Page 229 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

216 Lagrange’s Equation Chap. 7

A brief development of Lagrange’s equation is presented in Appendix E. We
now demonstrate the use of Lagrange’s equation as applied to some simple
examples.
Example 7.3-1
Using Lagrange’s method, determine the equation of motion for the 3-DOF system
shown in Fig. 7.3-1.
Solution: The kinetic energy here is not a function of so that the term dT/dq- is zero.
We have the following for the kinetic and potential energies:
T = 1 • ? 1 1 • 2 1 1 -2

T-
+ 2W39 3
U = \k^qj + {k2(q2 ~ Q\f + 'i^^iq^ ~ qzf
and T for this problem is a function of only q- and not of q^.
By substituting into Lagrange’s equation for / = 1,
dT . d
Tt
dU
= ^,<7, - ^ 2(^2 - ii)
dq^
and the first equation is
W,<7, + (^, + ^2)91 - ^2^2 = 0
For / = 2, we have
dT d I dT\
dq2
dU
^ = ^2(92 - <?i) - ^3(^3 - di)
and the second equation becomes

^ 2^2 ~ ^2^1 (^2 ■*" ^3)^2 ~ ^3^3
Similarly for / = 3,
dT

dq^ "^3^3 dt \ 5^3 ) “■ '” 3^3
^ = ^3(^3 - di)

with the third equation
- ^3^2 + ^3^3 = 0

|V v v { ^ r ^ A A A A r Q - A A A A . Q Figure 7.3-1.