Vibrations and Impedances of Rectangular Plates with Free Boundaries

Prévia do material em texto

Lecture Notes in 
Engineering 
Edited by C. A. Brebbia and S. A. Orszag 
23 
P. Hagedorn, K. Kelkel, 
J. Wallaschek 
Vibrations and Impedances 
of Rectangular Plates 
with Free Boundaries 
Spri nger-Verlag 
Berlin Heidelberg New York 
L--~-....l London Paris Tokyo 
Series Editors 
C. A. Brebbia . S. A. Orszag 
Consulting Editors 
J. Argyris . K.-J. Bathe· A. S. Cakmak . J. Connor· R. McCrory 
C. S. Desai· K.-P. Holz . F. A. Leckie· G. Pinder· A. R. S. Pont 
J. H. Seinfeld . P. Silvester· P. Spanos· W. Wunderlich· S. Yip 
Authors 
Peter Hagedorn 
Klaus Kelkel 
Jorg Wallaschek 
Institut fOr Mechanik 
TH Darmstadt 
D-6100 Darmstadt 
Federal Republic of Germany 
ISBN-13:978-3-540-17043-3 
001: 10.1007/978-3-642-82906-2 
e-ISBN-13:978-3-642-82906-2 
Library of Congress Cataloging-in-Publication Data 
Hagedorn, Peter. 
Vibrations and impedances of rectangular plates 
with free boundaries. 
(Lecture notes in engineering; 23) 
Bibliography: p. 
1. Plates (Engineering) 
2. Mechanical impedance. 
3. Vibration. 
I. Kelkel, Klaus. 
II. Wallaschek, J. (jorg). 
III. Title. 
IV. Series. 
TA660.P6H28 1986 624.1'7765 86-22100 
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© Springer-Verlag Berlin, Heidelberg 1986 
Softcover reprint of the hardcover 1 st edition 1986 
Printing: Mercedes-Druck, Berlin 
Binding: B. Helm, Berlin 
2161/3020-543210 
Abstract 
The aim of the present study is to derive expressions for impedances of 
rectangular plates with free boundaries by analytical and semi-analytical 
methods and to give physical interpretations of the mathematical formalism 
used and of the results. The impedances obtained can be used in studying the 
dynamics of more complicated systems via substructure techniques. 
First, some general considerations on plate impedances are given, it is 
shown that concentrated moments lead to infinite angular displacements in 
elastic plates and the concept of reduced multipoint impedance matrix is in-
troduced. The circular plate with free boundary supported at a rigid, circu-
lar, central hub can be solved in terms of Bessel functions. Also the results 
for the circular plate supported excentrically at a single point as well as 
the reduced multipoint impedances can be given in terms of Bessel functions 
and this was done in the ESA report 5683/83/NL/PP(SC). 
The rectangular plate with free boundary leads to a far more complicated 
problem, for which solutions are given in form of double series. The 
convergence of these series is good in all cases examined, and the obtained 
formulas were also tested numerically. The impedance expressions given for the 
rectangular and for the circular plate seem appropriate for usage in the 
computation of the dynamics of composite structures. Also these results were 
already given in the ESA report mentioned above; in the present publication 
they are complemented by a comparison with some experimental results. 
Most of the material of chapter 3 of the present publication is based on 
the work carried out by Mr. Wallaschek under the orientation of the first two 
authors. Chapters 1, 2 and 3 of this publication correspond to chapters 1, 2 
and 4 of the ESA report. 
The authors wish to thank the European Space Agency for permitting the 
use of material prepared under the ESTEC Contract No. 5683/83/NL/PP(SC) and in 
particular also to the technical contract manager Mr. Frank Janssens and to 
Mr. Schniewind for the numerous discussions which were extremely useful. 
CONTENTS 
1. Introduction 
1.1 Substructure techniques in the dynamics of large flexible 
structures 
1.2 Remarks on the mechanical impedance and on the dynamic stiff-
ness of elastic systems 
2. 
2.1 
2.2 
2.3 
2.4 
3. 
3.1 
3.1.1 
3.1.2 
3.1.3 
3.2 
3.2.1 
3.2.2 
3.2.3 
3.2.3.1 
3.2.3.2 
General considerations on the mechanical impedance and on the 
dynamic stiffness of plates 
The classical plate theory 
Plate impedances and the reduced multipoint impedance matrix 
Singularities in Kirchhoff plates 
Literature survey on plate vibrations 
Dynamic stiffness of rectangular plates 
Symmetric and antisymmetric vibrations 
Double symmetric vibrations Wss 
Symmetric-antisymmetric vibrations Wsa 
Double antisymmetric vibrations Waa 
The method of superposition 
LEVY-type solutions 
Determination of the beam functions 
Superposition of building blocks 
First approach: load developed along the x-axis 
Second approach: load expanded in a double FOURIER series 
3 
9 
9 
11 
24 
26 
28 
30 
31 
33 
34 
38 
38 
41 
51 
51 
60 
3.3 
3.3.1 
3.3.2 
3.3.2.1 
3.3.2.2 
3.3.2.3 
3.3.2.4 
Plate connected at center 
Analytical solution 
Numerical Tests 
v 
Comparison with known results for the free vibrations 
Comparison with the rigid plate 
Comparison with the beam 
Irregularities of the distribution of zeroes and poles for 
the square plate connected at center 
6B 
6B 
76 
76 
B3 
90 
92 
3.4 Plate connected at a point on a line of symmetry 94 
3.4.1 Analytical solution 94 
3.4.1.1 Double symmetric vibrations 94 
3.4.1.2 Symmetric-antisymmetric vibrations 95 
3.4.2 Numerical Tests 100 
3.5 Plate connected at an arbitrary point 112 
3.5.1 Analytical solution 112 
3.5.1.1 Double symmetric vibrations 112 
3.5.1.2 Symmetric-antisymmetric vibrations 116 
3.5.1.3 Double antisymmetric vibrations 11B 
3.5.2 Numerical tests 121 
3.5.2.1 Test for convergence 127 
3.6 On the reduced multipoint stiffness and impedance matrices 134 
3.7 Comparison with experiments 137 
3.B Conclusions 144 
4. Final remarks 145 
5. Literature 147 
1. Indroduction 
1.1 Substructure techniques in the dynamics of large flexible structures 
Modern spacecraft are equipped with large light-weight appendages which in 
many cases can be modelled as beams and in other cases as thin plates or 
shells (solar-arrays, antennas, etc.). These appendages cannot be regarded as 
rigid and they may have a marked effect on the attitude dynamics of the space-
craft. It is then mandatory to describe with sufficient precision the dynamic 
behaviour of such appendages and their interaction with the main structure of 
the spacecraft in the frequency range of interest. Similar problems may arise 
in marine structures and in other cases. 
One way to derive a linear dynamical model for a complex system and in 
particular for a spacecraft is to describe the dynamics of subsystems by means 
of certain input-output relations, which can e.g. be mechanical impedance or 
stiffness matrices. These substructure techniques have been used successfully 
in different areas, e.g. in the computation of the dynamics of ship structures 
(see GAUL) or in wind-excited vibrations (HAGEDORN 1982).1) Particularly, when 
there is a large main structure with additional light substructures, the dyna-
mical behaviour of the overall system can sometimes be calculated with 
sufficient accuracy using a simple perturbation approach (see GAUL). In 
satellite dynamics the substructure technique has been exposed for example by 
CRELLIN & JANSSENS, JANSSENS & CRELLIN and by POELAERT and the method has so 
far been used for satellites equipped with one-dimensional elastic appendages 
like beams or cables and also for membranes. 
The present report is concerned with appendages in the form of rectan-
gular plates, for which impedance matrices are derived; only plates describedby the classical Kirchhoff plate theory are studied. Basically these impedan-
ces (or equivalent input-output-relations) could be derived by any suitable 
numerical technique, like FEM modelling, etc.; in the present study however, 
1) References are listed at the end of this report in chapter 5. 
2 
mainly analytical methods will be used for obtaining expressions of the impe-
dances. Numerical calculations are only used to test the obtained analytical 
results. 
In the case of rectangular plates it is convenient to make extensive use 
of symmetry considerations and the results are given in series of trigonome-
tric and beam functions. In all cases, the formulas obtained can easily be 
explored numerically and the new impedance matrices should form a useful tool 
to be used in substructure dynamics programs. 
3 
1.2 Remarks on the mechanical impedance and on the dynamic stiffness of 
elastic systems 
Consider first the linearized equations of motion of a damped mechanical 
system with n degrees of freedom 
M; + 0 W + C w = get), ( 1.1) 
where MT = M is the positive definite mass matrix, CT = C the stiffness ma-
trix, which is at least positive semi-definite, and OT = 0 the positive 
semi-definite damping matrix, w is the n-dimensional vector of the generalized 
coordinates and g( t) the vector of the given generalized forces. We are 
interested in the case of harmonic excitation, with the forces not necessarily 
all in phase, so that the complex notation is convenient. 1) With 
(1 .2) 
we look for solutions of the type 
( 1 .3) 
(steady state) only, which leads to 
2 A A (c - .a: M + jill) ! = ~ (1.4) 
and 
i = [(.11) k (1.5) 
with 
( 1.6) 
1) Here and in what follows complex quantities are underlined and vectors and 
matrices are denoted by boldface characters. 
4 
In most systems however, not all the generalized coordinates w1,w2 , ••• ,wn 
are accessible, which means that part of the generalized forces g1,g2, .•• ,gn 
vanishes identically. Let 
.Y = (11' 1:.2 ' .. -, 
correspond to the accessible variables, while 
.. -, 
corresponds to "inner" variables, i.e. ~+1' ~+2' 
... , 
(1.5) can be written as 
k = O. With 
-f) 
(1.7) 
(1.8) 
(1.9) 
(1.10) 
Note that in the language of linear network theory (1.10) describes a system 
of m inputs (Q1' Q2' ••• , ~) and n outputs (.9.1 ' .9.2 ' ••. , .9.m, ';;'1' .•• , Y...2' 
••• , ~ ). If the attention is focussed only on the accessible variables, one 
-f)-m 
has a system with m inputs and m outputs described by 
(1.11) 
with 
R CO.) : = [rc.o.)] 
- - mxm 
(1.12 ) 
being the upper left mxm submatrix of [(.0.). For m = 2 this linear system is 
completely analogous to a four-terminal R-L-C electrical network (see RUPP-
RECHT). In the mechanical system ~(fr) is called complex dynamic receptance 
matrix, its inverse 
5 
(1.13) 
is. the complex dynamic stiffness matrix, while (1/ j.l1) I(.I1) is the complex impe-
dance matrix. Note, that I(.I1) also can be obtained from the inverse of ~(.I1) by 
(see CRELLIN & JANSSENS, ESA STR-209) 
= [ ~11 S = F- 1 
~21 
~12] • 
~22 
(1.13a) 
In ESA publications the matrix defined by (1.13) is also simply refered 
to as the "I-matrix". Usually in the literature and in technical standards 
(see ISO 2041 and also HARRIS & CREDE) the letter Z is reserved for impedance 
rather than dynamic stiffness; in the present report we will however employ 
the somewhat unusual notation for the complex dynamic stiffness matrix. 
With the Z-matrix one therefore has 
(1.14 ) 
note that in this representation, displacement amplitudes are always regarded 
as the inputs of the system, while the amplitudes of the generalized forces 
are the outputs. In particular, these matrices are real if the system is un-
damped. Several important properties of these matrices are analyzed in the 
paper by HAGEDORN & TALIAFERRO. Of course these properties also hold for a 
large class of continuous mechanical systems which upon discretization lead to 
(1 .1) • 
It is easily seen that different I-matrices which may also correspond to 
different values of m can be assocciated to one and the same mechanical 
system. If we simply change m in (1.12), the different receptance matrices are 
all submatrices of the same matrix ~(.I1). This does however not hold for the 
I-matrix, as follows easily from (1.13). Thus, for example, a different 
element ~11(.I1) will in general be obtained for m = 1, m = 2 and m = 3. 
For illustration, we discuss the case of a rigid body of Fig. 1.1, sub-
jected to forces and concentrated moments acting at the point P. For (small) 
linear oscillations about equilibrium we have n = 6. 
6 
p 
~ 
Fig.1.1 Scheme of a rigid body connected at a single point 
7 
Let a material cartesian frame of reference be attached to the body at 
this point, and consider the linear and angular harmonic displacements about 
these axes, as well as the corresponding forces and moments. If ~1' ~2' ~3 are 
the complex amplitudes of the (linear) displacement of P and g1' g2' g3 the 
complex amplitudes of the angular displacements of the rigid body, then the 
relation between these "displacements" and the complex amplitudes of the exci-
ting force components 1 1 , 1 2, 13 and moments fu1' fu2' fu3 is of the type (1.14) 
with 
(~1 ' A !!1 ' !!2 ' 9 )T (1.15) .9 = ~2' ~3' -3 
!l (11 ' i 2 , i 3 , 
A 
A )T (1.16) = !!!1 ' !!!2 ' !!!3 
and the 6x6 complex matrix Z = (z .. ) is the "driving point complex stiffness 
- -lJ - - -
matrix" or simply the complex stiffness matrix. This matrix is of course a 
function of the angular frequency fr. 
If the body is not rigid n will be larger than 6, in the case of a 
continuous system it will go to infinity. Nevertheless, a 6x6 I-matrix can 
still be defined in the same way. This I-matrix could be determined by giving 
the point P a harmonic motion in one of the generalized coordinates qi with 
complex amplitude .9.., maintaining the other coordinates equal to zero, and 
1 
measuring the corresponding generalized force's amplitudes Q1' Q2' ... , ~. In 
this manner, the i-th row of the Z-matrix is determined as 
z .. = Q,;;;., 
-J1 -J.::I.1 j = 1,2, ••• 6 (1.17) 
(remember that in (1.14) the coordinate amplitudes are the inputs, the force 
amplitudes the output). 
It could also be the case that e.g. the moments about all the three axis 
in Fig 1.1 vanish for arbitrary harmonic motions of the point P, as is the ca-
se if the point of connection is physically realized by an articulation. In 
this case, we have only three accessible coordinates and consequently a 3x3 
I-matrix (nonsingular) could be defined. This matrix will not be a submatrix 
of the 6x6 Z-matrix mentioned above! This does of course not mean that for a 
force excitation, e.g. along the x-axis, there is no angular motion of the 
8 
body (the internal coordinates are in general not zero). 
One can of course not apply a concentrated transverse force to an elastic 
membrane or concentrated moments to a string and it is also not possible to 
apply a concentrated moment to an elastic plate, as we shall see in chapter 2. 
In many problems, connections between di fferent physical systems or 
rather subsystems are made not only at one point but on n different points. In 
this case, for each frequency in general the complex amplitudes of the 
"generalized forces" at all points of connection are linearly related to the 
complex amplitudes of the "generalized displacements" at all points. Formula 
(1.14) holds again, where now however g is the 6n-dimensional vector 
(1.18) 
and g is given by 
(1.19) 
where the upper index refers to the point of connection under consideration. 
The 6nx6n Z-matrix is in this case called "multipoint complex dynamic 
sti ffnessmatrix". 
Of course in most instances in which the physical systems are such that 
no concentrated moments can act on them the subsystems will be interconnected 
not at a single point but at several points simultaneously, so that multipoint 
Z-matrices are then of interest. On the other hand, moments can be applied to 
the structure through some rigid hub and this leads to the concept of reduced 
multipoint complex dynamic stiffness matrix, which is defined in chapter 2. 
In the present report only undamped systems are considered, so that the 
I-matrix will always be real. It should also be noted that (1.1) can be solved 
for arbitrary generalized forces via Laplace or Fourier transform, once the 
problem has been solved for the case of harmonic excitation. 
2. General considerations on the mechanical impedance and on the dynamic 
stiffness of plates 
2.1 The classical plate theory 
In this work we consider plates whose dynamical behavior in transverse 
direction can be described by the Kirchhoff plate theory, unless specifically 
stated otherwise. This classical plate theory is described for example in 
LEISSA (1973), NADAl and TIMOSHENKO & WOINOWSKY-KRIEGER. The effect of in 
plane forces on the transverse vibrations and of damping will be disregarded 
in this paper unless otherwise stated. In what follows we first give a short 
review of the Kirchhoff plate equations. 
If the transverse displacement of the middle surface of the plate at a 
point (x, y) is denoted by w(x, y, t), then the forced vibrations are des-
cribed by the partial differential equation 
2 D.1 w + ph w = q(x, y, t) 
where D is the plate's bending stiffness 
D = 2 12(1-v) 
(2.1) 
(2.2) 
with Young's modulus E, the plate thickness h and Poisson's ratio v the 
operator .1 in cartesian coordinates is defined as 
.1w = w + w 
xx yy (2.3) 
p is the mass density of the plate material and q(x, y, t) is the exciting 
force per unit area acting in the w-direction; the dots denote differentiation 
with respect to the time t. 
The bending moment Mn' the twisting moment Mnt and the shear force Qn at 
the boundary are then related to w through 
M = - D(wnn + VWtt ) n (2.4a) 
Mnt = D( 1 - v)w nt (2.4b) 
Qn = - D(wnn + wtt)n (2.4c) 
For the KIRCHHOFF approximation, at the boundary the influences of a shear 
force and a twisting moment are not distinguishable; only the sum 
10 
V := Q - (M t)t = - D(w + (2- v)wtt ) n n n nn n (2.4d) 
called the effective transverse force is relevant, where on the right hand 
sides the indices nand t indicate partial differentiation with respect to the 
outer normal and to the tangent to the boundary, the notation of TIMOSHENKO & 
WOINOWSKY-KRIEGER being used. 
While geometric boundary conditions are formulated directly by means of w 
and its first derivatives, natural boundary conditions are formulated by means 
of (2.4), and at a free boundary 
M = 0 
n 
v = 0 
n 
(2.5) 
have to hold. Since in the present study plates with free boundaries are 
considered, (2.5) will usually be required at the (outer) boundary of the 
plate. 
For the computation of the impedances or dynamic stiffnesses the steady 
state response to harmonic excitation will be calculated, so that the function 
q(x, y, t) will be of the form 
q(x, y, t) = f(x, y) sin llt . (2.6) 
The steady state response is then also harmonic 
w(x, y, t) = W(x, y) sin llt , (2.7) 
so that 
(2.8) 
follows from (2.1) with 
(2.9) 
In what follows, the case of a concentrated load will be examined in de-
tail, and in this case f(x, y) will be of the type of a delta-distribution 
(2.10) 
Since the boundary conditions have to hold for all values of t, w(x, y, t) can 
simply be substituted by W(x, y) in the corresponding expressions. The problem 
of finding the steady state response to a harmonic excitation consists there-
fore in finding the solution to (2.8) under the appropriate boundary con-
ditions. 
11 
2.2 Plate impedances and the reduced multipoint impedance matrix 
For the elastic plates considered in this study, one of the three axes of 
the frame of reference is taken along the normal to the plate's middle plane, 
the other two axes lie in this plane. When referring to the plate, we will 
identify the displacement u3 with the variable w introduced in (2.1), see Fig. 
2.1, and the plates treated in this study have a free boundary. 
The plates are assumed rigid in the directions of their middle plane, so 
that displacements u1 and u2 in this plane and rotations Q3 only cause forces 
in this plane as well as a moment acting in the x3-direction. Similarly, 
displacements acting in the x3 direction as well as rotations Q1 and Q2 along 
the x1 and x2 axes are related only to f 3, m1 and m2. From this it follows 
that the 6x6 driving point complex stiffness matrix defined in (1.14) to 
(1.16) simplifies to 
Z11 z12 0 0 0 z16 
z21 z22 0 0 0 z26 
0 0 z33 z34 z35 0 
Z = (2.11) 
0 0 z43 z44 z45 0 
0 0 z53 z54 z55 0 
z61 z62 0 0 0 z66 
The nonzero elements z .. with i,j f. 3,4,5 can easily be calculated from the lJ 
equations describing the in-plane-motion of the plate. In the "inertial" frame 
of reference Ox1x2 (see Fig. 2.2), the equations of momentum and of moment of 
momentum give respectively 
m X = f1 (t) G1 
= (2.12) 
12 
Fig. 2.1 Scheme of a plate with free boundary connected at a single point 
13 
where xG1 , xG2 are the coordinates of the center of mass G in this frame of 
reference, i is the central radius of gyration for the in-plane-motion and d1, 
d2 are the components of the radius vector PG as shown in Fig. 2.2 (for the 
moment let us simply write 9 instead of 93 ) and m is the total mass of the 
plate. The coordinates of G and P are related by 
(2.13) 
so that the linearized relations between the accelerations are given by 
(2.14) 
The components of the displacements of the point of attachment Pare u1 = xp1 ' 
u2 = xp2 ' so that the linearized version of (Z.1Z) can be written as 
Z" 
m i 9 
(Z.15) 
and for harmonic motions with complex amplitudes ~1' ~Z' 9 for the displace-
ments one has 
2 A dz~) m.o. (~1 - = i1 
2 A 
m.o. (~Z + d1~) i2 (Z.16) 
miZ.o.Z9 
= !!l.3 + dzi1 d1iz· 
14 
G,m,i 
Fig. 2.2 On the in-plane-motion of the plate 
15 
If one assumes ~Z = 0, 9 = 0, the elements z11 = 11/~1' zZ1 = 1Z/~1 and z61 = 
~3/~1 can easily be calculated from (Z.16) and one obtains 
Z 0, Z z11 = mil, zZ1 = z61 = m dZil (Z.17) 
Similarly ~1 = 0, 9 = 
° 
gives 
0, Z d .o.Z z1Z = zzz = -m.o., z6Z = - m 1 (Z.18) 
and ~1 = 0, ~Z = ° leads to 
d .o.z .o.Z ( . Z dZ Z ).o.Z z16 = m Z ,zZ6 = - m d1 ,z66 = - m 1 + 1 + dz • (Z.19) 
Observe that besides the components already indicated in (Z.11) also z1Z and 
zZ1 vanish! 
Let us now look at the elements z.. for i, j = 3,4,5. They relate the IJ 
force component 13 and the moments ~1' ~Z to the generalized displacement 
amplitudes ~3' g1' gZ· As explained in more detail in section Z.3, a Kirchhoff 
plate cannot be subjected to a concentrated moment acting about an axis 
situated in its plane. This means that in 
h h 
z43 ~ + z44 g1 + z45 gz = ~1' 
} (2.20) 
h h 
z53 ~ + z54 g1 + z55 gz = ~Z 
the moment amplitudes ~1' ~Z vanish for arbitrary linear and angular 
displacement amplitudes ~,g1' gz' so that zks = ° for k = 4,5, s = 3, 4, 5. 
The Z-matrix (Z.11) does then contain two vanishing lines (and columns) and 
becomes singular. 
It is therefore convenient to drop the fourth and the fifth column and 
line of this Z-matrix in order to keep it regular. This does of course make 
sense physically, since in any technical application moments cannot be applied 
to a plate at a single point. In other words, the angles 91, 9Z are disregar-
ded and a regular 4x4 dynamic stiffness matrix16 
zll z12 0 z16 
z21 z22 0 z26 
~ = (2.21) 
0 0 z33 0 
z61 z62 0 z66 
is defined for the elastic plate. Only the term z33 of this matrix then 
remains to be calculated. It can be determined from the solution W( x, y) of 
(2.8) corresponding to a function f(x, y) = f 8 (x-xp ' y-yp)' according to 
(2.22) 
Due to the uncoupling of the in-plane and out-of-plane motions in the present 
problem, it is indifferent wether in (2.22) f is regarded as the input or the 
output, although in general the forces are always considered to be the outputs 
in the dynamic stiffness and impedance definitions. 
The whole problem of computing the driving point complex dynamic stiff-
ness matrix is then reduced to the solution of (2.8) for a free plate under a 
concentrated single transversal load. This is however a difficult mathematical 
problem which has not been solved for a plate with free boundary, except for 
some special cases. An analytical solution will be given for this problem in 
the present study for rectangular plates. Although in principle the problem 
can always be solved by expanding the solution in the eigenfunctions corres-
ponding to the free vibrations, this is frequently not the most convenient 
approach. Different series expansions will therefore be given in this study. 
From (2.22) some well-known properties of the dynamic stiffness follow in 
a direct way. Thus, z33 vanishes at the eigenfrequencies of the free plate, 
provided the corresponding eigenfunction does not vanish at the point of 
connection. On the other hand, z33 has a pole for a certain frequency, if this 
frequency is one of the eigenfrequencies of the plate free at the boundary but 
fixed at the unmovable point of connection, the corresponding force not being 
zero. 
17 
Fig. 2.3 On the calculation of the multipoint,dynamic stiffness matrix 
18 
The problem of finding z33 is therefore also equivalent to the following 
problem: find the eigenfrequencies (and eigenmodes) of a plate with free 
boundary attached to a linear spring at the point P, with stiffness -z33' In 
this case, instead of solving for z33(ll), one determines ll(z33) and this is 
done by solving an eigenvalue problem which is positive definite, at least as 
long as z33 < O. Note that this eigenvalue problem has to be solved for all 
values of z33' so that this procedure does not seem very practical. 
Of course in practical applications plates will have to be supported in 
such a way that also moments (out of plane) can be applied to it. From the 
vanishing of the dynamic stiffness coefficients related to the concentrated 
moment, it follows that this can only be done by fixing the plate not at a 
single point, but at a - possibly small - hub or at at least three distinct 
points. An analytic solution for the case of a circular plate centrally 
connected at a rigid hub is given in the ESA report 5683/83/NL/PP(SC). In this 
case, the full 6 x 6 I-matrix will in general be nonsingular, with the excep-
tion of discrete values of the frequency parameter. Another solution would be 
to support a plate along an edge or along a diameter in the case of a circular 
plate, where the diameter may be reinforced by a flexible or rigid beam. 
The case of a plate connected simultaneously at distinct points first 
leads to the problem of finding the multipoint dynamic stiffness matrix. 
Again, the plate's in-plane-motion can be completely separated from the trans-
versal motion, so that the problem shown in Fig. 2.3 has to be studied (in the 
case of connection at 3 points). If only the transverse motion of the plate is 
considered, we define a new 3x3 matrix I ~) which relates the complex amplitu-
des of the transversal exciting forces 1; ,1;2), 1;3) to the corresponding 
A(1) A(2) A(3) displacement amplitudes ~3 '~3 '~3 : 
= (r(1) r(2) r(3»T 
-3 '-3 '-3 (2.23) 
(in what follows we will drop the lower index "3", since it is clear that we 
are refering to the transverse displacements and transverse forces only). If 
the multipoint stiffness matrix is known, also the driving point stiffness 
matrix of a plate with respect to a point P attached to the plate by means of 
three (rigid) rods can be calculated (see Fig.2.4). This can be done (automa-
tically) by the standard methods used for composite structures and the resul-
ting I-matrix will be nonsingular. 
19 
Fig. 2.4 Point moments and forces acting on a plate through three (rigid) bars 
20 
The multipoint stiffness matrix ~p can be calculated from the transfer 
stiffnesses 
z(P.,P.) := ~(i)/W(i)(p.), 
- 1 J - - J i,j = 1,2,3. (2.24) 
In the definition (2.24) W(i)(p.) is the displacement amplitude at the point 
- J . 
caused by a concentrated harmonic force of amplitude ~(1) acting at the P. 
J 
point Pi. The transfer stiffness coefficients are therefore easily determined 
if the solution W(x,y) for the transverse vibrations of a plate excited at an 
arbitrary single point is known. Since superposition holds, one has 
and comparing (2.25) to (2.23) immediately leads to 
(where ( 
(Z M-p1) .. = 1/z(P., P.) 
- 1J - J 1 
) .. denotes the element i-j of the matrix), or 
1J 
[ 1 )T ]-1 . ! MP = (z(P.,P.) 
- 1 J 
(2.25) 
(2.26) 
(2.27) 
In certain applications it may be useful to have an expression for the 
stiffness matrix of a plate connected at a small rigid massless disc, fixed to 
the plate at three distinct points. The corresponding 6x6 matrix ~ will then 
be nonsingular (except for discrete frequencies), it will however depend on 
the particular configuration. This matrix follows easily from the multipoint 
stiffness matrix and we will name it the reduced multipoint stiffness matrix. 
In this report we always refer to it for the case of a rigid disc in the form 
of an equilateral triangle connected at its vertices, as shown in Fig. 2.5. 
21 
a) Z 
b) Y 
~ 
R p 
x 
l 
~ 
2b b 
Fig. 2.5 On the definition of the reduced multipoint stiffness matrix 
a) plate with small triangular rigid disc connected at its vertices 
b) configuration of the disc 
22 
The reduced multipoint stiffness matrix is then simply the stiffness 
matrix of the system, connected at the point P of the triangular disc. Due to 
the rigidity for the in-place-motions, this ~ matrix has the structure given 
in (2.11) and only the elements zRo ° with i,j = 3,4,5 remain to be calculated, IJ 
since the other terms are the same as before. Noting that the points P1,P2 and 
P3 have the coordinates (-2b,0), (b,-vJb) and (b,V3b) respectively, the 
relation between the transverse forces 1(1),1(2),1(3) acting at these points 
and the resulting force f acting at P and the moments m , m about the Px and 
the Py axes can be written as 
(1, ~, 
with 
5 = 
A T 
m ) 
-y 
o 
2b -b 
On the other hand, one also has 
-b 
s\~o g 
--x 
-x -y 
(2.28) 
(2.29) 
(2.30) 
where _u g , g are now the disc's displacement amplitude in z-direction and 
-x -y 
the angular amplitudes about the Px and the Py axes respectively. Taking into 
account (2.23) it follows from (2.28) and (2.30) that the relation 
(1, m 
--x 
A T 
m ) 
-y (2.31) 
holds. The part of the 6x6 (reduced multipoint) complex stiffness matrix still 
to be calculated is therefore 
23 
zR33 zR34 zR35 
T (2.32) zR43 zR44 zR45 . - S !:. MpS 
zR53 zR54 zR55 
and the complete 6x6 ~ matrix has the structure 
z11 z12 0 0 0 z16 
z21 z22 0 0 0 z26 
0 0 zR33 zR34 zR35 0 
~ = (2.33) 
0 0 zR43 zR44 zR45 0 
0 0 zR53 zR54 zR55 0 
z61 z62 0 0 0 z66 
It can easily be obtained by simple algebraic operations if only the problem 
of a plate driven at an arbitrary single point has been solved. Note that in 
general zR33 is different from z33' 
24 
2.3 Singularities in Kirchhoff plates 
In the theory of elasticityit is said that the state of stress has a 
singularity at some point, if any of the stress components at this point beco-
mes infinitely large. In three-dimensional elasticity an idealized concentrat-
ed force therefore always leads to a singularity in stress and therefore also 
in strain. In the elementary theory of membranes, plates and beams, the 
stresses normal to the middle plane (in the case of membranes and plates) and 
to the beam axis are disregarded. Although a transversal concentrated force 
acting on a beam always leads to unbounded stresses in this direction (on a 
section normal to the force) this is only a local effect which has no dramatic 
consequences on the displacement and can therefore be disregarded. In 
elementary beam theory, concentrated loads are therefore allowed. 
It is not difficult to show that also in a Kirchhoff plate the transverse 
displacement remains bounded at the point of application of a concentrated 
force. In a membrane on the other hand, it is physically clear that a concen-
trated force leads to infinite displacements. Therefore, the driving point 
complex sti ffness of a membrane is zero (arbitrarily large transverse dis-
placements lead to small forces), while in the case of a plate the complex 
stiffness for a concentrated force remains bounded, unless there is a 
resonance. 
In the classical literature on plate theory, also the type of singularity 
caused by a concentrated moment is studied. This case can be reduced to the 
case of a couple formed by two concentrated forces applied at a very small 
relative distance. It can be shown, that in this case the rotation angle of a 
normal to the middle plane assumes arbitrarily large values for arbitrarily 
small concentrated moments and the complex stiffness for moments vanishes! 
The problem of singularities has been studied extensively in the 
classical literature for the static case, due to its importance in civil and 
in mechanical engineering. In civil engineering for example, frequently plates 
are connected to columns at "single points", Le. with a small area of 
contact, and the resulting stresses, strains, as well as the linear and 
angular displacements are important. A good summary on plate singularities can 
be found in TIMOSHENKO & WOINOWSKI-KRIEGER (pp. 325) and a more detailed 
discussion is given in NADAl (pp. 162 and pp. 202). 
25 
The nature of the singularities in the dynamic case is the same as in the 
static case, since locally the influence of the inertia terms is negligible. 
This can also be confirmed by studying the rotational excitation of a circular 
plate through a rigid hub. An analytic solution can be given for this problem, 
and letting the hub radius tend to zero the problem of a circular plate 
excited by a harmonically varying concentrated moment is obtained. It is then 
seen that the complex dynamic stiffness for a concentrated moment vanishes. 
It should be mentioned that the nature of the singularities changes if 
Reissner's or Mindlin's plate theory are used, however the complex dynamic 
stiffness for moments will also vanish in this case. The literature on singu-
larities caused by concentrated forces in Reissner and Mindlin plates is 
rather scarce. The singularity corresponding to a concentrated force can how-
ever be examined using the fundamental solution given by VANDER WEEEN. The 
singularities are also mentioned e.g. in GIRKMAN and in MARGUERRE & WOERNLE. 
DYER claimes that the point moment impedance for Mindlin plates are finite but 
examines only infinite plates. 
26 
2.4 Literature survey on plate vibrations 
There is an extensive classical literature on Kirchhoff's plate theory 
and one of the best references for statical problems is still the book by 
TIMOSHENKO & WOINOWSKY-KRIEGER; another classical reference is NADAl's book. 
These relatively old works on statical problems are of importance also for the 
solution of vibration problems, due to the fact that many of the different 
methods of solution and series expansions can also be used to treat dynamical 
problems. Thus, the convergence and relative merits of different expansions 
for rectangular plates are discussed in TIMOSHENKO & WOINOWSKY-KRIEGER and 
also the problem of a circular plate under an excentric concentrated load is 
solved in section 64 of this book dividing the plate in an inner circular and 
an outer annular part. A similar procedure is used for the analogous vibration 
problem by different authors and also in section 3 of the ESA report 
5683/83/NL/PP(SC), the main difference being that the radial dependence is far 
more complicated in the vibration problem, since it involves Bessel functions 
as opposed to the logarithmic functions and polynomials of the static problem. 
No such comprehensive compendium seems to exist for the plate vibrations, 
although several vibration texts include chapters on these problems (see 
MEIROVITCH, MORSE, HAGEDORN & KELKEL). An extremely useful and important 
reference is LEISSA (1969), which is an outstanding review of the literature 
published until 1969 and constitutes a most valuable tool to the engineer and 
designer. Unfortunately no other similar and updated review has been published 
since this date. 
REISMANN and SNOWDON (1980) study the forced vibrations of clamped 
circular plates; the latter paper gives the impedance of a clamped plate for 
the case of a noncentral force. Both authors use the divisions into an inner 
circular and an outer annular plate, already mentioned above, and in the first 
paper also Green's function is derived and the solution is found by 
superposition of the general solution of the homogeneous problem and a 
particular solution to the inhomogeneous case. SNOWDON (1970) gives the 
impedance of a circular free plate driven at the center. IRIE & YAMADA & 
MURAMOTO using Green's function gave a solution to the problem of a circular 
plated excited by moments and forces distributed along a rigid diameter. This 
result can be used to find the corresponding dynamic stiffness matrix. 
27 
The forced vibrations of circular plates are also studied by MCLEOD & 
BISHOP for a number of different boundary conditions; their results for the 
cases of free boundary are however wrong, since the expression for Rn in table 
7d on page 27 is incorrect. The apparent dynamic mass is calculated by WARRE~ 
for circular plates excited at a massless rigid hub, and the case of vanishing 
hub radius is also considered (the boundary can be free or clamped). ITAO & 
CRANDALL (1978 & 1979) study the free and forced v ibrations of circular 
plates. In the first paper the 701 first eigenfunctions and eigenfrequencies 
for circular plates of free boundary are given. The second paper contains an 
expression for the impedances of centrally loaded free plates, which was used 
in connection with the study of wide-band random vibrations. 
There is a considerable amount of literature on the vibrations of rectan-
gular plates: IGUCHI (1937, 1940) studied plates with different boundary 
conditions, and in particular the free vibrations of the completely free plate 
are examined in IGUCHI (1953). In this paper, exact solutions are given for 
the eigenfunctions by writing them in a sum of two different Levy expansions 
and by considering the different symmetries and asymmetries. A similar 
approach is taken in GORMAN & SHARMA, GORMAN (1976, 1978) and in the book 
GORMAN (1982), which all deal exlusively with the free vibrations of rectan-
gular plates. Also LEISSA (1973) in his paper on the free vibration of rectan-
gular plates for different boundary conditions uses similar methods and refers 
to IGUCHI's results. SNOWDON (1974) treats forced vibrations of damped rectan-
gular plates with simply supported boundaries. Clamped rectangular plates are 
also treated by LAURA & DURAN and by WARBURTON & LAURA and platesof different 
forms are examined by NIKIFOROV and by IRIE & YAM,ADA & NARITA. DONALDSON 
presents a method for vibration problems in rectangular plates in which 
extended plates with different boundary conditions are used; his approach is 
related to GORMAN's and IGUCHI's work and is also analogous to procedures used 
in the static plate problems. 
The influence of an elastic support on the plate vibrations is discussed 
by AL-JUMAILY & FAULKNER, IRIE & YAMADA & TANAKA, KOVINSKAYA & KONEVALOV, 
LAURA & POMBO & LUISONI, LOMAS & HAYEK, and NARITA & LEISSA. Special aspects 
of plate impedances and vibrations together with their numerical calculation 
are treated by BAILY, and by NAGAMATSU & OOKUMA. VOINOV et a1. give an 
elementary relation between Green's function for the statical problem and the 
impedance of mechanical systems. Forced vibrations of simply supported 
orthotropic sandwich plates are studied by BHAT & SINHA, where also impedances 
are given. Some general notes on plate vibrations were also given in HAGEDORN. 
Examples of measured plate impedances can be found in SKUDRZYK. 
3. Dynamic stiffness of rectangular plates 
In what follows the dynamic stiffness z33 of a rectangular plate will be 
computed and to this end the problem 
(3.1) 
has to be solved in the domain -a ~ x ~ a, -b ~ Y ~ b with the boundary 
conditions corresponding to a free boundary. This problem is considerably more 
di fficul t than the corresponding problem for the circular plate, since even 
the eigenfunctions of the corresponding homogeneous problems have to be 
written in the form of double series. The solution of the eigenvalue problem 
no longer involves only the solution of a relatively simple transcendental 
equation, as in the case of circular plates, but requires the solution of an 
infinite-dimensional matrix eigenvalue problem. Consequently, the modal 
analysis approach does not seem to be very appropriate in the present case. It 
is more convenient to expand directly the solution to (3.1) in a series of the 
same types of functions which are used for the eigenvalue problem, so that the 
boundary conditions can easily be fulfilled. 
Two different approaches seem particularly feasable. In the first one, 
the central load is developed in a Fourier series along one of the axes of 
symmetry, e.g. the x-axis. Then the original problem is substituted by a 
sequence of problems in which a sinusoidal distributed force acts on this line 
and each of the problems may then be solved using symmetry considerations 
similar to the ones employed by GORMAN for the free vibrations. 
In the second approach, the solution to (3.1) with the respective bounda-
ry conditions is written as a sum 
(3.2) 
where Wp is a solution of (3.1) which does not satisfy the boundary conditions 
corresponding to the free boundary. Such a solution can e.g. be determined by 
expanding the concentrated load in an appropriate double generalized Fourier 
29 
series with respect to x and y, and using relations of orthogonality. The 
function WH is a general solution of the homogeneous equation corresponding to 
(3.1) and is chosen in such a way that the sum (3.2) satisfies the conditions 
required at the free boundary. 
In what follows, the case of the rectangular plate excited at an 
arbitrary point will be treated by using the second approach, while certain 
special cases will also be solved via the first approach. Of course, also in 
the second approach use will be made of symmetry considerations. 
First, in section 3.1 it is shown that the general problem can be split 
up into four different problems with certain symmetry properties. In section 
3.2 certain series expansions of solutions are given which are then used later 
in section 3.3. In this section the problem of the plate loaded centrally, 
loaded on a point of an axis of symmetry and the general problem are solved 
successively by using the superposition of certain "building blocks". Each 
class of symmetry is treated separately; in the general case all four classes 
are needed. 
30 
3.1 Symmetric and antisymmetric vibrations 
A rectangular plate with free boundaries has two lines of symmetry, which 
correspond to the x and y axis respectively. Now, if we split up the amplitude 
f(x,y) of the exciting force per unit area (see (2.1),(2.6» in one double 
symmetric, two symmetric-antisymmetric and one double antisymmetric - which is 
possible for every function f(x,y) - the corresponding vibrations will have 
the same properties of symmetry: 
2 Ll W .. (x,y) lJ 
W (x,y) = ss 
W (x,y) = sa 
W (x,y) = as 
W (x,y) = aa 
the whole deflection of 
W (x,-y) 
ss 
W (x,-y) 
sa 
-W (x,-y) 
as 
-W (x,-y) 
aa 
the plate 
i,j E ls,a} 
= W (-x,y) ss 
= -W (-x,y) sa 
= W (-x,y) as 
= -W (-x,y) , aa 
being 
W(x,y) = W (x,y) + W (x,y) + W (x,y) + W (x,y) . 
ss sa as aa 
(3.3) 
(3.4) 
(3.5) 
(3.6) 
(3.7) 
(3.8) 
Because of the symmetry, for each one of the above four parts of the load only 
a quarter segment of the fully free plate need be analyzed. 
In what follows, these different types of solution are discussed separa-
tely together with their boundary conditions. 
31 
3.1.1 Double symmetric vibrations Wss 
In the present case (3.4) leads to the relations 
w (x,y) = -W (-x,y), 
x x 
W (x,y) = -W (x,-y) y y 
for the first derivations of W, and in particular to 
W (x,D) = 0, y 
(3.9) 
(3.10) 
due to the continuity (Note that in this section the index ss is omitted since 
only the double symmetric case is studied). Analogously, from (2.4d) in 
cartesian coordinates we get the effective transverse force 
and 
v (x,y) 
x 
v (x,y) = -v (x -y) y y , (3.11) 
v (O,y) = ° (3.12) 
x 
- if there exists no external force p(y) distributed along the line x 0-
and 
(3.13) 
if there exists a force ps(x) (symmetric to the y-axis) acting on the x-axis. 
Note that the indices x and y in (3.9), (3.10) indicate differentiation with 
respect to x, y, while V and V stand for the amplitude of the effective 
x y 
transverse forces (and later M and M for the bendlng moments at the bounda-
x y 
I' ies). The quarter-plate segment, refered to here as the "quarter-plate", for 
the load shown in Fig. 3.1 then has the boundary conditions 
M (a,y) - 0, 
x 
M (x,b) - 0, y 
V (a,y) - 0, 
x 
V (x,b) - ° y 
(3.14) 
(3.15 ) 
32 
------:;oo-=-----"--b 
x 
z 
r------:'I"tF---"""'2""""~- y 
b 
x 
+ slip shear 
Fig. 3.1 Rectangular plate with free boundary loaded along the 
x-axis with symmetry to the y-axis 
at the free edges and 
W (o,y) - 0, 
x 
W (x,D) :: 0, y 
33 
v (o,y) - 0, 
x 
(3.16) 
(3.17) 
along the centerlines of the fully free plate. The boundary conditions (3.16) 
and (3.17) will be refered to as "slip shear" conditions. 
3.1.2 Symroetric-antisymmetric vibrations W 
sa 
In this case (3.5) gives 
W(x,y) = -W(-x,y), 
and therefore also 
W(o,y) :: 0, 
W (x,y) = -W (x,-y) y y 
W (x,D) :: 0, y 
(3.18) 
(3.19) 
since the deflection and the slopes of the plate are continuous functions of x 
and y. 
From (2.4) the bending moments and the effective transverse forces are 
now of the type 
and 
M (x,y) 
x 
= -M (-x,y), 
x 
-v (x,-y) y (3.20) 
(3.21 ) 
- if there exists a distributed external force p (x) (antisymmetric with 
a 
respect to the y-axis) acting on the x-axis and 
M (O,y) :: 0, 
x 
34 
(3.22) 
if there is no external moment, which always will be the case in this report. 
For the load shown in Fig. 3.2, the quarter plate segment is subjected to the 
boundary conditions 
M (a, y) 
-
0, Vx(a,y) - 0, x 0.23 ) 
M (x,b) :: 0, V (x ,b) 
-
0, y y (3.24) 
along the free edges and 
W(O,y) :: 0, M(O,y) 
-
° x 
(3.25) 
W (x,D) :: 0, V (x,D) 1 = - 2" Pa(x) y y (3.26) 
along the other edges. In other words the quarterplate is simply supported 
along the y-axis and has slip-shear boundary conditions along the x-axis. 
The boundary condi tions for the vibrations of a quarter plate with 
symmetry properties 0.6) can be obtained from (3.23) to 0.26) by exchanging 
x and y in (3.25) and (3.26). 
3.1.3 Double antisymmetric vibrations Waa 
Conditions (3.7) now give 
W(O,y) - 0, W(x,O) _ ° (3.27) 
and the bending moments follow from (2.4) as 
M (x,y) = -M (-x y) x x' , M (x,y) = -M (x,-y), y y (3.28 ) 
35 
x 
z 
b 
x 
Fig. 3.2 Rectangular plate with free boundary loaded along the 
x-axis with antisymmetry to the y-axis 
y 
in particular 
M (o,y) -= 0, 
x 
36 
M (x,D) -= 0, y (3.29) 
if there are no distributed forces and moments acting on the plate, as shown 
in Fig. 3.3. The quarter plate is then subjected to the boundary conditions 
M (a, y) 
-
0, Vx(a,y) - 0, x (3.30) 
M (x,b) -= 0, V (x,b) 
-
° y y 
(3.31 ) 
at the free edges and 
W(o,y) 
-
0, M (o,y) 
-
0, 
x 
(3.32 ) 
W(x,o) 
-
0, M (x,D) -= ° y (3.33 ) 
along the other edges, i.e. the quarter plate is simply supported along the x-
and y-axis. 
x 
1 
4 
x 
37 
f 
4 
---b 
z 
z f 
4 
f 
4 
b 
T 
f 
4 
y 
-'----+-----,-------:~y 
---.;~,/ 
Fig. 3.3 Rectangular plate with free boundary loaded with antisymmetry 
to the x- and y-axis 
38 
3.2 The method of superposition 
For the fully free plate and also fo~ the quarter plates discussed above, 
it is not possible to obtain directly a LEVY-type solution; in other words, it 
is not possible to reduce the solution of the plate equation to the solution 
of a proper beam equation (see GORMAN (1982, Chapter 1.» Therefore, we will 
solve the problem by means of superposition. This means, we consider two or 
more appropriate plate problems whose solutions are of LEVY-type and then 
superimpose these different solutions. We then adjust constants appearing in 
their boundary condition formulation so that their combination provides boun-
dary conditions that satisfy the requirements specified in the original 
problem. The plate problems that are superimposed are refered to as "building 
blocks". Typical building blocks are shown in Fig. 3.4 (they belong to 
di fferent problems). In 3.3 all the building blocks needed will be discussed 
in detail. 
3.2.1 LEVY-type solutions 
We turn to the first building block of Figure 3.4, where the solution is 
refered to as W1(x,y), seeking a LEVY-type solution. Here, we require the 
boundary conditions (3.25) and (3.26) 
W1 (x,O) - 0, 
,y 
1 
ZP(x)' 
(3.34) 
(3.35) 
along the edges x = 0 and y = 0 whereas, at the originally free edges, we take 
the slip shear conditions 
W1 (a, y) =- 0, 
,x 
(3.36) 
on x = a and zero vertical edge reaction with a distributed bending moment 
(3.37) 
39 
;---------..,..---- Y 
x 
Z.W2 (X.y) 
r-------~:___--- y 
x 
Fig. 3.4 Typical building blocks for analyzing forced vibrations of the 
completely free plate loaded along the x-axis 
40 
along y = b. The solution for this building block can be expressed in the form 
proposed by LEVY as 
CD () .. :-. ().mx W1 x,y = .. :~. Ym y Sln 271"8· (3.38) 
m=1,3, ••• 
One can develop this solution in the usual way, enforcing the first boundary 
conditions (3.34) and (3.36). 
For the second building block of Fig. 3.4, where the solution is refered 
to as W2(x,y), we have the boundary conditions 
W2 (x,D):: 0, 
,y 
W2 (x, b):: 0, 
,y 
The solution W2 is sought in the LEVY form as 
CD 
...... n Y.... 
::- X (x)cos -2 71" 
...... n b 
n=D,2,4, ••• 
(3.39) 
(3.40) 
(3.41 ) 
(3.42) 
(3.43) 
This equation differs from (3.38) in that now n assumes even values only and 
the cosine has replaced the sine. To simplify the notation we define the 
functions 
g (x) 
m . -
{ 
cos !!!.71"~ 
2 a' 
sin !!!.71"~ 2 a' I 
and analogously in y-direction 
h (y) := g (-ba y). 
n n 
m E IN[ :={0,2,4, ••• ,m} 
:= {1,3,5, ••• ,m} 
(3.44a) 
(3.44b) 
41 
3.2.2 Determination of the beam functions 
We go back to the first building block in Fig. 3.4. Substituting (3.38) 
into the plate equation (3.1), which is homogeneous for the building blocks 
considered here, leads to 
":::·fy - 2(m2n )2y + [(m2an )4 - (34]Ym}gm(X) = 0, ~_. m,YYYY a m,YY 0.45) 
with m E ~o and (34 = phU2/D. Later we will however also need a solution of the 
type (3.38) with the cosine instead of the sine function and with m even. We 
therefore consider at once the cases m E ~o and m E ~E respectively in 0.45), 
in what follows. Multiplying (3.45) by g (x) and integrating over the length 
n 
of the plate in x-direction gives 
0.46) 
if the orthogonality relations 
a with m = n = 0 
fa g (x)g (x)dx = m n I with m = n ~ 0 0.47) 
o 
o with m ~ n 
are used. Note that here and in the sequel, wherever orthogonality relations 
are used, the indices m and n always are taken from the same set ~E or ~o re-
spectively. The same operation, applied to the boundary conditions (3.35) and 
0.37), gives 
y (0) = 0, 
m,y (3.48) 
y (0) - (2_v)(mn )2y (0) 1 fa () ()d 
m,YYY 2a m,Y = aD p x gm x x, (3.49) 
o 
y (b) - (2_v)(mn )2y (b) = 0, 
m,yyy 2a m,y (3.50) 
42 
2 fa 
- aD (3.51) 
o 
with the relations between the deflection W(x,y) and the "effective transverse 
force" V (~, y) and bending moment M (x, y) written in cartesian coordinates 
n n 
(compare with (2.4» 
V = -D(W + (2-tJ)W ), 
n nn ss n 
M 
n 
D(W + tJ W ), 
nn ss 
(3.52 ) 
n,s = x,y, n ~ s. 
For m=O, in (3.49) and (3.51) the inhomogeneous terms are to be multiplied 
1 by 2. 
The homogeneous differential equation (3.46) and the inhomogeneous boun-
dary conditions (3.48) to (3.51) represent a boundary value problem with 
constant coefficients. (3.46) has solutions of the form 
(3.53) 
and substitution of (3.53) into (3.46) leads to 
(3.54) 
and A 2 
m 1,2 
the general solution can therefore be written as 
Y (y) = A cosh B+y + B sinh B+y + C cos B-y + D sin Bm-y, 
m m m m m m m m 
(3.55) 
a- ~ 
m E lNE : = 1 0, 2 ,4, ••• , - Ba - } m ~ 21t < m+2 or 
< ~+21 
43 
Ym(y) = A cosh B+y + B sinh B+y + C cosh B-y + D sinh B~y, (3.56) m m m m m m m 
a+ f m+2, m+4, .•. ,m}or mE INE . -
a+ { ~+2, ~+4, .•. ,m } mE INO . -
B+ = v'B2 + (mn) 2 B~ = v'IB2 - (~n:)21 , m = O,1,2,3, ... ,m. (3.57) m 2a 2a with 
The constants Am to Dm can be calculated from the inhomogeneous linear system 
formed by (3.48) to (3.51) with (3.55): 
o
 
o
 
n
+
 s
ir
lh
 s
 +
. 
m
 
m
 
2 
8+
 [
8+
 
m
 
m
 
8+
 
m
 (2
-v
) (
m
n)
2J
 
28
 
8+
co
sh
 s
+
· 
m
 
m
 
o
 
o
 
(l-
si
n 
s 
.
 
m
 
m
 
[ 
2 
J 
+
 
m
n
 
2 
8 
-
(2
-v
) (
-)
 
rn
 
28
 
[8
+2
 _
 
(Z
_V
)(
m
n)
2J
 
m
 
2a
 
2 
[f3~
 +
 
(2-1
1)(~
:)2J
 
c
o
s
h 
+
 
sm
" 
s
in
h 
s
+
· 
m
 
[ 
2 
(l+
 
,
rn
 
-
v 
(m
Jr
/J
 
28
 
[n~ 2
 -
t! 
(m
n)
2]
 
2a
 
[ 
2 
(l- m
 
w
it
h 
+
 
S 
In
 
a 
a 
m
 E
 IN
E 
D
r 
m
 E
 IN
o 
8+
b 
m
 
' 
s 
rn
 
(l~
b, 
m
 
0
,1
,2
,3
, .
.
.
 
,
C
D
 
a
n
d 
th
e 
,
-
(3
lJ
RI
ER
-c
oe
ff
ic
ie
nt
s 
Pil
i 
a
n
d 
M b
rn
 
d
ef
jn
ed
 b
y 
p
(x
) 
~:;:
. 
P 
'
l 
(x
) 
,
 
.
1..
.. 
mm
 
m
 
11
b ( 
x
) 
=
 .
\. 
Mb
 
g 
(x
) 
,
 
.
1..
.. 
rn
 
rn
 
rn
 
-
c
o
s
 
s 
.
 
m
 
+
 
,
,
(m
n)
21
 
2a
 
w
he
re
 
th
e 
su
m
s 
a
r
e
 
ta
k
en
 o
v
e
r 
IN
E 
o
r
 
IN
O 
r
e
s
p
ec
ti
v
el
y
. 
8 m
 
-
[ 
_
2 
m
n
 
2J
 
-
8 
(l 
+
 
(2
-v
)(
-2
 ) 
rn
 
m
 
8 
-
o
-c
o
s
 
m
 
S
m
" 
[
_
2 
m
n
2
J 
,(lm
 
+
 
(2
-v
)(
2a
) 
[(l~ 2
 -
s
in
 S
m
" 
+
 
"
 
(IT
I7f
)2J
 
2a
 
A
 rn B m
 
D
 III 
0 P r
n 
2D
 (3 M b
rn
 
D
 
,
 
(3
.5
8)
 
(3
.5
9)
 
(3
.6
0)
 
(3
.6
1)
 
.
j:>
 
.
j:>
 
45 
Multiplying the rows by b, a2b and a2 and subtracting the first row multiplied 
with [f3~2 _ (2-v) (~:)2] from the second row, the system (3.58) can be simpli-
fied to 
0 + 0 A s s 
m m m 
0 0 0 -s B 
m m 
+ -
sinh + + - cosh + - + - + C s u s s u s s u sin s -s u cos s 
m m m m m m m m m m m m m 
+ cosh + + sinh + - -u sin 0 u s u s -u cos s s 
m m m m m m m m m 
a a 
mE IN[ or mE lNo , 
with the abbreviations 
+ 2 2 2 2 2 2 u = s + (1-v)(I) "I" , u s (1-v)(I) 7r , m m 
and 
F = _b_ P m 4s2 m 
m = 0,1,2,3, ..• ,m. 
The solution of the linear system is then given by 
A 
m = 
a 2 1 1 
D+"N 
s m 
m 
2 
+ s + u - cosh s + cos s - ] + Mb s + s - u + sin 
m m m m m m m m s~ } , 
0 
2 
~F 
o m 
= , (3.62) 
0 
2 
a Mbm 
0 
s = f3a (3.63) 
(3.64) 
+ 
(3.65 ) 
C 
m = 
2 
8111 [-+-
--- \F suu D-N.mmmm 
s m 
m 
46 
2 
+ s-u+ cosh s+cos s 
m m m m 
2 
- s+u- sinh s+sin s-] - Mb s-s+u-sinh s+} , 
m m m m m m m m m 
2 
D = ~.LF 
m D m' 
s 
m 
8 8 
m E lNE or m E lNO 
Similarly, substituting (3.56) into (3.48)-(3.51) leads to 
A 
m 
~ F [s + U + u - + s + u - 2 cosh s + cosh s I m m m m m m m m 
2 
B =~.LF 
m D + m 
s 
m 
2 
+ s-u+ cosh s+cosh s 
m m m m 
2 
D ~.L F 
m D m' 
s 
m 
2 
N = s-u+ sinh s-cosh s+ 
m m m m m 
(3.66) 
47 
We nOw turn to the second building block of Figure 3.4. Substituting (3.43) 
into the homogeneous part of the plate equation (3.1), one obtains 
(3.67) 
if the orthogonality relations 
b with m n o 
b . t 2" Wl h m n f. 0 (3.68 ) 
o with m f. n 
for m,n from lNE or lNO are used. Again, only the case m,n E lNE corresponds to 
(3.43), but the analogous results for m, n E lNO will be needed later and both 
cases are therefore treated simultaneously. Similarly, the boundary conditions 
(3.39) and (3.41) give 
X(O)=O, 
n 
x (0) v(n27rb)2Xn(0) = 0, 
n,xx 
x ( a ) ( 2 - v)( n27rb) 2 X ( a) = 0, n,xxx n,x 
n7r 2 X ( a) - v ( 2b) Xn (a) n,xx 
with the FOURIER series 
M (y) = "':;:. M h (y) 
a ...... an n 
n 
(3.69 ) 
(3.70 ) 
(3.71 ) 
(3.72) 
(3.73) 
48 
The differential equation (3.67) has the general solution 
x (x) = G coshy+x + H sinh y+x + I cos Y x + J sin V-x, 
n n n n n n n n n 
(3.74) 
= G cosh y+x + H sinh y+x + I cosh V-x + Jnsinh Ynx, 
n n n n n n 
(3.75) 
with + __ v'f32 (nn)2 Yn + 2b (3.76) 
n = O,1,2,3, ... ,m. 
The sets lN~-, lNg-, lN~+ and lNg+ are defined as in (3.55), (3.56), a being 
substituted by b. The constants Gn to I n can be calculated from the inhomoge-
neous linear system formed by (3.69) to (3.72) with (3.74): 
2 
+
 
I'n
 
v(
~)
2 
2b
 
y+
si
nh
·t
+.
 
n
 
n
 
[ 
+
2 
n1
£ 
2
] 
Vn
 
-
(2
-v
)(
2b
) 
c
o
sh
 t
+
. 
n
 
2 ~~ 
-
v(~
:)2
J 
o
 
o
 
V
+c
os
h 
t+
. 
n
 
n
 
[V
+2
 
_
 
(2
_V
)(n
1£
)2
] 
n
 
2b
 
s
in
h 
t+
. 
n
 
[V
+2
 
_
 
v
(n
1l
:)2
] 
n
 
2b
 
b-
b-
n
 8
 lN
E 
o
r 
n
 8
 lN
O
 ' 
2 
[)l~
 +
 
l,(
~)2
] 
2b
 
v
-s
in
 t
-·
 
n
 
n
 
[V~2
 + 
(2_
v)(
~)2
] 
2b
 
-
c
o
s
 
t-
· 
n
 
[V~2
 + v
(~~)
2] 
w
it
h 
t+
 
n
 
+
 
V
na
, 
t n
 
Y 
n
a
,
 
n
 
=
 0
,1
,2
,3
, .
.
.
 
,
m
. 
o
 
o
 
-
I' 
c
o
s 
t -
.
 
n
 
n
 
[V~2
 + (
2_V)
(~~)
2] 
-
s
in
 t
 -
.
 
n
 
2 
[ 
-
n1
l: 
2
] 
Vn
 
+
 
v
(2
b)
 
G
 n II n
 
I n
 
J n
 
o
 
o
 
o
 
M
 an
 
-
0
 
Su
bt
ra
ct
in
g 
th
e 
fi
rs
t 
ro
w
 
o
f 
(3
.7
6)
 m
u
lt
ip
li
ed
 w
ith
 [V
~2 
-
v(~
~)2
J f
ro
m
 t
he
 s
e
c
o
n
d 
ro
w
, 
it
 i
s 
e
a
sy
 t
o
 s
e
e
, 
th
at
 n
ow
 
G n
 
In
 
0,
 
n
 
0,
'1
,2
,3
, .
.
.
 
,
m
 
ho
ld
s;
 
th
er
ef
or
e,
 t
he
 s
ys
te
m
 (
3.
77
) 
c
a
n
 
be
 s
im
pl
if
ie
d 
to
 (
co
mp
are
 w
it
h 
(3
.6
2)
) 
(3
.7
7)
 
(3
.7
8)
 
(3
.7
9)
 
.
1>0
 
co
 
50 
[ + - t+ - + t [ ::] 0 t v cosh -t v cos n n n n n n = (3.80 ) v+sinh t+ 
-v sin t b
2M 
an 
n n n n 
-D-
b- b-
n to lNE or n to lNo ' 
with 
+ 02 2 (1 )(n)2 2 02s 2 _ n 2 2 0 b (3.81 ) v s + -v "2 7l , V (1-v)("2) 7l , a' n n 
n 0,1,2,3, •.. ,Ul. 
The remaining constants are then given by 
H 
1 b2 _ + 
t 
- J 1 b
2 + - t+ 
= - - M t v cos n' = - - M t v cosh n E D an n n n E D an n n n 
n n 
with E 
n 
defined as (3.82 ) 
2 2 
E + - cosh t+sin t t-v+ sinh t+cos t = t v n n n n n n n n n 
Similarly, for the higher indices one obtains 
H 
1 b2 _ + 
t - J 
1 b2 + _ t+ 
= - - M t v cosh n' - - M t v cosh n E D an n n n E D an n n n 
n n (3.83 ) 
2 2 
E + - cosh t+sinh t - + sinh t+cosh = t v t v t n n n n n n n n n 
51 
3.2.3 Superposition of building blocks 
3.2.3.1 First approach: load developed along the x-axis 
In this case, we superpose two building blocks, which generalize W1(x,y) 
and W2(x,y) from Fig. 3.4, namely 
...... 
•• ;::. W1mn (y)gm(x), W1mn (y) = 
m 
, (3.84) 
where the functions Ym(y) are given by (3.55), (3.56) and Xm(y) is defined as 
X (y) = H sinh G+y + J sin G-y, 
m m m m m 
H }1 - + DE Mbmsmumcos s m' m 
m (3.85) 
J a
21 + - + DE Mbmsmumcosh s m' m 
m 
+ 2 + _ +2 + [ 
= s u cosh s sin s s u sinh s cos s m' m m m m m m m m 
a a 
m EO IN[ or m EO lNO , 
X (y) = H sinh G+ + J sinh G-y, m m mY m m 
H a
21 - + 
= DE Mbmsmumcosh s m' m 
m (3.86) 
J a
21 + - + 
= DE Mbmsmumcosh s m' m 
m 
52 
a+ a+ 
m E lNE or m E lNO ' 
(compare with (3.74), (3.75), (3.82) and (3.83» and 
--: ... 
• :. W2 (x)h (y), W2 (x) = 
...... nm n nm 
, (3.87) 
n 
where the functions X (x) are given by (3.74), (3.75) and Y (x) is defined as 
n n 
A b21 M t - + sin t -= 
-ON v n' n an n n 
n (3.88) 
C b21 M + - t+ = ON t v sinh n an n n n' 
r 
n 
N 
n 
Y (x) = A cosh y+x + C cosh -n n n n Ynx, 
A b
21 M t - + sinh t -= ON v n' n an n n 
n (3.89) 
C b
21 M t+v-sinh t+ = 
-ON n an n n n' 
n 
N 
n 
53 
_ +2 + 
= t v cosh t sinh t 
n n n n 
+ _2 + 
t v sinh t cosh t-, 
n n n n 
(compare with (3.55), (3.56), (3.65) and (3.66) with F = 0). 
m 
Since both solutions satisfy exactly the differential equation governing 
the plate vibration (the homogeneous part of (3.1», their sum W (x, y) will 
also satisfy this equation. We must, however, constrain the FOURIERcoeffi-
cients in the moment expressions (3.61) and (3.73) of the respective solutions 
so that the net effect gives a zero bending moment along the edges x = a and 
y = b. It is easily seen that all other boundary conditions are satisfied. 
We begin with the contributions to the bending moment along the edge 
x a: 
(3.90) 
Substituting (3.52), (3.84) and (3.73) into (3.90) gives 
= -D .. :::. [- (mn) 2W ( y) + v W 1 ( Y )] g (a) 
.. ::.. 2a 1mn mn,yy m 
m 
= 
...... 
':. M h (y) 
.. :._ an n 
(3.91) 
n 
Multiplying (3.91) by h (y) and integrating over the length of the quar-p 
ter-plate in y-direction gives with (3.55), (3.56) 
(m-1)m 
... -. --2- ~ 2 m 2 2 b+ 
':. (- 1 ) ( v S - (1- v) ( -2) n )( A e + 
..:... m mn 
m 
m 2 2 b- b- f + (1-v)(-2) n )(e e + D S ) = 
m mn m mn 
54 
n 0 
= 
D n=2,4, ... ,rn, 
m E lNE or m E lNO ' 
and with (3.85), (3.86) 
(m-1)m 
...... --2- ) 2 m 2 2 - b+ 2 
.:. (-1) ( vs - (1- v)( -2) n ) H 5 - (vs + 
•• :... m mn 
m 
with the abbreviations 
b 
= t J 
o 
= 
b 
= t [ 
= 
}M 
an 
= -D-
cosh G+y h (y) dy = 
m n 
sinh G+y h (y) dy = 
m n 
./.2 2 
I(J s + 
m,n O,1,2,3, ... ,rn , 
(3.92) 
(3.93 ) 
(3.94) 
55 
(n+1)n (n-1)n 
= 
(n-1)n - - nn(( 1)--2- _ (_1)--2-
(_1)---2-- _2_s~m~Sl_·n __ S~m_+~2~ __ - __ ~ ____ ~ ________ _ 
2 2 
V/s2 - (,62(I) + (¥-) )n2 
(3.95 ) 
a a 
m E IN[ or m E lNO ' n = 0,1,2,3, ... , ill , 
and finally 
(n+1)n (n-1)n 
(n-1)n 2s sinh sm- _ n2n ((_1)---2-- (_1)---2--
= _(_1)---2-- --~m~--------------~2----~2----------
,62s2 - (,62(I) + (¥-) )n2 
(3.96) 
a+ a+ 
m E IN[ 0 r m E lNO ' n = 0, 1 , 2 , 3, ... ,ill • 
56 
Substituting 0.65) and 0.66) into 0.92) or 0.85) and 0.86) into 0.93) 
leads to the inhomogeneous linear system of equations for the FOURI[R compo-
nents of the bending moments Ma and Mb 
...... .. .... 
.. ;~.AnmMbm + Man = f n' f n := 2~ Cnm F m 0.97) 
m m 
m E IN[ or m E lNO' n E IN[ or n E lNO' 
with (3.64), wherein the coefficients Anm and Cnm must be determined separate-
ly for each combination of the two sets IN[ and lNO for the indices m and n. 
The contributions to the bending moment along the edge y = b are (compare 
with 0.90)) 
or with (3.52), (3.87) and (3.61) 
, .... 
= -D 2:. [ -(n2b7l")2W2 (x) + vW2 (x)] h (b) nm nm,xx n 
n 
...... 
= - .:. M g (x) 
.. : ••• bm m 
m 
0.98) 
(3.99) 
Multiplying 0.99) by g (x) and integrating over the length of the quar-p 
ter-plate in x-direction gives with (3.74), (3.75) 
57 
(n-1)n 
... -. --2-~ 2 2 n 2 2 a+ 
::- (-1) (vJa s - (1-v)(-2) 7l )H 5 
... _. n nm 
n 
, me lNo' 
and with (3.88), (3.89) 
(n-1)n --2-~ 2 2 n 2 2 - a+ 2 2 :~~:: (-1) /(vll5 s - (1-v)(z) 7l )AnCnm - (vll5 s + 
n 
n 2 2 - a-I 
+ (1-v)(-2) 7l )C C \~= 
n nm 
2b2M bm 
--D-
= 
b2M bm 
-D-
m = 0 
m = 2,4, ••. ,OJ 
(3.100) 
(3.101) 
wherein the constants C~:, s~:, C~: and s~: are defined as in (3.94), (3.95) 
and (3.96) b being substituted by a, e.g. 
a+ C 
nm 
=~ 
a 
= 
(m+1)m (m-1) 
(m-1)m 2t+sinh t+ _ m71«_1)--2- _ (_1)-2-
(_1)--2- __ ~n~_~n_~2~_~_~~ ____ _ 
222 
s + «I) + _n_)712 
4115 2 
m,n = 0,1,2,3, ••• ,OJ • (3.102) 
58 
Substituting (3.82) and (3.83) into (3.100) or (3.88) and (3.89) into (3.101) 
leads to the homogeneous linear system for the Man' Mbm 
... -. 
~::. BmnMan + Mbm 
n 
o , (3.103 ) 
wherein now the coefficients Bmn must be determined separately for each combi-
nation of the two sets IN[ and lNo for the indices nand m. 
In summary, we can develop a set of 2k inhomogeneous algebraic equations 
relating the 2k moment coefficients Man and Mbm . The set of equations can best 
be handled by matrix techniques. A schematic representation is given in Fig. 
3.5 for m,n E IN[. For simplicity here the series (3.91) and (3.99) were trun-
cated and only three terms were taken into account in each series. In practi-
cal applications the number of terms to be considered does of course depend on 
the frequency range and on the desired precision. It should also be noted that 
two of the matrix blocks in Fig. 3.5 are unit matrices. 
0 0 I ADD A02 A04 MaO fo I 
I 
J 
0 0 I A20 A22 A24 Ma2 f2 I 
I 
0 0 
, 
A40 A42 A44 Ma4 f4 I 
I 
------------1 - -- ----- ----- = 
BOO B02 B04 0 0 MbO 0 
B20 B22 B24 0 0 Mb2 0 
B40 B42 B44 0 0 Mb4 0 
Fig.3.5 Structure of the coefficient matrix used in the analysis of the forced 
vibrations of the completely free rectangular plate loaded on a line 
of symmetry. 
59 
With the matrices 
A := (A ), B:= (B ) , 
nm mn 
(3.104) 
f ._ (f ) 
n 
(3.97) and (3.103) lead to 
} (3.105) 
and with the unit matrix I to 
(3.106) 
With known Man' Mbm the equations (3.84) to (3.89), (3.55), (3.56), (3.74) and 
(3.75) give the plate deflection 
(3.107) 
and at the point P(xp'O) of excitation the deflection is 
(3.107a) 
60 
3.2.3.2 Second approach: load expanded in a double fOURIER series 
Here, we superpose three building blocks, which generalize W1(x,y), 
W2(x,y) and W3(x,y) from Fig. 3.6. The first two are nearly the same as in 
chapter 3.2.3.1, but now with Fm = 0, i.e. 
° Ym(y), n E IN[ , 
W1(x,y) = ... -'. .. ' 
...... 
W1mn(y)gm(x), W1mn(y) = (3.108) 
m 
Xm(y) , n E lNO ' 
m E IN[ or m E lNO' 
where the functions X (y) are given by (3.85), (3.86) and yo(y) is defined as 
m m 
(compare (3.55), (3.56), (3.65) and (3.66), wherein N is unchanged) 
m 
yo( ) 
= 
AOcosh B+y + Cocos B~y, m y m m m 
AD a21 - + -
= 
-ON Mbmsmumsin s m' m 
m 
CO a21 + - + 
= ON Mbmsmumsinh s m' m 
m 
a a-
mE IN[ or mE lNO ' 
yO( ) 
= 
AOcosh B+y + COcosh B~y, m y m m m 
AD a21 - + -
= 
-ON Mbmsmumsinh s m' m 
m 
CO a21 + - + 
= ON Mbmsmumsinh s m' m 
m 
a+ a+ 
mE IN[ or mE lNO ' 
(3.109) 
(3.110) 
61 
z,W,(x.y) 
~----------~-----y 
o 
o 
x 
Z'W2 (X,y) 
y 
x 
y 
o 
x 
Fig. 3.6 Typical building blocks used in the analysis of the forced v ibra-
tions of the completely free plate loaded at an arbitrary point 
62 
whereas W2(x,y) can be taken unchanged from (3.87). It is easily seen, that 
the third building block has the solution 
., .......... . 
= .,;:: . .,;::. F g (x)h (y) , mn m n (3.111) 
m n 
with g (x) from (3.108) and h (y) from (3.87). 
m n 
If we insert the solution (3.111) into the inhomogeneous differential 
equation (3.1) for the quarter plate, we obtain 
., .... 
.. 
... 
.I •••• 
m 
:~~:: 1 [(~:)2 + (~~)2]2 - [34 I Fmngm(x)hn(y) = 468(x-Xp'y-yp) 
n 
(3.112) 
Multiplying (3.112) by g (x) and h (y) and integrating over the area of the p q 
plate gives 
F 
mn 
f 
4D 
g(x)h(y) 
m p n p 
[34 _ [34 
mn 
4 
ab ' 
2 
ab ' 
with m~O and n~O , 
with m=O,n~O or n=O,m~O , (3.113) 
with m=O and n=O 
m,n = 0,1,2, •.. ,rn , (3.114) 
if the orthogonality relations (3.47) and (3.68) are used. 
Since the solutions (3.108) and (3.87) satisfy the homogeneous part 0 f 
the differential equation (3.1) and the solution (3.111) satisfies the 
inhomogeneous differential equation (3.1), the sum 
(3.115) 
63 
will satisfy the inhomogeneous equation (3.1). Again we must constrain the 
FOURIER coefficients in the moment expressions (3.61) and (3.73), so that the 
net effect - now for the superposition of three solutions - gives a zero 
bending moment along the edges x = a and y = b. The other boundary conditions 
are satisfied automatically. 
We begin with the contributions to the bending moment along the edge 
x = a: 
(3.116) 
Substituting (3.52), (3.108), (3.73) and (3.111) into (3.116) gives 
DrW + vW ] I L 1,xx 1,yy x=a 
= - D :~~:: [- (~;)2W1mn(Y) + v W1mn,yy(y)Jgm(a) = 
m 
= D[lrJ +vW ]1 - ":;:' M h (y) = 
. 3,xx 3,yy x=a d.... an n 
n 
":"'{ .. :... [ mn 2 nn 2] } 
= - :. D :. F ( -) + v ( 2b ) gm (a) + M h (y) , d:... d:... mn 2a an n 
n m 
m E lNE 0 r m -E lNO' n E lNE 0 r n E lNO • (3.117) 
Multiplying (3.117) by h (y) and integrating over the length of the quarter p 
plate in y-direction gives with (3.109), (3.110) and (3.113) 
(m-1)m 
...... --2-) 2 2:. (-1) I (vs 
m 
...... 
:=-
.a •• _ 
m 
(m-1)m A 
2 f (-1) 7g(x)h(y)' 
~ m p n p 
(3.118) 
and with (3.85), (3.86) 
... -
.. 
... 
... _. 
m 
(m-1)m 
-2 \ 2 (-1) 1 (vs 
64 
2 m 2 2 - b-/ (vs + (1-v)(-2) 7C )J S {= 
m mn l 
= - M + .:. 1 a2 ... _. D an .. : .. 
(m-1)m A 
2 f (-1) 7g(x)h(y)· 
IU m p n p 
m 
(3.119) 
wherein the KRONECKER symbol 
with ih , 
(3.120) 
with i=j , 
and the abbreviations (3.94) to (3.96) are used. Substituting (3.109) and 
(3.110) into (3.118) or (3.85) and (3.86) into (3.119) leads to the inhomoge-
neous linear system of equations for the FOURIER components of the bending 
moments Ma and Mb 
with 
"': ... 
.. 
..:: .. 
m 
C 
nm 
f 
an 
·(1 - 12 8 )(1 1 [, ) om - 2 on ' 
"': ... 
.. 
. . 
...... 
m 
C F 
nm m 
(3.121) 
(3.122) 
F = 
m 
65 
LA 
8s2 f with m = 0 , 
Lr 
4s2 
with m = 1,2,3, ••• ,m 
(3.123 ) 
The coefficients Anm must be determined separately for each combination of the 
indices m and n. 
The contributions to the bending moment along the edge y = bare 
Mb(x) + My2 (x,b) + My3 (x,b) = 0, (3.124) 
or with (3.52), (3.87), (3.61) and (3.111) 
My2 (x,b) = - DrW2 + vW2 ] b = l ,yy ,xx y= 
= - D 
= D[W3 + v W3 ] b 
,yy ,xx y= 
-r. ... 
'. Mbg(x)= 
.. ::.. m m 
m 
= - :~. {D ::~ F [(nn)2 + v (mn)2]h (b) + Mbm}gm(x) , 
.I.M. ...... mn 2b 2a n 
m n 
(3.125 ) 
Multiplying (3.125) by g (x) and integrating over the length of the quarter p 
plate in x-direction gives with (3.88), (3.89) and (3.113) 
n 
(n-1)n 
( )--2-~(.L2 2 ( )(n)22)- a+ 22 n 2 2- a-I 
-1 VIOS - 1-v -2 n AC -(v~s +(1-v)(-2)n )CC \= 
n nm n nm 
(n-1)n 
(_1)--2 - r~ g (x )h (y ). 
m p n p 
(3.126) 
66 
and with (3.74), (3.75) 
...... 
.. ;::. 
n 
(n-1)n 
= !l~M + .. :::- (_1)--2 - f,l g (x )h (y ). 
D 1 bm "::N m p n p 
n 
n E lNE or n E lNo' m E lNo . (3.127) 
Substituting (3.88) and (3.89) into (3.126) or (3.82) and (3.83) into (3.127) 
leads to the inhomogeneous linear system for the Man' Mbm 
with 
D 
mn 
'"': ... 
'. 
.' 
...... 
n 
f bm .-
(n-1)n 
--2-
= - 4(-1) g (x )h (y ) 
m p n p 
.. .... 
'. 
.' 
.. : ... 
n 
D F 
mn n 
s2[<-~i + v(¢2 ~i]7f2 
~¢ I)2 + (·~i]27f4 _ ¢4s4 
(3.128) 
and F from (3,123) with n for m. Again the coefficients B must be determi-
n nm 
ned in each case separately. 
In summary, again we can develop a set of 2k inhomogeneous algebraic 
equations relating the 2k moment coefficients Man and Mbm' A schematic repre-
sentation with m,n E lNO is given in Fig. 3.7, where only three terms are taken 
into account in each series. 
67 
0 0 A11 A13 A15 Ma1 f a1 
0 0 A31 A33 A35 Ma3 fa3 
0 0 1 A51 A53 A55 Ma5 fa5 I 
----- ---1------ - = 
B11 B13 B15 I 0 0 Mb1 f b1 I 
I 
B31 B33 B35 I 0 0 Mb3 f b3 1 , 
B51 B53 B55 I 0 0 Mb5 fb5 1 
Fig.3.7 Structure of the coefficient matrix used in the analysis of the forced 
vibrations of the completely free rectangular plate loaded at an arbi-
trary point. 
With known Man' Mbm the equations (3.109), (3.110), (3.74), (3.75), (3.82), 
(3.83) and (3.85) to (3.89) give the plate deflection at the point P(x ,y ) of 
P P 
excitation. 
68 
3.3 Plate connected at center 
3.3.1 Analytical solution 
Let us consider first the completely free plate loaded at the center with 
the "double symmetric" transverse force (see Fig. 3.1) 
f(x,y) = p (x)~(y) = f(-x,y) = f(x,-y) , 
s 
p (x) = f8(x) = p (-x). 
s s 
} (3.130) 
Then, according to (3.3), (3.4), only the double symmetric vibrations W (x,y) 
ss 
occur. The corresponding quarter plate also is given in Fig. 3.1. We use the 
method of superposition, described in chapter 3.2.3.1, choosing the two 
building blocks shown in Fig. 3.B. 
We begin with the first building block of Figure 3.B, where the solution 
is refered to as W1(x,y), seeking a LEVY-type solution. Here, we require the 
boundary conditions (3.16) and (3.17) 
W1 (O,y) - 0, 
,x 
(3.131) 
W1 (x,O):: 0, 
,y (3.132) 
along the edges x = 0 and y = 0 whereas, at the originally free edges, we take 
the slip shear conditions 
W1 (a, y) :: 0, 
,x 
(3.133) 
on x = a and zero vertical edge reaction with a distributed bending moment 
Mb(x) 
(3.134) 
along y = b. Then the solution for this building block can be expressed in the 
form proposed by LEVY (see LEVY) as 
69 
Z,W(X,y) 
~-----=o~O~----~~~y 
x 
b -----..-.,/ 
x 
+ 
z,W2 (X,y) 
>-----~O=O~----__ ----y 
....... -. 
Mo{y) x 
Fig. 3.8 Building blocks used in the analysis of the doubly symmetric forced 
vibrations of the completely free plate 
or according to (3.86) 
70 
m 
... ". m x 
::- Y (y) cos -271"-a 
... ". m 
m=O,2, ••• 
"I •••• 
.. ;:~ W1 mn ( y) gm (x) , 
m 
m,n e IN[ 
(3.135) 
(3.136) 
We now turn to the second building block of Figure 3.8, where the solution is 
referred to as W2(x,y) and the boundary conditions are 
W 2,x (O,y) :: 0, Vx2 (O,y) - 0, (3.137) 
Vx2 (a,y) = 0, Mx2 (a,y) = Ma(y), (3.138 ) 
W 2,y (x,O) - 0, Vy2 (x,O) = ° (3.139 ) 
w 2,y (x, b) - 0, Vy2 (x,b) = 0. (3.140) 
The solution for this building block can be expressed in the LEVY form 
(compare (3.40), (3.42), (3.37) and (3.35) for p(x) = 0) 
m 
... _. n 'L 
::- Y (x)cos-2 71"b 
...... n 
(3.141) 
n=0,2, •.. 
or according to (3.89) 
"' .... 
= .. ;::. W2 (x)h (y), nm n (3.142 ) 
n 
71 
With (3.136) the coefficients A and C in (3.104), (3.105) result from 
nm nm 
(3.92) with (3.65) and (3.66) as 
and 
A 
nm 
A 
nm 
= (-1) 
(m-1)m+n 
2 
2 4 
vrO s + 
4 (rOs) -
m E lNr (the formula is also valid for m E lN~-), 
= (-1) 
(m-1)m+n 
2 
2 4 
vrO s + 
2 2 2 4 
(1-v) (I) (I) 7r 
4 (rOs) -
a+ a+ 
mE IN[ (valid also for mE lNO ), 
(3.143) 
(3.144) 
C = 
nm 
2 4 2 m 2 n 2 4 1 (- + - + - +)] 
• (vrO s + (1-1» (-) (-) 7r ) -N s u sin s - s u sinh s , 2 2 mm m mm m 
m 
m E lNr (valid also for m E lN~ ), (3.145) 
72 
as well as 
(m-1)m 
--2- 2 2 2 .Q. 
C = 
(-1 ) 4s [(~2(!!!) + v ("i) )n2 + (-1 )2. 
nm 2 2 2 4 (~2(!!!) + (.Q.) )2n4 (~s) - 2 2 
a+ a+ 
m E lNE (valid also for mE lNO ), n E lNE/O . (3.146) 
For n = 0 in (3.143), (3.144), (3.145) and (3.146) the right-hand sides are to 
1 be multiplied by 2 . 
Analogously with (3.142) the coefficients B in (3.104), (3.105) result 
mn 
from (3.101) with (3.88), (3.89) as 
For m 
B 
mn 
B 
mn 
= (-1) 
(n-1)n+m 
2 
= -(-1) 
(m-1)m+n 
2 
4 
N 
n 
4 
N 
n 
1 
= 0 the right-hand sides are to be multiplied by 2' 
(3.147) 
(3.148 ) 
73 
With known Man' Mbm from (3.105) the equations (3.135), (3.55), (3.56), 
(3.141), (3.88) and (3.89) give the double symmetric plate response 
k 
+ ::~ (A cosh B+y + B sinh B+y + C cosh B-y + 
.... " m m m m m m 
iiH-2 
+ D sinh B-y)cos mn ~ + 
m m 2 a 
n 
+ 
'1'"' - + _) n7l" v 
::- (A cosh y x + C cos Ymx cos -2 .L + 
... ". n m n b 
n::O,2,4, ... 
(3.149) 
with (3.65) to (3.68) and (3.88), (3.89), and this solution still holds for a 
plate excited by an arbitrary symmetric load on the x-axis. For the case of 
the concentric load considered here and in particular, at the point P(O,O) of 
excitation we have 
2(k-1) 
-r.; (A