Page 227 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

214 Lagrange’s Equation Chap. 7

Example 7.2-2
Using the virtual work method, determine the equations of motion for the system
shown in Fig. 7.2-2.
Solution: The generalized coordinates for the problem are x and 6. Sketch the system in
the displaced position with all the active forces and inertia forces. Giving x a virtual
displacement 8x, the virtual work equation is
8W = -\{m^ + m2)X ^ kx^8x - cos 0j 8x

|m 2^^^sin 0 j 8x + F{t) 5x = 0
Because 8x is arbitrary, the preceding equation leads to

(m^ + m2)x + m2^(0 cos 6 - O^sin 6) kx = F(t)
Next, allow a virtual displacement 86. 8W is then

8W = - 2^S^l^ 86 - ^m2Y^6^ 86 - {m2g 6)1^ 86

- {m2 'x cos 6) ^86 + [F(t)cos 6]l 86 = 0
from which we obtain
- I I
m2~j6 -I- m2 2 "^ cos 6 + m2g^sin 6 = F{t)l cos 6
These are nonlinear differential equations, which for small angles simplify to

(m^ + m2)x + kx = F(t)
/ /
m2-^6 + m2^x -I- m2g^6 = ^^(0
which can be expressed by the matrix equation
" / \ “
1 <
(mj + m2) ^ 2 2 X k 0
m 0 /
^22 23 r m 2 ^ 2 <

Figure 7.2-2.