CC BY
21DOI: 10.17516/1999-494X-0213 yflK 517.442+519.677
Assessment of Hybrid Method on Investigation of Dynamic Behaviour of isotropic Rectangular Plates Resting on Two-Parameters Foundation
Saheed Salawu and Gbeminiyi Sobamowo*
University of Lagos Akoka, Nigeria
Received 07.04.2019, received in revised form 01.11.2019, accepted 21.01.2020
Abstract. Dynamic behaviour of isotropic rectangular plate resting on two-parameter foundation is investigated. The governing partial differential equation is transformed to ordinary differential equation due to Galerkin method of separation. The hybrid method of Laplace transform and variation parameters method is used to analyze the ordinary differential equation. Introduction of exact method helps in fast convergence of the results. Obtained analytical solutions are compared with existing literature and confirmed as accurate. They are used to examine the effect of controlling parameters on the plate natural frequencies. Due to obtained results it is obvious that, the increase of both elastic foundation parameter and aspect ratio results in increasing the natural frequency. The solution is found immediately by means of a few iterations.
Keywords: dynamic analysis, natural frequency, deflection, Winkler and Pasternak, Laplace and variation parameters method.
Citation: Salawu S., Sobamowo G. Assessment of hybrid method on investigation of dynamic behaviour of isotropic rectangular plates resting on two-parameters foundation, J. Sib. Fed. Univ. Eng. & Technol., 2020, 13(2), 162-174. DOI: 10.17516/1999-494X-0213
© Siberian Federal University. All rights reserved
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0). Corresponding author E-mail address: safolu@gmail.com, mikegbeniyi@gmail.com
Оценка гибридного метода исследования динамического поведения изотропных прямоугольных пластин,
опирающихся на двухпараметрическое основание
Сахид Салаву, Гбеминийи Собамово
Университет Лагоса Нигерия, Акока
Аннотация. Исследовано динамическое поведение изотропной прямоугольной пластины, опирающейся на двухпараметрическое основание. Основное дифференциальное уравнение в частных производных преобразуется в обыкновенное дифференциальное уравнение с использованием метода разделения Галеркина. Для анализа обыкновенного дифференциального уравнения применен гибридный метод преобразования Лапласа и параметров вариации. Введение точного метода помогает в быстрой конвергенции результатов. Полученные аналитические решения сравниваются с существующей литературой и подтверждаются с хорошей точностью. Их используют для изучения влияния управляющих параметров на собственные частоты пластины. Из полученных результатов видно, что рост параметров упругого основания и соотношения сторон увеличивает собственную частоту. Решение находят быстро с несколькими итерациями.
Ключевые слова: динамический анализ, собственная частота, отклонение, Винклер и Пастернак, метод параметров Лапласа и вариации.
Цитирование: Салаву, С. Оценка гибридного метода исследования динамического поведения изотропных прямоугольных пластин, опирающихся на двухпараметрическое основание / С. Салаву, Г. Собамово // Журн. Сиб. федер. ун-та. Техника и технологии, 2020. 13(2). С. 162-174. DOI: 10.17516/1999-494Х-0213
1. Introduction
Recently the significant and wide application of thin rectangular plates in mechanical, civil, marine, naval, nuclear and aeronautic engineering, have increased various research interests to the study of dynamic analysis of thin rectangular plate. In the study of free vibration of rectangular plate of varying thickness Sundara et al. [1] applied finite element method. In another work, Cheung and Kong [2] used finite element method for analyzing dynamic response of rectangular plate of varying thickness. However, research into plate resting on two-parameter elastic foundation has gained much attention among researchers based on the various publications available due to the subject, importance and application in various branches of engineering. Singh and Adhikari [3] adopted finite element method in determining the dynamic response of functionally graded plate on two-parameter foundation. Finite element method is applied to analyze the vibration of rectangular plates resting on elastic foundation by Karasin [4]. In another study, Zenkour and Radwar [5] used hyperbolic shear deformation in studying of functionally graded plate resting on Winkler and Pasternak foundation.
Based on previous studies on vibration, it is realized that sorting out vibration problem is difficult to handle due to inherent non-trivial solution. Meanwhile, the most adequate method of solution for nonlinear problem is numerical, but this one has restrictions and limitations. The numerical method is not able to provide the closed form solutions. Exact analytical approximate
method is also limited to handle linear problem. While semi - analytical method is able to correct the limitation in both methods. However, to obtain closed form solution, Vgor and Eisen [6] adopted semi-analytic Kantorovich method for determining free vibration response of varying thickness rectangular plate. In a further study, Mustapha and Ajetumobi [7] utilized variation iteration method for solving some problems in vibration. Also, Attarnejad et al. [8] applied differential transform method for analyzing Timoshenko beam on two-parameter foundation. Variation of parameter method (VPM) [9-12] is a very reliable method of getting closed form solution without the restriction of small parameters in Homotopy perturbation method, conversion of governing equation to recursive relation in differential transform and calculating the langrage multiplier in variation of iteration method. By means of a few iterations the solution has been found. Combining Laplace transform and Variation of parameter methods means that, the effectiveness of the method increases further because of the combination of exact method for linear part of the equation and VPM for the other part of the governing equation.
After detail study of the literature review, the authors found out that no attention has been drawn to application of hybrid VPM to investigate the dynamic analysis of isotropic rectangular plates resting on Winkler and Pasternak foundations. Therefore, the present study focuses on the application of hybrid VPM dynamic investigation of isotropic thin rectangular plate resting on two-parameter nonlinear foundation. Obtained results are used for parametric investigations.
2. Problem formulation and mathematical analysis
Rectangular plate of uniform thickness and uniform density is considered. The plate is resting on combine linear Winkler and nonlinear Pasternak foundation under different edge conditions as shown in Fig. 1.
The following assumptions are made for the development of the governing equation [13]:
1) Plate is a member whose middle surface lies in plane.
2) Thickness of plate is smaller compared to the other dimensions.
3) Plate is of constant thickness.
4) Thickness is normal to the mid-surface plane.
The governing equation for thin isotropic rectangular plate as reported by [14] is;
( 94w(x, v,t) 94w(x, v,t) 34w(x, y,tn d2w(x, v,t) , . ,, .
* l—/-1+2wJ+—^j - kww (xy,^^ ^=0,
Eh3
where D is the flexural rigidity —-.-E is the modulus of elasticity, h is the plate thicknets, v is
12(1-k2 )
the Poissoh's ratio, Plate is resting on kw, kp Winkler foundation and Pasternak foundation respectively and p 2s 1he de,ooty. Offering a further general solution, the following dimensionless variables are defined:
w x n
W =-, X = -, Y = -. (2)
wmax a b
Forfree v ibration,the sb lution of Eq. (1) can be presented in Kantorovich type approximation
w (x,y,t) = w( x, y) em. (3)
0______________1
Fig. 1. Platerestingon two-parameter foundation
2 __ a4ph 2 ; _ct, __ a kpwrt
Ahrc.O^^W.k = ^=,k =—^^, X
D o D 2 D
d4W(x,y) 7d4W(x,y) c d(W(x,o) , / , , x 3/ a
-+ 21-x-n--x-kl -n2IV(x,y) + kwIV(x,y) + kIV3(x,y) = 0. (5)
dX4 dX2XY2 dY4 v ' o v '-1' p v '
A-urning tlie two 2pposite edges of Fig. 1, 7 = 0 and Y = 1 to be sioply suppoxtYd, keflection function can be repreoented ys follYws
W = W{ X (6)
OubsSituting tne der2^?etiveof Ec. (6) inCo gcveroing differential Eq. (5) gives:
YWgXt 2 2 tgtwex) i 2 r r D x , 3/ ,
-go - 2(0 m2 e2-y-r~r - (a2 -xw - ( W ejW gX) + XpW3 gX) u 0. (7)
dX dX)
2.1. Boundcry Condttions
Three bonodasy cogditiond C0mbinaii2ns are conssd^oey at X =0 and X = l. For brevity sake, notations are adgpted in identifying the odge conditions nmpy sup°ortYd aod Qamped edgx --C(, Simply sogported ard 0imily Suppor4eii )SS) an- -imidly ^up^oried axd orei i;dLt;c conditXns (SF).
rdW
dlompgd ed^: W = — = X. WW)
dx
Simpfy í5ulii(ostí;d c:(dg)(s: W = 12--vtWPe2) W = 0. (9)
m2
Free °doe: dm=W - v(n2Pel2-W = 0, eg-—s-ii -v-l m2Pm2- —do. (10)
o dX21 ; dX3 0 W ' dX
S. Pee^lloti ofsolution: li^iolace transfor2n ond goriatisn parameter method
3.X. Basic ideal of Lcplcce Ircnsi'idrm
lfdLi is a functionof a variable l. X j-7-2)}and m defined Xy th0 integral:
X ^X-^/)}^ f (d)e,jVdl F{d)dt. (11)
0
Someofthe propertiesusedinthtsstudyinclude:
• JT {1}o I (s >00), (12)
n d
• Zitn} = (s > 0), t)3)
• X{F{"\t)] =s"f(s)-s"-lF(0)-s"-2F'(0)----F'"-l\0), (14)
where Ln)(t) (epresents> the n-th derivative of F(t) and ri{F(t)} = f (s). If Laplace transform of F(t) is f(s), then the inverse Laplace transform off(s) is expressed by F(t) = L {f (s)}, where L is called inverse Lapiece operntop.
The inverse Laplace of Eqs. (12) and (13) are:
• 1 e L (, (15)
• tn = Z (^J. (16)
3.2.Basic ideal variation of parameter method
To Xemгnsnrate the principle of variation parameter method (VPM), this nonlinear differentia term is considered,
im
+ R( x) + N (x) = f (x). (17)
Wift thR mitiN cxndifiox,
wn)(0 x d wq , (181
where x = w(x\ R w«jjxi^^isent xhe lineae operator, N is the nonlinear operator, f(x)represent the no^
homone nxus tewm. is the highest order derivative.
w dxm
d^e{e) + Rw(x) + Nw(x) = g, (19)
L denotes highest linear operator, R the remaining linear operators aside the highest linear, N is the nonlinwar o p erato x
Wn+1 (x) = nг Cxc) + | il(x, ¿;) CR(wnrZ+N^^ - f (Z))d¿;, (202
0
ws(x-) = wwii) -(xw'(0) x) ^ztw"(tx^ + ^ w"'i0) + hs^«(0), (21)
2! 3! n-w
e f—wi n^d-,Xe-0
X is a general Wronskian technique A = Vn—n-, and n represents nth approximation, wn(x)
^ (i-1)!(e-¡')!
represent the solution of the linear operator. e in the Wronskian represent the order of the derivative. Having obtained the initial iteration wn(x) and the Wronskian, subsequent iterations is performed using Eq. (20).
3.u. Basic Concept cud Ike Proueduoe oYLap)cco Vaoiation Patametor Metho(
Yluo append helps in the rate 0i) 00300(0(00e, 0nc tdc linear p art of the operator is solved using exact Fatldoa o( 1ha tohttian. paole^ taap^^orn^ )f governing eqaotio n is taking as;
o^ a)=os) + ] ¿to, a aYOo^+RYo^+No^Y -m0 da
0
Y(\Wd+1(-)] = np [w^) [m)^ Lion) ]pu-R(wJ+)V(W))-f (o) ],
w^ (s)=Lo0)(de+-is ((.n^ (s^ (s)). own dwn dw) 0 '
iO^jUty re1stricte(0 variatio u 0w) = 0
[T a1ren]c[WmTs)] + WenT)]SsX(s)].
Tlie extremumoonditeon0^ W)+l(n) oead Swn+l{s) = 0
^(s) =1 -d0-.
O
Sub) equend tterationsobtainedthroughinverseLaplacetransforFofEq.(24);
w„+ ¡(o) = dr„(k) - w
• c-4
00 q
^ jSX (s) -S^w(p) -... - w'^11 (0) + ^ ) + N(wn) -/(*)]} J^-'m-CO) +... + M'{m-l) (0)} + ^r1 [-L jC[R(wh) + N{wn) - fix)]
W)+¡(k)=>+¿■-¡ -1X RaOO)) -0 0°(W)) - /(o)]
3.4.Applicclio( ofLH-VPM on the system governing equation Based on the basic rule of LH-VPM, the governingequation may be analyzed as:
—apa-2m2-() -m4PX'-k )w(x)-k ^3(x) = 0, X dx2 W w! ( ) p ( ) ,
dnW
cK"
= s"J?[w]~ s"-lW( 0) - W '(0) - s"-3W "(0) -......W"-1 (0),
(222
1)32
= + + ¿[R{wn)+N{wn) -/(x)]}. (24)
Eq. 02)) is malye rtationaey wWd,,
(25)
(262 (ati)
7)82
2 (30)
(t(t (32)
L[L W( x, y)] + L [ R W (o, y)] + L [ NW (o, y) ] = L[ f (o, y) ],
(333
(343 (35)
¿[w^ x)] = n[w„ (x )] + n [*]
s4)Tn(s) - s3)F(e) - sdW '(0) - s W "(0) - W "'(0)
-ne
dX2
(q2-wV,24 -kw)""(x) + kp)Vn3(x)
wee+o)s)^ws;(^ )+x(s-
s4ln (s) - s3W(0)-s2id '(0) - s FW "(0) - (- '"(0)
-L
2e2 JXd X^"04 + / q2 -w 4 j4X 4 -k )W (x) + k T3(x) dX v '
n36x
Lap)ace Teaeuform of Boundary condition
1
(38)
A(s) = —^
S
Taking nhe ienerse LapWam Transform let h unknown be a => ¡V'=0) end L =T W"')Q)
K^x) ^I) " sW]
s L J
111
2m—A2 d"!+(q2 -mV4/}4 -k)mn(x)+kp"Tn3(x)
dX2 n j4 " )
Wq( x) =-6 + 6a),
i
JiOo()^rl^0 -i l
T"1(x) = en1<j — l
2ww2VX2 d mX) +(Q2-wW-* )"(x) +k T3(x)
dX v ; p
2wWn2L2 ^m"? = (f = ww474^'!^ = ¿J""" (x) -) ^(""(x) 2=t4L4ww4 -sf2 =Tt-{ots1 = J "Qas2 = J
)(TwVtT > Q= a^as = + /{) 2 > '
(39)
(40)
(41)
(42)
(43)
(44)
s>1 = —i)/):2 + 6a) +
5 r-259))59200^4///ttw4x2 + /kpxm - 008972864007z-4a/l4w4 + 630a/2kPx8 + 98280 a2/krx6 }
' I
+3603600a3krx4 + 259459200Q2/x2 -259459200/ke,x2 + 00897286400Q2a-00897286400ak.
J
2807674368000
Using tiie defimtixn of VPM, the xna^^ti^);£d teries solгeion is obtained.
w(x) = 2X to = W0 to + (x) +W2 (x) +..
(45)
Table 1 provides parameters for validation of the hybrid method to confirm the accuracy of the results. First equation of the simultaneous equation obtained through introduction of parameters Table 1 to Eq. (45) to obtain values of the unknowns introduced into the boundary condition.
Table 1. Parameters for validation of the model
Pas te tnalt foundation Winkler foundation Poisson ratio Integer Aspect ratio
Kw )e V m X
0 0 6.3 1 1
dl^QK + '^Q) t2=0. (46)
The pofynomials are repreeented as f22 and ^^ Eq. (46) can be written in matrix
form as:
v! r ](n) 1(0) llt^O) nS^O
dnt foUowing Charadtetistsc determinant is obtai2ed applying the non-trivial condition
^(o) l>(o )
, - (47)
0
= 0. (48)
SoMngEq^^gives the natural frequencies.
i^ol11 ing the qua0r7lic Eo^ (48) givesthe natural frequencies
CI = 11.U9220774.
Subo1i(u)e (0(8 yahm of 0atural frequency obtained into Eq. (47) -41777 MBD " o2701 2)1703 =|01
^96^i)4 ne607 67?j ■(o
(49)
7691 1817755 Puttm= a= 1 and solving for 13
o] f 1
/?JW [-3.881269291 Deflection soh8i2n )f Ürn governingequationgives;
(49)
x5(-44326904170xl+479670396000) i f 40241x2 ^
w(x) = —i-L + _x--+ 6 I. (50)
1307674368000 6 I 10368 1
4. Results and discussions
The solution of Laplace transform and Variation parameter method is presented here. Table 2 presents comparisons of fundamental natural frequency results of the present study with that of the past works. It was observed from the Table 2 that, there are good agreements between the present study and the past results, while Table 3 shows effect of plate resting on elastic foundation. Figs. 2 and 3, show the fundamental modal shapes of the thin rectangular plate. Also, Figs. 4 and 5 show the variation
Table 2. Showingvalidation ofre suits
Edge Condition/ Dimensionless Natural frequency Simply-supported(SS) Simply supported- Clamped(SC) Simply- supported-Free (SF)
Bambilletal [15] Present Leissa [16] Present Leissa [16] Present
Q1 19.7392 19.7434 23.6463 23.6486 11.7195 11.7606
(a) (b)
Fig. 2. Fundamental mode shapesof Simply Supported Conditionat bothedges and Simply Supported with Clamped edge
Fig. 3.FundamentalmodeshapesofSimplySupportedConditionatoneedgeandfreeatotheredge
effects of foundation parameters and aspect ratio on natural frequency. It is observed that the natural frequency increases with increases in foundation parameters and aspect ratio. Variation of Combine effect of Winkler and Pasternak foundation also results into increases with natural frequency.
4.1. Effect offoundation Parameter on natural frequency
Figs. 4 and 5 illustrate the effects of foundation Parameter on natural frequency. It is clear from the Figures that the foundation parameter influences on natural frequency, increasing values of the foundation parameter has direct l effect on the natural frequency based on classical theory. Increasing stiffness results in increase in natural frequency
- 170 -
Table 3.Variation of Aspect ratio and foundation coefficient
Edge Condition Natural frequency X=0.5 X=1.5
kw=10 kw=50 kw=100 kw=10 kw=50 kw=100
SS Q1 12.7358 14.2198 15.8809 32.2317 32.8464 33.5989
SC Q1 17.6179 18.7187 20.0097 35.1935 351513 36.4497
SF Q1 5.1255 8.14 07 10.7829 24.2175 25.0297 26.0093
(a) (b)
Fig. 4. Variation of foundation parameter on SS-edge and SC edge condition
Fig. 5. Variation of foundationparameteronSFedgecondition
4.2.Effectofvariationofaspectratioonnaturalfrequency
According to Figs. 6 and 7 the aspect ratio influences on natural frequency. It is shown that, the natural frequency increases with the increase in aspect ratio. This is because, the aspect ratio here means variation in size of plate length divided by width. Increasing size of the plate increases the stiffness of theplateresulting into theplatebeingstiffer.lt results in increasing thenaturalfrequency oftheplate.
5. Conclusion
In this study, the investigation of dynamic analysis of isotropic rectangular plates resting on Winkler and Pasternak foundations using Laplace transform and variation parameters method is
- 171 -
0.5 1 1.5 2 2.5 3
Aspect ratio
0.5 1
1.5 2
Aspect ratio
2.5 3
Fig. 6. VariationofAspectratio on SSandSC edgecondition
1 fundamental mode I
1 1.5 2 2.5 3
Aspect ratio
Fig. 7. Variation of Aspect ratioonSFedgecondition
0.5
analyzed. The governing nonlinear partial differential equation is transformed to nonlinear ordinary differential equation by means of Galerkin method of separation. The nonlinear ordinary differential equations have been solved using Laplace transform and variation parameters. The accuracies of the obtained analytical solutions were ascertained with the results obtained by some other methods as presented in the past works. The obtainedamdyticalsolutions were used to examine the effectsof foundation parameterand aspect ratio.Basedonthe parametricstudies,thefollowing observations were made:
1) Increases inelastic foundation parameter increases the natural frequency.
2) Increases in aspect ratio increasesthe naturalfrequency.
3) Increases in combine foundationparametersincreasesthenatural frequency.
Abbreviations:Nomenclature
a: Length of the plate
b: Widthof theplate
C: Clamped edge plate;
E: Young's modulus
F: Free edgesupport
S: Simply supported edge —: Differential operator w: Dynamic deflection
X: Space coordinate along the length of thin plate
Symbol
h: Plate thickness p: Mass density D: Modulus of elasticity Q: Natural frequency
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors express sincere appreciation to the University of Lagos, Nigeria, for providing material supports and good environment for this work.
References
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