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Proceedings of the Institution of Mechanical
http://pme.sagepub.com/content/168/1/371The online version of this article can be found at:
DOI: 10.1243/PIME_PROC_1954_168_040_02
1954 168: 371Proceedings of the Institution of Mechanical Engineers G. B. Warburton
The Vibration of Rectangular Plates
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 Jun 1, 1954Version of Record>>
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371
The Vibration of Rectangular Plates
By G.
B.
Warburton, M.A.,
Ph.D.*
In
this
paper are considered the free transverse vibrations of rectangular plates with all possible
boundary conditions obtained by combining free, freelysupported, and fixed edges. The Rayleigh
method, assuming waveforms similar to those of beams, is used to derive a simple approximate
frequency expression for
all
modes of vibration. The terms
in this
expression depend on the nodal
pattern and the boundary conditions; they are tabulated for fifteen boundary con di ti on s4 four
edges free, freelysupported, or fixed and the twelve cases in which some of the edges have one
condition and the rest another. The expression can
also
be used to obtain frequencies for a plate
which has a combination of all three boundary conditions. The effect on frequency of an edge being
supported and partially restrained is discussed.
For some boundary conditions it is possible to compare derived frequencies with those obtained
by various methods of accurate analysis and by experiment; except for a few cases, the results from
the approximate expression and accurate analysis agree closely.
For rectangular plates and for most boundary conditions and modes of square plates, the nodal
pattern consists of lines approximately parallel to the sides of the plate. T he exceptions are discussed
and the gradual transition from these nonparallel patterns characteristic of a square plate to those
of a rectangular plate is traced.
Th e frequencies of extensional vibrations of rectangular plates are derived for two boundary con
ditions.
I N T R O D U C T I O N
In
this
paper are considered the frequencies of free transverse
vibrations of rectangular plates for all combinations of free,
freelysupported, and fixed edges. In transverse vibration, points
in the plate undergo small displacements in the direction per
pendicular to the plane of the plate.
All
four edges of the plate
may be free, freelysupported, or
fixed;
here are twelve mixed
boundary conditions
in
which some of the edges have one con
dition and the rest another. The author has obtained an approxi
mate formula, expressing frequency
in
terms of the boundary
conditions, the nodal pattern, the dimensions of the plate, and
the constants of the material;
this
formula is applicable to all
fifteen boundary conditions. For
six
of these the author has not
found any reference giving frequencies; or only one al l edges
freelysupportedis there a wellknown frequency expression.
For the remaining conditions, previous investigators, whose
work is referred to, have obtained frequencies for some modes
of vibration for specific values of the ratio of the sides of the
rectangle. For two of these eight conditions, Eschler (1950)t
has
obtained frequency equations similar to those found by the
author and including a l l modes of vibration. The author also
outlines a method of obtaining frequencies for a plate which has
a combination of all three boundary conditions, for example,
two free, one freelysupported, and one fixed edge.
The approximate frequency formula is derived by applying
the Rayleigh method (Rayleigh 1894, p. 109 et seq. ) and by
assuming that the waveforms of vibrating plates and beams are
similar;
thus for a plate with all its edges fixed, the waveform
assumed is the product of the characteristic functions for two
beams with fixed ends. In effect, the Rayleigh method imposes
additional constraints on a system, so that the resulting fre
quencies are higher than those given by an exact analysis.
Whenever possible, the approximate frequencies are compared
with the results of exact or more accurate analyses. For most
boundary conditions, the errors are very small, but for a few
they are appreciable. To use
this
method the nodal patterns must
consist of lines approximately parallel to the sides of the plate.
Usually
t h i s
condition is satisfied; the exceptions, and their
effect on
requency,
are
discussed.
The
MS.
f
t h i s
paper
was
first received at the
Institutionon.
21st
July 1953, and in its revised form, as accepted by the C o u n d for
publication,
on
24th November 1953.
* Lecturer, former1
I.C.I.
Research Fellow,Department of Engi
neering,
University o?Edinburgh.
t An
alphabetical list of reference8
is given
in Appendix
111.
In practice, one edge of
a
panel may be supported and
partially restraineda condition between that of freely
supported and fixed. For any mode the increase of frequency,
which is caused by the substitution of a fixed for a freely
supported edge, may be obtained from the approximate formula.
Thus limits are imposed on the required frequency and, from a
knowledge of the degree of restraint, an estimate of the frequency
can
be made. Some of the boundary conditions investigated
appear to have little practical significance, but those containing
various numbers of freelysupported edges are required in order
to
give lower limits for the frequencies of plates with partially
restrained edges.
Extensional vibrations can also
occur in
a plate; then motion
is only
in
the plane of the plate. AS
this
type of vibration is
associated with high frequencies and
has
ess practical significance
than transverse vibrations, it is only considered briefly for two
boundary conditions (Appendix I).
Throughout the paper, the plates are assumed to be isotropic,
elastic, free from applied loads and of uniform thickness, and
the analysis is based on the ordinary theory of th n plates.
Also
the thickness is assumed to be small in comparison with the
wavelength.
Notation.
Lengths of sides of rectangular plate. ,
Ratios
in
expressions for displacement.
Frequency.
Acceleration due to gravity.
Thickness of plate.
Number of nodal lines in X and
Y
directions
respectively.
b2h
d ( n  1)2
Y
directions.
rz =
ime.
Component displacements of a point in the
X
and
Transverse disulacement of a Doint.
x, Y
A , B, C
Component amplitudes.
Coorciinate distances in p h i f plate.
Eh3
D = 12 1
u2)'
E Youne's
modulus.
Gx,
Hx, Jx
Fun zon s of
m in
frequency expression.
Gy H,, Jy
Functions
of n
n
frequency expression.
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T H E V I B R A T I O N
O F
R E C T A N G U L A R P L A T E S
Ria integral.
Kinetic energy.
Strain
energy.
w =
wsinwt.
Indices.
Factors in amplitude expressions defining nodal
pattern.
form.
Functions
in
frequency expressions.
Functions of
x
and
y
respectively, defining wave
2rf circular
frequency.
G E N E R A L
METHOD
FOR T R A N S V E R S E
V I B R A T I O N S
Before considering the various boundary conditions, the basic
equations and methods of solution
will
be given. For a rectangular
plate of sides of lengths a and b, the vibration form must satisfy
Fig.
1.
Details of Rectangular Plate
the boundary conditions at the edges OA,
AB,
BC,
CO
(Fig.
1);
also it must satisfy the plate equation (Love
1927)
:
where w, the displacement at any point
x, y)
at time
t,
is given
by
w
=
wsin
wt
= Ae(x)+Cy) inwt
. 2)
In general, it is not possible to find a form for
w
to satisfy
equation 1) together with the boundary conditions. For these
cases, an infinite series
can
be assumed for
W;
each term of the
series satisfies equation
1)
and some of the boundary con
ditions and, by taking suitable values of the coefficients
A,
the
remaining conditions are satisfied.
Usually this
procedure leads
to an infinite determinant being zero, each term in the deter
minant being a different function of the frequency. The roots
of the determinant give the natural frequencies for the modes.
Normally, the roots obtained by taking
two,
three, four and
five rows and columns of the determinant converge to a
limit.
Accurate values of frequency
can
also be obtained by con
siderations
of
energy. For a rectangular plate, the potential
energy of bending
U
s given (Timoshenko
1937)
by
and the kinetic energy
T
by
The maximum values of potential and kinetic energy are given
by
Eh3
T m U = t g
h2 1:
W2 dy . .
.
.
.
(6)
whencc
By the Rayleigh principle (Rayleigh 1894, p. 109 et
seq.), if
a
suitable waveform W is assumed, satisfying approximately the
boundary conditions, the resulting frequency value is very near
to, but higher
than,
the true value, because the assumption of
an incorrect waveform is equivalent to the introduction of
constraints to the system. The characteristic beam functions in
the directions
OX
and
OY
are normally assumed. By this
method the approximate frequency expressions in
this
paper are
obtained.
The energy method gives more accurate values
if
the
RayleighRitz method*
is
used. In
ths
a
series is assumed for
W, of the form
where
(x) and
are the
various
characteristic
bcam
functions corresponding to different modes. The
coefl icients
Aab have values, which make w2, given by equation (7),a
minimum.
For any mode of vibration the nodal pattern is dciined by
m
and
n,
the
number of nodal lines in the
X
nd
Y
directions
respectively. This number includes the edge
if it
is supported;
thus, for
two
parallel
edges
supported,
m
(or
n) can
be
any
integer given by
m> 2.
For a plate with one edge supported and
the parallel edge free,
m =
1 is an additional value, and for
two
parallel
edges
free,
m
= 0 and m =
1 are
possible. The
method of denoting the mode of vibration by the numbers of
nodal lines is suitable, if the
nodal
lines are approximately
parallel to the sides of the rectangle. However, when the plate
is square and
similarly
supported in the
X
nd
Y
irections, the
nodal lines
may
not be parallel to the sides. Thus, for a square
plate with
all
edges fixed (or free) or with
two
neighbouring
edges lixed and
two
free (or freelysupported), the patterns
may
not be
similar
to those for rectangles; the case of a square
plate with
a l l
edges freelysupported is an exceptionthe nodal
lines are always parallel to the edges. These nonparallel
patterns were first observed experimentally for a square plate
with free edges by
Chladni 1802
and
1817).
For
this
plate, the
normal
modes of vibration are of the type
m/n
(that is,
m nodal
lines
in
the
X
direction and n nodal lines in the
Y
direction)
only when m = n or
m
s odd and n is even, or
v i c e
versa. When
mn = f2, f4, f6 he normal modes are of the types
m/n+n/m
and
m/ n n / m.
These patterns
do
not consist of lines
parallel to the sides of the plate; the amplitude of any point is
the sum of the amplitudes of that’point for the component
vibrations
m/n
and
n/ m.
Thus
two
new nodal patterns are
obtained; the plus and minus
signs
corresponding to
m/n
and
n/m
being in phase and out of phase respectively. However, the
normal modes are
m/nfn/m
and,
in
these
cases,
the modes
m/n, n / m
cannot exist separately. Simple examples of modes of
the
types
m/nfn/m
are given in Fig.
2
(first
column)
for
a / b
= 1 for various boundary conditions. The
two
types have
discrete frequencies as well
as
discrete
nodal
patterns. The
gradual transition from
normal
modes of the types
m/nfn/m
to normal modes of the types m/n and n/m,
as
a / b deviates
from
unity,
will
be considered later in
this
paper.
When
m
is even and
n
is odd (or
vim
versa)
for
a free square
plate, the normal modes of vibration have nodal
lines
parallel
This
method was first applied by Rim
1909)
to the transverse
vibration
of free plates; the
method
and its ap
licatian
to
various
plate problana
have
been described
in
detpil
by
&instock
1952).
W=z%A
O(x)[email protected])
.
.
.
8)
ab
ab a
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T H E V I B R A T I O N O F RECTANGULAR PLATES
373
are only approximate for free edges. The appropriate expressions
for e ( x ) are given below; corresponding expressions for
$(y) can
be obtained by substituting
y b, E,
and
c
for
x ,
a,
y,
and
k
respectively.
to the sides, and the modes m/n and
n/m
have, in theory,
identical frequencies. The
tw o
modes may vibrate together and
an
infinite number of nodal patterns
is
theoretically possible,
corresponding to combinations of the modes m/n and n/m,
\Iv’th
varying amplitude ratios and phase angles.
Such
combinauons (1) Freelysuppfled
at 0
and
= a.
are given by Chladni (1802) and are discussed
in
detail by
Waller (1939). In practice, the frequencies of modes mln and
n/m may not be identical, owing to small imperfections in the
plate, and the
nodal
pattern obtained
d
epend
on
he method
of
excitation.
Characteristic Beam Functions.
Using the
Rayleigh
method,
the characteristic
beam
functions are substituted for
O(x) and
for
m
= 5 4 ,
6, . .
(lOa)
&)
n equation
2).
These functions satisfy the
boundary
conditions for plates
wt
fixed or freelysupported edges, but
m
>nx
a
(x) = sin
or
m = 2,3,4,. .
(9)
2)
Fixed at
= 0
and
=
a.
e x )
=
cos
y( ; 
j
k
cosh
y z+)
X
Sin
fr
s l n h
4Y
here
k
nd
tan b+tanh fr =
0.
L a d
Bl
H
lo
w
p o p
2//1+52
C /Il/2
i/;
]
h  DO
albo.98
X
e( )
=
sin yt ;j) +K’
inh
yt(fj)
for m
= 3a5a
7,
(lob)
and tan w tanh w
0.
in
fr’
S i n h
r’
here k’ =

e(x) =
1 form
=
0 . . . .’ .
(lla)
e(x)
=
1
form
= 1 . . . . .
(llb)
d x ) = cosy( ; + ) + k
cosh
y(f+)
for
m
= 2,4,
6,
(11~)
and
tan fr+tanhfr = 0.
here
k
=

3) Free
at x
=
0 and
x
= a.
2
a
x
Sin
4Y
w r
€3
a l b 
0.95
. .
(lld)
En
I
Fig.
2.
Variation of
Nodal
Pattern with u / b
a All
CS
fixed, m =
4,
n = 2.
b
All
edges
free, m = 2, n = 0.
c
T w o
neighbouring edges fixed, two Erce,m = 2,
n
= 1.
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374 THE
VIBRATION
O F RECTANGULAR PLATES
(4)
Fixed at
x =
0,
free
at
x = u.
e(x )
=
cosFcosh$+k(sin$sinhy)
for
m
=
1 ,2 ,3 , . .
. . . .
12)
( 5 )
Fixed at x
=
0, freelysupported at
x
=
a.
e( )
= sin
yfZh+)
+ A ’
sinh
y t (  + )
for
m = 2,3,4,. . . . . . 13)
where y’ and
k’ are
defined in equation
(lob).
e( ) = I;
for m =
1 . . . . . . . . . .
14a)
(6) Free at
x
= 0, freelysupported at
x
= a.
x
e x>
= siny‘ ~f)+k‘sinhy‘ ~f)
form
=
Z, 3,4,
. . .
14b)
where y’ and k‘ are defined in equation (lld).
A P P R O X I M A T E F R E Q U E N C Y E X P R E S S I O N
For any boundary condition, expressions for
O(x)
and
+(y)
are obtained from equations
(9)(14),
and are substituted
in
equations (2) and (7) to give an expression for frequency. In
each case a nondimensional frequency factor
A
proportional to
frequency, is derived, where
pU4(2&Z12(1 02)
. . .
15)
ltoEh2g
2 =
It is found that for
all
possible boundary conditions consisting
of any combination of Mysupported,
free,
and
fixed edges,
frequencies
can
be determined from the expression
where the coefficients
Gxa Gya Hxa Hya
Jxa
and
Jy
depend
on
he
nodal pattern and the boundary conditions. Values of these
coefficients are given in Table
1
for the three pure boundary
conditions,
in
which
all
the plate edges have the same condition,
and for the twelve cases
in
which some of the plate edges have
one condition and the rest another.
Suppose the frequency
of
a rectangular plate with three edges
fixed and one freelysupported (parallel to the Yaxis) is required
for the modem
= 2, n
=
3.
Then from Table
1
and equation
(16)
+(l0). 1*252(1mT) ,2 52
d
UZ
=
2*44+39.0%+ lO45p
Thus the frequency factor
h
is obtained for any ratio a / b ; the
frequency is given by
f
In
this
case,
A
is independent
of
Poisson’s ratio
u,
because
H x
=
Jx
and
H, = J,; this
is true for any plate with no
free
edges.
Equation (1 6) and Table
1
can
also
be used to give frequencies
of plates with boundaries including
all
three edge conditions
free, freelysupported, and fixed. For instance,
if
a rectangular
plate has the side
x = 0
fixed,
y =
0 freelysupported, and
x = a
and
y = b
free, the values of G,,
Hx,
and
Jx
are idenucal
with those of cases
12, 13,
and
15
in Table
1
; hat is, only the
boundary
conditions for the edges
x
= 0 and
x = a and
the
number of nodes
in
the Xdirection affect
the
values of
G,, Hx,
and
Jx.
Similarly, the edge conditions for
y = 0
and
y = b
and
the value of
n
control the values of
Gy, H,,,
and Jy, and the latter
are identical
in this
case with those of
9 m
Table
1.
Thus for
the fundamental mode 111 the frequency factor
h
is given by
2a2 3
h2 =
05974+0+~[u(0*0870). O+(lu)O.471
a]
a2
= O.l27+0*287g(l~)
The last column of Table
1
gives the conditions for which
modes
of
the typesm/ n &n / m
occur
or square plates. Sometimes
the effect of these modes on requency is considerable;
this w i l l
be discussed in the
next
seaion, where the necessary modified
frequency expressions are given. In that section the frequencies
for modes
of
he
typem/n,
given by equation
(16),
are compared
with those obtained by previous investigators. However,
it
may
be noted that the errors in the frequencies obtained from
equation (16) for plates with no free edges are less
than
0.6 per
cent.
A C C U RA C Y O F A P P R O X I M A T E F R E Q U E N C Y E X P R E S S IO N
A N D D I S C U S S I O N O F M O D E S m / n f n / m
(1) Four Edges Freelysuppo lted.
This is the only boundary
condition for which h
can
be expressed exactly by a simple
formula. The frequency equation derived by the Rayleigh
method, equation (16), is identical with the wellknown exact
expression, Timoshenko
(1937).
For square or rectangular plates
the
nodal
lines are always parallel to the sides.
(2)
Four Edges Ftked.
Sezawa (1931) obtained frequencies for
several modes, but,
owing
to his assumptions, his values are
inaccurate by 35 per cent. Kato (1932),by progressive approxi
mation, obtained a value
h =
3.641 for the fundamental mode
for a square plate. A more accurate value for
this
mode,
h
= 3.646, has been obtained by Merent methods by Tomotika
(1935 and 1936) and Iguchi (1938). The latter also obtained
frequencies for various higher modes of square plates and for
the
fundamental for rectangular plates. By using a modified
minimal
problem, Weinstein
(1937
and
1951)
obtained upper
and lower bounds for the frequency of ten modes for a square
plate. Hopkins
(1945)
extended the theory to include transverse
load, the influence of stiffeners and aeolotropy of the material;
the only calculated frequency
in
his paper agrees with those of
Tomotika and Iguchi for the fundamental mode for a square
plate. The methods of Tomotika
(1936), Iguchi (1938),
and
Hopkins
(1945)
were all
similar
to that previously described,
whereW onsists of an infinite series considerable computation
is required to obtain a single frequency by these methods.
Young (1950) applied the RayleighRitz method to square
plates, and obtained frequencies agreeing with those of Iguchi
(to within
0.1
per cent). Eschler
(1950)
used the Rayleigh method
to obtain an approximate frequency expression,
similar
to but
more complicated
than
equation (16). Comparing values of
h
given by approximate and exact analyses, the maximum err01
is 0.5 per cent.
Existence
of Modes
m/nfn/m. When a clamped plate is
square or almost square, the nodal patterns
no
onger consist of
lines parallel
to
the sides.
Also
a square plate does not have two
normal modes
m/n, n/m
with identical frequencies, but
has
normal modes represented by
m/n n/m
with two discrete
frequencies. The nodal patterns of the modes 4/2&2/4 are
shown in Fig. 2u. The literature contains no record of the
changeover
in
pattern from a rectangle to a square. Young
(1950)
obtained discrete frequenciesfor the
two
modes
4/2&2/4,
but Iguchi (1938) obtained a frequency only for the mode
4/2+2/4. In order to
check
the existence of these two modes
with discrete frequencies, the author exteaded the analysis of
Tomotika (1936) to these modes, and found that they exist
separately with discrete frequencies.
The author investigated the transition of patterns as
a / h l
by using the RayleighRitz method, but
limiting
the analysis
to the two principal terms. For a rectangular plate
in
which
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T H E V I B R A TI O N OF R E C T A N G U L A R P L A T E S
TABU
. COEFFICIENTS
N FREQUENCY
QUATION
16)
Freelysupported edge,
m
ixed
edge.
     
375
Free edge.
~
Boundary
conditioi
a
1
2,3,4,. .
2
3,4,5, .
0
1
2
3,4,5,
.
2,3,4,
1,3,4,..
2
3,4,5, .
,
.4, .
1
, .4,
1
t,
3.4,.
0
1
2
I,4,5,
.,
1
I 3,4,. .
1
2
1,4,5,
I
2
I4.5,.
. I
2
,4,5,. .
1
2
,4.5,. .
2,3,4, .
2
I, 4.5,. .
0
1
2
I,
4,5,
2
1,4,5,
3,4,
.
I 3.4, .
r 3 r
4,. .
0
1
2
3 4,5,
1
8 3.4,. .
1
3.4, .
B3.4,. .
2
I
4,5,
.
1
2
4,5,. .
0
1
2
4,5,
0
1
2
4,5,
.

GY

n.
150
n
0
0
193
n1
160(
n4
n4
nI
n1
0
0
1*50(
n4
0
nf
n  1
01
L.506
*f
1.597
,494
1+
0
0
506
I+
0
0
506
1f

HY
(nl)z
0
0
2
1+)2 c y n  * ) n
1.248
I w1
An
(nl)a
(nl)1
(n1)*
0.0870
1.347
2
0
0
~
JY
1.248
2
0
(nl)2
(n 111
0.471
3.284
2
Modes
m/n&nlm
exist
for
a 
if,
None
mn= f2,4,6... .
rn = i2.4.6, ...
None
None
None
None
None
None
None
m i t n
None
None
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376 T H E V I B R A T IO N O F R E C T A N G U L A R P L A T E S
there are an even number of nodes in both the
X
nd
Y
directions,
the displacement may be expressed as
W
= A [ . . s y ( ~  ~ ) + k r o s h y ~  , ) ] [ 0 0 ~ € ( f  f )
+c cosh c(f+)]
+B[cosc~f)+ccosh~~))][cosY(%f)
+ k coshy($t)]
where the amplitude ratioA/B s chosen for each mode to make
the frequency a minimum for that mode. This is expressed
mathematically by
aR
aR
A = 0 , a T = O
where
R
is the Riu integral given by
a2w a2w
]&dy .
(17)
azw
2
12(1u2)pw2~
Eh2g
The evaluation of
t h i s
integral is simplified by noting that
provided that W = 0 at the four edges. This w a s proved by
Leggett (1941) in a paper on the buckling of plates.
Hence
aR
9 4 2y2czr) phw2
o =  

g + p + x  x ) A + & j (18)
. . (19)
where
2 2
[1 
‘I ;
sin y ) ] [1

l)n(cz; sin
€ I
and
Eh3
12(1
2 )
D=
Owing to the existence of the small term T]’he two modes
m/ n
and n / m are never completely independent
;
heir effect on each
other
wll
be investigated for particular values of
m
and n. If the
modes 412 and 214 (or the associated modes 4/2f2/4) be con
sidered, then substitutingy = 3.57~
nd
e = 1.506~n equations
(18) and (19), two frequency factorshl and
X2,
with corresponding
amplitude ratios
@/A ) ,
and (B/A)2, are obtained for any value
of
a / b . It
should be noted that (B/A)l
=
(A/&.
In Fig. 3a,
hl and h2 are plotted against a/b for these modes; the dotted
lines represent the values of X 1 and hz obtained by applying
equation (16) to these modes. I t
wil l
be observed that the
resulting difference in frequency, if a dotted line is used instead
of its associated full line, is negligible. Thus f it is erroneously
0:6 0 9
a/b
b
All edges free, m = 2, n = 0.
16
u
All
edges
fixed, m = 4, n = 2.
a 6
c
Two
neighbouringedges
fixed,
two free, m
= 2, n =
1.
Fig. 3. Variation of Frequency Factor X with a /b
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T H E V I B R A T I O N OF R E C T A N G U L A R P L A T E S
377
corrected the work of Goldmann and showed theo re ti dy and
experimentally that for
alb
= 1.5, the nodal patterns were
approximately parallel to the sides, conesponding to modes of
the type mln. By taking three plates for which alb = 1,1.04, and
1.07,
Pavlik showed theoretically and experimentally the transi
tion from the characteristic patterns of square plates to those of
rectangular plates. Waller's papers (1939 and 1949), which
include a large number of experimental patterns for square and
r e ~ . 3 u l a rree plates, showed the same transition.
Equation (16), with Table 1, gives frequencies for all modes of
the type
mln;
that
is,
it
gives all frequencies for rectangular
plates and frequencies for square (or almost square) plates when
m =
n
or m  n is an odd number. When the plate is square and
m  n 1s
an even number, the modes are of the types m/nfn/m.
Th e range of values of
a / b ,
in which the transition from modes
m / n to modes m l n f n l m occurs, is considerably greater for free
plates than for fixed plates.
Also,
the percentage difference in
frequency between two associated modes m/n+n/m and
mlnnlm is much greater for free plates than for fixed plates.
Thus it is necessary to modify the frequency expressions
previously given.
Modes
m10fOlmfor
Square Plates Where
m
s Even. Assuming
vibrauon forms
LV
=
A[cosy(~+)+k coshy(:+)]
assumed that the normal modes for a square plate with fixed
edges are
412
and
214,
instead of
4/2f2/4,
the frequency error
will be very small. The amplitude ratio ( B / A ) l is plotted in
Fig. 4 (curve
I);
these values have been used to construct the
0
fA[cos y e  i ) +k a s h y e  + )
Fig. 4. Variation of Amplitude Ratio with
a / b
I
All edges fixed, m =
4,
n = 2.
I1
All edges free,
m
=
2,
n =
0.
I11
TWO
eighbouring
edges
fixed,
two
free, m =
2,
n =
1.
nodal patterns for a / b = 1 and a / b = 0.98, shown in Fig. 2 ~ .
Fig. 2 shows the rapid change from patterns characteristic of
square plates to those for rectangular plates, as
a / b
deviates from W = A[sin
y E+)
+k' sinh
f(:f)](l?)
it is found that
A2
=
(m+)4f2o(m+)2;;,8 . . (20)
Modes m l l lm for Square Plates Where m = 3 ,
5,
7 . .
Assuming vibration forms
..n; .r
y . Y L , .
Similar results are obtained for modes
with
odd numbers of
nodes in the
X
and Y directions, provided m
n.
When
m
is
odd and
n
even, or vice versa, there is no term in
AB
in the
integral
R.
Hence the two equations, corresponding to equations
(18) and (19), are independent. Thus, in this case, m l n and n l m
are normal modes for
all
values of
a / b
and there is only one
frequency for the square plate, as the two equations are identical
for
a / b = 1.
Only one pair of modes has been investigated in detail, but
these modes have the lowest frequencies at which this pheno
menon occurs. From the nature of
7
[equation (19a)],
it
can
be seen that the percentage difference between the frequencies
for modes
m/n+n/m
and
m/nn/m
decreases as
m
and
n
increase. Thus the author has not attempted to allow for these
modes in his approximate frequency equation (16). When the
plate is square, and m and n are unequal but both even (or both
odd), there will be two frequencies, both nearly equal to that
given by the approximate formula. The nodal patterns for these
modes
can
be determined by addition
and
subtraction of the
relevant patterns for modes
m / n
and n/m.
Chladni
(1802
and
1817)
determined
experimentally the nodal patterns of square plates. The
theoretical determination of frequencies and nodal patterns of
free square plates was the first application by Ritz (1909) of
his method, which has since been applied to many other
problems. Lemke (1928) repeated the work of Ritz with greater
accuracy (in effect this means taking more terms in the series
for displacement) and she also obtained experimental frequencies
and nodal patterns; for the graver tones there was good agree
ment between theory and experiment. Ritz showed that for
square plates the normal modes were of the type mln with nodal
lines approximately parallel to the sides, if m
=
n or if m was
odd and n even (or vice versa). For m  n
=
5 2 , f4, f6, .
the normal modes were of the types
m/nfn/m.
Goldmann (1918) extended the work of Ritz to rectanguh
plates, but erroneously assumed that modes of the types
m/nfn/m existed in rectangular plates. Pavlik (1936 and 1937)
(3)
Four
Edges
Free.
it
is
found that
Modes
mlnfn lm
for Square Plates Where m and n are Like
Quantities but Unequal ( m > l ,
n>l). Assuming vibraaon
forms
W
=
A[cos
y ( z  4 )
+k cosh
y z+)] [ m s c ~  + )
+c coshce;'+)]
& A [ c o s c ~  + ) + c o s h ~ ( ~  f f ) ] [ c o s y ~  f )
+k coshy(:+)]
it is found that
A2 = (m. ) )4+(n+)4
6 6
160 1
m  n
,2 (m

)4{m
+ n 
1+ m )2+ (n )*
m+n1 2
. . (22)
1
m  n
32(1
)
1
2
(m
+)2(n+)2[
m+nl+ (m 
)2
+
n 121
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378 THE V I B R A T I O N OF R E C T A N G U L A R P L A T E S
It
will be seen that for
all
modes
m/n&n/m,
the expression for
A2 consists of the corresponding expression for the nonexistent
mode
m / n
plus (or minus) a ‘correction factor’. These factors are
not negligible; the difference in frequency for the modes
2/0+0/2
nd
2100/2
is approximately
25
per cent for
I
=
3.
However, in the correction factor
of
equations
(20)
nd
(22),
t
is sufficiently accurate to use the integral value m
= 2 or n =
2)
with no modification; the usual modifications
are
made in the
remainder of the expressions for hz Frequencies have been
calculated from these equations for various modes for square
plates and for rectangles, where
a / b = 1.5,
and have been
compared with the accurate values of
Ritz (1909),
Lemke
(1928),
and Pavlik
(1936).
For square plates, frequencies given by the
approximate formulae are
5
per cent high for modes
111,
211,
and 2/2,but for most other modes the error is less than 1 per
cent. For the rectangular plate the error for the fundamental
mode
1/1
is
6
per cent, but for other modes the errors in fre
quency are about
1
per cent.
The author has investigated the transition as a/ l from the
modes
210
and 0/2 f rectangular plates to the modes 2/0+0/2
and
2/00/2
of square plates. Th e method
has
been described
for plates with fixed edges. Th e waveform assumed is
:
W
=
A[cos
y(:+)
+ k
~~shy(:+)]
In Fig.
3b
the frequency factors,
A 1
and
h2,
for the
two
modes
are plotted against
a / b .
The frequency factors that would be
obtained by erroneously using equation
(16)
are shown by
dotted lines. Over the range
0.8< a / b
<
1.25, if
frequencies are
obtained from equation (16) or these modes, they are inaccurate.
In Fig.
4
(curve 11) the variation of the amplitude ratio
with
a / b
is shown;
this has
been used to plot the nodal patterns
for the two modes for various values of
a / b
n Fig. 2
t
will be
seen
from Figs.
2
and
4
hat
the transition from modes charac
teristic of rectangular plates to those of square plates
is
much
more gradual for free plates than for fixed plates. T he amplitude
ratios agree with those obtained theoretically and experimentally
by Pavlik
(1936
nd
1937)
his results show that for other modes
( B / A ) l
decreases as the difference between m and
n
increases,
for a particular value of
a / b .
Thus
it
is safe to conclude that for
all modes for which
( m  n )
is an even number, equation (16) s
inaccurate only over the range
0.8
< a / b
<
1.25.
(4,
Mixed Boundmy Conditions.
For
six
of the twelve mixed
boundary conditions listed
in
Table 1 accurate values of
frequency have been obtained for some modes by other investi
gators. For each of the conditions, in which
two
parallel edges
are freelysupported (conditions
6,7, 10,
and 1
,
there exists a
waveform, which satisfies the plate equation
(1)
and
all
the
boundary conditions. The resulting frequency equations are
transcendental and can be solved numerically only for particular
values of (nl)a/b. For a plate with two parallel edges fixed
and
two
freelysupported, Sezawa (1931) and Kato (1932)
obtained the fundamental frequency, and Iguchi (1938) btained
frequencies for several modes. Eschler
(1950)
olved the problem
with the addition of loads applied in the plane of the plate and
he also obtained an approximate frequency expression similar
to equation
(16).
Voigt
(1893)
irst derived the transcendental
equations for plates with
two
parallel edges free, two freely
supported; following
this
analysis Zeissig
(1898)
calculated
frequencies for various modes and performed experimental
investigations, which gave frequencies and nodal patterns
agreeing with
his
calculations. Eschler (1950) epeated Voigt’s
analysis, adding the effect on frequency of loads applied to the
plate in its plane, and he also obtained an approximate frequency
expression, but it included only modes with two or more nodal
lines parallel to the free edges. Iguchi (1938) calculated fre
quencies for several modes for a plate with one edge fixed, three
edges freelysupported. For the last of these four conditions
one edge free, three edges freelysupportedthe author has
derived the transcendental frequency equation (Appendix
11).
Young
(1950)
used the RayleighRim method to calculate
frequencies of the five gravest modes of vibration of square
plates with
two
neighbouring edges free and
two
fixed, and with
one edge fixed and three edges free. For the latter condition
Barton (1951) performed similar calculations for rectangular
plates and obtained experimental frequencies for these modes.
Eschler (1950) btained an approximate solution for plates with
two parallel edges free and two fixed, but his solution considered
only modes in which there were two or more nodal lines parallel
to the free edges; thus the
two
gravest families of modeswith
no nodal line and one nodal line parallel to the free edgeswere
ignored.
The accuracy of the frequencies obtained from equation (16)
depends on the boundary conditions.
As
has been mentioned,
the assumed waveforms do not satisfy exactly the conditions for
a free edge
;
ence the frequencies are not as accurate for plates
with one or more free edges. When there are no free edges, the
frequencies given by equation
(16)
and by exact analysis agree
to within 0.6 per cent. When free edges exist, the approximate
frequencies are usually within
1
per cent of the exact values,
but they are sometimes
5
per cent too high when the charac
teristic functions for freefree or fixedfree beams are used in
the analysis. The greatest errors occur for cantilever plates
(condition 15) or the modes I and 2/1; or a square plate the
approximate frequencies are
11
and 3 per cent too high and for
a / b
= 5
they are 27 and 12 per cent too high for these modes.
For both these modes the waveform
in
the Ydirection is
I
2
3
4
5
6
7
n t , NUMBER OF NODA L LINES
IN
S
DIRECTION
Fig.
5.
Frequencies of Cantilever Plate
a/b = 1.855.
o
Grinsted’s experimental points.
n
Number
of nodal
lines
in
Y
direction.
assumed to be linear. This assumption, which cannot be
expected to give good results, is also associated with the less
accurate frequencies obtained for free plates. I n Fig.
5
the
frequencies of a rectangular cantilever plate
( a / b = 1.855),
obtained from equation (16) and Table 1, have been plotted
against the number of nodal lines
in
the Xdirection, m, each
curve representing a different number
of
nodes in the Ydirec
tion, n. As
m
and n can only be integers, the curves connecting
the frequencies have no physical significance, but merely indicate
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379
HE VIBRATION OF RECTANGULAR PLATES
the different families of modes. Grinsted (1952) obtained experi
mental frequencies for
24
modes for a plate of these dimensions
and his experimental points are shown in Fig. 5. Apart from the
mode
111
the greatest difference between experimental and
theoretical frequencies is 6.4 per cent and the mean error is
25 per cent. The diagram shows that, in general, the frequency
error decreases as m and
n
increase.
For three of the mixed boundary conditions of Table
1
(5, 9, and 13) modes of the types
m/nfn/m
exist for square
plates, when
m
and n are unequal. When the plate has two
neighbouring edges fixed and two freelysupported (condition
5),
the difference in frequency between associated modes
is
negligible, and equation (16) with Table 1 can be used to give
approximate frequencies; the rapid transition from nodal
patterns characteristic of square plates to those of rectangular
plates is similar to that described for a plate with
all
its edges
fixed. A square plate with two neighbouring edges free and two
freelysupported (condition
9)
behaves
in
a manner similar to
that of a plate with free edges. Thus the percentage difference in
frequency for associated modes, m(n+n m .and mlnnlm, may
be considerable. Frequency expressions slrmlar to equations (21)
and
(22) can
be derived, and for modes
m / l
/m,
here
m> ,
The
behaviour of the modes
211
and
1/2
for a plate with two
neighbouring edges lixed and two free
(condition 13)
has been
investigated by the RayleighRitz method, assuming
W =
A [ c o s ~  c o s h ~ + k ( s i n ~  s i n h ~ ) ] [ c o s $

osh
?+ c(sin  sinh f ) ]
€X
w
( :
a
+B[cos%cosh EX
+c s i n sinh )][ax 7
a
 c o s h ~ + k ( s i n ~  s i n h p ) ] . .
,
(24)
In Fig.
4,
curve I11 gives the
variation of the
amplitude ratio
A/B with a/b. Fig. 3c shows the variation of frequency with a/b
for the two modes; the full lines represent the frequencies
calculated by the RayleighRitz method and the dotted lines
represent those calculated from equation
(16)
and Table
1 .
The points marked ere obtained for a square plate by
Young, using several terms
in
his series expression for
W,
nd
show that the use of two terms in the expression for W nstead
of the approximate frequency expression (16) may not, in fact,
increase the accuracy. The author considers that approximate
frequencies for all modes
can
be obtained from equation (16),
particularly as the percentage difference in frequency between
associated modes m/ n+ n/ m and m / n  n / m decreases as
m
and n
increase; the errors for modes
m/n&tn/mwill
not be greater than
those existing for modes m / n for
this
boundary condition. If
exact values are required for the higher modes, a series of
several terms must be used for
W.
he variation of nodal pattern
with
a/b
is shown
in
Fig.
2c
for the modes
2/1
and
1/2.
E F F F C T O F A P A R T I A L L Y R E S T R A I N E D E D G E
For a plate which has three edges fixed and the other supported
and partially restrained frequency limits can be found for any
mode by considering the fourth side to be (a) fixed and (b) freely
supported. The greatest percentage difference between these
limits will occur for the fundamental mode. For
t h i s
mode for
a square plate, taking values from Table
1,
with four edges fixed
h,2
=
1.5064+1.5064+2. 1.2482
= 13.41
A, = 3.66
with three edges fixed and one freelysupported
hb2 = 1.254+1*5064+2.
1.252 1=)1*248
= 10.49
& = 3.24
Therefore 3*24
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380
THE VIBRATION O F RECTANGULAR PLATES
where
A,
B, and
C
are component amplitudes. From the ampli
tude ratios for cylinders [Arnold and Warburton
1949,
equations
(17)
and
(re)]
it follows that for plates,
for
h1
A = B = O
for
h2
(m

)b
(n 1)a
=O a n d  A =B
(n 1)a
(m

)b
=Oand A=B
nd for
h3
Thus, the vibrations associated with hl are transverse vibra
tions, and the frequency expression for hl agrees with that pre
viously given [equation
(16)].
The other roots give much higher
frequencies, as
h / a
must be small for a
th n
plate, and are asso
ciated with extensional vibrations; that is, the motion occurs
only in the plane of the plate.
Love
(1927,
p.
497)
gives equations for extensional vibrations
of plates; by substituting
u
and w from equations
(26)
in his
equations, the expressions for
X2
and
h3,
given in equations
(25b
and
c),
can be
obtained. For transverse vibrations the
boundary conditions considered were freelysupported, free,
and fixed; for extensional vibrations new definitions are neces
sary. In general, there are
two
conditions for the edge
OY;
these
conditions depend on whether that edge is free to move in the
Xdirection. These conditions
will
be referred to as free and
restrained; in this section
a l l
edges of the plate are free. From
equation (26) it
can
be seen that there are m equallyspaced lines.
parallel to the Yaxis, including the edges, for which
u = 0;
between these lines there are
(m1)
lines for which
u
= 0.
Similarly, parallel to the Xaxis there are
n
lines, including the
edges, for which
u = 0
and
n1
lines for which u
= 0.
Thus
the only real nodes are a finite number of points, given
by
the
intersection of these two series of lines. Further investigation
shows the difference between the two types of mode associated
with
h2
and
A 3
respectively. For the former type the plate edges,
x = 0, x
=
a,y =
0,
and
y
= b, must be free from direct stress,
but a shear
stress
must
be
applied to the edges for the equili
brium equation to be satisfied. For the latter type no stresses
are
applied to the edges. Thus equation
(25c)
gives the frequency of
extensional vibrations of rectangular plates with perfectly free
edges.
Plates with
Two
Parallel
Sides
Fixed
and
Two Freelysupported.
From the results for cylinders with fixed ends
[Arnold
and
Warburton
1953,
equation
(23)]
a frequency equation is obtained
for plates of this type; the roots of the equation are
d
2aZ
b4 b2
h12
= (m+)4+(n11)4+(m+)2(nl)2p
.
+4(nrf)z(nl)$(1+~)2p] t . (27b
and c)
for m
= 3, 4, 5, . .
here
p
=
l
(m
T
2
; = 0.550
nd
m+
is replaced by 1.506 for m = 2.
The lowest mot agrees with that given by equation
(16)
for
transverse vibrations; this is to be expected, as the theories for
cylinders and
beams
are both based on assuming the beam
vibration form between the fixed ends. The two higher mots
give the frequencies of extensional vibrations for plates with two
parallel edges restrained and
two
free. As the waveform for a
cylinder with fixed ends is only approximate, the frequencies
given by equation (27) are also approximate, whilst those given
by equations
(25)
are exact, because the assumed waveform
[equation
(26)]
satisfies the plate equations.
A P P E N D I X
I 1
T R A N SV E R SE V I B R A T I O N OF R E C T A N G U L A R P L A T E W I T H
O N E E D G E F R EE A N D T H R E E E D G E S F R E E L Y  S U P P O R T E D
ConsideringOC (Fig.
1) to
be free and the other edges fieely
supported the waveform which satisfies the plate equation is
w
=
sinotsin n(nl)y[
Asrn
(n(n;1)2u rZl)+($+)}
From the boundary conditions
and the frequencies
are
given by
2fr2+l)f
coth
[ ~ ( r z + l ) + ]
( l>a
=
0.
(30)
This equation
has
to be solved numerically for particular values
of
(nl)a/b
and
U.
A P P E N D I X
I 1 1
R E F E R E N C E S
ARNOLD,
R. N.,
and
WARBURTON,
.
.
1949 P m .
OY.SOC.
A$ vol.
197,
p.
238,
‘Flexural Vibrations of the Walls of
Thin
Cylindrical Shells Having Freelysupported Ends’.
1953
Proc. 1.Mech.E. (A), vol.
167,
p.
62,
‘The Flexural
Vibrations of
Thin
Cylinders’.
BARTON,
M.
V. 1951
Jl. of Applied Mechanics,
Trans.
A.S.M.E., vol.
73,
p.
129,
‘Vibration of Rectangular and
Skew Cantilever Plates’.
CHADN
E.
F.
F.
1802
‘Die
Akustik‘.
ESCHLBR,
H.
1950
IngenieurArchiv, vol.
18, p. 330,
‘Zur
Ermittlung der Eigenschwingungmhlen der
in
ihrer
Mittelebene belasteten Rechteckplatte’.
GOLDMA”,E. 1918 ‘Anwendung der Ritzschen Methode auf
die Theorie der Tr an sv er sa ls ch wi ng n freischwingen
der Platten von rechteckiger, rhombischer, dreieckiger
und elliptischer B e g r m g ’ (Dissertation, Breslau).
GRINSTID,
B.
1952
Proc. 1.Mech.E.
(A),
vol.
166,
p.
309,
‘Nodal Pattern Analysis’.
HM
R. F. S.
1952
JI. of Applied Mechanics, Trans.
A.S.M.E., vol.
74,
p.
402,
‘The Frequency of Vibration of
Rectangular Isotropic Plates’.
1945
Aeronautical Research Council, Reports
and Memoranda No. 2234, ‘The Solution of
Small
Displacement, Stability or Vibration Problems Con
cerning a Flat Rectangular Plate when the Edges are
either Clamped or Simplysupported’.
IGUCHI, . 1938
Memoirs of the Faculty of Engineering,
Hokkaido Imperial University, ‘Die Eigenwertprobleme
f i r die elastische rechteckige Platte’.
1817
‘Neue Bietriige
zur
Akustik‘.
HOPKINS,. .
1937
IngedeurArchiv,
vol. 8,
p.
11.
1938a IngenieurArchiv, vol. 9, p.
1.
KATO,
H.
1932 J1. of the Society of Naval Architects, vol. 50,
p.
209.
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THE VIBRATION
OF
RECTANGULAR PLATES 381
LECGBTT,
. M. A. 1941 Aeronautical Research Council,
TOMOTIKA,.
1935 Report of the Aeronautical Research
Reports and Memoranda No. 1991, ‘The Buckling of a Institute, Tokyo Imperial University, No. 129, ‘Trans
Square Panel Under Shear when One Pair of Edges is verse Vibration of a Square Plate with Four Clamped
Clamped and the Other Pair is Simplysupported’.
Edges’.
LEMKE,LICE
1928 Annalen der Physik, Leipzig, fourth 1936 Phil. Mag., seventh series, vol. 21, p. 745, ‘The
series, vol.
86,
p.
717,
‘Experimentelle Untersuchungen Transverse Vibration of a Square Plate Clamped at Four
zur W. Ritzschen Theorie der Transversalschwingungen Edges’.
quadratischer Platten’. VOIGT,W. 1893 Nachrichten, Giittingen, p. 225, ‘Bemerkung
LOVE, A. E. H. 1927 ‘Mathematical Theory of Elasticity’, zu dem Problem der transversalen Schwingungen
fourth edition (Cambridge University Press).
rechteckiger Platten’.
PAVLIK,B.
1936
Annalen der Physik, Leipzig,
fifth
series,
WALLER,Mary
D.
1939 Proc. Phys. SOC., vol. 51, p. 831,
vol. 27, p. 532, ‘Beitrag
zur
theoretischen und experi ‘Vibrations of Free Square Plates’.
mentellen Untersuchung der Biegungsschwingungen bei 1949 Proc. Phys. SOC. B), vol. 62, p. 277, ‘Vibrations of Free
rechteckigen Platten mit freien Riindern’. Rectangular Plates’.
1937 Annalen der Physik, Leipzig,
fifth
series, vol. 28,
1937
Memorial des Sciences Mathematiques,
p. 632, ‘Beitrag zur Untersuchung des Zusammenhanges No. 88, ‘Etude des spectres des equations aux dtrivkes
der bei Biegungsschwingungen an rechteckigen und partielles de la thb rie des plaques elastiques’.
quadratischen Platten beobachteten Staubfiguren’. 1951 Jl. of Applied Mech. Trans. A.S.M.E., vol. 73, p. 229,
RAYLEIGH,ord 1894 ‘Theory of Sound’, vol. 1, second Contributions to the Discussion of paper ‘Vibration of
edition (Macmillan and
Co.,
London). Rectangular Plates by the Ritz Method’.
RITZ, W. 1909 Annalen der Physik, fourth series, vol. 28, 1952 ‘Calculus of Variations with Applications
p. 737, ‘Theorie der Transversalschwingungen einer to Physics and Engineering’ (McGrawHill, New York
quadratischen Platte
mit
freien Riindern’.
and London).
SEZAWA,
K 1931 Report of the Aeronautical Research YOUNG,
D.
1950 Jl. of Applied Mechanics, Trans. A.S.M.E.,
Institute, Tokyo Imperial University, No. 70, ‘On the vol. 72, p. 448, ‘Vibration of Rectangular Plates by the
Lateral Vibration of a Rectangular Plate Clamped at
Ritz
Method’.
Four Edges’. ZEISSIG,
C.
1898 Annalen der Physik, Leipzig (Wiedemann),
1937
‘Vibration Problems in Engineering’, vol.
64,
p.
360,
‘Ein einfacher
Fall
der transversalen
second edition (Van Nostrand). Schwingung einer rechteckigen elastischen Platte’.
WEINSTEIN, .
WEINSTOCK,.
TIMOSHENKO,
.
Communications
Mr. R.
F.
S.
HEARMON
Princes Risborough) wrote that the
author had shown that the systematic application of beam
functions not only simplified the approximate derivation of the
transverse frequencies of rectangular isotropic plates under all
possible boundary conditions, but also completely unified the
presentation
of
the results.
It was worth discussing briefly the possibility of applying
beam functions to plates which differed in specified ways from
those dealt with by the author and, in particular, with plates
(1) which were isotropic, but oblique;
2)
which were rectangular but anisotropic.
(1) Barton (1951) had calculated the frequencies of the first
two modes of oblique (or skew or swept) cantilever plates, having
all sides the same length. He had used a series formed from
suitable beam functions to represent the deflexion
W
nd, in his
treatment, had limited the series to eighteen terms, but even
so,
he had been unable to avoid somewhat complex calculations.
Bereuter (1946)* had obtained estimates for the fundamental
frequency of oblique plates with all edges either clamped or
supported and of the clamped oblique plate with a concentrated
mass at its midpoint. He had assumed straightforward trigono
metric or polynomial functions for
W
and,,by applying the
*BERBUTER,. 1946 ‘Theoretische Untersuchungen iiber die
Eigenfrequenz Parallelogramftrmiger Platten’. Ecole Polytechnique
Fedkrale Publications du Laboratoire de PhotdlasticitC No.
3.
Edition S.
A.
Leeman frkres et Cie, Stockerstrasse
64,
Zurich.
(Acknowledgement is made to Professor B. G. Johnston, University
of
Michigan, or drawing attention
to t h i s
reference.)
RayleighRitz method, had been able to derive for the funda
mental frequency of the clamped oblique plate a closed formula
which was substantially ndependent of the form assumed for W.
In the case of the supported oblique plate, Bereuter had been
unable to find any simple expression for W which satisfied the
boundary conditions, and, as a result, the RayleighRitz method
proved unreliable. Bereuter, however, developed a finitedif
ference method applicable to the supported plate, but the solution
was only slowly convergent with respect to the fineness of the net
into which the plate was divided, and thus the accuracy was not
high. The problem of buckling was closely allied to that of
frequency, and Wittrick (1953)t had studied the buckling of the
clamped oblique plate. It was evident that there w a s some
interest at the present time in oblique plates, and it seemed that
the application of the author’s beam function method might
yield results
of
considerable value.
(2) The differential equation and the potential energy
equation had been derived for a generally anisotropic plate (for
example
see
Hearmon 1948) , but the only case which had
so
far proved amenable to treatment was the ‘specially orthotropic’
plate (Hearmon and Adams 1952)9, that is, a plate made from a
t
WIITRICX, .
H. 1953
Aeronautical Quarterly,
vol.
4, p. 151,
‘Budding
of
Oblique Plates with Clamped Edges Under Uniform
Compression’.
*
HFARMON,. F. S.,and
ADAMS,
E.
H. 1952
British Jl. of Applied
Phys., vol. 3, p. 150, ‘The Bending and Twisting of Anisotropic
Plates’.
HEARMON,
. F. S. 1946
Proc. Phys.
SOC.
Lond.,
vol. 58, p. 78,
‘ T h e
Fundamental Frequency of Vibration
of
Rectangular Wood and
Plywood
Plates’.
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382
C O M M U N I C A T I O N S O N T H E V I B R A T I O N O F R EC TA N GU LA R PLA TES
material possessing three perpendicular axes of elastic symmetry,
two
of which lay in the plane of the plate,
in
directions parallel
and perpendicular to the plate edges. Expressions for the fre
quency of vibration had been derived for all the modes of the
simplysupported orthotropic plate, but
only
for the funda
mental mode of the clamped plate (Hearmon 1946)*. However,
using the appropriate beam functions (equations (1Oa) and (lob))
in conjunction with the potential energy expression from
reference Hearmon (1948) or Hearmon (1946), the frequencies of
a l l
the modes, except the fundamental of
a
clamped orthotropic
plate, were obtained as :
the
difiicult
question of ‘single’ vibrations which had no separate
physical existence and of true ‘combined’ modes.
The nodal systems for a square plate with fixed edges shown
in Fig. 2u were 3/1+ 1/3and 3/1
1
3 systems respectively, not
as stated, 4/2&2/4 systems. That might be made
clear
by con
sidering the analogous (and theoretically simpler) case of a
where
y
= (m+>rr
for
m =
3a4a 5
c = (n+>rr
for n
=
3,4, 5
. . .
Dl, z,
nd
D 3
were the
flexural
and torsional rigidities defined
in Hearmon
(1948)
and Hearmon (1946) and the other symbols
were defined as in the author’s paper, exceptp, which represented
mass per
unit
volume. For an isotropic material
and equation
31)
reduced to the equation given by the author
for the clamped isotropic plate.
The fundamental corresponded to
m = n
=
2; in
the case in
question
D,= D~=
o3
~h3/12(1*)
where the numerical factors were given to
three
significant
f igures.
The previous estimate(Hearmon1946) had been obtained by
assuming
and gave for the fundamental mode :
W = K x 3 2 ( u  x ) 2 ( b  y ) 2 sin w t
Evidently, thetwo estimates from equations (32) and (33) were in
good agreement, and M ered appreciably only in the num eri d
factor of the third term; the effect of that difference on the
calculated frequencies was, however, small.
The above example showed that it was possible to apply the
beam functions to calculating the frequency of vibration of
anisotropic plates. There was no reason to expect difficulties
with boundary conditions other
than
clamped, and
it
was thus
possible
in
principle to construct a table similar to Table 1, but
valid for anisotropic instead of isotropic plates.
I 2 3
m /n 
/m
.
Fig.
7.
Square Membrane (Theory)
Compare plate
wt
fixed edges.
0
I 2
c
t
I
Imlm
L
.
Fig. 8. Free Square Plate (Experimental)
Dr.
MARY
. WALLERLondon) wrote that it had been stated
in the paper that, ‘the analysis is based on the ordinary theory of
thin plates’. That assumed that the principle
of
superposition
was valid, namely, that a stationary wave
in
one direction was
unaffected in frequency
or
amplitude by the superposition of a
second stationary wave of the same frequency at right angles to
it. It then followed that, as the author had remarked, ‘When
m
is odd and
n
s even (or
wice
wersa)
for a free square plate
wo
modes
may
vibrate together and an infinite number of nodal
patterns is theoretically possible, corresponding to combinations
of the modes m / n and n/m, with varying amplitude ratios and
phase angles’. This
was orthodox
theory. She had, however,
shown (Waller 1952)t that Such additionalmodes were impossible
and that the displacements of freevibrating surfaces of geo
metrical shapes must
be
‘mechanicallybalanced’ about one
or
more lines
of
w e t r y . She had also discussed (W der 1953)$
*
HEARMON,. F.
S. 948
Forest Products Research Laboratory,
Stationery ice, London).
t WALLER,MARY.
1952 Proc.
ROY.
OC.
A),
V O ~ . 211, p. 265,
‘Vibrations
of
Free Plates :Line Symmetry: Corresponding Modes’.
w m m , y >.
1953 ACU(tjCa, ~01.
,
P. 370, ‘Concerning
Combined and Degenerate Vibrations
of
Plates’.
vibrating membrane some of whose nodal systems were
shown
in Fig.
7.
They corresponded
to
the formula
Special Re
rt
No. 7, Elasticity of
Wood and Plywood’
(H.M. m X
mx
z = s i n  s i n ~ & s i n  s i n ~ = O
.
(34)
U
U
a
where, for the case in question m = 3 and n = 1 and wice verso.
Fig.
7
might be compared with Fig.
8
(which was part of a
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A U T H O R ’ S
R E P L Y
O N T H E V I B R A T I O N O F R E C T A N G U L A R
P L A T E S
383
diagram in Waller
1939)
which related to the free square plate.
The nodal systems of a hypothetical free membrane could be
similarly arranged and their equation would read
e = c o s  c o s ~ f c o s n ~ c o s ~ = Ox . . .
(35)
a U
givingm =
2, n
=
0
r erice oersa, for the square nodal systems
so marked in Fig.
2b.
A U T H O R ’ S R E P L Y
Dr. G. B. WARBURTONrote,
in
reply to the communications,
that the possibility of applying beam functions to plates, other
than rectangular isotropic plates, as suggested by
Mr.
Hearmon,
was very interesting. He agreed with
Mr.
Hearmon that the
application of the beam function method might give valuable
results for some boundary conditions for oblique isotropic
plates. Probably an oblique cantilever plate was an exception;
there were discrepancies between the theoretical and experi
mental results of Barton
(1951)
or that case, although he had
used a series of eighteen terms to represent
W.
arton had
concluded that the RayleighRitz method, using beam deflexion
forms, was suitable for that type of plate only
if
the angles of the
plate were between
60
and
120
deg. Thus, the beam function
method was likely to be inaccurate for the oblique cantilever
plate. However, frequencies obtained for oblique plates
with
other boundary conditions might be accurate, as the work on
rectangular plates had shown that it was difficult to apply his
own method to a cantilever plate.
It seemed that his o w n method could be expected to give
frequencies for anisotropic rectangular plates with a fair degree
of accuracy. For the ‘specially orthotropic’ clamped plate,
mentioned by Mr. Hearmon, values of the fundamental natural
frequency had been given in a paper by Kanazawa and Kawai
(1952)*. The method was
similar
to that of Tomotika (1935),
successive approximations being obtained for M Y required
frequency. Unfortunately, for anisotropic plates, only the first
approximations for the frequency factor h had been given for
various values of DI/D and 0 2 / 0 3 between
+
and
3.
The author
had compared the values of
h
given by Kanazawa and Kawai with
those obtained from equations
(32)
and
(33)
for a square plate.
Over the above range of values of
D11D3
and
D2/D3
equation
(32),
based on he beam function method, always gave the lowest
value of
A;
the corresponding values of frequency given by
equation
(33)
were up to 1 per cent higher, and those of Kanazawa
and Kawai were up to
1.5
per cent higher. As the Rayleigh
method always gave frequencies higher than the exact values,
the frequencies obtained from equation
(32)
had the smallest
error. The fundamental frequencies of ‘specially orthotropic’
square plates with other boundary conditions (combinations of
fixed and freelysupported edges), had been given by Kanazawa
and Kawai.
They had also tabulated frequencies for two or three modes of
isotropic rectangular plates with one or more edges clamped and
the rest freelysupported. In some cases they had given only a
first approximation, but in others they had included better
approximations
;
hey had compared their values of frequency
with those of other Japanese workers (for example, Iguchi
(1938),
Sezawa
(1931)a
Kato
(1932)a
and Tomotika
(1935)),
where
available. His own values of frequency from equation (16)were
always lower than the first approximations of Kanazawa and
Kawai and sometimes lower than their second approximations.
Comparing the errors in the Mere nt methods applied to iso
tropic and orthotropic plates, it appeared that frequencies
obtained for orthotropic plates with combinations of fixed and
freelysupported edges would be accurate, if the appropriate
beam functions, equations
(9)a
(lo), and
(13),
were used for the
plate displacement.
Congress for Applied Mechanics, p. 333, ‘On he Lateral Vibration of
Anisotropic RectPngular Plates (Studed by the Integral Equation)’.
K A N A u W A a
Tea
and
KAWAIa
T. 952 pro^. Second
Japan
National
Dr.
Waller had misinterpreted
his
notation, when she had
stated that the nodal patterns for a square plate with fixed edges
shown in Fig.
2a
were not
4/2f2/4.
The author had defined
ma
n
as the number of nodal lines in the
X
and
Y
directions
respectively; if an edge were freelysupported or fixed, it was
included in the number of nodal ines in the relevant direction.
Thus
or a plate with all its edges fixed, there were at least
two
nodes in each direction, and patterns such as
3/1+1/3
nd
311

13
were impossible. Tha t notation could be applied to a l l
boundary conditions without any inconsistency occurring, .and
was convenient because the nodal patterns consisted of h e s
approximatelyparallel to the sides of the plate for most boundary
conditions. Dr. Waller had consideredm and n to be the number
of half waves in the X and Y directions respectively; that was
shown by the form of equations
(34)
and
(35).
Orthodox theory predicted that when
m
was odd and tz was
even (or vice aersa) for
a
free square plate,
two
modes of equal
frequency might vibrate together and an infinite number of
nodal patterns was possible. Dr. Waller had pointed out that
such patterns were impossible to obtain experimentally; she
called them ‘degenerate modes’ (Waller
1953).
However, in that
paper she had admitted that for particular values of m and
n
two
patterns might be obtained, similar o those of two degenerate
modes, owing to faulty experimentation or lack of uniformity
of the plate. That result was t o be expected from a considera
tion of the effects of small imperfectionson he vibrations of a
plate.
In the papers quoted
in
her contribution to the discussion
(Waller
1952
and
1953)
Dr. Waller had considered the classes of
symmetry of the nodal patterns for vibrations of free plates of
various forms. For a free square plate
six
classes of symmetry
had been given
; wo
of them included
all
the patterns for which
(mn)was an odd number, and the remaining four classes those
patterns for which
(mn)
was an even number and m and n
were equal. The four latter classes had been defined in terms of
their lines of symmetry, the two diagonals and the
two
medians
of the square, as given below :
(a) both diagonals and both medians antinodal (for
example, patterns for modes
2/0+0/2
nd
212,
Fig.
8);
(b)
both diagonals and both medians nodal;
(c) both diagonals antinodal and both medians nodal (for
example, pattern for mode
111,
Fig.
8)
( d ) both diagonals nodal and both medians antinodal (for
example, pattern for mode
210

12,
Fig.
8).
Dr. Waller had stated that if the patterns of a free square plate
were tabulated as in Fig.
8,
which could be extended indefinitely
downwards and to the right, the classes of symmetry repeated
themselvesin every four columns and every two ows. Thus the
class of symmetry for modes
2/00/2, 610016..
.a
212,
4/2+2/4,6/2+2/6
. .,should be the same. The nodal patterns
of a free square plate had been obtained experimentally and
analytically, and had been given for many modes by Chladni
(1802),Ritz (1909), Lemke (1928)a Waller (1939), and Grinsted
(1952).A
study of any of those papers showed that the classes of
symmetry, as defined above, did not repeat themselves
in
every
four columns and every
two
rows. For instance, it was obvious
from Fig.
8
hat the patterns for modes
210012
and
212
did
not belong to the same class of symmetry.
He did not find any law governing the repetition of classes of
symmetry; M erent laws were applicable, depending on whether
the mode was to the left or to the right of the diagonal line,
representing modes for which
m
= n, in Fig.
8.
However, the
various modes could be divided amongst the above four classes
of symmetry as given below :
Class a) modes
m / n + n / m
when both
m
and n were even; also
(b) modes
mlnnlm
when
m
and n were both odd, but
(c) modes
m / n + n / m
when both
m
and n were odd; also
( d ) modes mlnn lm when
m
and
n
were both even, but
modes 2/29 414,616 .
;
unequal;
modes 111,
313,515
. .;
unequal.
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384
He had stated that the value of the frequency factor, A,
depended on Poisson’s ratio u f one or more edges of the plate
were free. Th e value of
u
also affected the nodal patterns obtained
for modes of the types
m/nfn/m.
Those effects were small, but
it should be noted that a value of u
=
0.3 had been assumed in
A U T HO R ’S R E P L Y O N
THE
V I B R A T I O N O F R E C T A N G U L A R P L A T E S
obtaining the patterns of Fig. 2 b and c), the frequency factors
of Fig. 3 b and c), and curves I1 and I11 in Fig.
4.
However, any
other practical value of u would have little effect on the results
plotted
in
Figs. 2
and
4; the effect on frequency of varying values
of
u
was
usually
s m a l l and could be obtained from equation (16).