Page 478 - Dynamics of Mechanical Systems

0593_C13_fm Page 459 Monday, May 6, 2002 3:21 PM

Introduction to Vibrations 459

Specifically, if x and x are solutions to a linear equation, then a x + a x , where a and a 2
1
1 1
2 2
2
1
are constants, is also a solution.
Linear differential equations arise from assumptions made in the modeling of mechan-
ical systems. Typical of such assumptions are “small displacements” (as with the simple
pendulum) and forces proportional to the displacement (as with linear mass–spring
systems).
An assumption of small displacements means that the modeling, and hence the solution
of the ensuing linear equation, becomes more and more accurate the smaller the displace-
ment becomes. For large displacements, however, the solution becomes less accurate —
that is, less representative of the behavior of the system. In this section, we will explore
the extent to which large displacement affects the solution.
If the displacements are large, the use of linear differential equations to model the
displacement may no longer be appropriate. Instead, we need to use a nonlinear differ-
ential equation and attempt to find a solution to it — or at least an approximate solution.
Therein, however, lies the difficulty — known solutions to nonlinear differential equations
are very few, thus we generally have to resort to approximation methods to obtain a
solution.
We will illustrate such an approach with an analysis of large movements of a simple
pendulum as in Figure 13.9.1. Recall from Eq. (8.4.4) that the governing differential equa-
tion is:

˙˙ g sinθ = 0 (13.9.1)
θ +( ) l
where, as before, θ is the displacement angle and is the pendulum length. The presence
of sinθ in Eq. (13.9.1) makes the equation nonlinear and prevents us from obtaining a
solution by conventional methods; however, we can obtain what is called a first integral
θ
˙
of the equation by multiplying the equation by and integrating. That is,
˙˙˙
θθ +( ) l θ ˙ sinθ = 0 (13.9.2)
g
Thus, we have:

d θ 2) dt −( ) ( dt = 0 (13.9.3)
(
d cosθ)
˙ 2
g l
or

˙ 2
θ 2 −( ) l cosθ = C (13.9.4)
g

θ

FIGURE 13.9.1
The simple pendulum with large
displacement.