natural frequencies of rectangular plate ... · web viewconsider the rectangular plate in figure 1....
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RESPONSE OF A RECTANGULAR PLATE TO BASE EXCITATIONRevision E
By Tom IrvineEmail: [email protected]
April 10, 2013
Variables
amplitude coefficient
a length
b width
D plate stiffness factor
E elastic modulus
h plate thickness
km wavenumber
M plate mass
bending moment
m, n, u, v mode number indices
Poisson's ratio
mass per volume
z relative displacement, out-of-plane
normalized mode shape
participation factor
excitation frequency
natural frequency
modal damping
z relative displacement, out-of-plane
w(t) base input acceleration
W( ) Fourier transform of base input acceleration
Relative displacement
Absolute Acceleration
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Introduction
Consider the rectangular plate in Figure 1. Assume that it is simply-supported along each edge.
Figure 1.
Normal Modes Analysis
Note that the plate stiffness factor D is given by
(1)
The governing equation of motion is
(2)
Assume a harmonic response.
(3)
(4)
2
a
b
Y
X0
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(5)
The boundary conditions are
for x = 0, a (6)
for y = 0, b (7)
Assume the following displacement function which satisfies the boundary conditions, where c is an amplitude coefficient and m an n are integers.
(8)
The partial derivates are
(9)
(10)
(11)
(12)
Similarly,
(13)
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Also,
(14)
(15)
(16)
(17)
(18)
(19)
Note that the wave numbers are
(20)
(21)
Normalized Mode Shapes
The mode shapes are normalized such that
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(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
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(31)
(32)
(33)
(34)
(35)
Note that is considered to be dimensionless, although the units must be consistent within the analysis.
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Participation Factors
The mass density is constant. Thus
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
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(44)
(45)
Effective Modal Mass
(46)
The modes shapes are normalized such that
(47)
Thus
(48)
(49)
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Response to Base Excitation
The forced response equation for a plate with base motion is
(50)
where w is base excitation.
The term on the right-hand-side is the inertial force per unit area.
The displacement is
(51)
(52)
(53)
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(54)
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(55)
(56)
(57)
(58)
(59)
(60)
(61)
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(62)
(63)
(64)
(65)
(66)
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(67)
(68)
Multiply each term by .
(69)
Integrate with respect to surface area.
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(70)
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(71)
The eigenvectors are orthogonal such that
for u m or v n (72)
for u = m and v = n (73)
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(74)
(75)
(76)
Add a damping term.
(77)
(78)
The following solution is taken from Reference 3. The transfer function is
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(79)
(80)
The relative displacement is
(81)
(82)
The relative velocity is
(83)
(84)
The absolute acceleration is
(85)
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(86)
The bending moments are
(87)
(88)
Let be the distance from the centerline in the vertical axis.
The bending stresses from Reference 4 are
(89)
(90)
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(91)
(92)
(93)
(94)
(95)
(96)
(97)
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References
1. Dave Steinberg, Vibration Analysis for Electronic Equipment, Wiley-Interscience, New York, 1988.
2. Arthur W. Leissa, Vibration of Plates, NASA SP-160, National Aeronautics and Space Administration, Washington D.C., 1969.
3. T. Irvine, Steady-State Vibration Response of a Cantilever Beam Subjected to Base Excitation Revision A, Vibrationdata, 2009.
4. J.S. Rao, Dynamics of Plates, Narosa, New Delhi, 1999.
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APPENDIX A
Normal Stress-Velocity Relationship
The maximum absolute spatial stresses for a given mode m,n are
(A-1)
(A-2)
The maximum spatial velocity for a given mode m,n is
(A-3)
Thus
(A-4)
(A-5)
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