Datasets:
vnpthack
/
vnpt_data

Dataset card Data Studio Files
xet
 Community
1
text
stringlengths
0
6.73k
uọ = =(/2)(A+ A*) +ix(A— A*)
= —*#o/2 + i„x(2Ar),
where 4= Ag-+¿4;, and 4Ý = Ag — ¿Ar. Thus we ñnd
An — zo/2
--- Trang 435 ---
Ar = —(0o + +#o/2)/2„. (24.21)
This completely determines 4 and 4Ý, and therefore the complete curve of the
transient solution, in terms of how it begins. Incidentally, we can write the
solution another way if we note that
c® +e~?” =2cos8 and c8 — e~?# = 2isin 6.
W©e may then write the complete solution as
z—e 2 la COS(„Ý + to + 120/2 sỉn ¬ : (24.22)
where œ„ = +/œ — 2/4. Thịis is the mathematical expression for the way an
oscillation dies out. We shall not make direct use of it, but there are a number
of poïints we should like to emphasize that are true in more general cases.
First of all the behavior of such a system with no external force is expressed by a
sum, or superposition, of pure exponentials in tỉme (which we wrote as e?%!), 'This
is a good solution to try in such circumstances. The values of œ may be complex
in general, the imaginary parts representing damping. Finally the intimate
mathematical relation of the sinusoidal and exponentfial function discussed In
Chapter 22 often appears physically as a change from oscillatory to exponential
behavior when some physical parameter (in this case resistance, +) exceeds some
critical value.
--- Trang 436 ---
X}irnoer Sggséormes cn«Ï lïotosr
25-1 Linear diferential equations
In this chapter we shall discuss certain aspects of oscillating systems that are
found somewhat more generally than just in the particular systems we have been
discussing. For our particular system, the diferential equation that we have been
solving is
dỀz da 2
mu + m + Ta0+ = F). (25.1)
Now this particular combination of “operations” on the variable ø has the
interesting property that if we substitute (+) for z, then we get the sum of the
same operations on z and ø; or, if we multiply z by a, then we get just ø times
the same combination. This is easy to prove. Just as a “shorthand” notation,
because we get tired of writing down all those letters in (25.1), we shall use the
symbol (+) instead. When we see this, it means the left-hand side of (25.1),
with z substituted in. With this system oŸ writing, Ù(z + ) would mean the
following:
đˆ(x+ d(z +
L(x+ụ) =m “Œ T9) „mm đŒ $9) muà(z + g). (25.2)
(We underline the Ù so as to remind ourselves that it is not an ordinary function.)
W©e sometimes call this an operator no‡øtion, but 1t makes no diference what we
call it, it is just “shorthand”
Our frst statement was that
Lí +) = L(z) + LỤU): (25.3)
which of course follows from the fact that ø(# + ) = a# + aụ, đ(z + U) /dt —=
dz/dt + dụ/dt, etc.
--- Trang 437 ---
Our second statement was, for constant ø,
T(az) = aE(œ). (25.4)
[Actually, (25.3) and (25.4) are very closely related, because iŸ we put # + #
into (25.3), this is the same as setting ø = 2 in (25.4), and so on.
In more complicated problems, there may be more derivatives, and more
terms in Ù; the question of interest is whether the two equations (25.3) and (25.4)
are maintained or not. If they are, we call such a problem a iZ»eør problem. In
this chapter we shall discuss some of the properties that exist because the system
1s linear, to appreciate the generality of some of the results that we have obtained
in our special analysis of a special equation.
Now let us study some of the properties of linear differential equations, having
illustrated them already with the specific equation (25.1) that we have studied
so closely. "The first property of interest is this: suppose that we have to solve
the diferential equation for a transient, the free oscillation with no driving force.
'That is, we want to solve
L(z) =0. (25.5)
Suppose that, by some hook or crook, we have found a particular solution, which
we shall call z¡. That is, we have an #¡ for which L(z¡) =0. Now we notice
that øz, 1s also a solution to the same equation; we can multiply this special
solution by any constant whatever, and get a new solution. In other words, IŸ we
had a motion of a certain “size,” then a motion ©wice as “big” is again a solution.
Proof: L(a#1) = &E(#1) = a-0 =0.
Next, suppose that, by hook or by crook, we have not only found øøwe solu-
tion #, but also another solution, z¿. (Remember that when we substituted
œ = e?®† for finding the transients, we found f#+»o values for œ, that is, two solutions,
#¡ and #øa.) Now let us show that the combination (# + #a) is also a solution. In
other words, if we put #ø = #1 + #a, # is again a solution of the equation. Why?
Because, if U(z¡) = 0 and (4a) = 0, then E(zi+z2) = E(œi)+ E(z:) = 0+0 = 0.
So if we have found a number of solutions for the motion of a linear system we
can add them together.
Combining these two ideas, we see, of course, that we can also add six of
one and two of the other: IÝ ø is a solution, so is œ#. Therefore any sum of
these tEwo solutions, such as (œ#i + z2), is also a solution. If we happen to
be able to fnd three solutions, then we fñnd that any combination of the three
solutions is again a solution, and so on. Iỳ turns out that the number of what
--- Trang 438 ---
we call ?dependent solufions* that we have obtained for our oscillator problem
is only ưuo. The number of independent solutions that one finds in the general
case depends upon what is called the number of degrees oƒ freedom. We shall
not discuss this in detail now, but if we have a second-order difÑferential equation,
there are only two independent solutions, and we have found both of them; so
we have the most general solution.
Now let us go on to another proposition, which applies to the sibtuation in
which the system is subjected to an outside force. Suppose the equation 1s
L(z) = F(). (25.6)
and suppose that we have found a special solution of it. Let us say that Joe”s
solution is z;, and that E(z;) = Ƒ). Šuppose we want to find yet another
solution; suppose we add to Joe”s solution one of those that was a solution of the
free equation (25.5), say z¡. Then we see by (25.3) that
TE(z„ + #1) = L(z) + L(xì) = F() +0 = F0). (25.7)