Page 39 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

26 Free Vibration Chap. 2

2.5 PRINCIPLE OF VIRTUAL WORK
We now complement the energy method by another scalar method based on the
principle of virtual work. The principle of virtual work was first formulated by
Johann J. Bernoulli.^ It is especially important for systems of interconnected
bodies of higher DOF, but its brief introduction here will familiarize the reader
with its underlying concepts. Further discussion of the principle is given in later
chapters.
The principle of virtual work is associated with the equilibrium of bodies, and
may be stated as follows: If a system in equilibrium under the action of a set of
forces is given a virtual displacement, the virtual work done by the forces will be zero.
The terms used in this statement are defined as follows: (1) A virtual
displacement ôr is an imaginary infinitesimal variation of the coordinate given
instantaneously. The virtual displacement must be compatible with the constraints
of the system. (2) Virtual work is the work done by all the active forces in a
virtual displacement. Because there is no significant change of geometry associated
with the virtual displacement, the forces acting on the system are assumed to
remain unchanged for the calculation of SW,
The principle of virtual work as formulated by Bernoulli is a static procedure.
Its extension to dynamics was made possible by D’Alembert* (1718-1783), who
introduced the concept of the inertia force. Thus, inertia forces are included as
active forces when dynamic problems are considered.
Example 2.5-1
Using the virtual work method, determine the equation of motion for the rigid beam
of mass M loaded as shown in Fig. 2.5-1.
Solution: Draw the beam in the displaced position 6 and place the forces acting on it,
including the inertia and damping forces. Give the beam a virtual displacement 80
and determine the work done by each force.
/2
-

Inertia force ÔW — —\ m I 86
Spring force 8W = - j ^ 88
(
Damper force 8W = -{cl6)l 88
Uniform load 5IF = / {p^f{t) dx)x 88 p^f{t)~^ 88
•'0 ^
Summing the virtual work and equating to zero gives the differential equation of
motion;

|
2
/
/ yi/ /2 \ . . J 2 /2 722
\

je + (c/2)e + k-^e = pQ^f{t)
^Johann J. Bernoulli (1667-1748), Basel, Switzerland.
* D’Alembert, Traite de dynamique, 1743.