Page 226 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 7.2 Virtual Work 213

This later equation leads to Lagrange’s equation when the displacement r, is
expressed in terms of generalized coordinates.
The virtual displacements 8r^ in these equations are arbitrary variations of
the coordinates irrespective of time but compatible with the constraints of the
system. Being an infinitesimal quantity, 8r¡ obeys all the rules of differential
calculus. The difference between 8r¡ and dr¡ is that dr¡ takes place in the time dt,
whereas 8r¿ is an arbitrary number that may be equal to dr¡ but is assigned
instantaneously irrespective of time. Although the virtual displacement 8r is
distinguished from dr, the latter is often substituted for 8r to ensure compatibility
of displacement.
Example 7.2-1
We first illustrate the virtual work method for a problem of statie equilibrium. Figure
7.2-1 shows a double pendulum with generalized coordinates 0, and 6 2 - Determine its
static equilibrium position when a horizontal force P is applied to m 2 .
With the system in its equilibrium position, give 6 2 a virtual displacement 8 6 2
[Fig. 7.2-l(a)] and write the equation for the virtual work 8 W of all the applied forces:
8 W = - ( ^ 2 ^ sin ^ 2 ^ ^ 2 + ( T cos 6 2 )^ ^ ^ 2 ^ ^

From the equilibrium position (with 8 6 2 = 0), give 6 ^ a virtual displacement

50,, as in Fig. 7.2-l(b), and write the equation for 8W:
8 W = - ( m , g sin 0 ,)/5 0 , - (m 2 g s in 0 ,) /5 0 , + ( P cos 0 ,)/5 0 , = 0
These equations lead to the two equilibrium angles, given as
P
tan 0 9 =
m2g

tan 0, =
( m , -r m2)g