Vibration of Beams

1. Dynamics of Continuous Structures
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Vibration of Continuous
Structures

2. Dynamics of Continuous Structures
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Beam Vibration

3. Dynamics of Continuous Structures
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Course Contents
 SDOF
 M-DOF
 Cables/String
 Bars
 Shafts
• Beams
• Vibration Attenuation
• FEM for Vibration
• Plates
• Aeroelasticity

4. Dynamics of Continuous Structures
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Beam Vibration
• The beam element is the most famous
structural element as it presents a lot of
realistic structural elements
• It bears loads normal to its longitudinal
axis
• It resists deformations by inducing bending
stresses

5. Dynamics of Continuous Structures
Maged Mostafa
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Bending vibrations of a beam
2
2
),(
)(),(
aboutinertia
ofmomentareasect.-cross)(
modulusYoungs
)(stiffnessbending
x
txw
xEItxM
z
xI
E
xEI






Next sum forces in they- direction (up, down)
Sum moments about the pointQ
Use the moment given from
stenght of materials
Assume sides do not bend
(no shear deformation)
f (x,t)
w (x,t)
x
dx A(x)= h1h2
h1
h2
M(x,t)+Mx(x,t)dx
M(x,t)
V(x,t)
V(x,t)+Vx(x,t)dx
f(x,t)
w(x,t)
x x +dx
·Q

6. Dynamics of Continuous Structures
Maged Mostafa
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Summing forces and moments
0)(
2
),(),(
),(
),(
0
2
),(
),(
),(),(
),(
),(
),(
)(),(),(
),(
),(
2
2
2


































dx
txf
x
txV
dxtxVdx
x
txM
dx
dxtxf
dxdx
x
txV
txVtxMdx
x
txM
txM
t
txw
dxxAdxtxftxVdx
x
txV
txV













0

7. Dynamics of Continuous Structures
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A
EI
c
x
txw
c
t
txw
txf
x
txw
xEI
xt
txw
xA
t
txw
dxxAdxtxfdx
x
txM
x
txM
txV




























,0
),(),(
),(
),(
)(
),(
)(
),(
)(),(
),(
),(
),(
4
4
2
2
2
2
2
2
2
2
2
2
2
2
2
Substitute into force balance equation yields:
Dividing by dx and substituting for M yields
Assume constant stiffness to get:

8. Dynamics of Continuous Structures
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Boundary conditions (4)
0forceshear
0momentbending
endFree
2
2
2
2








x
w
EI
x
x
w
EI






0slope
0deflection
endfixed)(orClamped


x
w
w


0momentbending
0deflection
endsupported)simply(orPinned
2
2


x
w
EI
w


0forceshear
0slope
endSliding
2
2








x
w
EI
x
x
w







9. Dynamics of Continuous Structures
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Solution of the time equation:
)()0,(),()0,(
:conditionsinitialTwo
cossin)(
0)()(
)(
)(
)(
)(
00
2
22
xwxwxwxw
tBtAtT
tTtT
tT
tT
xX
xX
c
t












10. Dynamics of Continuous Structures
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Spatial equation (BVP)
xaxaxaxaxX
AexX
EI
A
c
xX
c
xX
x





coshsinhcossin)(
:getto)(Let
Define
.0)()(
4321
22
4
2
















Apply boundary conditions to get 3
constants and the characteristic equation

11. Dynamics of Continuous Structures
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Example: compute the mode shapes and
natural frequencies for a clamped-pinned
beam.
0)coshsinhcossin(
0)(
0coshsinhcossin
0)(
andend,pinnedAt the
0)(0)0(
00)0(
and0endfixedAt
4321
2
4321
31
42
















aaaa
XEI
aaaa
X
x
aaX
aaX
x

12. Dynamics of Continuous Structures
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

  






tanhtan
0)det(,
0
0
0
0
coshsinhcossin
coshsinhcossin
00
1010
4
3
2
1
2222








































BB
a
a
a
a
B
0a0a
a
The 4 boundary conditions in the 4 constants can be
written as the matrix equation:
The characteristic equation

13. Dynamics of Continuous Structures
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Solve numerically to obtain solution to
transcendental equation
4
)14(
5
493361.16351768.13
210176.10068583.7926602.3
54
321









n
n
n


Next solve Ba=0 for 3 of the constants:

14. Dynamics of Continuous Structures
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Solving for the eigenfunctions:

















xxxxaxX
aa
aa
aa
aa
B
nnnn
nn
nn
nn
nn
nn
nnnn












coscosh)sin(sinh
sinsinh
coscosh
)()(
sinsinh
coscosh
:yieldsSolving
equationfourth)(orthirdthefrom
0)cos(cosh)sinsinh(
equationsecondthefrom
equationfirstthefrom
:4ththeofin termsconstants3yields
4
43
43
42
31
0a

15. Dynamics of Continuous Structures
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Mode shapes
X ,n x .
cosh n cos n
sinh n sin n
sinh .n x sin .n x cosh .n x cos .n x
0 0.2 0.4 0.6 0.8 1
2
1.5
1
0.5
0.5
1
1.5
X ,3.926602 x
X ,7.068583 x
X ,10.210176 x
x
Mode 1
Mode 2
Mode 3
Note zero slope
Non zero slope

16. Dynamics of Continuous Structures
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Summary of the Euler-Bernoulli
Beam
• Uniform along its span and slender
• Linear, homogenous, isotropic elastic
material without axial loads
• Plane sections remain plane
• Plane of symmetry is plane of vibration so
that rotation & translation decoupled
• Rotary inertia and shear deformation
neglected

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Homework
• Get an expression for  for the cases of a
uniform Euler-Bernoulli beam with BC’s as
follows:
– Clamped-Clamped
– Pinned-Pinned
– Clamped-Free
– Free-Free