Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 1, pp. 123-139, Warsaw 2008

METHODS BASED ON THE DIFFERENTIAL QUADRATURE
IN VIBRATION ANALYSIS OF PLATES

Artur Krowiak

Institute of Computing Science, Cracow University of Technology, Cracow, Poland

e-mail: krowiak@mech.pk.edu.pl

The paper deals with the methods based on the differential quadrature and
their application to the free vibration analysis of plates. The spline-based dif-
ferential quadratureMethod (SDQM) is presented as an alternative to known
methods based on the interpolation polynomial (PDQM). The SDQMuses a
polynomial piecewise function to approximate the wanted solution of a go-
verning equation. The way of determining the spline functions as well as the
way of computing weighting coefficients for the method are presented in the
paper. Then the SDQM is applied to determine natural frequencies of plates.
The influence of the spline degree, number of nodes and grid point distribu-
tion on the accuracy, convergence and stability is investigated in an example.
All results are comparedwith values obtained by the conventional differential
quadraturemethod (PDQM).

Keywords:Differential quadraturemethod, spline interpolation, freevibration

1. Introduction

Most engineering problems are described by partial differential equations. It
is very difficult, if possible at all, to find closed-form solutions to them. A
great increase in the computational power in recent years enables one to use
numerical methods for solving very complex physical tasks. It is the reason
that stimulates the development of known methods and the search for new,
more efficient ones. Most of these methods rely on the conversion of a phy-
sical model of a system from continuous to discrete one. The approximate
solution is searched at some special points in the domain. Among currently
used discretisation methods, the most popular are: finite difference method,
finite element method and finite volume method. These discretisation tech-
niques use a low degree interpolation to determine function values at a node



124 A. Krowiak

and, therefore, they are numbered among low order methods. They allow one
to achieve high accuracy by using a large number of nodes. In many prac-
tical engineering problems, the numerical solution is required only at a few
discrete points in the domain. The modal analysis could be an example. By
discretizing apartial differential equation, an algebraic eigenvalue problemcan
be obtained. Next, the latter is solved giving approximation values of natural
frequencies (or/andnaturalmodes) of a continuous system.Thenumber of ob-
tained frequencies corresponds to the number of nodes of the imposed mesh.
Among these frequencies, only a few first are interesting from the practical
point of view. However, in order to achieve high accuracy for these values by
low order methods, one has to use a large number of nodes. It requires much
virtual storage and computational effort. This drawback can be overcome by
using so-called global methods, which take muchmore information than only
local neighborhood of a node to approximate the function. Itmakes the rate of
convergence of thesemethodsmuch higher than low ordermethods and allows
one to achieve very accurate results with only a few discrete points.
The differential quadraturemethod (DQM) falls under this category. The-

oretical foundations of the DQMwere given by Bellman and Casti (1971). In
recent years, onehas beenable tonotice significant development of themethod
and its application in many fields of mechanics. Somemain works are quoted
by Bert and Malik (1996). Higher efficiency of the DQM for linear problems
than the finite element and finite difference method was proved by Bert et al.
(1988, 1993),Wang andBert (1993), Malik andCiven (1994). In addition, the
DQM ismuchmore effective for nonlinear problems comparingwith low order
methods, which was presented by Bellman and Casti (1971), Bellman et al.
(1972), Bert et al. (1989), Feng and Bert (1992), Malik and Civen (1994).
The idea of the DQM is based on the approximation of spatial derivatives

of a function at each node by a linear combination of function values at all
discrete points in the domain along the coordinate lines. Considering a two-
dimensional case, where the domain of the function f(x,y) is a rectangular
area, partial derivatives with respect to spatial variables at each point (xi,yi)
can be expressed in the method as

∂nf

∂xn

∣

∣

∣

x=xi
y= yj

=
Nx
∑

k=1

a
(n)
k
(xi)f(xk,yj)=

Nx
∑

k=1

a
(n)
ik
f(xk,yj)

(1.1)

∂mf

∂ym

∣

∣

∣

x=xi
y= yj

=

Ny
∑

k=1

b
(m)
k
(yj)f(xi,yk)=

Ny
∑

k=1

b
(m)
jk
f(xi,yk)



Methods based on the differential quadrature... 125

for i= 1, . . . ,Nx, j = 1, . . . ,Ny, where Nx, Ny are the numbers of nodes in

the x and y directions, and a
(n)
ik
, b
(m)
jk
are the weighting coefficients for the

nth and mth order derivatives with respect to appropriate variables. In order
to approximate the mixed derivatives, the weighting coefficient matrices for
appropriate derivatives with respect to x and y have to bemultiplied. Using
formula (1.1), one has to collocate a governing equation at each grid point and
imposeboundary conditions in order to reduce the partial differential equation
to a system of algebraic or ordinary differential equations, respectively of the
case under consideration. Themain stage of the method is the determination
of the weighting coefficients.

2. Polynomial and Spline-based differential quadrature method

Values of the weighting coefficients depend on the way the solution is ap-
proximated (selection of trial functions), and they influence the accuracy, co-
nvergence and stability of the method. The interpolation polynomial is the
most often used to approximate the solution in the differential quadrature
method (PDQM). Using Lagrange base functions, the rth order derivatives
of the interpolation polynomial at the ith discrete point can be expressed as
follows

P(r)(xi)=
N
∑

j=1

l
(r)
j (xi)fj (2.1)

where lj(x) denotes Lagrange’s base polynomial of the (N −1)th degree.

It is easy to notice that the derivatives of the appropriate Lagrange base
functions are the weighting coefficients for the polynomial based differential
quadrature method.

Theanalytical formulas for the coefficients of thefirst order derivativewere
given by Quan and Chang (1989) and have following forms

a
(1)
ij =

1

xj −xi

N
∏

k=1,k 6=i,j

xi−xk
xj −xk

for j 6= i

(2.2)

a
(1)
ii =

N
∑

k=1,k 6=i

1

xi−xk



126 A. Krowiak

Due todifficultieswithderivation of explicit formulas for higher order derivati-
ves of Lagrange’s base functions, Shu andRichards (1992) gave recurrence re-
lationship (2.3) to overcome the problem, i,j =1,2, . . . ,N, r=2,3, . . . ,N−1

a
(r)
ij = r

(

a
(1)
ij a
(r−1)
ii −

a
(r−1)
ij

xi−xj

)

for i 6= j

(2.3)

a
(r)
ii =−

N
∑

j=1,j 6=i

a
(r)
ij

There are also otherways based on theFourier series expansion orB-spline
functions to approximate the wanted solution in the DQM. But especially
PDQM allows one to achieve very high accuracy by using only a few nodes.
However, this method is very sensitive to the type of imposed mesh and the
number of nodes.When a uniform grid distribution is imposed and too many
sampling points are used, the results are inaccurate or the method is not
convergent. The computational instability of the method is the result from
the manner of the interpolation polynomial. This polynomial oscillates much
when its degree N−1 (N –numberofnodes) is toohigh,particularlywhen the
uniform grid is imposed. In most problems of computational mechanics, the
only way to estimate the accuracy of results is to carry out the computation
again using larger number of nodes. Therefore, the computational stability
should be the important feature of a numerical method. To improve stability
of theDQM,Zhong(2004) usedquinticB-spline as trial functions todetermine
theweighting coefficients. It considerably improves stability of themethodbut
the convergence rate is less comparing to the PDQM.
In the present work, another way of the approximation of the solution is

shown. In the presentedmethod, the solution of a partial differential equation
is approximated by the nth degree polynomial piecewise function (SDQM).
The approximation was first applied in the DQM by Krowiak (2006). The
work, that has been done so far, indicates that using a suitably high spli-
ne degree the convergence rate of the method is similar to PDQM and the
computational stability is much better even when using uniformly spaced
nodes.

When the degree n of the spline is odd then the function is approximated
in the following way

f(x)≈{si(x), x∈ [xi,xi+1], i=1, . . . ,N −1} (2.4)

where N is the number of nodes.



Methods based on the differential quadrature... 127

When n is even, the auxiliary spline knots are defined at themidpoints of
the nodes in order to be able to meet the conditions for the determination of
the spline function

z1 =x1 zi+1 =
1

2
[xi+xi+1] i=1, . . . ,N−1 zN+1 =xN

(2.5)
and the interpolation function has the form

f(x)≈{si(x), x∈ [zi,zi+1], i=1, . . . ,N} (2.6)

In Equations (2.4) and (2.6), the ith spline section is defined as

si(x)=
n
∑

j=0

cijx
j (2.7)

The coefficients cij in Equation (2.7) are determined using the interpolation
conditions, the continuity conditions of the derivatives at the nodes and the
so-called natural end conditions at the domain boundaries. In the case of an
odd degree, these conditions have the following form

si(xi)= fi si(xi+1)= fi+1 i=1, . . . ,N−1

s
(k)
i (xi+1)= s

(k)
i+1(xi+1) i=1, . . . ,N−2, k=1, . . . ,n−1

s
(k)
1 (x1)= 0 s

(k)
N−1(xN)= 0 k=

n+1

2
, . . . ,n−1

(2.8)
Their number is (n+1)(N −1), which corresponds to the number of coeffi-
cients cij. In the case of an even spline degree, where the number of unknown
coefficients is (n+1)N, the auxiliary knots are also used in interpolation
conditions (2.9) and the continuity conditions for derivatives (2.10)

si(xi)= fi i=1, . . . ,N

si(zi+1)= si+1(zi+1) i=1, . . . ,N−1
(2.9)

and

s
(k)
i (zi+1)= s

(k)
i+1(zi+1) i=1, . . . ,N−1, k=1, . . . ,n−1 (2.10)

To complete the set of equations, the natural end conditions are introduced
at the end points

s
(k)
1 (x1)= 0 s

(k)
N (xN)= 0 k=

n

2
, . . . ,n−1 (2.11)



128 A. Krowiak

It insures that in every case of an even spline degree the number of conditions
matches the number of coefficients cij. These coefficients depend on the grid
distribution and unknown function values according to the formula

cij =
N
∑

k=1

Cijk(x1, . . . ,xN)fk i=1, . . . ,N, j=0, . . . ,n (2.12)

where N =N−1 when n is odd and N =N when n is even.

Taking advantage of the symbolic computation system, one can easily de-
termine the weighting coefficients for the differential quadrature method ba-
sed on the approximation given by Eqs. (2.4) and (2.6). Bymanipulating the
expressions in which the unknown function values f(xi) are defined as sym-
bols fi, it is possible to computederivatives of anarbitrarydegree r (r <n−1)
for function (2.4) or (2.6) at the ith discrete point

f(r)(xi)≈











s
(r)
i (xi) i=1, . . . ,N

s
(r)
i−1(xi) i=N when n is odd

(2.13)

In Eq. (2.13), s
(r)
i (xi) and s

(r)
N−1(xN) have the forms

s
(r)
i (xi)=

n
∑

j=r

[(

N
∑

k=1

Cijkfk
)

x
j−r
i

j
∏

l=j−r+1

l
]

i=1, . . . ,N

(2.14)

s
(r)
N−1(xN)=

n
∑

j=r

[(

N
∑

k=1

CN−1jkfk
)

x
j−r
N

j
∏

l=j−r+1

l
]

where the advantage has been taken of Equation (2.12).

Equations (2.14) can be expressed in other forms

s
(r)
i (xi)=

N
∑

k=1

[

n
∑

j=r

(

Cijkx
j−r
i

j
∏

l=j−r+1

l
)]

fk i=1, . . . ,N

(2.15)

s
(r)
N−1(xN)=

N
∑

k=1

[

n
∑

j=r

(

CN−1jkx
j−r
N

j
∏

l=j−r+1

l
)]

fk

ComparingEq. (1.1) to (2.15), it is clear that theexpressions in squarebrackets
in the above formula are theweighting coefficients for the rth order derivative



Methods based on the differential quadrature... 129

in the differential quadraturemethod based on the spline functions which can
be written as follows

a
(r)
ik
=
n
∑

j=r

(

Cijkx
j−r
i

j
∏

l=j−r+1

l
)

i=1, . . . ,N

(2.16)

a
(r)
Nk
=
n
∑

j=r

(

CN−1jkx
j−r
N

j
∏

l=j−r+1

l
)

when n is odd

3. Free vibration analysis of rectangular plates

Plates belong to basic structural elements in civil andmechanical engineering
and, therefore, they are often subjects of static and dynamic research. Many
numericalmethodshave beenused in dynamical analysis of plateswith various
boundary conditions. The obtained results have been used in real structures
or as the comparison of accuracy and convergence for applied methods. The
conventional differential quadraturemethodhas alsobeenapplied to thevibra-
tion analysis of plates (Shu andDu, 1997; Wang and Bert, 1993). The results
show that the convergence rate of the PDQM is very high. Very accurate re-
sults can be obtained applying a grid with points densely concentrated near
boundaries. The use of an arbitrary grid, for example a uniform one, makes
the results inaccurate or the method not convergent. This drawback can be
overcome by using themethod presented in this paper.
Since the results and conclusions coming from application of the PDQM

to vibration analysis of plates are common and well know, the SDQM has
been applied to check its versatility in using various grid point distributions.
In the paper, the SDQMhas been applied to determine natural frequencies of
a thin, isotropic, rectangular plate. The dimensionless governing equation for
free vibration of the plate is as follows

∂4W

∂X4
+2λ2

∂4W

∂X2∂Y 2
+λ4
∂4W

∂Y 4
=Ω2W (3.1)

In the above equation W denotes dimensionless mode shape function,
X = x/a and Y = y/b are dimensionless coordinates, a and b are leng-
ths of the plate edges, λ= a/b is the aspect ratio and Ω is the dimensionless
frequency. Its relation with the dimensional circular frequency is following

Ω=ωa2
√

ρ

D
(3.2)



130 A. Krowiak

where ρ is the density of the plate material and D=Eh3/[12(1−ν2)] is the
flexural rigidity (E, ν, h are Young’s modulus, Poisson’s ratio and the plate
thickness, respectively). Calculations have beendone for squareplates (λ=1)
with following configurations of boundary conditions:

(a) SS-F-SS-F

— for X =0 and X =1

W =0
∂2W

∂X2
=0 (3.3)

— for Y =0 and Y =1

λ2
∂2W

∂Y 2
+ν
∂2W

∂X2
=0 λ2

∂3W

∂Y 3
+(2−ν)

∂3W

∂X2∂Y
=0 (3.4)

(b) C-F-SS-F

— for X =0

W =0
∂W

∂X
=0 (3.5)

— for X =1

W =0
∂2W

∂X2
=0 (3.6)

— for Y =0 and Y =1

λ2
∂2W

∂Y 2
+ν
∂2W

∂X2
=0 λ2

∂3W

∂Y 3
+(2−ν)

∂3W

∂X2∂Y
=0 (3.7)

(c) C-C-C-C

— for X =0 and X =1

W =0
∂W

∂X
=0 (3.8)

— for Y =0 and Y =1

W =0
∂W

∂Y
=0 (3.9)

whereC denotes the clamped edge, SS – simply supported edge andF – free
edge.
Chebyshev-Gauss-Lobatto grid (3.10) and a uniform grid have been used

to discretize the area 0¬X ¬ 1, 0¬Y ¬ 1. In both cases, the same number
of points has been used in the X and Y directions

Xi =Yi =
1

2

[

1− cos
( i−1

N−1
π
)]

i=1, . . . ,N (3.10)



Methods based on the differential quadrature... 131

Equation (3.1),written inadiscrete form following fromtheuseof themethod,
is as follows
N
∑

k=1

a
(4)
ik
Wkj +2λ

2
N
∑

k1=1

N
∑

k2=1

a
(2)
ik1
a
(2)
jk2
Wk1k2 +λ

4
N
∑

k=1

a
(4)
jk
Wik =Ω

2Wij (3.11)

where a
(r)
ik
denote theweighting coefficients for the rth order derivative in the

SDQM and Wij are unknown nodal values. Identical coefficients a
(r)
ik
appro-

ximate the derivatives in both directions due to the use of the same grid and
number of nodes in these directions.
The implementation of boundary conditions is the important stage of the

method. Various approaches to this problem were presented by Shu and Du
(1997). In the present study, the boundary conditions have been used to cal-
culate function values at the boundary points and points adjacent to the bo-
undaries as a linear combination of the values at the interior points. Details
are presented for theSS-F-SS-F plate configuration. According to theDQM,
the discrete form of the boundary conditions described by Eq. (3.3) and (3.4)
is following

W1j =0 j=1, . . . ,N

N
∑

k=1

a
(2)
1kWkj =0 j=2, . . . ,N −1

(3.12)

WNj =0 j=1, . . . ,N

N
∑

k=1

a
(2)
Nk
Wkj =0 j=2, . . . ,N −1

(3.13)

λ2
N
∑

k=1

a
(2)
1kWik+ν

N
∑

k=1

a
(2)
ik
Wk1 =0 i=2, . . . ,N−1

λ2
N
∑

k=1

a
(3)
1kWik+(2−ν)

N
∑

k1=1

N
∑

k2=1

a
(2)
ik1
a
(1)
1k2
Wk1k2 =0 i=3, . . . ,N−2

(3.14)

λ2
N
∑

k=1

a
(2)
Nk
Wik+ν

N
∑

k=1

a
(2)
ik
WkN =0 i=2, . . . ,N −1

λ2
N
∑

k=1

a
(3)
Nk
Wik+(2−ν)

N
∑

k1=1

N
∑

k2=1

a
(2)
ik1
a
(1)
Nk2
Wk1k2 =0 i=3, . . . ,N −2

(3.15)



132 A. Krowiak

The function values at the plate edges parallel to the Y axis are defined by
the first equation from equation sets (3.12) and (3.13). The rest from (3.12)
and (3.13) are used to determine function values at points adjacent to the
boundaries in the Y direction as a linear combination of values at the interior
points.Thediscretisation of plate edges parallel to the X axis is similar.Using
function values at the boundary points and points adjacent to the boundaries
in Eq. (3.1) one have to solve a standard eigenvalue problem to obtain natural
frequencies of the plate.

Tabela 1. SS-F-SS-F plate – mesh of the Chebyshev-Gauss-Lobatto type

N Ω1 Ω2 Ω3 Ω4 Ω5

n=11

16 9.634 16.153 36.778 38.987 46.877
(0.031%) (0.112%) (0.142%) (0.108%) (0.297%)

20 9.632 16.139 36.740 38.957 46.778
(0.010%) (0.025%) (0.038%) (0.031%) (0.086%)

n=14

16 9.632 16.139 36.741 38.961 46.789
(0.010%) (0.025%) (0.041%) (0.041%) (0.109%)

20 9.631 16.136 36.729 38.948 46.748
(0.000%) (0.006%) (0.008%) (0.008%) (0.021%)

PDQM

16 9.631 16.135 36.726 38.945 46.738
(0.000%) (0.000%) (0.000%) (0.000%) (0.000%)

20 9.631 16.134 36.725 38.945 46.738
(0.000%) (−0.006%) (−0.003%) (0.000%) (0.000%)

The results obtained by the DQM based on spline functions of various
degrees n are presented in Table 1, where the grid described by Equation
(3.10) has been applied, and inTable 2, where the uniformgrid has been used.
The tables contain also the results obtained by the conventional differential
quadrature method (PDQM). When the uniform grid is used, the PDQM
shows computational instability. The application of too many nodes makes
the results very inaccurate. The percentage relative error

δ=
ΩSDQM −Ωreference

Ωreference
·100% (3.16)



Methods based on the differential quadrature... 133

Tabela 2. SS-F-SS-F plate – uniform grid

N Ω1 Ω2 Ω3 Ω4 Ω5

n=11

16 9.666 16.349 37.271 39.201 47.964
(0.363%) (1.326%) (1.484%) (0.657%) (2.623%)

20 9.651 16.254 37.039 39.147 47.508
(0.208%) (0 .738%) (0.852%) (0.519%) (1.647%)

24 9.643 16.208 36.923 39.088 47.242
(0.125%) (0.452%) (0.536%) (0.367%) (1.078%)

n=14

16 9.642 16.203 36.867 39.228 47.455
(0.114%) (0.421%) (0 .384%) (0 .727%) (1.534%)

20 9.636 16.165 36.809 39.072 47.084
(0.052%) (0.186%) (0.226%) (0.326%) (0.740%)

24 9.633 16.150 36.775 39.010 46.924
(0.021%) (0.093%) (0.133%) (0.167%) (0.398%)

PDQM

10 9.638 16.157 37.740 38.925 47.029
(0.067%) (0.136%) (2.761%) (−0.051%) (0.623%)

12 0 0 0 0 9.627
16 0 0 0 0 0

has been calculated for the frequencies on the basis of Leissa’s (1973) re-
sults, which are exact for the SS-F-SS-F plate. The remaining two sets of
Leissa’s results (C-F-SS-F, C-C-C-C) were obtained by the Rayleigh-Ritz
method with beam functions for the displacement, taking nine terms into
account.

The presented results show that the convergence rate of the SDQM is
satisfactory when the weighting coefficients determined from the high degree
spline functions are used. The accuracy can be improved by increasing the
number of grid points.

Analysing the results for the C-F-SS-F plate configuration (Table 3 and
Table 4), one can notice that the error of calculated frequencies does not de-
creasemonotonicallywhen the splinedegree is higher and the number of nodes
increases. It is especially noticeable when non-uniform grid (3.6) is applied to
insure high rate of convergence. The calculations for another plate with clam-
ped edges (Table 5 and Table 6) confirm that results obtained by the SDQM
and PDQM converge to lower values than the reference results obtained by



134 A. Krowiak

Leissa (1973). It seems that the results obtained by differential quadrature
methods are closer to exact values since the approximate solutions from the
Rayleigh-Ritz method are upper bounds on the exact values. It should be no-
ted that in the case of the SS-F-SS-F plate, where the reference results are
the exact solutions, the values from the differential quadrature methods are
in very good agreement.

Tabela 3. C-F-SS-F plate – mesh of the Chebyshev-Gauss-Lobatto type

N Ω1 Ω2 Ω3 Ω4 Ω5

n=8

16 15.434 21.682 43.083 50.286 58.753
(0.975%) (4.881%) (8.317%) (1.118%) (3.773%)

20 15.344 21.259 41.763 49.985 57.833
(0.386%) (2.835%) (4.998%) (0.513%) (2.148%)

24 15.297 21.043 41.099 49.821 57.346
(0.079%) (1.790%) (3.329%) (0.183%) (1.288%)

n=11

16 15.220 20.677 39.885 49.588 56.632
(−0.425%) (0.019%) (0.277%) (−0.286%) (0.026%)

20 15.201 20.617 39.793 49.497 56.404
(−0.550%) (−0.271%) (0.045%) (−0.469%) (−0.376%)

24 15.194 20.596 39.760 49.467 56.328
(−0.595%) (−0.372%) (−0.038%) (−0.529%) (−0.510%)

n=14

16 15.209 20.644 39.831 49.545 56.514
(−0.497%) (−0.140%) (0.141%) (−0.372%) (−0.182%)

20 15.197 20.604 39.771 49.477 56.353
(−0.576%) (−0.334%) (−0.010%) (−0.509%) (−0.466%)

24 15.193 20.591 39.750 49.458 56.304
(−0.602%) (−0.397%) (−0.063%) (−0.547%) (−0.553%)

PDQM

16 15.193 20.602 39.776 49.545 56.515
(−0.602%) (−0.343%) (0.003%) (−0.372%) (−0.180%)

20 15.190 20.585 39.748 49.484 56.372
(−0.622%) (−0.426%) (−0.068%) (−0.495%) (−0.433%)

24 15.191 20.582 39.738 49.464 56.320
(−0.615%) (−0.440%) (−0.093%) (−0.535%) (−0.525%)



Methods based on the differential quadrature... 135

Tabela 4. C-F-SS-F plate – uniform grid

N Ω1 Ω2 Ω3 Ω4 Ω5

n=8

16 16.114 25.226 52.439 55.282 65.627
(5.424%) (22.024%) (31.839%) (11.164%) (15.914%)

20 15.901 24.082 50.942 51.994 63.494
(4.030%) (16.490%) (28.075%) (4.553%) (12.147%)

24 15.773 23.395 48.721 51.416 62.185
(3.193%) (13.167%) (22.492%) (3.390%) (9.834%)

n=11

16 15.408 21.223 40.706 50.204 58.501
(0.805%) (2.660%) (2.341%) (0.953%) (3.328%)

20 15.333 21.002 40.359 50.011 57.777
(0.314%) (1.591%) (1.468%) (0.565%) (2.049%)

24 15.290 20.877 40.171 49.862 57.337
(0.033%) (0.987%) (0.996%) (0.265%) (1.271%)

n=14

16 15.359 21.023 40.252 50.221 57.951
(0.484%) (1.693%) (1.199%) (0.987%) (2.356%)

20 15.295 20.865 40.087 49.905 57.290
(0.065%) (0.929%) (0.784%) (0.352%) (1.189%)

24 15.260 20.778 39.987 49.749 56.953
(−0.164%) (0.508%) (0.533%) (0.038%) (0.593%)

PDQM

10 15.492 21.204 41.353 50.585 58.559
(1.354%) (2.569%) (3.967%) (1.719%) (3.430%)

12 0 0 15.406 21.051 27.976
16 0 0 0 0 0

4. Concluding remarks

The problem of free vibration analysis of plates has been undertaken by the
author inorder to examine the computational stability of theproposedmethod
on various grid distributions.

The presented results show that the convergence rate of the SDQM is high
when a high degree spline is used to approximate the solution. The accuracy
can be improved by increasing the number of grid points without concern for
losing stability of themethod.Unlike thePDQM, themethod is not limited to



136 A. Krowiak

Tabela 5. C-C-C-C plate – mesh of the Chebyshev-Gauss-Lobatto type

N Ω1 Ω2 Ω3 Ω4 Ω5

n=8

12 36.021 73.498 73.498 108.549 131.895
(0.081%) (0.116%) (0.116%) (0.258%) (0.194%)

15 35.995 73.422 73.422 108.308 131.653
(0.008%) (0.012%) (0.012%) (0.035%) (0.001%)

20 35.987 73.399 73.399 108.233 131.593
(−0.014%) (−0.019%) (−0.019%) (−0.034%) (−0.036%)

25 35.986 73.395 73.395 108.221 131.584
(−0.017%) (−0.025%) (−0.025%) (−0.045%) (−0.043%)

30 35.985 73.394 73.394 108.218 131.582
(−0.019%) (−0.026%) (−0.026%) (−0.048%) (−0.044%)

n=11

12 35.990 73.417 73.417 108.288 131.659
(−0.006%) (0.005%) (0.005%) (0.017%) (0.014%)

15 35.986 73.397 73.397 108.228 131.595
(−0.017%) (−0.022%) (−0.022%) (−0.039%) (−0.034%)

20 35.985 73.394 73.394 108.217 131.582
(−0.019%) (−0.026%) (−0.026%) (−0.049%) (−0.044%)

25 35.985 73.394 73.394 108.217 131.581
(−0.019%) (−0.026%) (−0.026%) (−0.049%) (−0.045%)

30 35.985 73.394 73.394 108.217 131.581
(−0.019%) (−0.026%) (−0.026%) (−0.049%) (−0.045%)

PDQM

12 35.986 73.399 73.399 108.230 131.418
(−0.017%) (−0.019%) (−0.019%) (−0.037%) (−0.169%)

15 35.985 73.394 73.394 108.217 131.580
(−0.019%) (−0.026%) (−0.026%) (−0.049%) (−0.046%)

20 35.985 73.394 73.394 108.217 131.581
(−0.019%) (−0.026%) (−0.026%) (−0.049%) (−0.045%)

25 35.985 73.394 73.394 108.216 131.581
(−0.019%) (−0.026%) (−0.026%) (−0.050%) (−0.045%)

30 35.985 73.394 73.394 108.217 131.581
(−0.019%) (−0.026%) (−0.026%) (−0.049%) (−0.045%)



Methods based on the differential quadrature... 137

Tabela 6. C-C-C-C plate – uniform grid

N Ω1 Ω2 Ω3 Ω4 Ω5

n=8

12 36.577 75.061 75.061 112.497 136.144
(1.625%) (2.245%) (2.245%) (3.904%) (3.421%)

15 36.308 74.271 74.271 110.668 133.710
(0.878%) (1.169%) (1.169%) (2.215%) (1.572%)

20 36.132 73.788 73.788 109.407 132.445
(0.389%) (0.511%) (0.511%) (1.050%) (0.612%)

25 36.065 73.608 73.608 108.888 132.029
(0.203%) (0.266%) (0.266%) (0.571%) (0.296%)

30 36.033 73.523 73.523 108.633 131.847
(0.114%) (0.150%) (0.150%) (0.335%) (0.157%)

n=11

12 36.109 73.680 73.680 109.419 128.917
(0.325%) (0.364%) (0.364%) (1.061%) (−2.069%)

15 36.037 73.602 73.602 108.887 131.528
(0.125%) (0.257%) (0.257%) (0.570%) (−0.085%)

20 36.002 73.475 73.475 108.466 131.795
(0.028%) (0.084%) (0.084%) (0.181%) (0.118%)

25 35.992 73.428 73.428 108.323 131.699
(0%) (0.020%) (0.020%) (0.049%) (0.045%)

30 35.988 73.410 73.410 108.267 131.642
(−0.011%) (−0.004%) (−0.004%) (−0.003%) (0.002%)

PDQM

10 36.021 72.996 72.996 108.348 130.891
(0.081%) (−0.568%) (−0.568%) (0.072%) (−0.569%)

12 35.992 73.474 73.474 78.331 78.331
(0%) (0.083%) (0.083%) (−27.651%) (−40.496%)

15 0 0 0 0 0

special typesof grids.Various typesof gridpointdistributionscanbeapplied in
theSDQMgivingdifferences only in the convergence rate of themethod.These
make the SDQMmore versatile than the PDQMand in author’s opinion it is
a good starting point to applying themethod tomore challenging engineering
problems, even nonlinear ones. In the latter, the high rate of convergence and
stability of the SDQM should be a great advantage.



138 A. Krowiak

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Metody oparte na kwadraturach różniczkowych w zastosowaniu
do drgań płyt

Streszczenie

Praca dotyczy metod opartych na kwadraturach różniczkowych i ich aplikacji
do zagadnienia drgań własnych płyt. W pracy, jako alternatywę do znanych me-
tod kwadratur różniczkowych, opartych na wielomianie interpolacyjnym (PDQM),
przedstawionometodę bazującą na funkcjach sklejanych (SDQM). W SDQM poszu-
kiwane rozwiązanie przybliżane jest funkcją wielomianową, przedziałami zmienną.W
pracy przedstawiono sposób wyznaczenia takiej funkcji interpolacyjnej, jak również
sposób obliczenia współczynnikówwagowych, używanychwmetodzie kwadratur róż-
niczkowych. Następnie SDQM użyto do wyznaczenia częstości drgań własnych płyt,
gdzie analizowano wpływ stopnia wielomianu, liczby węzłów i ich rozmieszczenia na
zbieżność, dokładność i stabilność metody. Otrzymane rezultaty porównano z wy-
nikami uzyskanymi przy pomocy konwencjonalnej metody kwadratur różniczkowych
(PDQM).

Manuscript received February 21, 2007; accepted for print April 4, 2007