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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
1
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
(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)
3
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
4
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
5
(31)
(32)
(33)
(34)
(35)
Note that is considered to be dimensionless, although the units must be consistent within the analysis.
6
Participation Factors
The mass density is constant. Thus
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
7
(44)
(45)
Effective Modal Mass
(46)
The modes shapes are normalized such that
(47)
Thus
(48)
(49)
8
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)
9
(54)
10
(55)
(56)
(57)
(58)
(59)
(60)
(61)
11
(62)
(63)
(64)
(65)
(66)
12
(67)
(68)
Multiply each term by .
(69)
Integrate with respect to surface area.
13
(70)
14
(71)
The eigenvectors are orthogonal such that
for u m or v n (72)
for u = m and v = n (73)
15
(74)
(75)
(76)
Add a damping term.
(77)
(78)
The following solution is taken from Reference 3. The transfer function is
16
(79)
(80)
The relative displacement is
(81)
(82)
The relative velocity is
(83)
(84)
The absolute acceleration is
(85)
17
(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)
18
(91)
(92)
(93)
(94)
(95)
(96)
(97)
19
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.
20
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)
21