Page 234 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 7.4 Kinetic Energy, Potential Energy, and Generalized Force 221

The equations of motion for the frame ean then be written as
(
+ m2) 0 0

0 0 Qi

0 0 h ^3
■
1
~ /
\2EL 6EI^ ( \
0
6£7, 4E/i 4 El 2 2 El 2

< >= <M, >
- 7 T ^ “ ? r h
2 EE 4 EE
M2

1 )
7.4 KINETIC ENERGY, POTENTIAL ENERGY, AND GENERALIZED
FORCE IN TERMS OF GENERALIZED COORDINATE q

In the previous section, the use of Lagrange’s equation was demonstrated for
simple problems. We now discuss the quantities T, U, and Q from a more general
point of view.

Kinetic energy. By representing the system by N particles, the instanta­
neous position of each particle can be expressed in terms of the N generalized
coordinates
«•y =

The velocity of the 7 th particle is

1

and the kinetic energy of the system becomes

2
j=\ / = i 7=1 \ / - i
By defining the generalized mass as

^—