Page 40 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 2.5 Principle of Virtual Work 27

Figure 2.5-1.
Example 2.5-2
Two simple pendulums are connected together with the bottom mass restricted to
vertical motion in a frictionless guide, as shown in Fig 2.5-2. Because only one
coordinate 6 is necessary, it represents an interconnected single-DOF system. Using
the virtual work method, determine the equation of motion and its natural frequency.
Solution: Sketch the system displaced by a small angle 6 and place on it all forces,
including inertia forces. Next give the coordinate 6 a virtual displacement 80. Due
to this displacement, and m2 will undergo vertical displacements of 186 sin 0
and 2180 sin 0, respectively. (The acceleration of m2 can easily be shown to be
21(0 sin 0 -(- 0^ cos 0), and its virtual work will be an order of infinitesimal, smaller
than that for the gravity force and can be neglected.) Equating the virtual work to
zero, we have
81V = - 80 - (m^g)l 80 sin 0 - (m2g)2l 80 sin 0 = 0

= - \m^l0 (m^ -\- 2m2)g sin ^1/ = 0

Figure 2.5-2. Virtual work of dou­
ble pendulum with motion of m2
restricted along vertical line.