Page 241 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

228 Lagrange’s Equation Chap. 7

The virtual work done due to dq2 is
Qi8q2= + A / 2 ^

••• Q2 = [-F2{i - ‘^ ) + m , ] \
It should be noted that the dimension of Qj and Q2 is that of a force.
Example 7.5-3
In Fig. 7.5-3, three forces, Fj, F2, and F3, act at discrete points, Xj, X2, and X3, of a
structure whose displacement is expressed by the equation
n
y(x,t) = E <?,(•«)'/,(0
/ = 1
Determine the generalized force Qj.

^’1
JE 2 1 :

Figure 7.5-3.
Solution: The virtual displacement is
= E <Pi(x) Sq,
/ = 1
and the virtual work due to this displacement is

L F j - l
y=1 \i=I

= E E Fj<Pi(Xj) = E g ,
,=i \y=i / <-i
The generalized force is then equal to SW/Sq^, or
3
G,=
j=i
= Fi9,(xx) + F2<P.{x2) + F^ip.iXi)

REFERENCES

[1] R a y l e i g h , J.W.S. Theory of Sound, Dover Publication, 1 9 4 6 .
[2] G o ldstein, H. Classical Mechanics, Reading, Mass: Addison-Wesley, 1951.
[3] L a n c z o s , C. The Variational Principles of Mechanics, Toronto, Canada: The Univ. of
Toronto Press, 1949.