04 vibration of bars

Dynamics of Continuous StructuresMaged Mostafa

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Vibration of Continuous Structures



Bar Vibration



Course Contents SDOF M-DOF Cables/String• Bars• Shafts• Vibration Attenuation• Beams• FEM for Vibration• Plates• Aeroelasticity



Objectives

• Derive the equation of motion for Bars• Estimate the Natural Frequencies• Understand the concept of mode shapes• Apply BC’s and IC’s to obtain structure

response



Objectives

• Derive the equation of motion for Bars• Apply BC’s and IC’s to obtain structure

response• Estimate the Natural Frequencies• Understand the concept of mode shapes



Bar Vibration

• The bar is a structural element that bears compression and tension loads

• It deflects in the axial direction only• Examples of bars may be the columns of

buildings, car shock absorbers, legs of chairs and tables, and human legs!



Vibration of Rods and Bars

• Consider a small element of the bar

• Deflection is now along x (called longitudinal vibration)

• F= ma on small element yields the following:

x x +dx

u(x,t) x

dx

F+dF F

Equilibrium position

Infinitesimal element

0 l



Remember!

xtxuxEAF

xtxu

ExσAF

From

),()(

),(

)(Law sHawk'



0

),( :end free At the

,0),0( :end clamped At the

),(

),( constant)(

),( )(

),()(

),()(

),()(

),( )(

2

2

2

2

2

2

2

2

xxtxuEA

twt

txux

txuExA

ttxuxA

xtxuxEA

x

dxx

txuxEAx

dFx

txuxEAF

ttxudxxAFdFF

Force balance:

Constitutive relation:



Objectives





Apply the boundary conditions to the spatial solution to get

asin(0) bcos(0)0acos( l ) bsin( l )0

b 0 and det0 1

cos( l ) sin( l )

0

cos l 0 n

2n 12l , n 1,2,3,L

Xn (x)an sin((2n 1) x

2l ), n 1,2, 3,L



Objectives





time response equation :

Tn (t) c2 2n 12l

2

T (t)0

Tn (t)An sin(2n 1)c

2l t Bn cos(2n 1)c

2l t

n (2n 1)c

2l (2n 1)

2lE

, n 1,2,3L (6.63)

Thus the solution implies oscillation with Frequencies:



Note

• The equation of motion of the bar is similar to that of the cable/string the response should have similar form

• The bar may have different boundary conditions



Given v0(l)=3 cm/s, =8x103 kg/m3 and E=20x1010 N/m2, compute the response.

w(x,t) (cn sin nctn1

dn cos nct)sin(2n 1)

2l x

dn 2l w0 (x)sin

0

l

(2n 1)

2l xdx 0

wt (x,0)0.03 (x l ) cn nccos(0)n1

sin(2n 1)

2l x

0

Multiply by the mode shape indexed m and integrate:



0.03 (sin0

l

(2m 1)

2l x) (x l )dx

cn0

l

ncsinn1

(2m 1)2l x sin

(2n 1)2l xdx

0.03sin(2m 1)

2

cml2

cm cm 1

E

0.06( 1)m1

(2m 1)

cn 8103

210109

0.12( 1)n1

(2n 1)7.45510-6 ( 1)n1

(2n 1) m

w(x, t)7.45510-6 ( 1)n1

(2n 1)sin

2n 110

x

n1

sin 512.348(2n 1)t m



Assignment

• Solve the equation of motion of a bar with constant cross-section properties with

1. Fixed-Fixed boundary conditions2. Free-Free boundary conditions

• Compare the natural frequencies for all three cases

04 vibration of bars

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