1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

12.3 • VIBRATIONS I N MECHANICAL SYSTEMS 531

We obtain the differential equation for motion by using Newton's second law,
which states that the resultant of the forces acting on the mass at any instant
satisfies

FR.= ma. (12-17)

The distance from equilibrium at time tis measured by U (t), so the acceleration

a is given by a= U" (t). Applying Equations (12-16) and (12-17) yields

Fn = -kU (t) = mU" (t).

Hence the undamped mechanical system is governed by the linear differential
equation

mU" (t) + kU (t) = 0.

The general solution for an undamped system is

U (t) = A cos wt+ B sin wt, where w = /"f;.

12.3.1 Dampe d System

If we consider frictional forces that slow the motion of the mass, then we say that

the system is damped. To help visualize this situation, we connect a dashpot to

the mass, as indicated in Figure 12.18. For sma.11 velocities we assume that the

frictional force Fs is proportional to the velocity; that is,

F3 = - cU' (t).

The damping constant c must be positive, for if U' (t) > 0, then the mass is

moving downward and hence F3 must point upward, which requires that F3 be

negative. The result of the three forces acting on the mass is given by

F 1 + Fz +Fa= -kU (t) -cU' (t) = mU" (t) = Fn.

k Spring

m Mass

c Dashpot

Figure 12.18 The spring- mass-dashpot system.