natural frequencies of rectangular plate ... · web viewconsider the rectangular plate in figure 1....

RESPONSE OF A RECTANGULAR PLATE TO BASE EXCITATIONRevision E

By Tom IrvineEmail: tom@vibrationdata.com

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