Page 235 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

222 Lagrange’s Equation Chap. 7

the kinetic energy can be written as
N N
2 ^
1=1y=i (7.4-1)
= j{QV[m]{Q)

Potential energy. In a conservative system, the forces can be derived from
the potential energy U, which is a function of the generalized coordinates
Expanding U in a Taylor series about the equilibrium position, we have for a
system of n degrees of freedom

d^U
U QjQi + * * *
E 37T '7; + 9 E
In this expression, Uq is an arbitrary constant that we can set equal to zero.

The derivatives of U are evaluated at the equilibrium position 0 and are constants
when the q/s are small quantities equal to zero at the equilibrium position.
Because U is a minimum in the equilibrium position, the first derivative (dU/dqj)^
is zero, which leaves only {d^U/dq-dq^)^ and higher-order terms.
In the theory of small oscillations about the equilibrium position, terms
beyond the second order are ignored and the equation for the potential energy
reduces to
d^u
^ji - dqj dqi

and the potential energy is written in terms of the generalized stiffness kji as
1
== 2 E E k jiq jQ i
y=i /=i
(7.4-2)
=-\{QV{k\{q)

Generalized force. For the development of the generalized force, we start
from the virtual displacement of the coordinate r^:
_

i
and the time t is not involved.
When the system is in equilibrium, the virtual work can now be expressed in
terms of the generalized coordinates q^\

8W= E F ; - 5 r , = E E F , -