Page 238 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 7.5 Assumed Mode Summation 225

where the generalized stiffness is

= ¡AE<pWjdx (7.5-8)

Example 7.5-1
Determine the equation of motion and the natural frequencies and normal modes of
a fixed-free uniform rod of Fig. 7.5-1 using assumed modes (p^ix) = x/l and iP2ix) =
{x/lY.
The equation for the displacement of the rod is

u{x,t) = <p,(Ar)i7,(0 + <P2(-i)i2(0

Qi
Note that the assumed modes chosen satisfy the only geometric boundary condition of
the problem, which is w(0, t) = 0. Thus, the generalized mass and the generalized
stiffness are evaluated from

"î/y = j^<Pi(x)<Pj(x)mdx

kij = jEA(p'i(x)<Pj(x) dx
EA
= EAÇ j • j dx
~ 1
2x , EA
2, = m /J (y ) dx = jml ^12 ~ ^21 ^ / T ^ d x = —
rl( X 1 , , ^ riAx^ , AEA
/o ( l ) ^ ^ = 5 ^ ^ ki2 = EAl ' dx = 31
•0 1
which can be assembled into the following matrices:
1 1 ■
3 4 1 1 '
M = ml 1 1
4 5
The equation of motion for the normal mode vibration then becomes
1 j_ 1 1
o^ml 3 I 4 I EA
4 5

Figure 7.5-1.