56

Example A4.19

Given: Consider the bending beam shown below with a concentrated harmonic force acting at themid-length point C on the beam.

Find: For this problem:a) Use the modal uncoupling approach to determine the particular solution wP (t) = W (x)sin⌦t

of the beam‘s EOM.b) Make a hand sketch of |W (x = L/2)| vs. ⌦ for a range of values of ⌦ covering up through the

first four resonances of the response. Provide details on how you arrived at this hand sketch.

w(x,t)

x = 0 x = L

EI , ρA

ME 563 – Fall 2014

Hom

ework Problem

8.4 C

onsider the bending beam show

n below w

ith a concentrated harmonic force acting at

the mid-length point C

on the beam.

a) U

se the modal uncoupling approach to determ

ine the particular solution wP (x,t)=

W(x)sinΩ

t of the beam’s EO

M. Feel free to use natural frequency and

modal function results from

a previous homew

ork assignment in arriving at your

solution here.

b) M

ake a hand sketch of W(x

=L/2) vs. Ω

for a range of values of Ω covering

up through the first four resonances of the response. Provide details on how you

arrived at this hand sketch.

c) R

epeat b) above for W(x

=L)

.

or,

pj +ω j2pj = f̂ j sinΩt

The particular solutions for these modally-uncoupled EOMs are given by:

pjP (t) =f̂ j

ω j2 −Ω2

sinΩt

Therefore, the beam response is:

wP (x, t) = Wj( )(x)pjP (t)

j=1

∞

∑

= W j( )(x)f̂ j

ω j2 −Ω2

⎛

⎝⎜

⎞

⎠⎟

j=1

∞

∑⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪sinΩt

=W (x)sinΩt

From this, we have:

W L2

⎛⎝⎜

⎞⎠⎟ = W

j( ) L2

⎛⎝⎜

⎞⎠⎟

f̂ jω j2 −Ω2

⎛

⎝⎜

⎞

⎠⎟

j=1

∞

∑

= f0W j( ) L

2⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥2

ρA W ( j)2 dx0

L

∫⎛

⎝⎜

⎞

⎠⎟

1ω j2 −Ω2

⎛

⎝⎜

⎞

⎠⎟

j=1

∞

∑

Comments:

• Resonances will occur for the jth mode when Ω =ω j , unless W( j)(L / 2) = 0 .

• Anti-resonances will occur between every pair of resonances since the coefficient of 1/ ω j

2 −Ω2( ) in the above sum is positive for all modes.