Page 469 - Dynamics of Mechanical Systems

0593_C13_fm Page 450 Monday, May 6, 2002 3:21 PM

450 Dynamics of Mechanical Systems

Equation (13.6.1) then takes the form:

+
+
mx cx kx = F cos pt (13.6.3)
˙
˙˙
0
From Eqs. (13.2.27) through (13.2.31), we see that the solution of Eq. (13.6.3) may be
written as:
x = x + x (13.6.4)
h p

where x and x are:
h
p
−µ
x = e [ cos ω t Bsin ω t] (13.6.5)
+
t
A
h
and
F ) ( [ 2 ]
−
+
p
x = ( 0 ∆ k mp ) cos pt cpsin pt (13.6.6)
where µ, ω, and ∆ are defined by:

/
 k c 2  12
D
D
2
µ = cm , ω =  − 
 m 4 m 2  (13.6.7)
2
∆ = (kmp 2 ) + c p
−
D
22
2
where we assume that c < 4km. As before, the constants A and B in Eq. (13.6.5) are to be
evaluated by auxiliary (initial) conditions on the system.
Observe that the last term of Eq. (13.6.6) can become quite large if the damping coefficient
c is small and if the frequency p of the forcing function is nearly equal to the natural
frequency km of the undamped system. Note, however, that unlike the undamped
system, the presence of damping assures that the amplitude of the oscillation remains
finite. Thus, we see that damping (or effects of friction and viscosity) can have a beneficial
effect in preventing harmful or unbounded vibration of a mechanical system.
The phenomenon of damping in physical systems, however, is generally more complex
than our relatively simple model of Eq. (13.5.1). Indeed, damping is generally a nonlinear
phenomenon that varies from system to system and is generally not well understood.
Theoretical and experimental research on damping is currently a major interest of vibration
analysts.

13.7 Systems with Several Degrees of Freedom

We consider next mechanical systems where more than one body can oscillate. An example
of such a system might be a double mass–spring system as in Figure 13.7.1. This system
has two degrees of freedom as represented by the displacements x and x of the masses.
1 2
Accordingly, we expect to obtain two governing differential equations that must be solved