The Journal of Engineering Research (TJER), Vol. 15, No. 1 (2018) 14-25

DOI: 10.24200/tjer.vol15iss1pp14-25

Vibrations of Circular Plates Resting on Elastic Foundation
with Elastically Restrained Edge Against Translation

L.B. Raoa,* and C.K. Raob

a School of Mechanical and Building Sciences, VIT University, Chennai Campus, Vandalur-Kelambakkam Road, Chennai-
600127, Tamil Nadu, India.

b Department of Mechanical Engineering, School of Engineering, Chaudari guda, Korremulla ‘X’ Roads, Near Narapalli,
Ghatkesar (M), Hyderabad-500 088, India.

Received 6 July 2016; Accepted 26 January 2017

Abstract: The present paper deals with exact solutions for the free vibration characteristics of thin
circular plates resting on Winkler-type elastic foundation based on the classical plate theory
elastically restrained against translation. Parametric investigations are carried out for estimating the
influence of edge restraint against translation and stiffness of the elastic foundation on the natural
frequencies of circular plates. The elastic edge restraint against translation and the presence of elastic
foundation has been found to have a profound influence on vibration characteristics of the circular
plate undergoing free transverse vibrations. Computations are carried out for natural frequencies of
vibrations for varying values of translational stiffness ratio and stiffness parameter of Winkler-type
foundation. Results are presented for twelve modes of vibration both in tabular and graphical form
for use in the design. Extensive data is tabulated so that pertinent conclusions can be arrived at on the
influence of translational edge restraint and the foundation stiffness ratio of the Winkler foundation
on the natural frequencies of uniform isotropic circular plates.

Keywords: Plate; Frequency; Elastic edge; Translational stiffness; Elastic foundation.

اثناء احلركةاملرونة اهتزازات االلواح الدائرية املثبتة على قاعدة مرنة مع حافة مقيدة

ب، تشيالبيال كامسوارا راو*،أ لوكافارابو بهاسكارا راو

املثبتة على قاعدة قيقةلاللواح الدائرية الر ةاحلر اتخلصائ االهنزاز صحيحةالبحثية احللول ال : تتناول الدراسة لص امل

معاملية حباث ا. مت إجراء لتقليديةا لواحمبوجب نظرية االاحلركة اثناءرونة امل مقيدة ذات حافة مرنة من نوع وينكلر

و الدائرية. لأللواحصالبة القاعدة املرنة على الرتددات الطبيعية و احلركة اثناءيد احلافة يتأثري تق مدىلتقدير (بارامرتية)

مع وجود القاعدة املرنة هلا تأثري كبري على خصائ اهتزاز ركةاحلعملية أثناءاىل أن احلافة مقيدة املرونة توصل البحث

يعية لالهتزازات لقيم متفاوتة من تعرض الهتزازات عرضية حرة. ومت إجراء احلسابات للرتددات الطبيوح الدائري اليت لال

اتاالهتزاز أمناطالنتائج الثين عشر منطا من و قدم البحثمعدل الصالبة املتعدية ومعامل صالبة القاعدة من نوع وينكلر.

إىل منها حبيث ميكن التوصل شاملةيف شكل جداول و رسومات بيانية الستصدامها يف التصميم. كما مت جدولة بيانات

نوع وينكلر على الرتددات الطبيعية للوحات من استنتاجات ذات صلة بتأثري تقييد احلافة املتنقلة ونسبة صالبة القاعدة

الدائرية املتماثلة املوحدة.

قاعدة مرنة ،نقلةصالبة مت ،حافة مرنة،الرتدد ،: لوح فتاحيةملا كلماتال

* Corresponding author’s e-mail: bhaskarbabu_20@yahoo.com

mailto:bhaskarbabu_20@yahoo.com

L. B. Rao and C.K. Rao

Nomenclature

h Thickness of a plate, mm

a Radius of a plate, mm

 Poisson’s ratio

E Young’s modulus, N/mm2

 Density of a material, kg/mm3

),( rW Transverse deflection of the plate, mm

D Flexural rigidity of a plate, N.mm2

T
K Translational spring stiffness, N/mm

w
K Stiffness of Winkler foundation, N/mm2/mm

T Translational spring stiffness ratio

 Foundation stiffness ratio

mn
 Natural frequency, rad/sec

mn
 Eigenvalue without foundation, cycles/sec

mn
 Eigenvalue with Winkler foundation, cycles/sec nm, positive integers correspond-

ing to the number of concentric circles and nodal diameters in each flexural mode.

Vibrations of Circular Plates Resting on Elastic Foundation with Elastically Restrained Edge Against Translation

1. Introduction

Circular plates resting on elastic foundation
have a wide range of applications in the static
and dynamic design of linear/nonlinear
vibration absorbers, dynamic exciters, telephone
receiver diaphragms, computer discs, printed
circuit boards etc. (Leissa 1969). Due to the
essential use of vibration data in the
computation of stresses in such structures,
reliable prediction of vibration data is of great
importance. In view of its importance in
engineering design, the problem of vibration of
circular plates on elastic foundation has
attracted the focus and attention of many
researchers.
Some of the recent studies have reinstated
the classical approach efficiency in analyzing
the vibrations of variety of structures. Circular
plate problems allow for significant
simplification in view of their symmetry, but
still many difficulties arise when the plate
boundary conditions become complex involving
linear and rotational restraints. A recent review
of literature shows that very few studies exist on
the study of circular plates resting on elastic
foundations. Wang and Wang (2003), who
observed the switching between axisymmetric
and asymmetric vibration modes, have recently
investigated the effect of internal elastic
translational supports.
The vibration characteristics of plates resting
on an elastic medium are different from those of
the plates supported only on the boundary.
Leissa (1993) discussed the vibration of a plate
supported laterally by an elastic foundation.
Leissa deduced that the effect of Winkler
foundation merely increases the square of the
plate natural frequency by a constant. Salari et
al. (1987) speculated the same conclusion.

Ascione and Grimaldi (1984) studied unilateral
frictionless contact between a circular plate and
a Winkler foundation using a variational
formulation. Leissa (1969), who tabulated a
frequency parameter for four vibration modes
of a simply supported circular plate with
varying rotational stiffness, presented one of the
earliest formulations of this problem. Kang and
Kim (1996) presented an extensive review of the
modal properties of the elastically restrained
beams and plates.
Zheng and Zhou (1988) studied the large
deflection of a circular plate resting on Winkler
foundation. Ghosh (1997) studied the free and
forced vibration of circular plates on Winkler

foundation by an exact analytical method.
(Chang and Wickert (2001); Kim et al. (2001) and
Tseng and Wickert (1994) studied the dynamic
characteristics of bolted flange connections
involving circular plates displaying beating type
of repeat frequencies and typical mode shapes
of vibration. Bolted flange connections are
practically the best examples for the elastically
restrained boundary conditions of circular
plates on partial or continuous Winkler type
elastic foundation.
The most general soil model used in practical
applications is the Winkler (1867) model in
which the elastic medium below a structure is
represented by a system of identical but
mutually independent elastic linear springs.
Recent investigations have reiterated the
efficiency of the classical approach (Soedel 1993)

in analyzing the behavior of structures under
vibrations. There are other works (Weisman
1970; Dempsey et al. 1984; Celep et al. 1988)
dealing with the study of plates on a Winkler
foundation. In general, those dealing with
vibrating plates, shells and beams are concerned
with the determination of eigenvalues and
mode shapes (Leissa 1969).
A good number of studies was conduced
(Wang and Lin 1996; Kim and Kim 2001; Yayli
et al. 2014) using the method of Fourier series for
estimating the frequencies of beams with
generally restrained end conditions including
the effect of elastic soil foundation. The method
includes the use of Stoke’s transformation in
suitably modifying the complex boundary
conditions. Very much similar to the dynamic
stiffness matrix approach, the elements of the
matrix involving infinite Fourier series are
explicitly obtained in these studies. The
determinant of this matrix for each case
considered leads to the frequency equation and
the same can be solved using well known
numerical methods. The results obtained for
various elastically restrained beam cases in
these studies tallied well with those available in
the literature establishing the efficiency of this
method.
In view of the necessity of using complex
combinations of rotational and translational
springs at the circular plate boundary to
suitably simulate the practical non-classical
boundary connections being adopted in a wide
range of industrial applications (Bhaskara Rao et
al. 2009; Bhaskara Rao et al. 2010; Lokavarapu
and Chellapilla 2013; Bhaskara Rao et al. 2015;
Lokavarapu et al. 2015; Rao et al. 2016), the use
of exact method of solution becomes imperative

L. B. Rao and C.K. Rao

and hence the same is adopted in this paper.
Even though the method adopted here is
classical, the particular case of vibration of
elastically restrained circular plate resting on
elastic foundation considered here is not
addressed within the available literature.
Utilizing the classical plate theory, this paper
deals with an exact method of solution for the
analysis of thin circular plate free transverse
vibrations that is elastically restrained against
translation and resting on Winkler-type elastic
foundation. For estimating the influence of edge
restraint against the elastic foundation
translation and stiffness on circular plates
natural frequencies, parametric investigations
are carried out by varying the values of elastic
edge restraint stiffness against the elastic
foundation translation and stiffness. The results
obtained on natural frequencies of vibration
clearly show that the vibration characteristics of
the circular plate undergoing free transverse
vibrations are found to be profoundly
influenced by these variations. Computations
are carried out for natural frequencies of
vibrations for varying values of translational
stiffness ratio and stiffness parameter of
Winkler-type foundation. The results that are
presented for twelve modes of vibration both in
tabular and graphical forms are believed to be
quite useful for designers in this area.

2. Mathematical Formulation of the
System

The considered elastic thin circular plate is
supported on a Winkler foundation as shown in
Fig. 1. In the classical plate theory (Leissa 1969),

the following fourth order differential equation
describes free flexural vibrations of a thin
circular uniform plate:

0
2

t/)t,,r(w
2

h)t,,r(w
4

.D  (1)

where )
2

1(12/
3

EhD  is the flexural

rigidity of a plate and  ,E,,h,a are the plate’s

radius, thickness, density, Young’s modulus
and Poisson’s ratio, respectively.
The homogeneous equation for Kirchhoff’s

plate on one parameter elastic foundation is

given by the following equation:

 )t,,r(wwK)t,,r(w
4

.D (2)

2
t/)t,,r(w

2
h  =0

Displacement in (2) can be presented as a

combination of spatial and time dependent

components as follows:

Let
ti

erWtrw


 ),(),,(  (3)

Now substitute (3) in (2)

0),r(W).
2

hwK(),r(W
4

.D   (4)

The solution of the equation takes the
following form






 ncos.

rmn

nImnC
a

rmn
nJ

mnA),r(mnW







































 , n > 0

(5)

where mnA and mnC are constants, nJ is Bessel

function of the first kind of first order and nI is

modified Bessel function of the first kind of first
order, indexes m and n are positive integers
and correspond to the number of concentric
circles and nodal diameters in each flexural
mode. Considering an elastically supported
plate as shown in Fig. 1, boundary conditions
can be formulated at ar  , in terms of
translational stiffness ( TK ) as follows:

0),a(rM  (6)

    ,aWTK,arV  (7)

where the Kelvin-Kirchhoff and bending
moment are defined as follows:

Figure 1. A thin circular plate with translational
elastic edge restraint and supported
on elastic foundation.

Vibrations of Circular Plates Resting on Elastic Foundation with Elastically Restrained Edge Against Translation






















































),a(W
2

2
r

),a(W

2
r

),a(W
2

.D),a(rM






 (8)

























































),a(W

2
r

),a(W
2

1
)1(),a(W

r
.D),a(rV

(9)

By applying Eqs. (6) and (8), we obtain the

following equation

0
2

),a(W
2

2
r

),a(W

2
r

),a(W
2

















































(10)

From Eqs. (5) and (10), we derive the following

equation

)(
4

2
2

)(
4

2
2

mnn

I
n

J
n















































(11)

where
mn

P = )mn(1nJ)mn(1nJ   ;

)mn(2nJ)mn(2nJmnQ   ;

)mn(1nI)mn(1nImnS   ;

)mn(2nI)mn(2nImnT   ;

From Eqs. (7) and (9), we get the following

),a(W.TK
),a(W

2
r

),a(W
2

1
)1(),a(W

r
.D 


























































(12)

From Eqs. (5) and (12), we derived the

following equation

)mn(nIT
3

3
mn

2
n)3(8

mnS
2

2
n)2(44

3
mn

mnT2
mnU

)mn(nJT
3

2
n)3(8

mnP
2

2
n)2(44

3
mn

mnQ2
mnR

mnC


























































































 


 (13)

where,
D

Ka
T T



)()(
11 mnnmnnmn

JJP 


 ; )mn(2nJ)mn(2nJmnQ   ; )()( 33 mnnmnnmn JJR    ;

)()(
11 mnnmnnmn

IIS 


 ; )mn(2nI)mn(2nImnT   ; )()( 33 mnnmnnmn IIU    ;

If 
T

K then this case becomes a simply

supported boundary condition as shown in Fig.
2. The frequency equation can be calculated
From Eqs. (11) and (13), which allows

determining eigenvalues
mn

 . The mode shape

parameters mnC can be determined corres-

ponding to these eigenvalues by using either Eq.
(11) or Eq. (13). The amplitude of each vibration
mode in Eq. (5) is set by the normalization
constant mnA determined from the following

condition.

nqmpmnpq

mn
MrdrdrWrW 



 ),().,(
2

0 0

(14)

where,
mn

M is a mass of the plate,

1
nqmp

 if qnpm  , and nqmp =

0 if m ≠ p or n ≠ .q

Figure 2. A simply supported thin circular plate

resting on elastic foundation.

L. B. Rao and C.K. Rao

The normalization constant
mn

A can be
derived using Eqs. (5) and (14) as given below:

rdrd

ncos.

rmn
nI.mnC

rmn
nJ

.
2

1
mnA

























 





































































(15)

In Eq (4), the natural frequency is defined as

mn




























2
a

2
mn




(16)

It is clear from Eq. (16) that the natural

frequency parameter
mn

 is dependent on the

plate radius ‘a’.

From Eq. (16) we can express

2
mn

4
ha4

mn


  (17)

244*

 
mnmn

(18)

where
D

aK
w

4
2
 (19)

 4
1

24*
 

mnmn
(20)

3. Solution

Using Matlab programming, computer

software with symbolic capabilities, solves the

above set of equations. The program determines

eigenvalues ( *mn ), for a given range of

boundary conditions. The boundary’s linear

translational non-dimensional restraint

parameter can be defined as follows:

aK
T T

 (21)

aK
w

4
2
 (22)

The following represent the input parameters

to the program; (i) Translational stiffness ratio

( T ); (ii) Foundation ratio (  ); (iii) Poisson’s

ratio ( ); (iv) Upper bound for eigenvalues

( N ); (v) Suggested accuracy for eigenvalues

( d ); (vi) Number of mode shape parameters

( n );. The program finds eigenvalues *mn by

using Matlab root finding function.

4. Results and Discussion

The code developed is used to determine

eigenvalues of any set or range of translational

and foundation constraints. This code is also

implemented for various plate materials by

adjusting Poisson’s ratio. Such a wide range of

results is not available in the literature. The

eigenvalues for the plate edge, which is

elastically restrained against translation and

fully resting on the elastic foundation, at various

values of the translational stiffness ratios, are

computed and the results are given in Table 1.

The effects of the translational stiffness ratios

are plotted in Fig. 3. As seen from Fig. 3,

eigenvalues increase with an increment in the

translational stiffness ratio, and the plates

become unstable in the region when the

translational stiffness ratio exceeds a certain

value. Twelve vibration modes are presented in

Fig. 3. The smoothened stepped variation is

observed in Fig. 3. The stepped region increases

with increase in translational stiffness ratio and

vibration modes. The location of the stepped

region with respect to T changed gradually

from the range of 0.01526 † [9.9997] ‡ – 5587.5316

[10] to 16.62296 [14.6739] – 611824.96917

[16.75055]. Here †.represents translational

stiffness ratio and ‡ represents Eigen values

throughout the text. Here the value in the

bracket represents eigenvalue. The simply

supported boundary conditions (Fig. 2) could be

accounted for by setting ( 
T

K ) shown in

Fig. 1. The frequency in this case is 2.23175 and

this is in good agreement with the results

published by Wang (2005). Another result,

considered for comparison, is from Rao and Rao

(2009) on a study of the case of vibrations of

elastically restrained circular plates supported

on partial Winkler foundation. When the

Table 1. Eigenvalues for different Translational stiffness ratio for  =100 & =0.33.

log10 𝑇
𝜆00

𝜆10 𝜆20 𝜆01 𝜆11 𝜆21 𝜆02 𝜆12 𝜆22 𝜆03 𝜆13 𝜆23

-3 10 10.0205 10.3517 10.10358 10.79568 12.46774 10.29701 11.43857 13.52984 10.63672 12.2505 14.67332

-2 10 10.02051 10.35171 10.10359 10.79568 12.46775 10.29702 11.43857 13.52985 10.63673 12.25051 14.67332

-1 10.00005 10.02061 10.3518 10.10368 10.79576 12.46779 10.29711 11.43864 13.52989 10.63682 12.25056 14.67335

0 10.00047 10.02162 10.35263 10.10462 10.79649 12.46827 10.29803 11.43927 13.53026 10.63769 12.25109 14.67365

1 10.003 10.03287 10.36125 10.11468 10.804 12.47302 10.30745 11.44568 13.53403 10.64649 12.25645 14.67667

2 10.00568 10.1199 10.46887 10.2319 10.89095 12.52351 10.42003 11.5759 13.57333 10.74585 12.31326 14.70772

3 10.00614 10.203 11.00421 10.49407 11.61576 13.1793 10.91134 12.27823 14.08459 11.41839 12.99719 15.0966

4 10.00619 10.21322 11.14468 10.53794 11.94459 14.15323 11.03916 12.88272 15.33948 11.70347 13.90786 16.53545

5 10.00619 10.21423 11.15725 10.54212 11.97273 14.2438 11.05093 12.93527 15.47979 11.72934 13.99516 16.74092

6 10.00619 10.21433 11.15849 10.54254 11.97546 14.25215 11.05209 12.94027 15.49231 11.73186 14.00329 16.75866

12 10.00619 10.21434 11.15862 10.54258 11.97576 14.25307 11.05222 12.94082 15.49369 11.73214 14.00418 16.7606

2
0

V
ib

ration
s of C

ircu
lar P

lates R
estin

g
on

E
lastic F

ou
n

d
ation

w
ith

E
lastically

R
estrain

ed
E

d
g

e A
g

ain
st T

ran
slation

Table 2. Eigenvalues for different Foundation stiffness ratio for T =100 &  =0.33.

Table 3. Eigenvalues for different Translation and Foundation stiffness ratios for  =0.33.

log10 𝜉

log10 𝑇

𝜆00

𝜆10 𝜆20 𝜆01 𝜆11 𝜆21 𝜆02 𝜆12 𝜆22 𝜆03 𝜆13 𝜆23

-3 0.2115 3.0115 6.2054 4.52915 7.73687 10.9091 5.93655 9.18564 12.3826 7.27469 10.57846 13.80849

-2 0.37648 3.01187 6.20544 4.52925 7.73689 10.90911 5.93659 9.18566 12.38261 7.27471 10.57846 13.8085

-1 0.67602 3.0156 6.20584 4.53032 7.73709 10.90918 5.93708 9.18578 12.38266 7.27499 10.57855 13.80854

0 1.30374 3.067 6.21074 4.54336 7.73963 10.91008 5.94302 9.18733 12.38328 7.27834 10.57959 13.80899

1 3.25317 3.90325 6.34932 4.87899 7.81237 10.93614 6.10341 9.23131 12.40119 7.36838 10.60869 13.82199

2 10.00568 10.1199 10.46887 10.2319 10.89095 12.52351 10.42003 11.5152 13.57333 10.74585 12.31326 14.70772

3 31.62297 31.62939 31.65958 31.63958 31.68744 31.78104 31.65573 31.72292 31.85232 31.67796 31.76831 31.94926

Log10


𝜆00

𝜆10 𝜆20 𝜆01 𝜆11 𝜆21 𝜆02 𝜆12 𝜆22 𝜆03 𝜆13 𝜆23

-3 2.18341 4.70075 6.69703 5.56683 7.98678 10.99197 6.50356 9.33292 12.43922 7.59879 10.67534 13.84974

-2 2.18341 4.70075 6.69703 5.56683 7.98678 10.99197 6.50356 9.33292 12.43922 7.59879 10.67534 13.84973

-1 2.18365 4.70077 6.69704 5.56684 7.98679 10.99197 6.50357 9.33292 12.43922 7.59879 10.67535 13.84974

0 2.20704 4.70315 6.69786 5.56827 7.98727 10.99216 6.50447 9.33323 12.43935 7.59936 10.67555 13.84983

1 3.3284 4.92488 6.77875 5.7064 8.03541 11.01074 6.5926 9.36352 12.45218 7.65514 10.69583 13.84914

2 10.00568 10.1199 10.46887 10.2319 10.89095 12.52351 10.42003 11.5159 13.57333 10.74585 12.31326 14.70772

3 31.62296 31.62664 31.63867 31.63037 31.6549 31.73756 31.63691 31.68259 31.81038 31.6491 31.72496 31.90972

L
. B

. R
ao an

d
C

.K
. R

2
1

Vibrations of Circular Plates Resting on Elastic Foundation with Elastically Restrained Edge Against Translation

support position is in full span, which means

that when 1b , the case becomes a circular

plate having full foundation support with

elastically restrained edge against translation.

For this case, the frequency is 2.1834 and that is

in good agreement with the frequency of

2.18341 obtained from the present study.

The eigenvalues at various values of the
foundation stiffness ratios for

33.0&100  T are computed and the
results are given in Table 2. The effects of the
foundation stiffness ratio on eigenvalues are
plotted in Fig. 4. As seen from Fig. 4, the
eigenvalue increases with increase in the
foundation stiffness ratio, and the plate becomes
stiffer and stronger as the value of foundation
stiffness becomes greater than 102. As seen from
the Tables 1 and 2, the influence of foundation
stiffness ratio on eigenvalue is relatively greater
than that of the translation stiffness ratio in
increasing the overall natural frequencies of the
plate support system. As seen from Fig. 4, for all
the modes considered here, up to a value of 10
the eigenvalues stay constant and beyond this
value all the curves tend to converge to a
constant eigenvalue as the foundation stiffness

ratio increases up to 103. The convergence starts
from 1.07897 [2.0325779] and continues up to a
constant value of 9.63274 [13.84796].
The eigenvalues at various values of the

translational stiffness ratios and foundation

stiffness ratios are computed and the results are

given in Table 3. The effects of the translation

and foundation stiffness ratios on eigenvalues

are clearly observed in Fig. 5, eigenvalues

increases with an increment in both the

translational and foundation stiffness ratios. As

observed from the Table 1 and 3, the influence

of foundation stiffness ratio on eigenvalue is

more predominant than that of translation

stiffness ratio alone. From the results presented

in Tables 1 and 3, we can see that the influence

of foundation stiffness ratio on eigenvalues is

more predominant than that of translation

stiffness ratio. From the results given in Tables 1

to 3, one can easily find that the eigenvalues

become lower for lower values of foundation

and translation stiffness ratios. As seen from

Fig. 5, all the curves are stable up to a certain

region beyond which the curves tend to

converge for increasing values of translation

and foundation stiffness ratios.

Figure 3. Effect of translational stiffness ratio T on eigenvalues,

mn
 .

L. B. Rao and C.K. Rao

Figure 4. Effect of foundation stiffness ratio,  on eigenvalues, .mn

Figure 5. Effect of translational, T and foundation,  stiffness ratios on eigenvalues, .mn

Table 4. Eigenvalues for different Poisson ratios.

ν 1000T 10,100  T 1000,10  T 10,1  T 50,50  T

0 31.62925 4.92456 10.03027 3.62553 7.28924

0.1 31.62929 4.92466 10.03111 3.64904 7.2896

0.2 31.62934 4.92476 10.0319 3.6708 7.28993

0.3 31.62938 4.92485 10.03265 3.69097 7.29025

0.4 31.62942 4.92494 10.03336 3.70974 7.29056

-4 -3 -2 -1 0 1 2 3 4

E
ig

e
n

v
a
lu

e
s
,
λ

m
n

Foundation parameter,Log10 ξ

λ00

λ10

λ20

λ01

λ11

λ21

λ02

λ12

λ22

λ03

λ13

λ23

Vibrations of Circular Plates Resting on Elastic Foundation with Elastically Restrained Edge Against Translation

Figure 6. Effect of Poisson ratio,  on eigenvalues, .mn

5. Conclusion

This work deals with a method of computation

of eigenvalues of flexural vibrations of a circular

plate with translational edge supported and

resting on Winkler foundation using a

specifically written MATLAB code. The

computed numerical results are presented in a

tabular format to enable estimating the accuracy

of approximate methods being used by other

researchers for solving such problems. Two-

dimensional plots of eigenvalues are drawn for

a wide range of translational and foundation

stiffness ratios facilitating their use in design.

From the numerical and graphical results

presented in this paper, it can be easily observed

that the eigenvalues remain constant only for a

limited range of constraints specific to each

vibration mode and then steeply increase with

the increasing values of foundation stiffness

ultimately converging towards a constant value.

It is also observed that the influence of

foundation stiffness ratio on eigenvalues is

more predominant than that of translational

stiffness ratio.

Conflict of Interest

The authors declare no conflicts of interest.

Funding

No funding was received for this research.

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