Download - CM Lecture 10
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2012/5/24
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高瞻計畫_振動學課程
Lecture 10: Continuous
Vibration (II)
Prof. Kuo-Shen Chen
Department of Mechanical Engineering
National Cheng-Kung University
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Outline
Beam problem analysis
Membrane / Plate analysis
Approximation method
Simple problems
Finite Element Analysis
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Part I. Beam Problem Analysis
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Introduction Bending vibration of beam is the most seen
vibration in real applications
Tall buildings subject to earthquake
Motion induced vibrations of robotic arms
Wings of airplane
MEMS sensors
approximation methods provide effective ways to
estimate vibration properties
Natural frequencies
Rayleigh method, Rayleigh-Ritz method, Galerkin
method
Finite element analysis of continuous structures 4
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Equation of Motion (I)
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Equation of Motion (II)
If uniform/
homogeneous
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Natural Frequency / Mode Shape Separation of variables
where
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Natural Frequency / Mode Shape
where
Finally
And
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Typical Boundary Conditions
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Typical Boundary Conditions
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Flexural Bending Mode Shapes and
Boundary Conditions
Free-Free Beam
Clamped-Clamped Beam
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Flexural Bending Mode Shapes and
Boundary Conditions
Clamped-Free Beam
Simply Supported Beam
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Example Find the boundary conditions of the following system
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Mode Decomposition
Assume
Known mode
shapes
Unknown time-
dependent
weighting function
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Modal Decomposition
Convolution
solution
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Effect of Axial Tension
Beams with axial force: the natural frequencies will be changed
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Effect of Axial Tension
If homogeneous
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Effect of Axial Tension: Example
Natural frequency of a S-S beam subjected to
axial force
BCs
P >0 n ; P<0 n If P=2EI/l2 n =0
Critical load for buckling
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Timoshenko Beam
Thicker beam
Considering the shear
deformation and
rotary inertia effect
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Timoshenko Beam
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Timoshenko Beam
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Part II. Membrane/Plate Analysis
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Modeling of Membrane Vibration
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Modeling of Membrane Vibration
Membrane is the 2D
version of string
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Rectangular Membrane Vibration
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Vibrational Modes of a Rectangular
Membrane
(1,1) mode
(1,2) mode
(2,1) mode (2,2) mode 26
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Degenerate Modes for a Square Membrane
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Circular Membrane
(0,1) (0,2) (0,3)
(1,1) (1,2) (2,1)
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Plate Vibrations
29 Source Dr. Y-K Lee, HKUST
Boundary Conditions
30 Source Dr. Y-K Lee, HKUST
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Typical Plate Vibration Formula
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Steinberg, Vibration of Electronic Equipment, Wiley
Part III. Approximation Methods
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Rayleigh Method
Fundamental natural frequency estimation
Upper bound
Rayleigh Quotient
Equal max KE & PE X: assumed motion mode
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Rayleigh Method (Example)
Refer to the figure in previous page
assume
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Dunkerley Method
Fundamental natural frequency estimation
Lower bound
Starting from the characteristic equation
We have
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Dunkerley Method: Example mainly used for reconstruct the results from experimental data
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Rayleigh-Ritz Method
A basic extension of Rayleigh method
Motion is assumed to be composed by several
mode functions with different weighting
Minimize Rayleigh Quotient
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Rayleigh-Ritz Method
We have (n equations)
In matrix form
Solve 2 and mode shape weighting ratio
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Galerkin Method
Similar to RR method for finding approximated
solution of a continuous structure vibration
By minimizing the error in satisfying a differential
equation over the system range
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1
( ) ( )n
i i
j
z x c z x
( ) 2( ) ( ) ( ) 0ivL z EIz x z x
( ) ( ) 0 1,2,...L
i io
R z x L z dx i n
Assume
zi(x): assumed mode
ci: weighting
For beam L(z): operator
Weighted residual
N linear equations to find ci
Galerkin Method Example Find the first two natural frequencies of a fixed-fixed beam (length l,
mass/length , flexural rigidity EI)
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Trial modes
Weighted residual
Where
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Galerkin Method Example (Cont’d)
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2
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1
4
2
2
4
1
44
1
)8(
)2
(
ccR
ccR
0
82det
444
444
1 69.0 ,)(1.124
1 0.23 ,)(48.22
1
2/1
22
1
2/1
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cEI
l
cEI
l
l
x
l
xx
l
x
l
xx
4cos1)
2cos1(69.0)(
4cos1)
2cos1(23)(
2
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Formulas for Natural Frequency and
Mode Shape by R. D. Blevins
A handbook to list
natural frequency and
vibration modes of
elastic structures
Suitable for engineering
design
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Part IV. Simple Problems
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Problem 1. Natural Frequencies/Modes
of a Fixed-Pinned Beam (Rao. 8.7)
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Boundary Conditions:
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Problem 2. Forced Vibration of a
Simply Supported Beam (Rao 8.8)
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Problem 3. Vibration of a Railroad on
Elastic Foundation
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一個列車於軌道上運動 (速度 v0)所引發之振動問題. 我們可以將該題目簡化成 infinite beam on elastic foundation (with foundation (soil) modulus k)受到一個 moving loading的模型. 其運動方程式可以表示成
4 2
0 04 2( )
w wEI A kw F x v t
x t
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Part V. Flexible Beam-Hub Example
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Problem Statement
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Finite Element Frequency and Modal Dynamics
Analysis of the Hub-Beam Experiment
Modal Dynamics Analysis
Mode superposition method mentioned earlier
Hub
accelerometer (y2)
motor
accelerometer (y1) accelerometer (y
3)
4.5" 8.0"
Y1 Y
2Y
3
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FE MESHES
300 C3D20 elements; 45 C3D15 elements; 1500 nodes
modes
extration
modal
dynamics
linear dynamics w/
subspace projection
torque input
(pulse or
sinusoidal)
outputpower
spectrum
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Responses (II)
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Power Spectrum Density
PS
D
100
101
102
103
10-6
10-4
10-2
100
102
104
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POWER SPECTRUM DENSITY
Frequency Hz
Sensor Y3
Sensor Y2
Sensor Y1
PS
D
100
101
102
103
10-8
10-6
10-4
10-2
100
102
104
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POWER SPECTRUM DENSITY
Frequency Hz
Sensor Y3
Sensor Y2
Sensor Y1
Power spectrum density plots of system response. (a) without and (b) with input shaping
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Part VI. Youtube Demonstration
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