Page 422 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 12 Problems 409

M^ =2M^ El- constant

M

I ^
n 3 ■ Figure P12-2.
12-3 Estimate the fundamental frequency of the lumped-mass cantilever beam shown in
Fig. P12-3.
1.5/W, kg

r E I = constant
Figure P12-3.

12-4 Verify the results of Example 12.1-4 by using Eq. (12.1-3).
12-5 Another form of Rayleigh’s quotient for the fundamental frequency can be obtained
by starting from the equation of motion based on the flexibility influence coefficient
- aMX
= oi^aMX
Premultiplying by X^M, we obtain
X^MX = (o^X^MaMX
and the Rayleigh quotient becomes
2 _ X^MX
~ X^MaMX
Solve for Wj in Example 12.1-4 by using the foregoing equation and compare the
results with those of Prob. 12-4.
12-6 Using the curve

= ¿ 7 ( 7 )
solve Prob. 12-3 by using the method of integration. Hint: Draw shear and moment
diagrams based on inertia loads.
12-7 Using the deflection
y{x) =y„„^ sin(iTJ:/0.
determine the fundamental frequency of the beam shown in Fig. P12-7 (a) if
EI2 = EI^ and (b) if EI2 = 4F/,.

£ /,,m EIz,2m El^.m
A
V7/7//
Figure P12-7.