safety
Glass fiber has increased in popularity since the discovery that asbestos causes cancer and its subsequent removal from most products. Following this increase in popularity, the safety of glass fiber has also been called into question. Research shows that the composition of glass fiber can cause similar toxicity as asbestos since both are silicate fibers.
Studies on rats conducted during the 1970s found that fibrous glass of less than 3 μm in diameter and greater than 20 μm in length is a "potent carcinogen". Likewise, the International Agency for Research on Cancer found it "may reasonably be anticipated to be a carcinogen" in 1990. The American Conference of Governmental Industrial Hygienists, on the other hand, says that there is insufficient evidence, and that glass fiber is in group A4: "Not classifiable as a human carcinogen".
The North American Insulation Manufacturers Association (NAIMA) claims that glass fiber is fundamentally different from asbestos, since it is man-made instead of naturally occurring. They claim that glass fiber "dissolves in the lungs", while asbestos remains in the body for life. Although both glass fiber and asbestos are made from silica filaments, NAIMA claims that asbestos is more dangerous because of its crystalline structure, which causes it to cleave into smaller, more dangerous pieces, citing the U.S. Department of Health and Human Services: | Glass fiber | Wikipedia | 501 | 157606 | https://en.wikipedia.org/wiki/Glass%20fiber | Technology | Materials | null |
A 1998 study using rats found that the biopersistence of synthetic fibers after one year was 0.04–13%, but 27% for amosite asbestos. Fibers that persisted longer were found to be more carcinogenic.
Glass-reinforced plastic (fiberglass)
Glass-reinforced plastic (GRP) is a composite material or fiber-reinforced plastic made of a plastic reinforced by fine glass fibers. The glass can be in the form of a chopped strand mat (CSM) or a woven fabric.
As with many other composite materials (such as reinforced concrete), the two materials act together, each overcoming the deficits of the other. Whereas the plastic resins are strong in compressive loading and relatively weak in tensile strength, the glass fibers are very strong in tension but tend not to resist compression. By combining the two materials, GRP becomes a material that resists both compressive and tensile forces well. The two materials may be used uniformly or the glass may be specifically placed in those portions of the structure that will experience tensile loads.
Uses
Uses for regular glass fiber include mats and fabrics for thermal insulation, electrical insulation, sound insulation, high-strength fabrics or heat- and corrosion-resistant fabrics. It is also used to reinforce various materials, such as tent poles, pole vault poles, arrows, bows and crossbows, translucent roofing panels, automobile bodies, hockey sticks, surfboards, boat hulls, and paper honeycomb. It has been used for medical purposes in casts. Glass fiber is extensively used for making FRP tanks and vessels.
Open-weave glass fiber grids are used to reinforce asphalt pavement. Non-woven glass fiber/polymer blend mats are used saturated with asphalt emulsion and overlaid with asphalt, producing a waterproof, crack-resistant membrane. Use of glass-fiber reinforced polymer rebar instead of steel rebar shows promise in areas where avoidance of steel corrosion is desired.
Potential uses
Glass fiber has recently seen use in biomedical applications in the assistance of joint replacement where the electric field orientation of short phosphate glass fibers can improve osteogenic qualities through the proliferation of osteoblasts and with improved surface chemistry. Another potential use is within electronic applications as sodium based glass fibers assist or replace lithium in lithium-ion batteries due to its improved electronic properties.
Role of recycling in glass fiber manufacturing
Manufacturers of glass-fiber insulation can use recycled glass. Recycled glass fiber contains up to 40% recycled glass. | Glass fiber | Wikipedia | 496 | 157606 | https://en.wikipedia.org/wiki/Glass%20fiber | Technology | Materials | null |
A composite or composite material (also composition material) is a material which is produced from two or more constituent materials. These constituent materials have notably dissimilar chemical or physical properties and are merged to create a material with properties unlike the individual elements. Within the finished structure, the individual elements remain separate and distinct, distinguishing composites from mixtures and solid solutions. Composite materials with more than one distinct layer are called composite laminates.
Typical engineered composite materials are made up of a binding agent forming the matrix and a filler material (particulates or fibres) giving substance, e.g.:
Concrete, reinforced concrete and masonry with cement, lime or mortar (which is itself a composite material) as a binder
Composite wood such as glulam and plywood with wood glue as a binder
Reinforced plastics, such as fiberglass and fibre-reinforced polymer with resin or thermoplastics as a binder
Ceramic matrix composites (composite ceramic and metal matrices)
Metal matrix composites
advanced composite materials, often first developed for spacecraft and aircraft applications.
Composite materials can be less expensive, lighter, stronger or more durable than common materials. Some are inspired by biological structures found in plants and animals.
Robotic materials are composites that include sensing, actuation, computation, and communication components.
Composite materials are used for construction and technical structures such as boat hulls, swimming pool panels, racing car bodies, shower stalls, bathtubs, storage tanks, imitation granite, and cultured marble sinks and countertops. They are also being increasingly used in general automotive applications.
History
The earliest composite materials were made from straw and mud combined to form bricks for building construction. Ancient brick-making was documented by Egyptian tomb paintings.
Wattle and daub might be the oldest composite materials, at over 6000 years old. | Composite material | Wikipedia | 366 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Woody plants, both true wood from trees and such plants as palms and bamboo, yield natural composites that were used prehistorically by humankind and are still used widely in construction and scaffolding.
Plywood, 3400 BC, by the Ancient Mesopotamians; gluing wood at different angles gives better properties than natural wood.
Cartonnage, layers of linen or papyrus soaked in plaster dates to the First Intermediate Period of Egypt c. 2181–2055 BC and was used for death masks.
Cob mud bricks, or mud walls, (using mud (clay) with straw or gravel as a binder) have been used for thousands of years.
Concrete was described by Vitruvius, writing around 25 BC in his Ten Books on Architecture, distinguished types of aggregate appropriate for the preparation of lime mortars. For structural mortars, he recommended pozzolana, which were volcanic sands from the sandlike beds of Pozzuoli brownish-yellow-gray in colour near Naples and reddish-brown at Rome. Vitruvius specifies a ratio of 1 part lime to 3 parts pozzolana for cements used in buildings and a 1:2 ratio of lime to pulvis Puteolanus for underwater work, essentially the same ratio mixed today for concrete used at sea. Natural cement-stones, after burning, produced cements used in concretes from post-Roman times into the 20th century, with some properties superior to manufactured Portland cement.
Papier-mâché, a composite of paper and glue, has been used for hundreds of years.
The first artificial fibre reinforced plastic was a combination of fiber glass and bakelite, performed in 1935 by Al Simison and Arthur D Little in Owens Corning Company
One of the most common and familiar composite is fibreglass, in which small glass fibre are embedded within a polymeric material (normally an epoxy or polyester). The glass fibre is relatively strong and stiff (but also brittle), whereas the polymer is ductile (but also weak and flexible). Thus the resulting fibreglass is relatively stiff, strong, flexible, and ductile.
Composite bow
Leather cannon, wooden cannon
Examples
Composite materials | Composite material | Wikipedia | 441 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Concrete is the most common artificial composite material of all. , about 7.5 billion cubic metres of concrete are made each year.
Concrete typically consists of loose stones (construction aggregate) held with a matrix of cement. Concrete is an inexpensive material resisting large compressive forces, however, susceptible to tensile loading. To give concrete the ability to resist being stretched, steel bars, which can resist high stretching (tensile) forces, are often added to concrete to form reinforced concrete.
Fibre-reinforced polymers include carbon-fiber-reinforced polymers and glass-reinforced plastic. If classified by matrix then there are thermoplastic composites, short fibre thermoplastics, long fibre thermoplastics or long-fiber-reinforced thermoplastics. There are numerous thermoset composites, including paper composite panels. Many advanced thermoset polymer matrix systems usually incorporate aramid fibre and carbon fibre in an epoxy resin matrix.
Shape-memory polymer composites are high-performance composites, formulated using fibre or fabric reinforcements and shape-memory polymer resin as the matrix. Since a shape-memory polymer resin is used as the matrix, these composites have the ability to be easily manipulated into various configurations when they are heated above their activation temperatures and will exhibit high strength and stiffness at lower temperatures. They can also be reheated and reshaped repeatedly without losing their material properties. These composites are ideal for applications such as lightweight, rigid, deployable structures; rapid manufacturing; and dynamic reinforcement.
High strain composites are another type of high-performance composites that are designed to perform in a high deformation setting and are often used in deployable systems where structural flexing is advantageous. Although high strain composites exhibit many similarities to shape-memory polymers, their performance is generally dependent on the fibre layout as opposed to the resin content of the matrix.
Composites can also use metal fibres reinforcing other metals, as in metal matrix composites (MMC) or ceramic matrix composites (CMC), which includes bone (hydroxyapatite reinforced with collagen fibres), cermet (ceramic and metal), and concrete. Ceramic matrix composites are built primarily for fracture toughness, not for strength. Another class of composite materials involve woven fabric composite consisting of longitudinal and transverse laced yarns. Woven fabric composites are flexible as they are in form of fabric. | Composite material | Wikipedia | 495 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Organic matrix/ceramic aggregate composites include asphalt concrete, polymer concrete, mastic asphalt, mastic roller hybrid, dental composite, syntactic foam, and mother of pearl. Chobham armour is a special type of composite armour used in military applications.
Additionally, thermoplastic composite materials can be formulated with specific metal powders resulting in materials with a density range from 2 g/cm3 to 11 g/cm3 (same density as lead). The most common name for this type of material is "high gravity compound" (HGC), although "lead replacement" is also used. These materials can be used in place of traditional materials such as aluminium, stainless steel, brass, bronze, copper, lead, and even tungsten in weighting, balancing (for example, modifying the centre of gravity of a tennis racquet), vibration damping, and radiation shielding applications. High density composites are an economically viable option when certain materials are deemed hazardous and are banned (such as lead) or when secondary operations costs (such as machining, finishing, or coating) are a factor.
There have been several studies indicating that interleaving stiff and brittle epoxy-based carbon-fiber-reinforced polymer laminates with flexible thermoplastic laminates can help to make highly toughened composites that show improved impact resistance. Another interesting aspect of such interleaved composites is that they are able to have shape memory behaviour without needing any shape-memory polymers or shape-memory alloys e.g. balsa plies interleaved with hot glue, aluminium plies interleaved with acrylic polymers or PVC and carbon-fiber-reinforced polymer laminates interleaved with polystyrene.
A sandwich-structured composite is a special class of composite material that is fabricated by attaching two thin but stiff skins to a lightweight but thick core. The core material is normally low strength material, but its higher thickness provides the sandwich composite with high bending stiffness with overall low density. | Composite material | Wikipedia | 410 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Wood is a naturally occurring composite comprising cellulose fibres in a lignin and hemicellulose matrix. Engineered wood includes a wide variety of different products such as wood fibre board, plywood, oriented strand board, wood plastic composite (recycled wood fibre in polyethylene matrix), Pykrete (sawdust in ice matrix), plastic-impregnated or laminated paper or textiles, Arborite, Formica (plastic), and Micarta. Other engineered laminate composites, such as Mallite, use a central core of end grain balsa wood, bonded to surface skins of light alloy or GRP. These generate low-weight, high rigidity materials.
Particulate composites have particle as filler material dispersed in matrix, which may be nonmetal, such as glass, epoxy. Automobile tire is an example of particulate composite.
Advanced diamond-like carbon (DLC) coated polymer composites have been reported where the coating increases the surface hydrophobicity, hardness and wear resistance.
Ferromagnetic composites, including those with a polymer matrix consisting, for example, of nanocrystalline filler of Fe-based powders and polymers matrix. Amorphous and nanocrystalline powders obtained, for example, from metallic glasses can be used. Their use makes it possible to obtain ferromagnetic nanocomposites with controlled magnetic properties.
Products
Fibre-reinforced composite materials have gained popularity (despite their generally high cost) in high-performance products that need to be lightweight, yet strong enough to take harsh loading conditions such as aerospace components (tails, wings, fuselages, propellers), boat and scull hulls, bicycle frames, and racing car bodies. Other uses include fishing rods, storage tanks, swimming pool panels, and baseball bats. The Boeing 787 and Airbus A350 structures including the wings and fuselage are composed largely of composites. Composite materials are also becoming more common in the realm of orthopedic surgery, and it is the most common hockey stick material. | Composite material | Wikipedia | 429 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Carbon composite is a key material in today's launch vehicles and heat shields for the re-entry phase of spacecraft. It is widely used in solar panel substrates, antenna reflectors and yokes of spacecraft. It is also used in payload adapters, inter-stage structures and heat shields of launch vehicles. Furthermore, disk brake systems of airplanes and racing cars are using carbon/carbon material, and the composite material with carbon fibres and silicon carbide matrix has been introduced in luxury vehicles and sports cars.
In 2006, a fibre-reinforced composite pool panel was introduced for in-ground swimming pools, residential as well as commercial, as a non-corrosive alternative to galvanized steel.
In 2007, an all-composite military Humvee was introduced by TPI Composites Inc and Armor Holdings Inc, the first all-composite military vehicle. By using composites the vehicle is lighter, allowing higher payloads. In 2008, carbon fibre and DuPont Kevlar (five times stronger than steel) were combined with enhanced thermoset resins to make military transit cases by ECS Composites creating 30-percent lighter cases with high strength.
Pipes and fittings for various purpose like transportation of potable water, fire-fighting, irrigation, seawater, desalinated water, chemical and industrial waste, and sewage are now manufactured in glass reinforced plastics.
Composite materials used in tensile structures for facade application provides the advantage of being translucent. The woven base cloth combined with the appropriate coating allows better light transmission. This provides a very comfortable level of illumination compared to the full brightness of outside.
The wings of wind turbines, in growing sizes in the order of 50 m length are fabricated in composites since several years.
Two-lower-leg-amputees run on carbon-composite spring-like artificial feet as quick as non-amputee athletes.
High-pressure gas cylinders typically about 7–9 litre volume x 300 bar pressure for firemen are nowadays constructed from carbon composite. Type-4-cylinders include metal only as boss that carries the thread to screw in the valve.
On 5 September 2019, HMD Global unveiled the Nokia 6.2 and Nokia 7.2 which are claimed to be using polymer composite for the frames.
Overview | Composite material | Wikipedia | 457 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Composite materials are created from individual materials. These individual materials are known as constituent materials, and there are two main categories of it. One is the matrix (binder) and the other reinforcement. A portion of each kind is needed at least. The reinforcement receives support from the matrix as the matrix surrounds the reinforcement and maintains its relative positions. The properties of the matrix are improved as the reinforcements impart their exceptional physical and mechanical properties. The mechanical properties become unavailable from the individual constituent materials by synergism. At the same time, the designer of the product or structure receives options to choose an optimum combination from the variety of matrix and strengthening materials.
To shape the engineered composites, it must be formed. The reinforcement is placed onto the mould surface or into the mould cavity. Before or after this, the matrix can be introduced to the reinforcement. The matrix undergoes a melding event which sets the part shape necessarily. This melding event can happen in several ways, depending upon the matrix nature, such as solidification from the melted state for a thermoplastic polymer matrix composite or chemical polymerization for a thermoset polymer matrix.
According to the requirements of end-item design, various methods of moulding can be used. The natures of the chosen matrix and reinforcement are the key factors influencing the methodology. The gross quantity of material to be made is another main factor. To support high capital investments for rapid and automated manufacturing technology, vast quantities can be used. Cheaper capital investments but higher labour and tooling expenses at a correspondingly slower rate assists the small production quantities.
Many commercially produced composites use a polymer matrix material often called a resin solution. There are many different polymers available depending upon the starting raw ingredients. There are several broad categories, each with numerous variations. The most common are known as polyester, vinyl ester, epoxy, phenolic, polyimide, polyamide, polypropylene, PEEK, and others. The reinforcement materials are often fibres but also commonly ground minerals. The various methods described below have been developed to reduce the resin content of the final product, or the fibre content is increased. As a rule of thumb, lay up results in a product containing 60% resin and 40% fibre, whereas vacuum infusion gives a final product with 40% resin and 60% fibre content. The strength of the product is greatly dependent on this ratio. | Composite material | Wikipedia | 492 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Martin Hubbe and Lucian A Lucia consider wood to be a natural composite of cellulose fibres in a matrix of lignin.
Cores in composites
Several layup designs of composite also involve a co-curing or post-curing of the prepreg with many other media, such as foam or honeycomb. Generally, this is known as a sandwich structure. This is a more general layup for the production of cowlings, doors, radomes or non-structural parts.
Open- and closed-cell-structured foams like polyvinyl chloride, polyurethane, polyethylene, or polystyrene foams, balsa wood, syntactic foams, and honeycombs are generally utilized core materials. Open- and closed-cell metal foam can also be utilized as core materials. Recently, 3D graphene structures ( also called graphene foam) have also been employed as core structures. A recent review by Khurram and Xu et al., have provided the summary of the state-of-the-art techniques for fabrication of the 3D structure of graphene, and the examples of the use of these foam like structures as a core for their respective polymer composites.
Semi-crystalline polymers
Although the two phases are chemically equivalent, semi-crystalline polymers can be described both quantitatively and qualitatively as composite materials. The crystalline portion has a higher elastic modulus and provides reinforcement for the less stiff, amorphous phase. Polymeric materials can range from 0% to 100% crystallinity aka volume fraction depending on molecular structure and thermal history. Different processing techniques can be employed to vary the percent crystallinity in these materials and thus the mechanical properties of these materials as described in the physical properties section. This effect is seen in a variety of places from industrial plastics like polyethylene shopping bags to spiders which can produce silks with different mechanical properties. In many cases these materials act like particle composites with randomly dispersed crystals known as spherulites. However they can also be engineered to be anisotropic and act more like fiber reinforced composites. In the case of spider silk, the properties of the material can even be dependent on the size of the crystals, independent of the volume fraction. Ironically, single component polymeric materials are some of the most easily tunable composite materials known. | Composite material | Wikipedia | 483 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Methods of fabrication
Normally, the fabrication of composite includes wetting, mixing or saturating the reinforcement with the matrix. The matrix is then induced to bind together (with heat or a chemical reaction) into a rigid structure. Usually, the operation is done in an open or closed forming mould. However, the order and ways of introducing the constituents alters considerably. Composites fabrication is achieved by a wide variety of methods, including advanced fibre placement (automated fibre placement), fibreglass spray lay-up process, filament winding, lanxide process, tailored fibre placement, tufting, and z-pinning.
Overview of mould
The reinforcing and matrix materials are merged, compacted, and cured (processed) within a mould to undergo a melding event. The part shape is fundamentally set after the melding event. However, under particular process conditions, it can deform. The melding event for a thermoset polymer matrix material is a curing reaction that is caused by the possibility of extra heat or chemical reactivity such as an organic peroxide. The melding event for a thermoplastic polymeric matrix material is a solidification from the melted state. The melding event for a metal matrix material such as titanium foil is a fusing at high pressure and a temperature near the melting point.
It is suitable for many moulding methods to refer to one mould piece as a "lower" mould and another mould piece as an "upper" mould. Lower and upper does not refer to the mould's configuration in space, but the different faces of the moulded panel. There is always a lower mould, and sometimes an upper mould in this convention. Part construction commences by applying materials to the lower mould. Lower mould and upper mould are more generalized descriptors than more common and specific terms such as male side, female side, a-side, b-side, tool side, bowl, hat, mandrel, etc. Continuous manufacturing utilizes a different nomenclature.
Usually, the moulded product is referred to as a panel. It can be referred to as casting for certain geometries and material combinations. It can be referred to as a profile for certain continuous processes. Some of the processes are autoclave moulding, vacuum bag moulding, pressure bag moulding, resin transfer moulding, and light resin transfer moulding. | Composite material | Wikipedia | 501 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Other fabrication methods
Other types of fabrication include casting, centrifugal casting, braiding (onto a former), continuous casting, filament winding, press moulding, transfer moulding, pultrusion moulding, and slip forming. There are also forming capabilities including CNC filament winding, vacuum infusion, wet lay-up, compression moulding, and thermoplastic moulding, to name a few. The practice of curing ovens and paint booths is also required for some projects.
Finishing methods
The composite parts finishing is also crucial in the final design. Many of these finishes will involve rain-erosion coatings or polyurethane coatings.
Tooling
The mould and mould inserts are referred to as "tooling". The mould/tooling can be built from different materials. Tooling materials include aluminium, carbon fibre, invar, nickel, reinforced silicone rubber and steel. The tooling material selection is normally based on, but not limited to, the coefficient of thermal expansion, expected number of cycles, end item tolerance, desired or expected surface condition, cure method, glass transition temperature of the material being moulded, moulding method, matrix, cost, and other various considerations.
Physical properties
Usually, the composite's physical properties are not isotropic (independent of the direction of applied force) in nature. But they are typically anisotropic (different depending on the direction of the applied force or load). For instance, the composite panel's stiffness will usually depend upon the orientation of the applied forces and/or moments. The composite's strength is bounded by two loading conditions, as shown in the plot to the right.
Isostrain rule of mixtures
If both the fibres and matrix are aligned parallel to the loading direction, the deformation of both phases will be the same (assuming there is no delamination at the fibre-matrix interface). This isostrain condition provides the upper bound for composite strength, and is determined by the rule of mixtures:
where EC is the effective composite Young's modulus, and Vi and Ei are the volume fraction and Young's moduli, respectively, of the composite phases. | Composite material | Wikipedia | 462 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
For example, a composite material made up of α and β phases as shown in the figure to the right under isostrain, the Young's modulus would be as follows:where Vα and Vβ are the respective volume fractions of each phase.
This can be derived by considering that in the isostrain case, Assuming that the composite has a uniform cross section, the stress on the composite is a weighted average between the two phases, The stresses in the individual phases are given by Hooke's Law, Combining these equations gives that the overall stress in the composite is Then it can be shown that
Isostress rule of mixtures
The lower bound is dictated by the isostress condition, in which the fibres and matrix are oriented perpendicularly to the loading direction:and now the strains become a weighted averageRewriting Hooke's Law for the individual phases This leads toFrom the definition of Hooke's Lawand, in general,
Following the example above, if one had a composite material made up of α and β phases under isostress conditions as shown in the figure to the right, the composition Young's modulus would be: The isostrain condition implies that under an applied load, both phases experience the same strain but will feel different stress. Comparatively, under isostress conditions both phases will feel the same stress but the strains will differ between each phase. A generalized equation for any loading condition between isostrain and isostress can be written as:
where X is a material property such as modulus or stress, c, m, and r stand for the properties of the composite, matrix, and reinforcement materials respectively, and n is a value between 1 and −1.
The above equation can be further generalized beyond a two phase composite to an m-component system: | Composite material | Wikipedia | 370 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Though composite stiffness is maximized when fibres are aligned with the loading direction, so is the possibility of fibre tensile fracture, assuming the tensile strength exceeds that of the matrix. When a fibre has some angle of misorientation θ, several fracture modes are possible. For small values of θ the stress required to initiate fracture is increased by a factor of (cos θ)−2 due to the increased cross-sectional area (A cos θ) of the fibre and reduced force (F/cos θ) experienced by the fibre, leading to a composite tensile strength of σparallel /cos2 θ where σparallel is the tensile strength of the composite with fibres aligned parallel with the applied force.
Intermediate angles of misorientation θ lead to matrix shear failure. Again the cross sectional area is modified but since shear stress is now the driving force for failure the area of the matrix parallel to the fibres is of interest, increasing by a factor of 1/sin θ. Similarly, the force parallel to this area again decreases (F/cos θ) leading to a total tensile strength of τmy /sin θ cos θ where τmy is the matrix shear strength.
Finally, for large values of θ (near π/2) transverse matrix failure is the most likely to occur, since the fibres no longer carry the majority of the load. Still, the tensile strength will be greater than for the purely perpendicular orientation, since the force perpendicular to the fibres will decrease by a factor of 1/sin θ and the area decreases by a factor of 1/sin θ producing a composite tensile strength of σperp /sin2θ where σperp is the tensile strength of the composite with fibres align perpendicular to the applied force.
The majority of commercial composites are formed with random dispersion and orientation of the strengthening fibres, in which case the composite Young's modulus will fall between the isostrain and isostress bounds. However, in applications where the strength-to-weight ratio is engineered to be as high as possible (such as in the aerospace industry), fibre alignment may be tightly controlled.
Panel stiffness is also dependent on the design of the panel. For instance, the fibre reinforcement and matrix used, the method of panel build, thermoset versus thermoplastic, and type of weave. | Composite material | Wikipedia | 491 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
In contrast to composites, isotropic materials (for example, aluminium or steel), in standard wrought forms, possess the same stiffness typically despite the directional orientation of the applied forces and/or moments. The relationship between forces/moments and strains/curvatures for an isotropic material can be described with the following material properties: Young's Modulus, the shear modulus, and the Poisson's ratio, in relatively simple mathematical relationships. For the anisotropic material, it needs the mathematics of a second-order tensor and up to 21 material property constants. For the special case of orthogonal isotropy, there are three distinct material property constants for each of Young's Modulus, Shear Modulus and Poisson's ratio—a total of 9 constants to express the relationship between forces/moments and strains/curvatures.
Techniques that take benefit of the materials' anisotropic properties involve mortise and tenon joints (in natural composites such as wood) and pi joints in synthetic composites.
Mechanical properties of composites
Particle reinforcement
In general, particle reinforcement is strengthening the composites less than fiber reinforcement. It is used to enhance the stiffness of the composites while increasing the strength and the toughness. Because of their mechanical properties, they are used in applications in which wear resistance is required. For example, hardness of cement can be increased by reinforcing gravel particles, drastically. Particle reinforcement a highly advantageous method of tuning mechanical properties of materials since it is very easy implement while being low cost.
The elastic modulus of particle-reinforced composites can be expressed as,
where E is the elastic modulus, V is the volume fraction. The subscripts c, p and m are indicating composite, particle and matrix, respectively. is a constant can be found empirically.
Similarly, tensile strength of particle-reinforced composites can be expressed as,
where T.S. is the tensile strength, and is a constant (not equal to ) that can be found empirically. | Composite material | Wikipedia | 418 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Continuous fiber reinforcement
In general, continuous fiber reinforcement is implemented by incorporating a fiber as the strong phase into a weak phase, matrix. The reason for the popularity of fiber usage is materials with extraordinary strength can be obtained in their fiber form. Non-metallic fibers are usually showing a very high strength to density ratio compared to metal fibers because of the covalent nature of their bonds. The most famous example of this is carbon fibers that have many applications extending from sports gear to protective equipment to space industries.
The stress on the composite can be expressed in terms of the volume fraction of the fiber and the matrix.
where is the stress, V is the volume fraction. The subscripts c, f and m are indicating composite, fiber and matrix, respectively.
Although the stress–strain behavior of fiber composites can only be determined by testing, there is an expected trend, three stages of the stress–strain curve. The first stage is the region of the stress–strain curve where both fiber and the matrix are elastically deformed. This linearly elastic region can be expressed in the following form.
where is the stress, is the strain, E is the elastic modulus, and V is the volume fraction. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively.
After passing the elastic region for both fiber and the matrix, the second region of the stress–strain curve can be observed. In the second region, the fiber is still elastically deformed while the matrix is plastically deformed since the matrix is the weak phase. The instantaneous modulus can be determined using the slope of the stress–strain curve in the second region. The relationship between stress and strain can be expressed as,
where is the stress, is the strain, E is the elastic modulus, and V is the volume fraction. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. To find the modulus in the second region derivative of this equation can be used since the slope of the curve is equal to the modulus.
In most cases it can be assumed since the second term is much less than the first one.
In reality, the derivative of stress with respect to strain is not always returning the modulus because of the binding interaction between the fiber and matrix. The strength of the interaction between these two phases can result in changes in the mechanical properties of the composite. The compatibility of the fiber and matrix is a measure of internal stress. | Composite material | Wikipedia | 509 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
The covalently bonded high strength fibers (e.g. carbon fibers) experience mostly elastic deformation before the fracture since the plastic deformation can happen due to dislocation motion. Whereas, metallic fibers have more space to plastically deform, so their composites exhibit a third stage where both fiber and the matrix are plastically deforming. Metallic fibers have many applications to work at cryogenic temperatures that is one of the advantages of composites with metal fibers over nonmetallic. The stress in this region of the stress–strain curve can be expressed as,
where is the stress, is the strain, E is the elastic modulus, and V is the volume fraction. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. and are for fiber and matrix flow stresses respectively. Just after the third region the composite exhibit necking. The necking strain of composite is happened to be between the necking strain of the fiber and the matrix just like other mechanical properties of the composites. The necking strain of the weak phase is delayed by the strong phase. The amount of the delay depends upon the volume fraction of the strong phase.
Thus, the tensile strength of the composite can be expressed in terms of the volume fraction.
where T.S. is the tensile strength, is the stress, is the strain, E is the elastic modulus, and V is the volume fraction. The subscripts c, f, and m are indicating composite, fiber, and matrix, respectively. The composite tensile strength can be expressed as
for is less than or equal to (arbitrary critical value of volume fraction)
for is greater than or equal to
The critical value of volume fraction can be expressed as,
Evidently, the composite tensile strength can be higher than the matrix if is greater than .
Thus, the minimum volume fraction of the fiber can be expressed as,
Although this minimum value is very low in practice, it is very important to know since the reason for the incorporation of continuous fibers is to improve the mechanical properties of the materials/composites, and this value of volume fraction is the threshold of this improvement.
The effect of fiber orientation
Aligned fibers
A change in the angle between the applied stress and fiber orientation will affect the mechanical properties of fiber-reinforced composites, especially the tensile strength. This angle, , can be used predict the dominant tensile fracture mechanism. | Composite material | Wikipedia | 492 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
At small angles, , the dominant fracture mechanism is the same as with load-fiber alignment, tensile fracture. The resolved force acting upon the length of the fibers is reduced by a factor of from rotation. . The resolved area on which the fiber experiences the force is increased by a factor of from rotation. . Taking the effective tensile strength to be and the aligned tensile strength .
At moderate angles, , the material experiences shear failure. The effective force direction is reduced with respect to the aligned direction. . The resolved area on which the force acts is . The resulting tensile strength depends on the shear strength of the matrix, .
At extreme angles, , the dominant mode of failure is tensile fracture in the matrix in the perpendicular direction. As in the isostress case of layered composite materials, the strength in this direction is lower than in the aligned direction. The effective areas and forces act perpendicular to the aligned direction so they both scale by . The resolved tensile strength is proportional to the transverse strength, .
The critical angles from which the dominant fracture mechanism changes can be calculated as,
where is the critical angle between longitudinal fracture and shear failure, and is the critical angle between shear failure and transverse fracture.
By ignoring length effects, this model is most accurate for continuous fibers and does not effectively capture the strength-orientation relationship for short fiber reinforced composites. Furthermore, most realistic systems do not experience the local maxima predicted at the critical angles. The Tsai-Hill criterion provides a more complete description of fiber composite tensile strength as a function of orientation angle by coupling the contributing yield stresses: , , and .
Randomly oriented fibers
Anisotropy in the tensile strength of fiber reinforced composites can be removed by randomly orienting the fiber directions within the material. It sacrifices the ultimate strength in the aligned direction for an overall, isotropically strengthened material.
Where K is an empirically determined reinforcement factor; similar to the particle reinforcement equation. For fibers with randomly distributed orientations in a plane, , and for a random distribution in 3D, .
Stiffness and Compliance Elasticity
For real application, most composite is anisotropic material or orthotropic material. The three-dimension stress tensor is required for stress and strain analysis. The stiffness and compliance can be written as follows
and
In order to simplify the 3D stress direction, the plane stress assumption is apply that the out–of–plane stress and out–of–plane strain are insignificant or zero. That is and . | Composite material | Wikipedia | 505 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
The stiffness matrix and compliance matrix can be reduced to
and
For fiber-reinforced composite, the fiber orientation in material affect anisotropic properties of the structure. From characterizing technique i.e. tensile testing, the material properties were measured based on sample (1-2) coordinate system. The tensors above express stress-strain relationship in (1-2) coordinate system. While the known material properties is in the principal coordinate system (x-y) of material. Transforming the tensor between two coordinate system help identify the material properties of the tested sample. The transformation matrix with degree rotation is
for for
Types of fibers and mechanical properties
The most common types of fibers used in industry are glass fibers, carbon fibers, and kevlar due to their ease of production and availability. Their mechanical properties are very important to know, therefore the table of their mechanical properties is given below to compare them with S97 steel. The angle of fiber orientation is very important because of the anisotropy of fiber composites (please see the section "Physical properties" for a more detailed explanation). The mechanical properties of the composites can be tested using standard mechanical testing methods by positioning the samples at various angles (the standard angles are 0°, 45°, and 90°) with respect to the orientation of fibers within the composites. In general, 0° axial alignment makes composites resistant to longitudinal bending and axial tension/compression, 90° hoop alignment is used to obtain resistance to internal/external pressure, and ± 45° is the ideal choice to obtain resistance against pure torsion.
Mechanical properties of fiber composite materials
Carbon fiber & fiberglass composites vs. aluminum alloy and steel
Although strenth and stiffness of steel and aluminum alloys are comparable to fiber composites, specific strength and stiffness of composites (i.e. in relation to their weight) are significantly higher.
Failure
Shock, impact of varying speed, or repeated cyclic stresses can provoke the laminate to separate at the interface between two layers, a condition known as delamination. Individual fibres can separate from the matrix, for example, fibre pull-out. | Composite material | Wikipedia | 434 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
Composites can fail on the macroscopic or microscopic scale. Compression failures can happen at both the macro scale or at each individual reinforcing fibre in compression buckling. Tension failures can be net section failures of the part or degradation of the composite at a microscopic scale where one or more of the layers in the composite fail in tension of the matrix or failure of the bond between the matrix and fibres.
Some composites are brittle and possess little reserve strength beyond the initial onset of failure while others may have large deformations and have reserve energy absorbing capacity past the onset of damage. The distinctions in fibres and matrices that are available and the mixtures that can be made with blends leave a very broad range of properties that can be designed into a composite structure. The most famous failure of a brittle ceramic matrix composite occurred when the carbon-carbon composite tile on the leading edge of the wing of the Space Shuttle Columbia fractured when impacted during take-off. It directed to the catastrophic break-up of the vehicle when it re-entered the Earth's atmosphere on 1 February 2003.
Composites have relatively poor bearing strength compared to metals.
Testing
Composites are tested before and after construction to assist in predicting and preventing failures. Pre-construction testing may adopt finite element analysis (FEA) for ply-by-ply analysis of curved surfaces and predicting wrinkling, crimping and dimpling of composites. Materials may be tested during manufacturing and after construction by various non-destructive methods including ultrasonic, thermography, shearography and X-ray radiography, and laser bond inspection for NDT of relative bond strength integrity in a localized area. | Composite material | Wikipedia | 335 | 157616 | https://en.wikipedia.org/wiki/Composite%20material | Technology | Material and chemical | null |
The peregrine falcon (Falco peregrinus), also known simply as the peregrine, is a cosmopolitan bird of prey (raptor) in the family Falconidae. A large, crow-sized falcon, it has a blue-grey back, barred white underparts, and a black head. The peregrine is renowned for its speed. It can reach over during its characteristic hunting stoop (high-speed dive), making it the fastest animal on the planet. According to a National Geographic TV program, the highest measured speed of a peregrine falcon is . However, radar tracks have never confirmed this, the maximum speed reliably measured is , but nobody has been able to present unimpeachable measurements of speeds even close to the "well-known" . As is typical for bird-eating (avivore) raptors, peregrine falcons are sexually dimorphic, with females being considerably larger than males. Historically, it has also been known as "black-cheeked falcon" in Australia, and "duck hawk" in North America.
The breeding range includes land regions from the Arctic tundra to the tropics. It can be found nearly everywhere on Earth, except extreme polar regions, very high mountains, and most tropical rainforests; the only major ice-free landmass from which it is entirely absent is New Zealand. This makes it the world's most widespread raptor and one of the most widely found wild bird species. In fact, the only land-based bird species found over a larger geographic area owes its success to human-led introduction; the domestic and feral pigeons are both domesticated forms of the rock dove, a major prey species for Eurasian Peregrine populations. Due to their abundance over most other bird species in cities, feral pigeons support many peregrine populations as a staple food source, especially in urban settings. | Peregrine falcon | Wikipedia | 386 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The peregrine is a highly successful example of urban wildlife in much of its range, taking advantage of tall buildings as nest sites and an abundance of prey such as pigeons and ducks. Both the English and scientific names of this species mean "wandering falcon", referring to the migratory habits of many northern populations. A total of 18 or 19 regional subspecies are accepted, which vary in appearance; disagreement existed in the past over whether the distinctive Barbary falcon was represented by two subspecies of Falco peregrinus or was a separate species, F. pelegrinoides, and several of the other subspecies were originally described as species. The genetic differential between them (and also the difference in their appearance) is very small, only about 0.6–0.8% genetically differentiated, showing the divergence is relatively recent, during the time of the Last Ice Age; all the major ornithological authorities now treat the barbary falcon as a subspecies.
Although its diet consists almost exclusively of medium-sized birds, the peregrine will sometimes hunt small mammals, small reptiles, or even insects. Reaching sexual maturity at one year, it mates for life and nests in a scrape, normally on cliff edges or, in recent times, on tall human-made structures. The peregrine falcon became an endangered species in many areas because of the widespread use of certain pesticides, especially DDT. Since the ban on DDT from the early 1970s, populations have recovered, supported by large-scale protection of nesting places and releases to the wild.
The peregrine falcon is a well-respected falconry bird due to its strong hunting ability, high trainability, versatility, and availability via captive breeding. It is effective on most game bird species, from small to large. It has also been used as a religious, royal, or national symbol across multiple eras and areas of human civilization.
Description
The peregrine falcon has a body length of and a wingspan from . The male and female have similar markings and plumage but, as with many birds of prey, the peregrine falcon displays marked sexual dimorphism in size, with the female measuring up to 30% larger than the male. Males weigh and the noticeably larger females weigh . In most subspecies, males weigh less than and females weigh more than , and cases of females weighing about 50% more than their male breeding mates are not uncommon. The standard linear measurements of peregrines are: the wing chord measures , the tail measures and the tarsus measures . | Peregrine falcon | Wikipedia | 510 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The back and the long pointed wings of the adult are usually bluish black to slate grey with indistinct darker barring (see "Subspecies" below); the wingtips are black. The white to rusty underparts are barred with thin clean bands of dark brown or black. The tail, coloured like the back but with thin clean bars, is long, narrow, and rounded at the end with a black tip and a white band at the very end. The top of the head and a "moustache" along the cheeks are black, contrasting sharply with the pale sides of the neck and white throat. The cere is yellow, as are the feet, and the beak and claws are black. The upper beak is notched near the tip, an adaptation which enables falcons to kill prey by severing the spinal column at the neck. An immature bird is much browner, with streaked, rather than barred, underparts, and has a pale bluish cere and orbital ring.
A study shows that their black malar stripe exists to reduce glare from solar radiation, allowing them to see better. Photos from The Macaulay Library and iNaturalist showed that the malar stripe is thicker where there is more solar radiation. That supports the solar glare hypothesis.
Taxonomy and systematics
Falco peregrinus was first described under its current binomial name by English ornithologist Marmaduke Tunstall in his 1771 work Ornithologia Britannica. The scientific name Falco peregrinus is a Medieval Latin phrase that was used by Albertus Magnus in 1225. Peregrinus is Latin, meaning "one from abroad" or "coming from foreign parts". It is likely the name was used as juvenile birds were taken while journeying to their breeding location (rather than from the nest), as falcon nests are often difficult to get at. The Latin term for falcon, , is related to , meaning "sickle", in reference to the silhouette of the falcon's long, pointed wings in flight. | Peregrine falcon | Wikipedia | 418 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The peregrine falcon belongs to a genus whose lineage includes the hierofalcons and the prairie falcon (F. mexicanus). This lineage probably diverged from other falcons towards the end of the Late Miocene or in the Late Pliocene, about 3–8 million years ago (mya). As the peregrine-hierofalcon group includes both Old World and North American species, it is likely that the lineage originated in western Eurasia or Africa. Its relationship to other falcons is not clear, as the issue is complicated by widespread hybridization confounding mtDNA sequence analyses. One genetic lineage of the saker falcon (F. cherrug) is known to have originated from a male saker ancestor producing fertile young with a female peregrine ancestor, and the descendants further breeding with sakers.
Subspecies
Numerous subspecies of Falco peregrinus have been described, with 18 accepted by the IOC World Bird List, and 19 accepted by the 1994 Handbook of the Birds of the World, which considers the Barbary falcon of the Canary Islands and coastal North Africa to be two subspecies (F. p. pelegrinoides and F. p. babylonicus) of Falco peregrinus, rather than a distinct species, F. pelegrinoides. The following map shows the general ranges of these 19 subspecies. | Peregrine falcon | Wikipedia | 276 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
Falco peregrinus anatum, described by Bonaparte in 1838, is known as the American peregrine falcon or "duck hawk"; its scientific name means "duck peregrine falcon". At one time, it was partly included in F. p. leucogenys. It is mainly found in the Rocky Mountains. It was formerly common throughout North America between the tundra and northern Mexico, where current reintroduction efforts are being made to restore the population. Most mature F. p. anatum, except those that breed in more northern areas, winter in their breeding range. Most vagrants that reach western Europe seem to belong to the more northern and strongly migratory F. p. tundrius, only considered distinct since 1968. It is similar to the nominate subspecies but is slightly smaller; adults are somewhat paler and less patterned below, but juveniles are darker and more patterned below. Males weigh , while females weigh . It became regionally extinct in eastern North America in the mid 20th century, and populations there now are hybrids as a result of reintroductions of birds from elsewhere.
Falco peregrinus babylonicus, described by P.L. Sclater in 1861, is found in eastern Iran along the Hindu Kush and the Tian Shan to the Mongolian Altai ranges. A few birds winter in northern and northwestern India, mainly in dry semi-desert habitats. It is paler than F. p. pelegrinoides and similar to a small, pale lanner falcon (Falco biarmicus). Males weigh , while females weigh .
, described by Sharpe in 1873, is also known as the Mediterranean peregrine falcon or the Maltese falcon. It includes F. p. caucasicus and most specimens of the proposed race F. p. punicus, though others may be F. p. pelegrinoides (Barbary falcons), or perhaps the rare hybrids between these two which might occur around Algeria. They occur from the Iberian Peninsula around the Mediterranean, except in arid regions, to the Caucasus. They are non-migratory. It is smaller than the nominate subspecies and the underside usually has a rusty hue. Males weigh around , while females weigh up to . | Peregrine falcon | Wikipedia | 449 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
, described by John Latham in 1790, it was formerly called F. p. leucogenys and includes F. p. caeruleiceps. It breeds in the Arctic tundra of Eurasia from Murmansk Oblast to roughly Yana and Indigirka Rivers, Siberia. It is completely migratory and travels south in winter as far as South Asia and sub-Saharan Africa. It is often seen around wetland habitats. It is paler than the nominate subspecies, especially on the crown. Males weigh , while females weigh .
Falco peregrinus cassini, described by Sharpe in 1873, is also known as the austral peregrine falcon. It includes F. p. kreyenborgi, the pallid falcon, a leucistic colour morph occurring in southernmost South America, which was long believed to be a distinct species. Its range includes South America from Ecuador through Bolivia, northern Argentina and Chile to Tierra del Fuego and the Falkland Islands. It is non-migratory. It is similar to the nominate subspecies, but slightly smaller with a black ear region. The pallid falcon morph F. p. kreyenborgi is medium grey above, has little barring below and has a head pattern like the saker falcon (Falco cherrug), but the ear region is white.
Falco peregrinus ernesti, described by Sharpe in 1894, is found from the Sunda Islands to the Philippines and south to eastern New Guinea and the nearby Bismarck Archipelago. Its geographical separation from F. p. nesiotes requires confirmation. It is non-migratory. It differs from the nominate subspecies in the very dark, dense barring on its underside and its black ear coverts.
Falco peregrinus furuitii, described by Momiyama in 1927, is found on the Izu and Ogasawara Islands south of Honshū, Japan. It is non-migratory. It is very rare and may only remain on a single island. It is a dark form, resembling F. p. pealei in colour, but darker, especially on the tail. | Peregrine falcon | Wikipedia | 429 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
Falco peregrinus japonensis, described by Gmelin in 1788, includes F. p. kleinschmidti, F. p. pleskei, and F. p. harterti, and seems to refer to intergrades with F. p. calidus. It is found from northeast Siberia to Kamchatka (though it is possibly replaced by F. p. pealei on the coast there) and Japan. Northern populations are migratory, while those of Japan are resident. It is similar to the nominate subspecies, but the young are even darker than those of F. p. anatum.
, described by Swainson in 1837, is the Australian peregrine falcon or "black-cheeked falcon". It is found in Australia in all regions except the southwest, where replaced by F. p. submelanogenys; some authorities treat the latter as a synonym of F. p. macropus. It is non-migratory. It is similar to F. p. brookei in appearance, but is slightly smaller and the ear region is entirely black. The feet are proportionally large.
Falco peregrinus madens, described by Ripley and Watson in 1963, is unusual in having some sexual dichromatism. If the Barbary falcon (see below) is considered a distinct species, it is sometimes placed therein. It is found in the Cape Verde Islands and is non-migratory; it is also endangered, with only six to eight pairs surviving. Males have a rufous wash on the crown, nape, ears and back; the underside is conspicuously washed pinkish-brown. Females are tinged rich brown overall, especially on the crown and nape.
Falco peregrinus minor, first described by Bonaparte in 1850. It was formerly often known as F. p. perconfusus. It is sparsely and patchily distributed throughout much of sub-Saharan Africa and widespread in Southern Africa. It apparently reaches north along the Atlantic coast as far as Morocco. It is non-migratory and dark-coloured. This is the smallest subspecies, with smaller males weighing as little as approximately .
, described by Mayr in 1941, is found in Fiji and probably also Vanuatu and New Caledonia. It is non-migratory. | Peregrine falcon | Wikipedia | 466 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
, described by Ridgway in 1873, is Peale's falcon and includes F. p. rudolfi. It is found in the Pacific Northwest of North America, northwards from Puget Sound along the British Columbia coast (including the Haida Gwaii), along the Gulf of Alaska and the Aleutian Islands to the far eastern Bering Sea coast of Russia, and may also occur on the Kuril Islands and the coasts of Kamchatka. It is non-migratory. It is the largest subspecies and it looks like an oversized and darker tundrius or like a strongly barred and large F. p. anatum. The bill is very wide. Juveniles occasionally have pale crowns. Males weigh , while females weigh .
Falco peregrinus pelegrinoides, first described by Temminck in 1829, is found in the Canary Islands through North Africa and the Near East to Mesopotamia. It is most similar to F. p. brookei, but is markedly paler above, with a rusty neck, and is a light buff with reduced barring below. It is smaller than the nominate subspecies; females weigh around .
Falco peregrinus peregrinator, described by Sundevall in 1837, is known as the Indian peregrine falcon, black shaheen, Indian shaheen or shaheen falcon. It was formerly sometimes known as Falco atriceps or Falco shaheen. Its range includes South Asia from across the Indian subcontinent to Sri Lanka and southeastern China. In India, the shaheen falcon is reported from all states except Uttar Pradesh, mainly from rocky and hilly regions. The shaheen falcon is also reported from the Andaman and Nicobar Islands in the Bay of Bengal. It has a clutch size of 3 to 4 eggs, with the chicks fledging time of 48 days with an average nesting success of 1.32 chicks per nest. In India, apart from nesting on cliffs, it has also been recorded as nesting on man-made structures such as buildings and cellphone transmission towers. A population estimate of 40 breeding pairs in Sri Lanka was made in 1996. It is non-migratory and is small and dark, with rufous underparts. In Sri Lanka this species is found to favour the higher hills, while the migrant calidus is more often seen along the coast. | Peregrine falcon | Wikipedia | 471 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
, the nominate (first-named) subspecies, described by Tunstall in 1771, breeds over much of temperate Eurasia between the tundra in the north and the Pyrenees, Mediterranean region and Alpide belt in the south. It is mainly non-migratory in Europe, but migratory in Scandinavia and Asia. Males weigh , while females weigh . It includes F. p. brevirostris, F. p. germanicus, F. p. rhenanus and F. p. riphaeus.
Falco peregrinus radama, described by Hartlaub in 1861, is found in Madagascar and the Comoros. It is non-migratory.
, described by Mathews in 1912, is the Southwest Australian peregrine falcon. It is found in southwestern Australia and is non-migratory. Some authorities consider it a synonym of the widespread Australian subspecies F. p. macropus.
, described by C. M. White in 1968, was at one time included in F. p. leucogenys. It is found in the Arctic tundra of North America to Greenland, and migrates to wintering grounds in Central and South America. Most vagrants that reach western Europe belong to this subspecies, which was previously considered synonymous with F. p. anatum. It is the New World equivalent to F. p. calidus. It is smaller and paler than F. p. anatum; most have a conspicuous white forehead and white in ear region, but the crown and "moustache" are very dark, unlike in F. p. calidus. Juveniles are browner and less grey than in F. p. calidus and paler, sometimes almost sandy, than in F. p. anatum. Males weigh , while females weigh . Despite its current recognition as a valid subspecies, a population genetic study of both pre-decline (i.e., museum) and recovered contemporary populations failed to distinguish F. p. anatum and F. p. tundrius genetically. | Peregrine falcon | Wikipedia | 412 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
Barbary falcon
The Barbary falcon is a subspecies of the peregrine falcon that inhabits parts of North Africa, from the Canary Islands to the Arabian Peninsula. There was discussion concerning the taxonomic status of the bird, with some considering it a subspecies of the peregrine falcon and others considering it a full species with two subspecies.
Compared to the other peregrine falcon subspecies, Barbary falcons have a slimmer body and a distinct plumage pattern. Despite numbers and range of these birds throughout the Canary Islands generally increasing, they are considered endangered, with human interference through falconry and shooting threatening their well-being. Falconry can further complicate the speciation and genetics of these Canary Islands falcons, as the practice promotes genetic mixing between individuals from outside the islands with those originating from the islands. Population density of the Barbary falcons on Tenerife, the biggest of the seven major Canary Islands, was found to be 1.27 pairs/100 km2, with the mean distance between pairs being 5869 ± 3338 m. The falcons were only observed near large and natural cliffs with a mean altitude of 697.6 m. Falcons show an affinity for tall cliffs away from human-mediated establishments and presence.
Barbary falcons have a red neck patch, but otherwise differ in appearance from the peregrine falcon proper merely according to Gloger's rule, relating pigmentation to environmental humidity. The Barbary falcon has a peculiar way of flying, beating only the outer part of its wings as fulmars sometimes do; this also occurs in the peregrine falcon, but less often and far less pronounced. The Barbary falcon's shoulder and pelvis bones are stout by comparison with the peregrine falcon and its feet are smaller. Barbary falcons breed at different times of year than neighboring peregrine falcon subspecies, but they are capable of interbreeding. There is a 0.6–0.7% genetic distance in the peregrine falcon-Barbary falcon ("peregrinoid") complex.
Ecology and behaviour
The peregrine falcon lives mostly along mountain ranges, river valleys, coastlines, and increasingly in cities. In mild-winter regions, it is usually a permanent resident, and some individuals, especially adult males, will remain on the breeding territory. Only populations that breed in Arctic climates typically migrate great distances during the northern winter. | Peregrine falcon | Wikipedia | 492 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The peregrine falcon reaches faster speeds than any other animal on the planet when performing the stoop, which involves soaring to a great height and then diving steeply at speeds of over , hitting one wing of its prey so as not to harm itself on impact. The air pressure from such a dive could possibly damage a bird's lungs, but small bony tubercles on a falcon's nostrils are theorized to guide the powerful airflow away from the nostrils, enabling the bird to breathe more easily while diving by reducing the change in air pressure. To protect their eyes, the falcons use their nictitating membranes (third eyelids) to spread tears and clear debris from their eyes while maintaining vision. The distinctive malar stripe or 'moustache', a dark area of feathers below the eyes, is thought to reduce solar glare and improve contrast sensitivity when targeting fast moving prey in bright light condition; the malar stripe has been found to be wider and more pronounced in regions of the world with greater solar radiation supporting this solar glare hypothesis. Peregrine falcons have a flicker fusion frequency of 129 Hz (cycles per second), very fast for a bird of its size, and much faster than mammals. A study testing the flight physics of an "ideal falcon" found a theoretical speed limit at for low-altitude flight and for high-altitude flight. In 2005, Ken Franklin recorded a falcon stooping at a top speed of .
The life span of peregrine falcons in the wild is up to 19 years 9 months. Mortality in the first year is 59–70%, declining to 25–32% annually in adults. Apart from such anthropogenic threats as collision with human-made objects, the peregrine may be killed by larger hawks and owls. | Peregrine falcon | Wikipedia | 360 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The peregrine falcon is host to a range of parasites and pathogens. It is a vector for Avipoxvirus, Newcastle disease virus, Falconid herpesvirus 1 (and possibly other Herpesviridae), and some mycoses and bacterial infections. Endoparasites include Plasmodium relictum (usually not causing malaria in the peregrine falcon), Strigeidae trematodes, Serratospiculum amaculata (nematode), and tapeworms. Known peregrine falcon ectoparasites are chewing lice, Ceratophyllus garei (a flea), and Hippoboscidae flies (Icosta nigra, Ornithoctona erythrocephala).
Feeding
The peregrine falcon's diet varies greatly and is adapted to available prey in different regions. However, it typically feeds on medium-sized birds such as pigeons and doves, waterfowl, gamebirds, songbirds, parrots, seabirds, and waders. Worldwide, it is estimated that between 1,500 and 2,000 bird species, or roughly a fifth of the world's bird species, are predated somewhere by these falcons. The peregrine falcon preys on the most diverse range of bird species of any raptor in North America, with over 300 species and including nearly 100 shorebirds. Its prey can range from hummingbirds (Selasphorus and Archilochus ssp.) to the sandhill crane, although most prey taken by peregrines weigh between (small passerines) and (ducks, geese, loons, gulls, capercaillies, ptarmigans and other grouse). Smaller hawks (such as sharp-shinned hawks) and owls are regularly predated, as well as smaller falcons such as the American kestrel, merlin and, rarely, other peregrines. | Peregrine falcon | Wikipedia | 400 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
In urban areas, where it tends to nest on tall buildings or bridges, it subsists mostly on a variety of pigeons. Among pigeons, the rock dove or feral pigeon comprises 80% or more of the dietary intake of peregrines. Other common city birds are also taken regularly, including mourning doves, common wood pigeons, common swifts, northern flickers, eurasian collared doves, common starlings, American robins, common blackbirds, and corvids such as magpies, jays or carrion, house, and American crows. Coastal populations of the large subspecies pealei feed almost exclusively on seabirds. In the Brazilian mangrove swamp of Cubatão, a wintering falcon of the subspecies tundrius was observed successfully hunting a juvenile scarlet ibis.
Among mammalian prey species, bats in the genera Eptesicus, Myotis, Pipistrellus and Tadarida are the most common prey taken at night. Though peregrines generally do not prefer terrestrial mammalian prey, in Rankin Inlet, peregrines largely take northern collared lemmings (Dicrostonyx groenlandicus) along with a few Arctic ground squirrels (Urocitellus parryii). Other small mammals including shrews, mice, rats, voles, and squirrels are more seldom taken. Peregrines occasionally take rabbits, mainly young individuals and juvenile hares. Additionally, remains of red fox kits and adult female American marten were found among prey remains. Insects and reptiles such as small snakes make up a small proportion of the diet, and salmonid fish have been taken by peregrines. | Peregrine falcon | Wikipedia | 331 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The peregrine falcon hunts most often at dawn and dusk, when prey are most active, but also nocturnally in cities, particularly during migration periods when hunting at night may become prevalent. Nocturnal migrants taken by peregrines include species as diverse as yellow-billed cuckoo, black-necked grebe, virginia rail, and common quail. The peregrine requires open space in order to hunt, and therefore often hunts over open water, marshes, valleys, fields, and tundra, searching for prey either from a high perch or from the air. Large congregations of migrants, especially species that gather in the open like shorebirds, can be quite attractive to a hunting peregrine. Once prey is spotted, it begins its stoop, folding back the tail and wings, with feet tucked. Prey is typically struck and captured in mid-air; the peregrine falcon strikes its prey with a clenched foot, stunning or killing it with the impact, then turns to catch it in mid-air. If its prey is too heavy to carry, a peregrine will drop it to the ground and eat it there. If they miss the initial strike, peregrines will chase their prey in a twisting flight.
Although previously thought rare, several cases of peregrines contour-hunting, i.e., using natural contours to surprise and ambush prey on the ground, have been reported and even rare cases of prey being pursued on foot. In addition, peregrines have been documented preying on chicks in nests, from birds such as kittiwakes. Prey is plucked before consumption. A 2016 study showed that the presence of peregrines benefits non-preferred species while at the same time causing a decline in its preferred prey. As of 2018, the fastest recorded falcon was at 242 mph (nearly 390 km/h). Researchers at the University of Groningen in the Netherlands and at Oxford University used 3D computer simulations in 2018 to show that the high speed allows peregrines to gain better maneuverability and precision in strikes.
Reproduction | Peregrine falcon | Wikipedia | 417 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The peregrine falcon is sexually mature at one to three years of age, but in larger populations they breed after two to three years of age. A pair mates for life and returns to the same nesting spot annually. The courtship flight includes a mix of aerial acrobatics, precise spirals, and steep dives. The male passes prey it has caught to the female in mid-air. To make this possible, the female actually flies upside-down to receive the food from the male's talons.
During the breeding season, the peregrine falcon is territorial; nesting pairs are usually more than apart, and often much farther, even in areas with large numbers of pairs. The distance between nests ensures sufficient food supply for pairs and their chicks. Within a breeding territory, a pair may have several nesting ledges; the number used by a pair can vary from one or two up to seven in a 16-year period.
The peregrine falcon nests in a scrape, normally on cliff edges. The female chooses a nest site, where she scrapes a shallow hollow in the loose soil, sand, gravel, or dead vegetation in which to lay eggs. No nest materials are added. Cliff nests are generally located under an overhang, on ledges with vegetation. South-facing sites are favoured. In some regions, as in parts of Australia and on the west coast of northern North America, large tree hollows are used for nesting. Before the demise of most European peregrines, a large population of peregrines in central and western Europe used the disused nests of other large birds. In remote, undisturbed areas such as the Arctic, steep slopes and even low rocks and mounds may be used as nest sites. In many parts of its range, peregrines now also nest regularly on tall buildings or bridges; these human-made structures used for breeding closely resemble the natural cliff ledges that the peregrine prefers for its nesting locations. | Peregrine falcon | Wikipedia | 398 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The pair defends the chosen nest site against other peregrines, and often against ravens, herons, and gulls, and if ground-nesting, also such mammals as foxes, wolverines, felids, bears, wolves, and mountain lions. Both nests and (less frequently) adults are predated by larger-bodied raptorial birds like eagles, large owls, or gyrfalcons. The most serious predators of peregrine nests in North America and Europe are the great horned owl and the Eurasian eagle-owl. When reintroductions have been attempted for peregrines, the most serious impediments were these two species of owls routinely picking off nestlings, fledglings and adults by night. Peregrines defending their nests have managed to kill raptors as large as golden eagles and bald eagles (both of which they normally avoid as potential predators) that have come too close to the nest by ambushing them in a full stoop. In one instance, when a snowy owl killed a newly fledged peregrine, the larger owl was in turn killed by a stooping peregrine parent.
The date of egg-laying varies according to locality, but is generally from February to March in the Northern Hemisphere, and from July to August in the Southern Hemisphere, although the Australian subspecies F. p. macropus may breed as late as November, and equatorial populations may nest anytime between June and December. If the eggs are lost early in the nesting season, the female usually lays another clutch, although this is extremely rare in the Arctic due to the short summer season. Generally three to four eggs, but sometimes as few as one or as many as five, are laid in the scrape. The eggs are white to buff with red or brown markings. They are incubated for 29 to 33 days, mainly by the female, with the male also helping with the incubation of the eggs during the day, but only the female incubating them at night. The average number of young found in nests is 2.5, and the average number that fledge is about 1.5, due to the occasional production of infertile eggs and various natural losses of nestlings. | Peregrine falcon | Wikipedia | 450 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
After hatching, the chicks (called "es") are covered with creamy-white down and have disproportionately large feet. The male (called the "") and the female (simply called the "falcon") both leave the nest to gather prey to feed the young. The hunting territory of the parents can extend a radius of from the nest site. Chicks fledge 42 to 46 days after hatching, and remain dependent on their parents for up to two months.
Relationship with humans
Use in falconry
The peregrine falcon is a highly admired falconry bird, and has been used in falconry for more than 3,000 years, beginning with nomads in central Asia. Its advantages in falconry include not only its athleticism and eagerness to hunt, but an equable disposition that leads to it being one of the easier falcons to train. The peregrine falcon has the additional advantage of a natural flight style of circling above the falconer ("waiting on") for game to be flushed, and then performing an effective and exciting high-speed diving stoop to take the quarry. The speed of the stoop not only allows the falcon to catch fast flying birds, it also enhances the falcon's ability to execute maneuvers to catch highly agile prey, and allows the falcon to deliver a knockout blow with a fist-like clenched talon against game that may be much larger than itself.
Additionally the versatility of the species, with agility allowing capture of smaller birds and a strength and attacking style allowing capture of game much larger than themselves, combined with the wide size range of the many peregrine subspecies, means there is a subspecies suitable to almost any size and type of game bird. This size range, evolved to fit various environments and prey species, is from the larger females of the largest subspecies to the smaller males of the smallest subspecies, approximately five to one (approximately 1500 g to 300 g). The males of smaller and medium-sized subspecies, and the females of the smaller subspecies, excel in the taking of swift and agile small game birds such as dove, quail, and smaller ducks. The females of the larger subspecies are capable of taking large and powerful game birds such as the largest of duck species, pheasant, and grouse. | Peregrine falcon | Wikipedia | 467 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
Peregrine falcons handled by falconers are also occasionally used to scare away birds at airports to reduce the risk of bird-plane strikes, improving air-traffic safety. They were also used to intercept homing pigeons during World War II.
Peregrine falcons have been successfully bred in captivity, both for falconry and for release into the wild. Until 2004 nearly all peregrines used for falconry in the US were captive-bred from the progeny of falcons taken before the US Endangered Species Act was enacted and from those few infusions of wild genes available from Canada and special circumstances. Peregrine falcons were removed from the United States' endangered species list in 1999. The successful recovery program was aided by the effort and knowledge of falconers – in collaboration with The Peregrine Fund and state and federal agencies – through a technique called hacking. Finally, after years of close work with the US Fish and Wildlife Service, a limited take of wild peregrines was allowed in 2004, the first wild peregrines taken specifically for falconry in over 30 years.
The development of captive breeding methods has led to peregrines being commercially available for falconry use, thus mostly eliminating the need to capture wild birds for support of falconry. The main reason for taking wild peregrines at this point is to maintain healthy genetic diversity in the breeding lines. Hybrids of peregrines and gyrfalcons are also available that can combine the best features of both species to create what many consider to be the ultimate falconry bird for the taking of larger game such as the sage-grouse. These hybrids combine the greater size, strength, and horizontal speed of the gyrfalcon with the natural propensity to stoop and greater warm weather tolerance of the peregrine.
Today, peregrines are regularly paired in captivity with other species such as the lanner falcon (F. biarmicus) to produce the "perilanner", a bird popular in falconry as it combines the peregrine's hunting skill with the lanner's hardiness, or the gyrfalcon to produce large, strikingly coloured birds for the use of falconers. | Peregrine falcon | Wikipedia | 446 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
Decline due to pesticides
The peregrine falcon became an endangered species over much of its range because of the use of organochlorine pesticides, especially DDT, during the 1950s, '60s, and '70s. Pesticide biomagnification caused organochlorine to build up in the falcons' fat tissues, reducing the amount of calcium in their eggshells. With thinner shells, fewer falcon eggs survived until hatching. In addition, the PCB concentrations found in these falcons is dependent upon the age of the falcon. While high levels are still found in young birds (only a few months old) and even higher concentrations are found in more mature falcons, further increasing in adult peregrine falcons. These pesticides caused falcon prey to also have thinner eggshells (one example of prey being the Black Petrels). In several parts of the world, such as the eastern United States and Belgium, this species became locally extinct as a result. An alternate point of view is that populations in the eastern North America had vanished due to hunting and egg collection. Following the ban of organochlorine pesticides, the reproductive success of Peregrines increased in Scotland in terms of territory occupancy and breeding success, although spatial variation in recovery rates indicate that in some areas Peregrines were also impacted by other factors such as persecution.
Recovery efforts
Peregrine falcon recovery teams breed the species in captivity. The chicks are usually fed through a chute or with a hand puppet mimicking a peregrine's head, so they cannot see to imprint on the human trainers. Then, when they are old enough, the rearing box is opened, allowing the bird to train its wings. As the fledgling gets stronger, feeding is reduced, forcing the bird to learn to hunt. This procedure is called hacking back to the wild. To release a captive-bred falcon, the bird is placed in a special cage at the top of a tower or cliff ledge for some days or so, allowing it to acclimate itself to its future environment.
Worldwide recovery efforts have been remarkably successful. The widespread restriction of DDT use eventually allowed released birds to breed successfully. The peregrine falcon was removed from the U.S. Endangered Species list on 25 August 1999. | Peregrine falcon | Wikipedia | 463 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
Some controversy has existed over the origins of captive breeding stock used by the Peregrine Fund in the recovery of peregrine falcons throughout the contiguous United States. Several peregrine subspecies were included in the breeding stock, including birds of Eurasian origin. Due to the local extinction of the eastern population of Falco peregrinus anatum, its near-extinction in the Midwest, and the limited gene pool within North American breeding stock, the inclusion of non-native subspecies was justified to optimize the genetic diversity found within the species as a whole.
During the 1970s, peregrine falcons in Finland experienced a population bottleneck as a result of large declines associated with bio-accumulation of organochloride pesticides. However, the genetic diversity of peregrines in Finland is similar to other populations, indicating that high dispersal rates have maintained the genetic diversity of this species.
Since peregrine falcon eggs and chicks are still often targeted by illegal poachers, it is common practice not to publicise unprotected nest locations.
Current status
Populations of the peregrine falcon have bounced back in most parts of the world. In the United Kingdom, there has been a recovery of populations since the crash of the 1960s. This has been greatly assisted by conservation and protection work led by the Royal Society for the Protection of Birds. The RSPB estimated that there were 1,402 breeding pairs in the UK in 2011. In Canada, where peregrines were identified as endangered in 1978 (in the Yukon territory of northern Canada that year, only a single breeding pair was identified), the Committee on the Status of Endangered Wildlife in Canada declared the species no longer at risk in December 2017.
Peregrines now breed in many mountainous and coastal areas, especially in the west and north, and nest in some urban areas, capitalising on the urban feral pigeon populations for food. Additionally, falcons benefit from artificial illumination, which allows the raptors to extend their hunting periods into the dusk when natural illumination would otherwise be too low for them to pursue prey. In England, this has allowed them to prey on nocturnal migrants such as redwings, fieldfares, starlings, and woodcocks.
In many parts of the world peregrine falcons have adapted to urban habitats, nesting on cathedrals, skyscraper window ledges, tower blocks, and the towers of suspension bridges. Many of these nesting birds are encouraged, sometimes gathering media attention and often monitored by cameras. | Peregrine falcon | Wikipedia | 497 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
In England, peregrine falcons have become increasingly urban in distribution, particularly in southern areas where inland cliffs suitable as nesting sites are scarce. The first recorded urban breeding pair was observed nesting on the Swansea Guildhall in the 1980s. In Southampton, a nest prevented restoration of mobile telephony services for several months in 2013, after Vodafone engineers despatched to repair a faulty transmitter mast discovered a nest in the mast, and were prevented by the Wildlife and Countryside Act – on pain of a possible prison sentence – from proceeding with repairs until the chicks fledged.
In Oregon, Portland houses ten percent of the state's peregrine nests, despite only covering around 0.1 percent of the state's land area.
Cultural significance
Due to its striking hunting technique, the peregrine has often been associated with aggression and martial prowess. The Ancient Egyptian solar deity Ra was often represented as a man with the head of a peregrine falcon adorned with the solar disk, although most Egyptologists agree that it is most likely a Lanner falcon. Native Americans of the Mississippian culture (c. 800–1500) used the peregrine, along with several other birds of prey, in imagery as a symbol of "aerial (celestial) power" and buried men of high status in costumes associating to the ferocity of raptorial birds. In the late Middle Ages, the Western European nobility that used peregrines for hunting, considered the bird associated with princes in formal hierarchies of birds of prey, just below the gyrfalcon associated with kings. It was considered "a royal bird, more armed by its courage than its claws". Terminology used by peregrine breeders also used the Old French term , "of noble birth; aristocratic", particularly with the peregrine.
Since 1927, the peregrine falcon has been the official mascot of Bowling Green State University in Bowling Green, Ohio. The 2007 U.S. Idaho state quarter features a peregrine falcon. The peregrine falcon has been designated the official city bird of Chicago. | Peregrine falcon | Wikipedia | 423 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The Peregrine, by J. A. Baker, is widely regarded as one of the best nature books in English written in the twentieth century. Admirers of the book include Robert Macfarlane, Mark Cocker, who regards the book as "one of the most outstanding books on nature in the twentieth century" and Werner Herzog, who called it "the one book I would ask you to read if you want to make films", and said elsewhere "it has prose of the calibre that we have not seen since Joseph Conrad". In the book, Baker recounts, in diary form, his detailed observations of peregrines (and their interaction with other birds) near his home in Chelmsford, Essex, over a single winter from October to April.
An episode of the hour-long TV series Starman in 1986 titled "Peregrine" was about an injured peregrine falcon and the endangered species program. It was filmed with the assistance of the University of California's peregrine falcon project in Santa Cruz. | Peregrine falcon | Wikipedia | 211 | 157626 | https://en.wikipedia.org/wiki/Peregrine%20falcon | Biology and health sciences | Accipitriformes and Falconiformes | null |
The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is defined relative to a rotational axis. It is the ratio between the torque applied and the resulting angular acceleration about that axis. It plays the same role in rotational motion as mass does in linear motion. A body's moment of inertia about a particular axis depends both on the mass and its distribution relative to the axis, increasing with mass & distance from the axis.
It is an extensive (additive) property: for a point mass the moment of inertia is simply the mass times the square of the perpendicular distance to the axis of rotation. The moment of inertia of a rigid composite system is the sum of the moments of inertia of its component subsystems (all taken about the same axis). Its simplest definition is the second moment of mass with respect to distance from an axis.
For bodies constrained to rotate in a plane, only their moment of inertia about an axis perpendicular to the plane, a scalar value, matters. For bodies free to rotate in three dimensions, their moments can be described by a symmetric 3-by-3 matrix, with a set of mutually perpendicular principal axes for which this matrix is diagonal and torques around the axes act independently of each other.
In mechanical engineering, simply "inertia" is often used to refer to "inertial mass" or "moment of inertia".
Introduction
When a body is free to rotate around an axis, torque must be applied to change its angular momentum. The amount of torque needed to cause any given angular acceleration (the rate of change in angular velocity) is proportional to the moment of inertia of the body. Moments of inertia may be expressed in units of kilogram metre squared (kg·m2) in SI units and pound-foot-second squared (lbf·ft·s2) in imperial or US units. | Moment of inertia | Wikipedia | 416 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
The moment of inertia plays the role in rotational kinetics that mass (inertia) plays in linear kinetics—both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, and will vary depending on the chosen axis. For a point-like mass, the moment of inertia about some axis is given by , where is the distance of the point from the axis, and is the mass. For an extended rigid body, the moment of inertia is just the sum of all the small pieces of mass multiplied by the square of their distances from the axis in rotation. For an extended body of a regular shape and uniform density, this summation sometimes produces a simple expression that depends on the dimensions, shape and total mass of the object.
In 1673, Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, known as a compound pendulum. The term moment of inertia ("momentum inertiae" in Latin) was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Euler's second law.
The natural frequency of oscillation of a compound pendulum is obtained from the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. Comparison of this natural frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body.
The moment of inertia also appears in momentum, kinetic energy, and in Newton's laws of motion for a rigid body as a physical parameter that combines its shape and mass. There is an interesting difference in the way moment of inertia appears in planar and spatial movement. Planar movement has a single scalar that defines the moment of inertia, while for spatial movement the same calculations yield a 3 × 3 matrix of moments of inertia, called the inertia matrix or inertia tensor. | Moment of inertia | Wikipedia | 444 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
The moment of inertia of a rotating flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. The moment of inertia of an airplane about its longitudinal, horizontal and vertical axes determine how steering forces on the control surfaces of its wings, elevators and rudder(s) affect the plane's motions in roll, pitch and yaw.
Definition
The moment of inertia is defined as the product of mass of section and the square of the distance between the reference axis and the centroid of the section.
The moment of inertia is also defined as the ratio of the net angular momentum of a system to its angular velocity around a principal axis, that is
If the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their outstretched arms or divers curl their bodies into a tuck position during a dive, to spin faster.
If the shape of the body does not change, then its moment of inertia appears in Newton's law of motion as the ratio of an applied torque on a body to the angular acceleration around a principal axis, that is
For a simple pendulum, this definition yields a formula for the moment of inertia in terms of the mass of the pendulum and its distance from the pivot point as,
Thus, the moment of inertia of the pendulum depends on both the mass of a body and its geometry, or shape, as defined by the distance to the axis of rotation.
This simple formula generalizes to define moment of inertia for an arbitrarily shaped body as the sum of all the elemental point masses each multiplied by the square of its perpendicular distance to an axis . An arbitrary object's moment of inertia thus depends on the spatial distribution of its mass.
In general, given an object of mass , an effective radius can be defined, dependent on a particular axis of rotation, with such a value that its moment of inertia around the axis is
where is known as the radius of gyration around the axis.
Examples
Simple pendulum
Mathematically, the moment of inertia of a simple pendulum is the ratio of the torque due to gravity about the pivot of a pendulum to its angular acceleration about that pivot point. For a simple pendulum, this is found to be the product of the mass of the particle with the square of its distance to the pivot, that is
This can be shown as follows: | Moment of inertia | Wikipedia | 510 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
The force of gravity on the mass of a simple pendulum generates a torque around the axis perpendicular to the plane of the pendulum movement. Here is the distance vector from the torque axis to the pendulum center of mass, and is the net force on the mass. Associated with this torque is an angular acceleration, , of the string and mass around this axis. Since the mass is constrained to a circle the tangential acceleration of the mass is . Since the torque equation becomes:
where is a unit vector perpendicular to the plane of the pendulum. (The second to last step uses the vector triple product expansion with the perpendicularity of and .) The quantity is the moment of inertia of this single mass around the pivot point.
The quantity also appears in the angular momentum of a simple pendulum, which is calculated from the velocity of the pendulum mass around the pivot, where is the angular velocity of the mass about the pivot point. This angular momentum is given by
using a similar derivation to the previous equation.
Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield
This shows that the quantity is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values for all of the elements of mass in the body.
Compound pendulums
A compound pendulum is a body formed from an assembly of particles of continuous shape that rotates rigidly around a pivot. Its moment of inertia is the sum of the moments of inertia of each of the particles that it is composed of. The natural frequency () of a compound pendulum depends on its moment of inertia, ,
where is the mass of the object, is local acceleration of gravity, and is the distance from the pivot point to the center of mass of the object. Measuring this frequency of oscillation over small angular displacements provides an effective way of measuring moment of inertia of a body.
Thus, to determine the moment of inertia of the body, simply suspend it from a convenient pivot point so that it swings freely in a plane perpendicular to the direction of the desired moment of inertia, then measure its natural frequency or period of oscillation (), to obtain
where is the period (duration) of oscillation (usually averaged over multiple periods). | Moment of inertia | Wikipedia | 490 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
Center of oscillation
A simple pendulum that has the same natural frequency as a compound pendulum defines the length from the pivot to a point called the center of oscillation of the compound pendulum. This point also corresponds to the center of percussion. The length is determined from the formula,
or
The seconds pendulum, which provides the "tick" and "tock" of a grandfather clock, takes one second to swing from side-to-side. This is a period of two seconds, or a natural frequency of for the pendulum. In this case, the distance to the center of oscillation, , can be computed to be
Notice that the distance to the center of oscillation of the seconds pendulum must be adjusted to accommodate different values for the local acceleration of gravity. Kater's pendulum is a compound pendulum that uses this property to measure the local acceleration of gravity, and is called a gravimeter.
Measuring moment of inertia
The moment of inertia of a complex system such as a vehicle or airplane around its vertical axis can be measured by suspending the system from three points to form a trifilar pendulum. A trifilar pendulum is a platform supported by three wires designed to oscillate in torsion around its vertical centroidal axis. The period of oscillation of the trifilar pendulum yields the moment of inertia of the system.
Moment of inertia of area
Moment of inertia of area is also known as the second moment of area and its physical meaning is completely different from the mass moment of inertia.
These calculations are commonly used in civil engineering for structural design of beams and columns. Cross-sectional areas calculated for vertical moment of the x-axis and horizontal moment of the y-axis .
Height (h) and breadth (b) are the linear measures, except for circles, which are effectively half-breadth derived,
Sectional areas moment calculated thus
Square:
Rectangular: and;
Triangular:
Circular:
Motion in a fixed plane
Point mass
The moment of inertia about an axis of a body is calculated by summing for every particle in the body, where is the perpendicular distance to the specified axis. To see how moment of inertia arises in the study of the movement of an extended body, it is convenient to consider a rigid assembly of point masses. (This equation can be used for axes that are not principal axes provided that it is understood that this does not fully describe the moment of inertia.) | Moment of inertia | Wikipedia | 505 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
Consider the kinetic energy of an assembly of masses that lie at the distances from the pivot point , which is the nearest point on the axis of rotation. It is the sum of the kinetic energy of the individual masses,
This shows that the moment of inertia of the body is the sum of each of the terms, that is
Thus, moment of inertia is a physical property that combines the mass and distribution of the particles around the rotation axis. Notice that rotation about different axes of the same body yield different moments of inertia.
The moment of inertia of a continuous body rotating about a specified axis is calculated in the same way, except with infinitely many point particles. Thus the limits of summation are removed, and the sum is written as follows:
Another expression replaces the summation with an integral,
Here, the function gives the mass density at each point , is a vector perpendicular to the axis of rotation and extending from a point on the rotation axis to a point in the solid, and the integration is evaluated over the volume of the body . The moment of inertia of a flat surface is similar with the mass density being replaced by its areal mass density with the integral evaluated over its area.
Note on second moment of area: The moment of inertia of a body moving in a plane and the second moment of area of a beam's cross-section are often confused. The moment of inertia of a body with the shape of the cross-section is the second moment of this area about the -axis perpendicular to the cross-section, weighted by its density. This is also called the polar moment of the area, and is the sum of the second moments about the - and -axes. The stresses in a beam are calculated using the second moment of the cross-sectional area around either the -axis or -axis depending on the load.
Examples
The moment of inertia of a compound pendulum constructed from a thin disc mounted at the end of a thin rod that oscillates around a pivot at the other end of the rod, begins with the calculation of the moment of inertia of the thin rod and thin disc about their respective centers of mass. | Moment of inertia | Wikipedia | 447 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
The moment of inertia of a thin rod with constant cross-section and density and with length about a perpendicular axis through its center of mass is determined by integration. Align the -axis with the rod and locate the origin its center of mass at the center of the rod, then where is the mass of the rod.
The moment of inertia of a thin disc of constant thickness , radius , and density about an axis through its center and perpendicular to its face (parallel to its axis of rotational symmetry) is determined by integration. Align the -axis with the axis of the disc and define a volume element as , then where is its mass.
The moment of inertia of the compound pendulum is now obtained by adding the moment of inertia of the rod and the disc around the pivot point as, where is the length of the pendulum. Notice that the parallel axis theorem is used to shift the moment of inertia from the center of mass to the pivot point of the pendulum.
A list of moments of inertia formulas for standard body shapes provides a way to obtain the moment of inertia of a complex body as an assembly of simpler shaped bodies. The parallel axis theorem is used to shift the reference point of the individual bodies to the reference point of the assembly.
As one more example, consider the moment of inertia of a solid sphere of constant density about an axis through its center of mass. This is determined by summing the moments of inertia of the thin discs that can form the sphere whose centers are along the axis chosen for consideration. If the surface of the sphere is defined by the equation
then the square of the radius of the disc at the cross-section along the -axis is
Therefore, the moment of inertia of the sphere is the sum of the moments of inertia of the discs along the -axis,
where is the mass of the sphere.
Rigid body
If a mechanical system is constrained to move parallel to a fixed plane, then the rotation of a body in the system occurs around an axis parallel to this plane. In this case, the moment of inertia of the mass in this system is a scalar known as the polar moment of inertia. The definition of the polar moment of inertia can be obtained by considering momentum, kinetic energy and Newton's laws for the planar movement of a rigid system of particles. | Moment of inertia | Wikipedia | 488 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
If a system of particles, , are assembled into a rigid body, then the momentum of the system can be written in terms of positions relative to a reference point , and absolute velocities :
where is the angular velocity of the system and is the velocity of .
For planar movement the angular velocity vector is directed along the unit vector which is perpendicular to the plane of movement. Introduce the unit vectors from the reference point to a point , and the unit vector , so
This defines the relative position vector and the velocity vector for the rigid system of the particles moving in a plane.
Note on the cross product: When a body moves parallel to a ground plane, the trajectories of all the points in the body lie in planes parallel to this ground plane. This means that any rotation that the body undergoes must be around an axis perpendicular to this plane. Planar movement is often presented as projected onto this ground plane so that the axis of rotation appears as a point. In this case, the angular velocity and angular acceleration of the body are scalars and the fact that they are vectors along the rotation axis is ignored. This is usually preferred for introductions to the topic. But in the case of moment of inertia, the combination of mass and geometry benefits from the geometric properties of the cross product. For this reason, in this section on planar movement the angular velocity and accelerations of the body are vectors perpendicular to the ground plane, and the cross product operations are the same as used for the study of spatial rigid body movement.
Angular momentum
The angular momentum vector for the planar movement of a rigid system of particles is given by
Use the center of mass as the reference point so
and define the moment of inertia relative to the center of mass as
then the equation for angular momentum simplifies to
The moment of inertia about an axis perpendicular to the movement of the rigid system and through the center of mass is known as the polar moment of inertia. Specifically, it is the second moment of mass with respect to the orthogonal distance from an axis (or pole).
For a given amount of angular momentum, a decrease in the moment of inertia results in an increase in the angular velocity. Figure skaters can change their moment of inertia by pulling in their arms. Thus, the angular velocity achieved by a skater with outstretched arms results in a greater angular velocity when the arms are pulled in, because of the reduced moment of inertia. A figure skater is not, however, a rigid body. | Moment of inertia | Wikipedia | 511 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
Kinetic energy
The kinetic energy of a rigid system of particles moving in the plane is given by
Let the reference point be the center of mass of the system so the second term becomes zero, and introduce the moment of inertia so the kinetic energy is given by
The moment of inertia is the polar moment of inertia of the body.
Newton's laws
Newton's laws for a rigid system of particles, , can be written in terms of a resultant force and torque at a reference point , to yield
where denotes the trajectory of each particle.
The kinematics of a rigid body yields the formula for the acceleration of the particle in terms of the position and acceleration of the reference particle as well as the angular velocity vector and angular acceleration vector of the rigid system of particles as,
For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors from the reference point to a point and the unit vectors , so
This yields the resultant torque on the system as
where , and is the unit vector perpendicular to the plane for all of the particles .
Use the center of mass as the reference point and define the moment of inertia relative to the center of mass , then the equation for the resultant torque simplifies to
Motion in space of a rigid body, and the inertia matrix
The scalar moments of inertia appear as elements in a matrix when a system of particles is assembled into a rigid body that moves in three-dimensional space. This inertia matrix appears in the calculation of the angular momentum, kinetic energy and resultant torque of the rigid system of particles.
Let the system of particles, be located at the coordinates with velocities relative to a fixed reference frame. For a (possibly moving) reference point , the relative positions are
and the (absolute) velocities are
where is the angular velocity of the system, and is the velocity of .
Angular momentum
Note that the cross product can be equivalently written as matrix multiplication by combining the first operand and the operator into a skew-symmetric matrix, , constructed from the components of :
The inertia matrix is constructed by considering the angular momentum, with the reference point of the body chosen to be the center of mass :
where the terms containing () sum to zero by the definition of center of mass. | Moment of inertia | Wikipedia | 501 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
Then, the skew-symmetric matrix obtained from the relative position vector , can be used to define,
where defined by
is the symmetric inertia matrix of the rigid system of particles measured relative to the center of mass .
Kinetic energy
The kinetic energy of a rigid system of particles can be formulated in terms of the center of mass and a matrix of mass moments of inertia of the system. Let the system of particles be located at the coordinates with velocities , then the kinetic energy is
where is the position vector of a particle relative to the center of mass.
This equation expands to yield three terms
Since the center of mass is defined by
, the second term in this equation is zero. Introduce the skew-symmetric matrix so the kinetic energy becomes
Thus, the kinetic energy of the rigid system of particles is given by
where is the inertia matrix relative to the center of mass and is the total mass.
Resultant torque
The inertia matrix appears in the application of Newton's second law to a rigid assembly of particles. The resultant torque on this system is,
where is the acceleration of the particle . The kinematics of a rigid body yields the formula for the acceleration of the particle in terms of the position and acceleration of the reference point, as well as the angular velocity vector and angular acceleration vector of the rigid system as,
Use the center of mass as the reference point, and introduce the skew-symmetric matrix to represent the cross product , to obtain
The calculation uses the identity
obtained from the Jacobi identity for the triple cross product as shown in the proof below:
Thus, the resultant torque on the rigid system of particles is given by
where is the inertia matrix relative to the center of mass.
Parallel axis theorem
The inertia matrix of a body depends on the choice of the reference point. There is a useful relationship between the inertia matrix relative to the center of mass and the inertia matrix relative to another point . This relationship is called the parallel axis theorem.
Consider the inertia matrix obtained for a rigid system of particles measured relative to a reference point , given by
Let be the center of mass of the rigid system, then
where is the vector from the center of mass to the reference point . Use this equation to compute the inertia matrix,
Distribute over the cross product to obtain | Moment of inertia | Wikipedia | 472 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
The first term is the inertia matrix relative to the center of mass. The second and third terms are zero by definition of the center of mass . And the last term is the total mass of the system multiplied by the square of the skew-symmetric matrix constructed from .
The result is the parallel axis theorem,
where is the vector from the center of mass to the reference point .
Note on the minus sign: By using the skew symmetric matrix of position vectors relative to the reference point, the inertia matrix of each particle has the form , which is similar to the that appears in planar movement. However, to make this to work out correctly a minus sign is needed. This minus sign can be absorbed into the term , if desired, by using the skew-symmetry property of .
Scalar moment of inertia in a plane
The scalar moment of inertia, , of a body about a specified axis whose direction is specified by the unit vector and passes through the body at a point is as follows:
where is the moment of inertia matrix of the system relative to the reference point , and is the skew symmetric matrix obtained from the vector .
This is derived as follows. Let a rigid assembly of particles, , have coordinates . Choose as a reference point and compute the moment of inertia around a line L defined by the unit vector through the reference point , . The perpendicular vector from this line to the particle is obtained from by removing the component that projects onto .
where is the identity matrix, so as to avoid confusion with the inertia matrix, and is the outer product matrix formed from the unit vector along the line .
To relate this scalar moment of inertia to the inertia matrix of the body, introduce the skew-symmetric matrix such that , then we have the identity
noting that is a unit vector.
The magnitude squared of the perpendicular vector is
The simplification of this equation uses the triple scalar product identity
where the dot and the cross products have been interchanged. Exchanging products, and simplifying by noting that and are orthogonal:
Thus, the moment of inertia around the line through in the direction is obtained from the calculation
where is the moment of inertia matrix of the system relative to the reference point .
This shows that the inertia matrix can be used to calculate the moment of inertia of a body around any specified rotation axis in the body.
Inertia tensor | Moment of inertia | Wikipedia | 500 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
For the same object, different axes of rotation will have different moments of inertia about those axes. In general, the moments of inertia are not equal unless the object is symmetric about all axes. The moment of inertia tensor is a convenient way to summarize all moments of inertia of an object with one quantity. It may be calculated with respect to any point in space, although for practical purposes the center of mass is most commonly used.
Definition
For a rigid object of point masses , the moment of inertia tensor is given by
Its components are defined as
where
, is equal to 1, 2 or 3 for , , and , respectively,
is the vector to the point mass from the point about which the tensor is calculated and
is the Kronecker delta.
Note that, by the definition, is a symmetric tensor.
The diagonal elements are more succinctly written as
while the off-diagonal elements, also called the , are
Here denotes the moment of inertia around the -axis when the objects are rotated around the x-axis, denotes the moment of inertia around the -axis when the objects are rotated around the -axis, and so on.
These quantities can be generalized to an object with distributed mass, described by a mass density function, in a similar fashion to the scalar moment of inertia. One then has
where is their outer product, E3 is the 3×3 identity matrix, and V is a region of space completely containing the object.
Alternatively it can also be written in terms of the angular momentum operator :
The inertia tensor can be used in the same way as the inertia matrix to compute the scalar moment of inertia about an arbitrary axis in the direction ,
where the dot product is taken with the corresponding elements in the component tensors. A product of inertia term such as is obtained by the computation
and can be interpreted as the moment of inertia around the -axis when the object rotates around the -axis.
The components of tensors of degree two can be assembled into a matrix. For the inertia tensor this matrix is given by,
It is common in rigid body mechanics to use notation that explicitly identifies the , , and -axes, such as and , for the components of the inertia tensor. | Moment of inertia | Wikipedia | 473 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
Alternate inertia convention
There are some CAD and CAE applications such as SolidWorks, Unigraphics NX/Siemens NX and MSC Adams that use an alternate convention for the products of inertia. According to this convention, the minus sign is removed from the product of inertia formulas and instead inserted in the inertia matrix:
Determine inertia convention (Principal axes method)
If one has the inertia data without knowing which inertia convention that has been used, it can be determined if one also has the principal axes. With the principal axes method, one makes inertia matrices from the following two assumptions:
The standard inertia convention has been used .
The alternate inertia convention has been used .
Next, one calculates the eigenvectors for the two matrices. The matrix whose eigenvectors are parallel to the principal axes corresponds to the inertia convention that has been used.
Derivation of the tensor components
The distance of a particle at from the axis of rotation passing through the origin in the direction is , where is unit vector. The moment of inertia on the axis is
Rewrite the equation using matrix transpose:
where E3 is the 3×3 identity matrix.
This leads to a tensor formula for the moment of inertia
For multiple particles, we need only recall that the moment of inertia is additive in order to see that this formula is correct.
Inertia tensor of translation
Let be the inertia tensor of a body calculated at its center of mass, and be the displacement vector of the body. The inertia tensor of the translated body respect to its original center of mass is given by:
where is the body's mass, E3 is the 3 × 3 identity matrix, and is the outer product.
Inertia tensor of rotation
Let be the matrix that represents a body's rotation. The inertia tensor of the rotated body is given by:
Inertia matrix in different reference frames
The use of the inertia matrix in Newton's second law assumes its components are computed relative to axes parallel to the inertial frame and not relative to a body-fixed reference frame. This means that as the body moves the components of the inertia matrix change with time. In contrast, the components of the inertia matrix measured in a body-fixed frame are constant. | Moment of inertia | Wikipedia | 487 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
Body frame
Let the body frame inertia matrix relative to the center of mass be denoted , and define the orientation of the body frame relative to the inertial frame by the rotation matrix , such that,
where vectors in the body fixed coordinate frame have coordinates in the inertial frame. Then, the inertia matrix of the body measured in the inertial frame is given by
Notice that changes as the body moves, while remains constant.
Principal axes
Measured in the body frame, the inertia matrix is a constant real symmetric matrix. A real symmetric matrix has the eigendecomposition into the product of a rotation matrix and a diagonal matrix , given by
where
The columns of the rotation matrix define the directions of the principal axes of the body, and the constants , , and are called the principal moments of inertia. This result was first shown by J. J. Sylvester (1852), and is a form of Sylvester's law of inertia. The principal axis with the highest moment of inertia is sometimes called the figure axis or axis of figure.
A toy top is an example of a rotating rigid body, and the word top is used in the names of types of rigid bodies. When all principal moments of inertia are distinct, the principal axes through center of mass are uniquely specified and the rigid body is called an asymmetric top. If two principal moments are the same, the rigid body is called a symmetric top and there is no unique choice for the two corresponding principal axes. If all three principal moments are the same, the rigid body is called a spherical top (although it need not be spherical) and any axis can be considered a principal axis, meaning that the moment of inertia is the same about any axis.
The principal axes are often aligned with the object's symmetry axes. If a rigid body has an axis of symmetry of order , meaning it is symmetrical under rotations of about the given axis, that axis is a principal axis. When , the rigid body is a symmetric top. If a rigid body has at least two symmetry axes that are not parallel or perpendicular to each other, it is a spherical top, for example, a cube or any other Platonic solid.
The motion of vehicles is often described in terms of yaw, pitch, and roll which usually correspond approximately to rotations about the three principal axes. If the vehicle has bilateral symmetry then one of the principal axes will correspond exactly to the transverse (pitch) axis. | Moment of inertia | Wikipedia | 508 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
A practical example of this mathematical phenomenon is the routine automotive task of balancing a tire, which basically means adjusting the distribution of mass of a car wheel such that its principal axis of inertia is aligned with the axle so the wheel does not wobble.
Rotating molecules are also classified as asymmetric, symmetric, or spherical tops, and the structure of their rotational spectra is different for each type.
Ellipsoid
The moment of inertia matrix in body-frame coordinates is a quadratic form that defines a surface in the body called Poinsot's ellipsoid. Let be the inertia matrix relative to the center of mass aligned with the principal axes, then the surface
or
defines an ellipsoid in the body frame. Write this equation in the form,
to see that the semi-principal diameters of this ellipsoid are given by
Let a point on this ellipsoid be defined in terms of its magnitude and direction, , where is a unit vector. Then the relationship presented above, between the inertia matrix and the scalar moment of inertia around an axis in the direction , yields
Thus, the magnitude of a point in the direction on the inertia ellipsoid is | Moment of inertia | Wikipedia | 253 | 157700 | https://en.wikipedia.org/wiki/Moment%20of%20inertia | Physical sciences | Classical mechanics | null |
A hybrid vehicle is one that uses two or more distinct types of power, such as submarines that use diesel when surfaced and batteries when submerged. Other means to store energy include pressurized fluid in hydraulic hybrids.
Hybrid powertrains are designed to switch from one power source to another to maximize both fuel efficiency and energy efficiency. In hybrid electric vehicles, for instance, the electric motor is more efficient at producing torque, or turning power, while the combustion engine is better for maintaining high speed. Improved efficiency, lower emissions, and reduced running costs relative to non-hybrid vehicles are three primary benefits of hybridization.
Vehicle types
Two-wheeled and cycle-type vehicles
Mopeds, electric bicycles, and even electric kick scooters are a simple form of a hybrid, powered by an internal combustion engine or electric motor and the rider's muscles. Early prototype motorcycles in the late 19th century used the same principle.
In a parallel hybrid bicycle human and motor torques are mechanically coupled at the pedal or one of the wheels, e.g. using a hub motor, a roller pressing onto a tire, or a connection to a wheel using a transmission element. Most motorized bicycles, mopeds are of this type.
In a series hybrid bicycle (SHB) (a kind of chainless bicycle) the user pedals a generator, charging a battery or feeding the motor, which delivers all of the torque required. They are commercially available, being simple in theory and manufacturing.
The first published prototype of an SHB is by Augustus Kinzel (US Patent 3'884'317) in 1975. In 1994 Bernie Macdonalds conceived the Electrilite SHB with power electronics allowing regenerative braking and pedaling while stationary. In 1995 Thomas Muller designed and built a "Fahrrad mit elektromagnetischem Antrieb" for his 1995 diploma thesis. In 1996 Jürg Blatter and Andreas Fuchs of Berne University of Applied Sciences built an SHB and in 1998 modified a Leitra tricycle (European patent EP 1165188). Until 2005 they built several prototype SH tricycles and quadricycles. In 1999 Harald Kutzke described an "active bicycle": the aim is to approach the ideal bicycle weighing nothing and having no drag by electronic compensation.
A series hybrid electric–petroleum bicycle (SHEPB) is powered by pedals, batteries, a petrol generator, or plug-in charger—providing flexibility and range enhancements over electric-only bicycles. | Hybrid vehicle | Wikipedia | 508 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
A SHEPB prototype made by David Kitson in Australia in 2014 used a lightweight brushless DC electric motor from an aerial drone and small hand-tool sized internal combustion engine, and a 3D printed drive system and lightweight housing, altogether weighing less than 4.5 kg. Active cooling keeps plastic parts from softening. The prototype uses a regular electric bicycle charge port.
Heavy vehicle
Hybrid power trains use diesel–electric or turbo-electric to power railway locomotives, buses, heavy goods vehicles, mobile hydraulic machinery, and ships. A diesel/turbine engine drives an electric generator or hydraulic pump, which powers electric/hydraulic motors—strictly an electric/hydraulic transmission (not a hybrid), unless it can accept power from outside. With large vehicles, conversion losses decrease and the advantages in distributing power through wires or pipes rather than mechanical elements become more prominent, especially when powering multiple drives—e.g. driven wheels or propellers. Until recently most heavy vehicles had little secondary energy storage, e.g. batteries/hydraulic accumulators—excepting non-nuclear submarines, one of the oldest production hybrids, running on diesel while surfaced and batteries when submerged. Both series and parallel setups were used in World War II-era submarines.
Rail transport
Europe
The new Autorail à grande capacité (AGC or high-capacity railcar) built by the Canadian company Bombardier for service in France is diesel/electric motors, using 1500 or 25,000 V on different rail systems. It was tested in Rotterdam, the Netherlands with Railfeeding, a Genesee & Wyoming company.
China
The First Hybrid Evaluating locomotive was designed by rail research center Matrai in 1999 and built in 2000. It was an EMD G12 locomotive upgraded with batteries, a 200 kW diesel generator, and four AC motors.
Japan
Japan's first hybrid train with significant energy storage is the KiHa E200, with roof-mounted lithium-ion batteries.
India
Indian railway launched one of its kind CNG-Diesel hybrid trains in January 2015. The train has a 1400 hp engine which uses fumigation technology. The first of these trains is set to run on the 81 km long Rewari-Rohtak route. CNG is less-polluting alternative for diesel and petrol and is popular as an alternative fuel in India. Already many transport vehicles such as auto-rickshaws and buses run on CNG fuel. | Hybrid vehicle | Wikipedia | 485 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
North America
In the US, General Electric made a locomotive with sodium–nickel chloride (Na-NiCl2) battery storage. They expect ≥10% fuel economy.
Variant diesel electric locomotive include the Green Goat (GG) and Green Kid (GK) switching/yard engines built by Canada's Railpower Technologies, with lead acid (Pba) batteries and 1000 to 2000 hp electric motors, and a new clean-burning ≈160 hp diesel generator. No fuel is wasted for idling: ≈60–85% of the time for these types of locomotives. It is unclear if regenerative braking is used; but in principle, it is easily utilized.
Since these engines typically need extra weight for traction purposes anyway the battery pack's weight is a negligible penalty. The diesel generator and batteries are normally built on an existing "retired" "yard" locomotive's frame. The existing motors and running gear are all rebuilt and reused. Fuel savings of 40–60% and up to 80% pollution reductions are claimed over a "typical" older switching/yard engine. The advantages hybrid cars have for frequent starts and stops and idle periods apply to typical switching yard use. "Green Goat" locomotives have been purchased by Canadian Pacific, BNSF, Kansas City Southern Railway and Union Pacific among others.
Cranes
Railpower Technologies engineers working with TSI Terminal Systems are testing a hybrid diesel–electric power unit with battery storage for use in Rubber Tyred Gantry (RTG) cranes. RTG cranes are typically used for loading and unloading shipping containers onto trains or trucks in ports and container storage yards. The energy used to lift the containers can be partially regained when they are lowered. Diesel fuel and emission reductions of 50–70% are predicted by Railpower engineers. First systems are expected to be operational in 2007.
Road transport, commercial vehicles | Hybrid vehicle | Wikipedia | 375 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
Hybrid systems are regularly in use for trucks, buses and other heavy highway vehicles. Small fleet sizes and installation costs are compensated by fuel savings, with advances such as higher capacity, lowered battery cost, etc. Toyota, Ford, GM and others are introducing hybrid pickups and SUVs. Kenworth Truck Company recently introduced the Kenworth T270 Class 6 that for city usage seems to be competitive. FedEx and others are investing in hybrid delivery vehicles—particularly for city use where hybrid technology may pay off first. FedEx is trialling two delivery trucks with Wrightspeed electric motors and diesel generators; the retrofit kits are claimed to pay for themselves in a few years. The diesel engines run at a constant RPM for peak efficiency.
In 1978 students at Minneapolis, Minnesota's Hennepin Vocational Technical Center, converted a Volkswagen Beetle to a petro-hydraulic hybrid with off-the shelf components. A car rated at 32 mpg was returning 75 mpg with the 60 hp engine replaced by a 16 hp engine, and reached 70 mph.
In the 1990s, engineers at EPA's National Vehicle and Fuel Emissions Laboratory developed a petro-hydraulic powertrain for a typical American sedan car. The test car achieved over 80 mpg on combined EPA city/highway driving cycles. Acceleration was 0-60 mph in 8 seconds, using a 1.9-liter diesel engine. No lightweight materials were used. The EPA estimated that produced in high volumes the hydraulic components would add only $700 to the cost. Under EPA testing, a hydraulic hybrid Ford Expedition returned 32 mpg (7.4 L/100 km) City, and 22 mpg (11 L/100 km) highway. UPS currently has two trucks in service using this technology.
Military off-road vehicles
Since 1985, the US military has been testing serial hybrid Humvees and have found them to deliver faster acceleration, a stealth mode with low thermal signature, near silent operation, and greater fuel economy.
Ships
Ships with both mast-mounted sails and steam engines were an early form of a hybrid vehicle. Another example is the diesel–electric submarine. This runs on batteries when submerged and the batteries can be recharged by the diesel engine when the craft is on the surface.
, there are 550 ships with an average of 1.6 MWh of batteries. The average was 500 kWh in 2016. | Hybrid vehicle | Wikipedia | 479 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
Newer hybrid ship-propulsion schemes include large towing kites manufactured by companies such as SkySails. Towing kites can fly at heights several times higher than the tallest ship masts, capturing stronger and steadier winds.
Aircraft
The Boeing Fuel Cell Demonstrator Airplane has a Proton-Exchange Membrane (PEM) fuel cell/lithium-ion battery hybrid system to power an electric motor, which is coupled to a conventional propeller. The fuel cell provides all power for the cruise phase of flight. During takeoff and climb, the flight segment that requires the most power, the system draws on lightweight lithium-ion batteries.
The demonstrator aircraft is a Dimona motor glider, built by Diamond Aircraft Industries of Austria, which also carried out structural modifications to the aircraft. With a wingspan of , the airplane will be able to cruise at about on power from the fuel cell.
Hybrid FanWings have been designed. A FanWing is created by two engines with the capability to autorotate and landing like a helicopter.
Engine type
Hybrid electric-petroleum vehicles
When the term hybrid vehicle is used, it most often refers to a Hybrid electric vehicle. These encompass such vehicles as the Saturn Vue, Toyota Prius, Toyota Yaris, Toyota Camry Hybrid, Ford Escape Hybrid, Ford Fusion Hybrid, Toyota Highlander Hybrid, Honda Insight, Honda Civic Hybrid, Lexus RX 400h, and 450h, Hyundai Ioniq Hybrid, Hyundai Sonata Hybrid, Hyundai Elantra Hybrid, Kia Sportage Hybrid, Kia Niro Hybrid, Kia Sorento Hybrid and others. A petroleum-electric hybrid most commonly uses internal combustion engines (using a variety of fuels, generally gasoline or Diesel engines) and electric motors to power the vehicle. The energy is stored in the fuel of the internal combustion engine and an electric battery set. There are many types of petroleum-electric hybrid drivetrains, from Full hybrid to Mild hybrid, which offer varying advantages and disadvantages.
William H. Patton filed a patent application for a gasoline-electric hybrid rail-car propulsion system in early 1889, and for a similar hybrid boat propulsion system in mid 1889. There is no evidence that his hybrid boat met with any success, but he built a prototype hybrid tram and sold a small hybrid locomotive. | Hybrid vehicle | Wikipedia | 455 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
In 1899, Henri Pieper developed the world's first petro-electric hybrid automobile. In 1900, Ferdinand Porsche developed a series-hybrid using two motor-in-wheel-hub arrangements with an internal combustion generator set providing the electric power; Porsche's hybrid set two-speed records.
While liquid fuel/electric hybrids date back to the late 19th century, the braking regenerative hybrid was invented by David Arthurs, an electrical engineer from Springdale, Arkansas, in 1978–79. His home-converted Opel GT was reported to return as much as 75 mpg with plans still sold to this original design, and the "Mother Earth News" modified version on their website.
The plug-in-electric-vehicle (PEV) is becoming more and more common. It has the range needed in locations where there are wide gaps with no services. The batteries can be plugged into house (mains) electricity for charging, as well being charged while the engine is running.
Continuously outboard recharged electric vehicle
Some battery electric vehicles can be recharged while the user drives. Such a vehicle establishes contact with an electrified rail, plate, or overhead wires on the highway via an attached conducting wheel or other similar mechanisms (see conduit current collection). The vehicle's batteries are recharged by this process—on the highway—and can then be used normally on other roads until the battery is discharged. For example, some of the battery-electric locomotives used for maintenance trains on the London Underground are capable of this mode of operation.
Developing an infrastructure for battery electric vehicles would provide the advantage of virtually unrestricted highway range. Since many destinations are within 100 km of a major highway, this technology could reduce the need for expensive battery systems. However, private use of the existing electrical system is almost universally prohibited. Besides, the technology for such electrical infrastructure is largely outdated and, outside some cities, not widely distributed (see Conduit current collection, trams, electric rail, trolleys, third rail). Updating the required electrical and infrastructure costs could perhaps be funded by toll revenue or by dedicated transportation taxes.
Hybrid fuel (dual mode) | Hybrid vehicle | Wikipedia | 439 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
In addition to vehicles that use two or more different devices for propulsion, some also consider vehicles that use distinct energy sources or input types ("fuels") using the same engine to be hybrids, although to avoid confusion with hybrids as described above and to use correctly the terms, these are perhaps more correctly described as dual mode vehicles:
Some trolleybuses can switch between an onboard diesel engine and overhead electrical power depending on conditions (see dual-mode bus). In principle, this could be combined with a battery subsystem to create a true plug-in hybrid trolleybus, although , no such design seems to have been announced.
Flexible-fuel vehicles can use a mixture of input fuels mixed in one tank—typically gasoline and ethanol, methanol, or biobutanol.
Bi-fuel vehicle: Liquified petroleum gas and natural gas are very different from petroleum or diesel and cannot be used in the same tanks, so it would be challenging to build an (LPG or NG) flexible fuel system. Instead vehicles are built with two, parallel, fuel systems feeding one engine. For example, some Chevrolet Silverado 2500 HDs can effortlessly switch between petroleum and natural gas, offering a range of over 1000 km (650 miles). While the duplicated tanks cost space in some applications, the increased range, decreased cost of fuel, and flexibility where LPG or CNG infrastructure is incomplete may be a significant incentive to purchase. While the US Natural gas infrastructure is partially incomplete, it is increasing and in 2013 had 2600 CNG stations in place. Rising gas prices may push consumers to purchase these vehicles. In 2013 when gas prices traded around US, the price of gasoline was US, compared to natural gas's . On a per unit of energy comparative basis, this makes natural gas much cheaper than gasoline.
Some vehicles have been modified to use another fuel source if it is available, such as cars modified to run on autogas (LPG) and diesels modified to run on waste vegetable oil that has not been processed into biodiesel.
Power-assist mechanisms for bicycles and other human-powered vehicles are also included (see Motorized bicycle).
Fluid power hybrid | Hybrid vehicle | Wikipedia | 440 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
Hydraulic hybrid and pneumatic hybrid vehicles use an engine or regenerative braking (or both) to charge a pressure accumulator to drive the wheels via hydraulic (liquid) or pneumatic (compressed gas) drive units. In most cases the engine is detached from the drivetrain, serving solely to charge the energy accumulator. The transmission is seamless. Regenerative braking can be used to recover some of the supplied drive energy back into the accumulator.
Petro-air hybrid
A French company, MDI, has designed and has running models of a petro-air hybrid engine car. The system does not use air motors to drive the vehicle, being directly driven by a hybrid engine. The engine uses a mixture of compressed air and gasoline injected into the cylinders. A key aspect of the hybrid engine is the "active chamber", which is a compartment heating air via fuel doubling the energy output. Tata Motors of India assessed the design phase towards full production for the Indian market and moved into "completing detailed development of the compressed air engine into specific vehicle and stationary applications".
Petro-hydraulic hybrid
Petro-hydraulic configurations have been common in trains and heavy vehicles for decades. The auto industry recently focused on this hybrid configuration as it now shows promise for introduction into smaller vehicles.
In petro-hydraulic hybrids, the energy recovery rate is high and therefore the system is more efficient than electric battery charged hybrids using the current electric battery technology, demonstrating a 60% to 70% increase in energy economy in US Environmental Protection Agency (EPA) testing. The charging engine needs only to be sized for average usage with acceleration bursts using the stored energy in the hydraulic accumulator, which is charged when in low energy demanding vehicle operation. The charging engine runs at optimum speed and load for efficiency and longevity. Under tests undertaken by the US Environmental Protection Agency (EPA), a hydraulic hybrid Ford Expedition returned City, and highway. UPS currently has two trucks in service using this technology. | Hybrid vehicle | Wikipedia | 402 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
Although petro-hydraulic hybrid technology has been known for decades and used in trains as well as very large construction vehicles, the high costs of the equipment precluded the systems from lighter trucks and cars. In the modern sense, an experiment proved the viability of small petro-hydraulic hybrid road vehicles in 1978. A group of students at Minneapolis, Minnesota's Hennepin Vocational Technical Center, converted a Volkswagen Beetle car to run as a petro-hydraulic hybrid using off-the-shelf components. A car rated at was returning with the 60 hp engine replaced by a 16 hp engine. The experimental car reached .
In the 1990s, a team of engineers working at EPA's National Vehicle and Fuel Emissions Laboratory succeeded in developing a revolutionary type of petro-hydraulic hybrid powertrain that would propel a typical American sedan car. The test car achieved over 80 mpg on combined EPA city/highway driving cycles. Acceleration was 0-60 mph in 8 seconds, using a 1.9 L diesel engine. No lightweight materials were used. The EPA estimated that produced in high volumes the hydraulic components would add only $700 to the base cost of the vehicle.
The petro-hydraulic hybrid system has a faster and more efficient charge/discharge cycling than petro-electric hybrids and is also cheaper to build. The accumulator vessel size dictates total energy storage capacity and may require more space than an electric battery set. Any vehicle space consumed by a larger size of accumulator vessel may be offset by the need for a smaller sized charging engine, in HP and physical size. | Hybrid vehicle | Wikipedia | 321 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
Research is underway in large corporations and small companies. The focus has now switched to smaller vehicles. The system components were expensive which precluded installation in smaller trucks and cars. A drawback was that the power driving motors were not efficient enough at part load. A British company (Artemis Intelligent Power) made a breakthrough introducing an electronically controlled hydraulic motor/pump, the Digital Displacement® motor/pump. The pump is highly efficient at all speed ranges and loads, giving feasibility to small applications of petro-hydraulic hybrids. The company converted a BMW car as a test bed to prove viability. The BMW 530i gave double the mpg in city driving compared to the standard car. This test was using the standard 3,000 cc engine, with a smaller engine the figures would have been more impressive. The design of petro-hydraulic hybrids using well sized accumulators allows downsizing an engine to average power usage, not peak power usage. Peak power is provided by the energy stored in the accumulator. A smaller more efficient constant speed engine reduces weight and liberates space for a larger accumulator.
Current vehicle bodies are designed around the mechanicals of existing engine/transmission setups. It is restrictive and far from ideal to install petro-hydraulic mechanicals into existing bodies not designed for hydraulic setups. One research project's goal is to create a blank paper design new car, to maximize the packaging of petro-hydraulic hybrid components in the vehicle. All bulky hydraulic components are integrated into the chassis of the car. One design has claimed to return 130 mpg in tests by using a large hydraulic accumulator which is also the structural chassis of the car. The small hydraulic driving motors are incorporated within the wheel hubs driving the wheels and reversing to claw-back kinetic braking energy. The hub motors eliminate the need for friction brakes, mechanical transmissions, driveshafts, and U-joints, reducing costs and weight. Hydrostatic drive with no friction brakes is used in industrial vehicles. The aim is 170 mpg in average driving conditions. The energy created by shock absorbers and kinetic braking energy that normally would be wasted assists in charging the accumulator. A small fossil-fuelled piston engine sized for average power use charges the accumulator. The accumulator is sized at running the car for 15 minutes when fully charged. The aim is a fully charged accumulator that will produce a 0-60 mph acceleration speed of under 5 seconds using four wheel drive. | Hybrid vehicle | Wikipedia | 508 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
In January 2011 industry giant Chrysler announced a partnership with the US Environmental Protection Agency (EPA) to design and develop an experimental petro-hydraulic hybrid powertrain suitable for use in large passenger cars. In 2012 an existing production minivan was adapted to the new hydraulic powertrain for assessment.
PSA Peugeot Citroën exhibited an experimental "Hybrid Air" engine at the 2013 Geneva Motor Show. The vehicle uses nitrogen gas compressed by energy harvested from braking or deceleration to power a hydraulic drive which supplements power from its conventional gasoline engine. The hydraulic and electronic components were supplied by Robert Bosch GmbH. Mileage was estimated to be about on the Euro test cycle if installed in a Citroën C3 type of body. PSA Although the car was ready for production and was proven and feasible delivering the claimed results, Peugeot Citroën were unable to attract a major manufacturer to share the high development costs and are shelving the project until a partnership can be arranged.
Electric-human power hybrid vehicle
Another form of a hybrid vehicle are the human-powered electric vehicles. These include such vehicles as the Sinclair C5, Twike, electric bicycles, electric skateboards, and Electric motorcycles and scooters
Hybrid vehicle power train configurations
Parallel hybrid
In a parallel hybrid vehicle, an electric motor and an internal combustion engine are coupled such that they can power the vehicle either individually or together. Most commonly the internal combustion engine, the electric motor and gearbox are coupled by automatically controlled clutches. For electric driving, the clutch between the internal combustion engine is open while the clutch to the gearbox is engaged. While in combustion mode the engine and motor run at the same speed.
The first mass-production parallel hybrid sold outside Japan was the 1st generation Honda Insight.
The Mercedes-Benz E 300 BlueTEC HYBRID released in 2012 only in European markets is a very rare mass-produced diesel hybrid vehicle powered by a Mercedes-Benz OM651 engine developing paired with a electric motor, positioned between the engine and the gearbox, for a combined output of . The vehicle has a fuel consumption rate of .
Mild parallel hybrid
These types use a generally compact electric motor (usually <20 kW) to provide auto-stop/start features and to provide extra power assist during the acceleration, and to generate on the deceleration phase (also known as regenerative braking). | Hybrid vehicle | Wikipedia | 468 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
On-road examples include Honda Civic Hybrid, Honda Insight 2nd generation, Honda CR-Z, Honda Accord Hybrid, Mercedes Benz S400 BlueHYBRID, BMW 7 Series hybrids, General Motors BAS Hybrids, Suzuki S-Cross, Suzuki Wagon R and Smart fortwo with micro hybrid drive.
Power-split or series-parallel hybrid
In a power-split hybrid electric drive train, there are two motors: a traction electric motor and an internal combustion engine. The power from these two motors can be shared to drive the wheels via a power split device, which is a simple planetary gear set. The ratio can be from 100% for the combustion engine to 100% for the traction electric motor, or anything in between. The combustion engine can act as a generator charging the batteries.
Modern versions such as the Toyota Hybrid Synergy Drive have a second electric motor/generator connected to the planetary gear. In cooperation with the traction motor/generator and the power-split device, this provides a continuously variable transmission.
On the open road, the primary power source is the internal combustion engine. When maximum power is required, for example, to overtake, the traction electric motor is used to assist. This increases the available power for a short period, giving the effect of having a larger engine than actually installed. In most applications, the combustion engine is switched off when the car is slow or stationary thereby reducing curbside emissions.
Passenger car installations include Toyota Prius, Ford Escape and Fusion, as well as Lexus RX400h, RX450h, GS450h, LS600h, and CT200h.
Series hybrid
A series- or serial-hybrid vehicle is driven by an electric motor, functioning as an electric vehicle while the battery pack energy supply is sufficient, with an engine tuned for running as a generator when the battery pack is insufficient. There is typically no mechanical connection between the engine and the wheels, and the primary purpose of the range extender is to charge the battery. Series-hybrids have also been referred to as extended range electric vehicle, range-extended electric vehicle, or electric vehicle-extended range (EREV/REEV/EVER).
The BMW i3 with range extender is a production series-hybrid. It operates as an electric vehicle until the battery charge is low, and then activates an engine-powered generator to maintain power, and is also available without the range extender. The Fisker Karma was the first series-hybrid production vehicle. | Hybrid vehicle | Wikipedia | 503 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
When describing cars, the battery of a series-hybrid is usually charged by being plugged in—but a series-hybrid may also allow for a battery to only act as a buffer (and for regeneration purposes), and for the electric motor's power to be supplied constantly by a supporting engine. Series arrangements have been common in diesel-electric locomotives and ships. Ferdinand Porsche effectively invented this arrangement in speed-record-setting racing cars in the early 20th century, such as the Lohner–Porsche Mixte Hybrid. Porsche named his arrangement "System Mixt" and it was a wheel hub motor design, where each of the two front wheels was powered by a separate motor. This arrangement was sometimes referred to as an electric transmission, as the electric generator and driving motor replaced a mechanical transmission. The vehicle could not move unless the internal combustion engine was running.
In 1997 Toyota released the first series-hybrid bus sold in Japan. GM introduced the Chevy Volt series plug-in hybrid in 2010, aiming for an all-electric range of , though this car also has a mechanical connection between the engine and drivetrain. Supercapacitors combined with a lithium-ion battery bank have been used by AFS Trinity in a converted Saturn Vue SUV vehicle. Using supercapacitors they claim up to 150 mpg in a series-hybrid arrangement.
Nissan Note e-power is an example of a series hybrid technology since 2016 in Japan.
Plug-in hybrid electric vehicle
Another subtype of hybrid vehicles is the plug-in hybrid electric vehicle. The plug-in hybrid is usually a general fuel-electric (parallel or serial) hybrid with increased energy storage capacity, usually through a lithium-ion battery, which allows the vehicle to drive on all-electric mode a distance that depends on the battery size and its mechanical layout (series or parallel). It may be connected to mains electricity supply at the end of the journey to avoid charging using the on-board internal combustion engine.
This concept is attractive to those seeking to minimize on-road emissions by avoiding—or at least minimizing—the use of ICE during daily driving. As with pure electric vehicles, the total emissions saving, for example in CO2 terms, is dependent upon the energy source of the electricity generating company. | Hybrid vehicle | Wikipedia | 460 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
For some users, this type of vehicle may also be financially attractive so long as the electrical energy being used is cheaper than the petrol/diesel that they would have otherwise used. Current tax systems in many European countries use mineral oil taxation as a major income source. This is generally not the case for electricity, which is taxed uniformly for the domestic customer, however that person uses it. Some electricity suppliers also offer price benefits for off-peak night users, which may further increase the attractiveness of the plug-in option for commuters and urban motorists.
Road safety for cyclists, pedestrians
A 2009 National Highway Traffic Safety Administration report examined hybrid electric vehicle accidents that involved pedestrians and cyclists and compared them to accidents involving internal combustion engine vehicles (ICEV). The findings showed that, in certain road situations, HEVs are more dangerous for those on foot or bicycle. For accidents where a vehicle was slowing or stopping, backing up, entering, or leaving a parking space (when the sound difference between HEVs and ICEVs is most pronounced), HEVs were twice as likely to be involved in a pedestrian crash than ICEVs. For crashes involving cyclists or pedestrians, there was a higher incident rate for HEVs than ICEVs when a vehicle was turning a corner. However, there was no statistically significant difference between the types of vehicles when they were driving straight.
Several automakers developed electric vehicle warning sounds designed to alert pedestrians to the presence of electric drive vehicles such as hybrid electric vehicle, plug-in hybrid electric vehicles and all-electric vehicles (EVs) travelling at low speeds. Their purpose is to make pedestrians, cyclists, the blind, and others aware of the vehicle's presence while operating in all-electric mode.
Vehicles in the market with such safety devices include the Nissan Leaf, Chevrolet Volt, Fisker Karma, Honda FCX Clarity, Nissan Fuga Hybrid/Infiniti M35, Hyundai ix35 FCEV, Hyundai Sonata Hybrid, 2012 Honda Fit EV, the 2012 Toyota Camry Hybrid, 2012 Lexus CT200h, and all the Prius family of cars.
Environmental issues | Hybrid vehicle | Wikipedia | 424 | 157736 | https://en.wikipedia.org/wiki/Hybrid%20vehicle | Technology | Basics_7 | null |
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