1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

532 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM

Hence the damped mechanical system is governed by the differential equation

mU" (t) + cU' (t) + kU (t) = 0.

12.3.2 Forced Vibrations

The vibrations discussed earlier are called free vibrations because all the forces

that affect the motion of the system are internal to the system. We extend our

analysis to cover the case in which an external force F 4 = F (t) acts on the

mass, as depicted in Figure 12.19. Such a force might occur from vibrations of

the support to which the top of the spring is attached or from the effect of a

magnetic field on a mass made of iron. As before, we sum the forces F1, F2, F3,

and F 4 and set this sum equal to the resultant force FR, obtaining

F1 + F2 + F3 + F4 =FR= -KU (t) -cU' (t) + F(t) = mU" (t).

Therefore, the forced motion of t he mechanical system satisfies the nonho-

mogenous linear differential equation

mU" (t) + cU' (t) + kU (t) = F (t). (12-18)

The function F (t) is called the input, or driving force, and the solution U (t)

is called the output, or response. Of particular interest are periodic inputs

F ( t) that can be represented by Fourier series.

F 4 = F(r) t

External force. ~

Figure 12.19 The spring-ma.ss-<lashpot system with a.n external force.