Page 240 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)
P. 240
Sec. 7.5 Assumed Mode Summation 227
Generalized Stiffness (Beams)
Determine the generalized stiffness for a beam of cross-sectional property El
when the displacement y{x,t) is represented by the sum
(7.5-9)
/-I
The potential energy of a beam in bending is
(7.5-10)
Substituting for
d^y
dx^ / = 1
we obtain
u = jLL<li^jjEl<pW;dx
2 ^ijQiQj (7.5-11)
and the generalized stiffness is
k ,j = f E i< p y ’ dx (7.5-12)
Example 7.5-2 Generalized Force
The frame of Fig. 7.1-3 with rigid members is acted upon by the forces and moments
shown in Fig. 7.5-2. Determine the generalized forces.
Solution: We let be the virtual displacement of the upper left corner and 8q2 be the
translation of the right support hinge. Due to 8q^, the virtual work done is
2,5^1 =F|5^1 - F2jdqj + (Ml - M2)jSq^
/
■■■Qx = E > / |(M , -M ^)
