Page 225 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

212 Lagrange’s Equation Chap. 7

a system of a finite number of degrees of freedom by eonsidering its deflection to be
the sum of its normal modes multiplied by generalized coordinates:
y ( x , t ) = </.|(x)(:/|(i) + 4 > 2 { x ) q 2 ( t ) + 4 > 2 ( x ) Q 2 . { t ) + ■ ■ ■
In many problems, only a finite number of normal modes are sufficient, and the series
can be terminated at n terms, thereby reducing the problem to that of a system of n
DOF. For example, the motion of a slender free-free beam struck by a force P at
point (a) can be described in terms of two rigid-body motions of translation and
rotation plus its normal modes of elastic vibration, as shown in Fig. 7.1-6.
y(x, 0 - -f (¡)\{x)q^ + "

P I
I (a)

0 T
0R

01

0 2

03 Figure 7.1-6.

7.2 VIRTUAL WORK

In Chapter 2, the method of virtual work was briefly introduced with examples for
single-DOF problems. The advantage of the virtual work method over the vector
method is considerably greater for multi-DOF systems. For interconnected bodies
of many degrees of freedom, Newton’s vector method is burdened with the
necessity of accounting for all joint and constraint forces in the free-body dia­
grams, whereas these forces are excluded in the virtual work method.
In reviewing the method of virtual work, we summarize the virtual work
equation as
514" = = 0 (7.2-1)
i
where are the applied forces excluding all constraint forces and internal forces
of frictionless joints and are the virtual displacements. By including D ’Alem­
bert’s inertia forces, the procedure is extended to dynamical problems by
the equation
s w = E(^, 8r. = 0 (7.2-2)