4.7 Vibration of Membranes and Plates

Back

The cables, rods, shafts, and beams whose vibration we have studied thus far in this chapter are

one-dimensional members or line structures. These continuous members need one spatial variable ðxÞ;

in addition to the time variable ðtÞ; as an independent variable to represent their governing equation of

motion. Membranes and plates are two-dimensional members or planar structures. They need two

independent spatial variables (x and y) in addition to time ðtÞ; for representing their dynamics.

A membrane may be interpreted as a two-dimensional extension of a string or cable. In particular, it

has to be in tension and cannot support any bending moment. A plate is a two-dimensional extension of

a beam. It can support a bending moment. Their governing equations will, therefore, resemble twodimensional

versions of their respective one-dimensional counterparts. Modal analysis will also follow

the familiar steps, after accounting for the extra dimension. In this section, we will give an introduction

to the modal analysis of membranes and plates. For simplicity, only special cases of rectangular members

with relatively simple BCs will be considered. Analysis of more complicated boundary geometries and

conditions will follow analogous procedures, but requires a greater effort and produces more complicated

results.

4.7.1 Transverse Vibration of Membranes

Consider a stretched membrane (in tension) that lies on the x – y plane, as shown in Figure 4.19.

Transverse vibration vðx; y; tÞ in the z-direction is of interest. By following a procedure that is somewhat

analogous to the derivation of the cable equation, we can obtain the governing equation as

›2vðx; y; tÞ

›t2 ¼ c2 ›2

›x2 þ

›2

›y2

" #

vðx; y; tÞ ð4:233Þ

4-52 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

with

c ¼

ffiffiffiffiffi

T0

m0

s

ð4:234Þ

where

T0 ¼ tension per unit length of membrane section (assumed constant)

m0 ¼ mass per unit surface area of membrane

For modal analysis, we seek a separable solution of the form

vðx; y; tÞ ¼ Y ðxÞZðyÞqðtÞ ð4:235Þ

Substitute Equation 4.235 into Equation 4.233. We obtain

YðxÞZðyÞq€ðtÞ ¼ c2 1

Y ðxÞ

d2Y ðxÞ

dx2 þ

1

ZðyÞ

d2ZðyÞ

dy2

" #

ð4:236Þ

Since Equation 4.236 is true for all possible values of t; x; and y that are independent, the three function

groups should separately equal the constants; thus

1

Y ðxÞ

d2Y

dx2 ¼ 2a2 or

d2Y ðxÞ

dx2 þ a2Y ðxÞ ¼ 0 ð4:237Þ

1

ZðyÞ

d2ZðyÞ

dy2 ¼ 2b2 or

d2ZðyÞ

dy2 þ b2ZðyÞ ¼ 0 ð4:238Þ

q€ðtÞ

qðtÞ ¼ 2v2 or q€ðtÞ þv2qðtÞ ¼ 0 ð4:239Þ

with

v2 ¼ c2ða2 þ b2Þ ð4:240Þ

The argument for using positive constants a2; b2; and v2 is similar to that we gave for the onedimensional

case. Next, Equation 4.237 and Equation 4.238 have to be solved using two end conditions

for each direction, as usual. This will provide an infinite number of solutions ai and bj; and the

corresponding natural frequencies

v2

ij ¼ cða2i

þ b2j

Þ1=2 for i ¼ 1; 2; 3; … and j ¼ 1; 2; 3; … ð4:241Þ

along with the mode shape components YiðxÞ and ZjðyÞ for the two dimensions.

x

b

a

z, v(x,t)

0

y

FIGURE 4.19 A membrane or a plate in Cartesian coordinates.

Distributed-Parameter Systems 4-53

© 2005 by Taylor & Francis Group, LLC

4.7.2 Rectangular Membrane with Fixed Edges

Consider a rectangular membrane of length a and width b as shown in Figure 4.19 and with the four

edges fixed. The BCs are

vð0; y; tÞ ¼ 0; vða; y; tÞ ¼ 0; vðx; 0; tÞ ¼ 0; vðx; b; tÞ ¼ 0

Using these in solving Equation 4.237 and Equation 4.238, as usual, we obtain

YiðxÞ ¼ sin aix with ai ¼

ip

a ð4:242Þ

ZjðyÞ ¼ sin bjy with bj ¼

jp

b ð4:243Þ

vij ¼ c½a2i

þ b2j

􀀉1=2 ¼ pc

i2

a2 þ

j2

b2

" #1=2

for i ¼ 1; 2; 3; … and j ¼ 1; 2; 3 ð4:244Þ

Note that the spatial mode shapes are given by

YjðxÞZjðyÞ ¼ sin

ipx

a

sin

jpy

b ð4:245Þ

4.7.3 Transverse Vibration of Thin Plates

Consider a thin plate of thickness h in a Cartesian coordinate system as shown in Figure 4.19. The usual

assumptions as for the derivation of the Bernoulli – Euler beam equation are used. In particular, h is

assumed small compared with the surface dimensions (a and b for a rectangular plate). Then, shear

deformation and rotatory inertia can be neglected, and also normal stresses in the transverse direction ðzÞ

can be neglected. Furthermore, any end forces in the planar directions (x and y) are neglected. The

governing equation is

›2vðx; y; tÞ

›t2 þ c2 ›2

›x2 þ

›2

›y2

" #2

vðx; y; tÞ ¼ 0 ð4:246Þ

with

c2 ¼

E0I0

rA0 ¼

Eh2

12ð1 2 n2Þr ð4:247Þ

where

E0 ¼

E

ð1 2 n2Þ ð4:248Þ

I0 ¼

h3

12 ¼ second moment of area per unit length of section ð4:249Þ

A0 ¼ h ¼ area per unit length of section

r ¼ mass density of material

E ¼ Young’s modulus of elasticity of the plate material

n ¼ Poisson’s ratio of the plate material

If we attempt modal analysis by assuming a completely separable solution of the form vðx; y; tÞ ¼

Y ðxÞZðyÞqðtÞ in Equation 4.246, a separable grouping of functions of x and y will not be achieved in

general. However, the space and the time will be separable in modal motions. Hence, we seek a solution of

the form

vðx; y; tÞ ¼ Y ðx; yÞqðtÞ ð4:250Þ

4-54 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

We will obtain

q€ðtÞ þv2qðtÞ ¼ 0 ð4:251Þ

and

7272Y ðx; yÞ 2 l4Y ðx; yÞ ¼ 0 ð4:252Þ

with the natural frequencies v given by

v ¼ l2c ð4:253Þ

and 72 is the Laplace operator given by

72 ¼

›2

›x2 þ

›2

›y2 ð4:254Þ

x

y

Mode 1

i=1 , j=1

Mode 2

i=2 , j=1

Mode 3

i=1 , j=2

Mode 4

i=2 , j=2

Mode 5

i=3 , j=1

Mode 6

i=3 , j=2

FIGURE 4.20 Mode shapes of transverse vibration of a simply supported rectangular plate.

Distributed-Parameter Systems 4-55

© 2005 by Taylor & Francis Group, LLC

and, hence, 7272 is the biharmonic operation 74 given by

74 ¼

›4

›x4 þ 2

›4

›x2 ›y2 þ

›4

›y4 ð4:255Þ

The solution of Equation 4.252 will require two sets of BCs for each edge of the plate (as for a beam),

but will be mathematically involved. Instead of a direct solution, a logical trial solution that satisfies

Equation 4.252 and the BCs is employed next for a simply supported rectangular plate. The solution tried

is in fact the correct solution for the particular problem.

4.7.4 Rectangular Plate with Simply Supported Edges

As a special case, we now consider a thin rectangular plate of length a; width b; and thickness h; as shown

in Figure 4.19, whose edges are simply supported. For each edge, the BCs are the displacement in zero and

the bending moment about the edge zero. Specifically, we have

vðx; y; tÞ ¼ 0 and Mx ¼ E0I0 ›2v

›x2 þ n

›2v

›y2

􀁻 !

¼ 0 for x ¼ 0 and a; 0 # y # b

vðx; y; tÞ ¼ 0 and My ¼ E0I0 ›2v

›y2 þ n

›2v

›x2

􀁻 !

¼ 0 for y ¼ 0 and b; 0 # x # a ð4:256Þ

where E0 and I0 are as given in Equation 4.248 and Equation 4.249. In this case, the mode shapes are found

to be

Yijðx; yÞ ¼ sin

ipx

a

sin

jpy

b

for i ¼ 1; 2; … and j ¼ 1; 2; … ð4:257Þ

which clearly satisfy the BCs (Equation 4.256) and the governing model equation 4.252, with an infinite

set of solutions for l given by

l2

ij ¼ p2 i2

a2 þ

j2

b2

􀁻 !

ð4:258Þ

Hence, from Equation 4.253, the natural frequencies are

vij ¼ p2c

i2

a2 þ

j2

b2

􀁻 !

ð4:259Þ

where c is given by Equation 4.247. The overall response, then, is given by

vðx; y; tÞ ¼

X1

i¼1

X1

j¼1

􀀒

Aij sin vijt þ Bij cos vijt

􀀓

sin

ipx

a

sin

ipy

b ð4:260Þ

The unknown constants Aij and Bij are determined by the system initial conditions vðx; y; 0Þ and v_ðx; y; 0Þ:

The first six mode shapes of transverse vibration of a rectangular plate are sketched in Figure 4.20.

Навигация