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u" + 1 u' + 2u = 2cosrnt, u(0) = 0, u'(0) = 2.
(a) Determine the steady-state part of the solution of this problem.
(b) Find the amplitude A of the steady-state solution in terms of rn.
(c) Plot A versus rn.
(d) Find the maximum value of A and the frequency rn for which it occurs.
? 18. Consider the forced but undamped system described by the initial value problem
u" + u = 3 cos a>t, u(0) = 0, u'(0) = 0.
(a) Find the solution u(t) for rn = 1.
(b) Plot the solution u(t) versus t for rn = 0.7, rn = 0.8, and rn = 0.9. Describe how the
response u(t) changes as rn varies in this interval. What happens as rn takes on values closer and closer to 1? Note that the natural frequency of the unforced system is rn0 = 1.
3.9 Forced Vibrations
? 19. Consider the vibrating system described by the initial value problem
u" + u = 3 cos at, u(0) = 1, u'(0) = 1.
(a) Find the solution for a = 1.
(b) Plot the solution u(t) versus t for a = 0.7, a = 0.8, and a = 0.9. Compare the results with those of Problem 18, that is, describe the effect of the nonzero initial conditions.
? 20. For the initial value problem in Problem 18 plot d versus u for a = 0.7, a = 0.8, and
a = 0.9; that is, draw the phase plot of the solution for these values of a. Use a t interval
that is long enough so the phase plot appears as a closed curve. Mark your curve with arrows to show the direction in which it is traversed as t increases.
Problems 21 through 23 deal with the initial value problem
u" + 0.125u'+ u = F (t), u(0) = 2, d(0) = 0.
In each of these problems:
(a) Plot the given forcing function F(t) versus t and also plot the solution u(t) versus t on the same set of axes. Use a t interval that is long enough so the initial transients are substantially eliminated. Observe the relation between the amplitude and phase of the forcing term and the amplitude and phase of the response. Note that a0 = k/m = 1.
(b) Draw the phase plot of the solution, that is, plot u' versus u.
? 21. F(t) = 3 cos(0.3t)
? 22. F(t) = 3 cos t
? 23. F (t) = 3cos3t
? 24. A spring-mass system with a hardening spring (Problem 32 of Section 3.8) is acted on by
a periodic external force. In the absence of damping, suppose that the displacement of the mass satisfies the initial value problem
u" + u + 1 u3 = cos at, u (0) = 0, u'(0) = 0.
(a) Let a = 1 and plot a computer-generated solution of the given problem. Does the system exhibit a beat?
(b) Plot the solution for several values of a between 1/2 and 2. Describe how the solution changes as a increases.
? 25. Suppose that the system of Problem 24 is modified to include a damping term and that the
resulting initial value problem is
u" + 1F + u + 1 u3 = cos at, u(0) = 0, u'(0) = 0.
(a) Plot a computer-generated solution of the given problem for several values of a between 1/2 and 2 and estimate the amplitude R of the steady response in each case.
(b) Using the data from part (a), plot the graph of R versus a. For what frequency a is the amplitude greatest?
(c) Compare the results of parts (a) and (b) with the corresponding results for the linear spring.
Coddington, E. A., An Introduction to Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice Hall, 1961; New York: Dover, 1989).
There are many books on mechanical vibrations and electric circuits. One that deals with both is: Close, C. M., and Frederick, D. K., Modeling and Analysis of Dynamic Systems (2nd ed.) (Boston: Houghton-Mifflin, 1993).
Chapter 3. Second Order Linear Equations
A classic book on mechanical vibrations is:
Den Hartog, J. P., Mechanical Vibrations (4th ed.) (New York: McGraw-Hill, 1956; New York; Dover, 1985).
A more recent intermediate-level book is:
Thomson, W. T., Theory of Vibrations with Applications (3rd ed.) (Englewood Cliffs, NJ: Prentice Hall, 1988).
An elementary book on electrical circuits is:
Bobrow, L. S., Elementary Linear Circuit Analysis (New York: Oxford University Press, 1996).
Higher Order Linear Equations
The theoretical structure and methods of solution developed in the preceding chapter for second order linear equations extend directly to linear equations of third and higher order. In this chapter we briefly review this generalization, taking particular note of those instances where new phenomena may appear, due to the greater variety of situations that can occur for equations of higher order.
4.1 General Theory of nth Order Linear Equations
An nth order linear differential equation is an equation of the form
dn y dn-1 y dy
W dy + P() -yyi + + Pn-() dy + Pn (t )y = G(t). (1)
We assume that the functions P0,, Pn and G are continuous real-valued functions on some interval I: a < t < ft, and that P0 is nowhere zero in this interval. Then, dividing Eq. (1) by P0(t), we obtain
dny dn-1 y dy
L [y] = dn+Pi (t) dt + ???+pn-() dt + pn(t) y=g(t)- (2)
The linear differential operator L of order n defined by Eq. (2) is similar to the second order operator introduced in Chapter 3. The mathematical theory associated with Eq. (2) is completely analogous to that for the second order linear equation; for this reason we