4. Dynamics of Continuous Structures
Maged Mostafa
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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
5. Dynamics of Continuous Structures
Maged Mostafa
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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
6. Dynamics of Continuous Structures
Maged Mostafa
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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!
7. Dynamics of Continuous Structures
Maged Mostafa
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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+dFF
Equilibrium
position
Infinitesimal
element
0 l
8. Dynamics of Continuous Structures
Maged Mostafa
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Remember!
x
txu
xEAF
x
txu
E
xσAF
From
),(
)(
),(
)(
LawsHawk'
9. Dynamics of Continuous Structures
Maged Mostafa
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0
),(
:endfreeAt the
,0),0(:endclampedAt the
),(),(
constant)(
),(
)(
),(
)(
),(
)(
),(
)(
),(
)(
2
2
2
2
2
2
2
2
xx
txu
EA
tw
t
txu
x
txuE
xA
t
txu
xA
x
txu
xEA
x
dx
x
txu
xEA
x
dF
x
txu
xEAF
t
txu
dxxAFdFF
Force balance:
Constitutive relation:
10. Dynamics of Continuous Structures
Maged Mostafa
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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
11. Dynamics of Continuous Structures
Maged Mostafa
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Apply the boundary conditions to the
spatial solution to get
asin(0) bcos(0) 0
acos(l ) bsin(l ) 0
b 0 and det
0 1
cos(l ) sin(l )
0
cosl 0 n
2n 1
2l
, n 1,2,3,L
Xn (x) an sin(
(2n 1)x
2l
), n 1,2,3,L
12. Dynamics of Continuous Structures
Maged Mostafa
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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
13. Dynamics of Continuous Structures
Maged Mostafa
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time response equation :
&&Tn (t) c2 2n 1
2l
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)
2l
E
, n 1,2,3L (6.63)
Thus the solution implies oscillation with Frequencies:
14. Dynamics of Continuous Structures
Maged Mostafa
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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
15. Dynamics of Continuous Structures
Maged Mostafa
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Given v0(l)=3 cm/s, =8x103 kg/m3 and
E=20x1010 N/m2, compute the response.
w(x,t) (cn sinnct
n1
dn cosnct)sin
(2n 1)
2l
x
dn
2
l
w0 (x)sin
0
l
(2n 1)
2l
xdx 0
wt (x,0) 0.03(x l ) cnnccos(0)
n1
sin
(2n 1)
2l
x
0
Multiply by the mode shape indexed m and integrate:
16. Dynamics of Continuous Structures
Maged Mostafa
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0.03 (sin
0
l
(2m 1)
2l
x)(x l )dx
cn
0
l
ncsin
n1
(2m 1)
2l
xsin
(2n 1)
2l
xdx
0.03sin
(2m 1)
2
cml
2
cm cm
1
E
0.06(1)m1
(2m 1)
cn
8 103
210 109
0.12(1)n1
(2n 1)
7.455 10-6 (1)n1
(2n 1)
m
w(x,t) 7.455 10-6 (1)n1
(2n 1)
sin
2n 1
10
x
n1
sin 512.348(2n 1)t m
17. Dynamics of Continuous Structures
Maged Mostafa
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Assignment
• Solve the equation of motion of a bar
with constant cross-section properties
with
1. Fixed-Fixed boundary conditions
2. Free-Free boundary conditions
• Compare the natural frequencies for all
three cases