Full page photo The Journal of Engineering Research (TJER),Vol. 13, No. 2 (2016) 187-196 Vibrations of Circular Plates with Elastically Restrained Edge against Translation and Resting on Elastic Foundation L.B. Rao*, a and C.K. Raob a School of Mechanical and Building Sciences, VIT University, Chennai-600127, Tamil Nadu, India. b Department of Mechanical Engineering, School of Engineering, Hyderabd-500 088, A.P. India. Received 17 June 2014; accepted 23 May 2016 Abstract: The present paper deals with exact solutions for the free vibration characteristics of thin circular plates elastically restrained against translation and resting on Winkler-type elastic foundation based on the classical plate theory. 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 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. اعدة اھتزازات األلواح الدائریة ذات الحافة المقیدة المرنة مقابل االنتقال والسكون على ق مرنة بوشیالبیال كامیسواره راو، ∗أ،وكافرابو بھاسكارا راول تتناول ھذه الورقة الحلول الدقیقة لخصائص االھتزاز الحر أللواح دائریة رقیقة مرنة ومقیدة مقابل االنتقال :الملخص والسكون على قاعدة مرنة من نوع وینكلر باالعتماد على نظریة اللوح التقلیدیة. وتم إجراء استقصاء بارامتري من أجل لقاعدة مرنة على الترددات الطبیعیة أللواح دائریة. وتبین أن حافة المرونة تقدیر تأثیر حافة مقیدة مقابل االنتقال والصالبة المقیدة مقابل االنتقال وبوجود القاعدة المرنة لھا تأثیر شدید على خصائص اھتزاز اللوح الدائري الخاضع الى اھتزازات تنوعة من نسبة الصالبة االنتقالیة ومتغیر عرضیة حرة. تم إجراء العملیات الحسابیة للترددات الطبیعیة لالھتزازات لقیم م الصالبة لقاعدة نوع وینكلر. وقد تم عرض النتائج ألثنى عشر طریقة لالھتزاز سویًا على شكل جداول ورسوم بیانیة وذلك الستخدامھا في التصمیم. وتم جدولة بیانات واسعة النطاق بحیث أن االستنتاجات ذات الصلة منھا یمكن التوصل الیھا معرفة تأثیر الحافة المقیدة االنتقالیة ونسبة الصالبة لقاعدة نوع وینكلر على الترددات الطبیعیة لأللواح الدائریة ذات ل الخواص الموحدة. صالبة انتقالیة، قاعدة مرنة، تردد، حافة مرنة، لوح: الكلمات المفتاحیة * Corresponding author’s e-mail: bhaskarbabu_20@yahoo.com L.B. Rao and C.K. Rao 188 Nomenclature h Thickness of a plate a Radius of a plate ν Poisson’s ratio E Young’s modulus ρ Density of a material ),( θrW Transverse deflection of the plate D Flexural rigidity of a plate TK Translational spring stiffness wK Stiffness of Winkler foundation T Translational spring stiffness ratio ξ Foundation stiffness ratio mnω Is the natural frequency of vibrations mnλ Eigenvalue without foundation * mnλ Eigenvalue with Winkler foundation nm, Positive integers corresponding to the number of concentric circles and nodal diameters in each flexural mode Vibrations of Circular Plates with Elastically Restrained Edge against Translation and Resting on Elastic Foundation 189 1. Introduction Circular plates resting on elastic foundation have wide range of application 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 reestablished the efficiency of the classical approach 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 boundary conditions of the plate become complex involving linear and rotational restraints. A recent survey of literature shows that very few studies exist on the study of circular plates resting on elastic foundation. Wang and Wang (2003), who observed the switching between axisymmetric and asymmetric vibration modes, 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 natural frequency of the plate 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 vibrational formulation. Leissa (1969), who tabulated a frequency parameter for four vibration modes of 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 exact analytical method. Chang and Wickert (2001), Kim et al. (2000) 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 papers (Weisman 1970; Dempsey et al. 1984; Celep 1988) dealing with the study of plates on a Winkler foundation. In general, papers dealing with vibrating plates, shells and beams are concerned with the determination of eigenvalues and mode shapes (Leissa 1969). A good number of studies are made by investigators (Wang and Lin 1996; Kim et al. 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 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 boundary of the circular plate to suitably simulate the practical non-classical boundary connections being adopted in a wide range of industrial applications (Bhaskara and Kameswara 2009; Bhaskara and Kameswara 2010; Lokavarapu and Chellapilla 2013), the use of exact method of solution becomes imperative and hence the same is adopted in this paper. Even though the method adopted here is L.B. Rao and C.K. Rao 190 classical, the particular case of vibration of elastically restrained circular plate resting on elastic foundation considered here is not dealt with in the available literature. Utilizing the classical plate theory, this paper deals with exact method of solution for the analysis of free transverse vibrations of thin circular plate that is elastically restrained against translation and resting on Winkler-type elastic foundation. For estimating the influence of edge restraint against translation and stiffness of the elastic foundation on the natural frequencies of circular plates, parametric investigations are carried out varying the values of elastic edge restraint stiffness against translation and the stiffness of the elastic foundation. 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 have been 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. Results presented for twelve modes of vibration both in tabular and graphical form 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/),,(),,(. 224 =∂∂+∇ ttrwhtrwD θρθ (1) Figure 1. A thin circular plate with translational elastic edge restraint and supported on elastic foundation. where )1(12/ 23 ν−= EhD is the flexural rigidity of a plate and νρ ,,,, Eha are the plate’s radiuses, thickness, density, Young’s modulus and Poisson ratio respectively. The homogeneous equation for Kirchhoff’s plate on one parameter elastic foundation is given by the following equation. ++∇ ),,(),,(. 4 trwKtrwD w θθ (2) 22 /),,( ttrwh ∂∂ θρ =0 Displacement in equation (2) can be presented as a combination of spatial and time dependent components as follows; Let tierWtrw ωθθ ),(),,( = (3) Now substitute the Eq. (3) in Eq. (2) 0),().(),(. 24 =−+∇ θωρθ rWhKrWD w (4) The solution of the equation takes the following form where3, 2, 1, 0,= ;…3 2, 1, 0,= cos. ),( θ λλ θ n a r IC a r JArW mnnmn mn nmnmn             +      = (5) where Amn and Cmn 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. 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),( =θaM r (6) ),(.),( θθ aWKaV Tr −= (7) where the Kelvin-Kirchhoff and bending moment are defined as follows                         ∂ ∂ + ∂ ∂ + ∂ ∂ −= 2 2 2 2 2 ),(1 ),(1 ),( .),( θ θ θ ν θ θ aW r r aW r r aW DaM r (8)                       ∂ ∂ − ∂∂ ∂ ∂ ∂ −+∇ ∂ ∂ −= θ θ θ θ θ νθ θ ),(1),(1 1 )1(),( .),( 2 2 2 aW rr aW r r aW r DaVr (9) Vibrations of Circular Plates with Elastically Restrained Edge against Translation and Resting on Elastic Foundation 191 By applying Eqs. (6) and (8), we obtain the following equation 0 ),(1 ),(1 ),( 2 2 2 2 2 =                         ∂ ∂ + ∂ ∂ + ∂ ∂ θ θ θ ν θ aW r r aW r r aW (10) From Eqs. (5) and (10), we derive the following equation )( 4 2 2 )( 4 2 2 2 2 2 2 mnn mn mn mn mn mnn mn mn mn mn mn I n ST J n PQ C λ λ ν λ ν λ λ ν λ ν         −++         +++− = (11) where mnP = )()( 11 mnnmnn JJ λλ −+ − ; )()( 22 mnnmnnmn JJQ λλ −+ += ; )()( 11 mnnmnnmn IIS λλ −+ += ; )()( 22 mnnmnnmn IIT λλ −+ += ; From Eqs. (7) and (9), we get the following ),(. ),(1 ),(1 1 )1(),(. 2 2 2 θ θ θ θ θ θ νθ aWK aW r r aW r r aW r D T−=                         ∂ ∂ − ∂∂ ∂ ∂ ∂ −+∇ ∂ ∂ − (12) From Eqs. (5) and (12), we derived the following equation )( 84)3(8)2(44 3 2 )( 84)3(8)2(44 3 2 33 2 2 2 33 2 2 2 mnn mnmnmn mn mnmn mn mn mnn mnmn mn mnmn mn mn mn IT n S nT U JT n P nQ R C λ λλλ ν λ ν λ λ λλλ ν λ ν λ         −+ − +         −− −++         −− − −         −+ +−− = (13) where, D Ka T T 3 = )()( 11 mnnmnnmn JJP λλ −+ −= ; )()( 22 mnnmnnmn JJQ λλ −+ += ; )()( 33 mnnmnnmn JJR λλ −+ −= ; )()( 11 mnnmnnmn IIS λλ −+ += ; )()( 22 mnnmnnmn IIT λλ −+ += ; )()( 33 mnnmnnmn IIU λλ −+ += ; If ∞→TK then this case becomes simply supported boundary condition as shown in Fig. 2. The frequency equation can be calculated from Eqs. (11) and (13), which allows determining eigenvalues mnλ . The mode shape parameters mnC can be determined corresponding 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 a mn MrdrdrWrW δδθθθ π =∫ ∫ ),().,( 2 0 0 (14) where, mnM is a mass of the plate, 1== nqmp δδ if qnpm == , and nqmpδδ = 0 if m ≠ p or n ≠ .q The normalization constant mnA can be derived using Eqs. (5) and (14) as given below: 1 2 0 0 2 2 cos. . . 1 −                 ∫ ∫                                   +       = π θθ λ λ π a mn nmn mn n mn rdrdn a r IC a r J a A (15) In Eq (4), mnω is the natural frequency of vibrations               = h D a mn ρ λ 2 2 (16) It is clear from the Eq. (16) the natural frequency of vibrations is dependent on the plate radius and eigenvalues. From Eq. (16) we can express D ha mn mn 24 4 ωρλ = (17) L.B. Rao and C.K. Rao 192 244* ξλλ += mnmn (18) where D aK w 4 2 =ξ (19) [ ]4 1 24* ξλλ += mnmn (20) where mnλ is eigenvalue without foundation and *mnλ is eigenvalue with Winkler’s foundation. 3. Solution Using Matlab programming, computer software with symbolic capabilities, solves the above set of equations. The program determines eigenvalues ( *mnλ ), for a given range of boundary conditions. The boundary linear translational non-dimensional restraint parameter can be defined as follows: D aK T T 3 = (21) D aK w 4 2 =ξ (22) The following are the input parameters to the program; (i) Translational stiffness ratio ( T ) (ii) Foundation ratio ( ξ ) (iii) Poisson ratio (ν ) (iv) Upper bound for eigenvalues (N) (v) Suggested for eigenvalues (d) (vi) Number of mode shape parameters (n). The program finds eigenvalues * mnλ 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 also implanted for various plate materials by adjusting Poisson ratio. Such a wide range of results is not available in the literature yet. 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 increases 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]. The simply supported boundary conditions (Fig. 2) could be accounted for by setting ( ∞→TK ) shown in Fig.1. The translational edge supports becomes simply supported (or hinged) for very high values (close to infinity) of translational stiffness parameter i.e. ∞→TK . 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 study of the case of vibrations of elastically restrained circular plates supported on partial Winkler foundation. When the support position is full span which means that when b = 1, the case becomes a circular plate having full foundation support with elastically restrained edge against translation. For this case, the frequency is 2.1834 which is in good agreement with the frequency of 2.18341 obtained from the present study. Here †.represents translational stiffness ratio and ‡ represents Eigen values throughout the text. The eigenvalues at various values of the foundation stiffness ratios [ ]33.0&100 == νT 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 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]. Vibrations of Circular Plates with Elastically Restrained Edge against Translation and Resting on Elastic Foundation 193 0 2 4 6 8 10 12 14 16 18 -5 0 5 10 15 E ig en va lu e, λ m n Transverse stiffness ratio, Log10(T) λ00 λ10 λ20 λ01 λ11 λ21 λ02 λ12 λ22 λ03 λ13 λ23 Table 1. Eigenvalues for different Translational stiffness ratio for ξ =100 and ν =0.33. Table 2. Eigenvalues for different Foundation stiffness ratio for T =100 and ν =0.33. Figure 2. A simply supported thin circular plate resting on elastic foundation. Figure 3. Effect of translational stiffness ratio ξ on eigenvalues, λmn. 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 Log10(ξ) 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. Rao and C.K. Rao 194 Figure 4. Effect of Foundation stiffness ratio, ξ on eigenvalues, λmn. Figure 5. Effect of translational, T and foundation, ξ stiffness ratio son eigenvalues, λmn. 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. As observed from Table 1, 2 and 3, in Table 3, lower eigenvalues are recorded for lower values of foundation and translation stiffness ratios together. As seen from Fig. 5, all the curves are stable up to certain region and beyond this all the curves tend to converge as the value of translation and foundation stiffness ratios increases. The eigenvalues for different plate materials and various values of translational, foundation stiffness ratios are computed, and the results are given in Table 4. It was observed that for high ξ , eigenvalues are independent of Poisson ratio, as shown in Fig. 6. In addition, it was observed that for any value of T, eigenvalues are independent on Poisson ratio. 5. Conclusion This paper deals with a method of computation of eigenvalues of axi-symmetric flexural vibrations of a circular plate with translational edge supports and resting on Winkler foundation using a specifically written MATLAB code. In this paper, the computed numerical results are presented in a tabular format to enable an estimating the accuracy of 0 5 10 15 20 25 30 35 -4 -3 -2 -1 0 1 2 3 4 Ei ge n va lu e( λm n) Foundation parameter, Log10( ξ) λ00 λ10 λ20 λ01 λ11 λ21 λ02 λ12 λ22 λ03 λ13 λ23 0 5 10 15 20 25 30 35 -4 -3 -2 -1 0 1 2 3 4 E ig en va lu e( λm n) Transverse stiffness, Log10 (T) and Foundation ratos, Log10(ξ) λ00 λ10 λ20 λ01 λ11 λ21 λ02 λ12 λ22 λ03 λ13 λ23 Vibrations of Circular Plates with Elastically Restrained Edge against Translation and Resting on Elastic Foundation 195 Table 4. Eigenvalues for different Poisson ratios. ν 1000== ξT 10,100 == ξT 1000,10 == ξT 10,1 == ξT 50,50 == ξT 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 0.5 31.62946 4.92503 10.03404 3.72725 7.29085 Figure 6. Effect of Poisson ratio, ν on eigenvalues, λmn. 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. It has been observed that the eigenvalues remain constant without change only in a limited range of constraints (0 to 10) specific to each vibration mode and then steeply increase with increasing values of foundation ultimately converging to 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. The effects of various parameters such as translational stiffness, foundation stiffness and Poisson ratio parameters on natural frequencies of the plate with elastic edge and resting on elastic foundation are studied in detail. References Ascione L, Grimaldi A (1984), Unilateral contact between a plate and an elastic foundation. Mechanica 19(3): 223-233. Bhaskara RL, Kameswara RC (2009), Buckling of circular plates with a ring support and elastically restrained edge. Advances in Vibration Engineering 8(1): 61-69. Bhaskara RL, Kameswara RC (2010), Buckling analysis of circular plates with elastically restrained edges and resting on internal elastic ring support. Mechanics Based Design of Structures and Machines 38(4): 440-452. 0 5 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 E ig en va lu e( λ1 0) Poisson ratio(ν) T=1000, ξ=1000 T=100, ξ=10 T=10, ξ=1000 T=1, ξ=10 T=50, ξ=50 L.B. Rao and C.K. Rao 196 Celep Z (1988), Circular plate on tensionless Winkler foundation. Journal of Engineering Mechanics 114(10): 1723-1739. Chang JY, Wickert JA (2001), Response of modulated doublet modes to traveling wave excitation. Journal of Sound and Vibration 242(1): 69-83. Dempsey JP, Keer LM, Patel NB, Glasser ML (1984), Contact between plates and unilateral supports. Journal of Applied Mechanics 51(2): 324-328 Ghosh AK (1997), Axisymmetric dynamic response of a circular plate on an elastic foundation. Journal of Sound and Vibration 205(1): 112-120. Kang KH, Kim KJ (1996), Modal properties of beams and plates on resilient supports with rotational and translational complex stiffness. Journal of Sound and Vibration 190(2): 207-220. Kim M, Moon J, Wickert JA (2000), Spatial modulation of repeated vibration modes in rotationally periodic structures. Journal of Vibration and Acoustics, Transactions of ASME 122(1): 62-68. Kim KM, Kim MS (2001), Vibration of Beams with Generally Restrained Boundary Conditions using Fourier Series. Journal of Sound and Vibration 245(5): 771-784. Leissa AW (1969), Vibration of Plates (NASA SP-160). Office of Technology Utilization, Washington, DC. Leissa AW (1993), Vibration of Plates. Acoustical Society of America, Sewickley, PA. Lokavarapu BRL Chellapilla KR (2013), Fundamental buckling of circular plates with elastically restrained edges and resting on concentric rigid ring support. Frontiers of Mechanical Engineering 8(3): 291-297. Rao CK, Rao LB (2009), Vibrations of elastically restrained circular plates resting on partial winkler foundation. The Open Acoustics Journal 2: 68-74. Salari M, Bert CM, Striz AG (1987), Free vibration of a solid circular plate free at its edge and attached to a Winkler foundation. Journal of Sound and Vibration 118(1): 188-191. Soedel W (1993), Vibrations of shells and plates. Mareel Dekker, Inc. Tseng JG, Wickert JA (1994), On the vibration of bolted plate and flange assemblies. Journal of Vibration and Acoustics, Transactions of ASME 116(4): 468-473. Wang JTS, Lin CC (1996), Dynamic analysis of generally supported beams using fourier series. Journal of Sound and Vibration 196(3): 285-293. Wang CY, Wang CM (2003), Fundamental frequencies of circular plates with internal elastic ring support. Journal of Sound and Vibration 263: 1071-1078. Wang CY (2005), Fundamental frequency of a circular plate supported by a partial elastic foundation. Journal of Sound and Vibration 285(4-5): 1203-1209. Weisman Y (1970), On foundations that react in compression only. Journal of Applied Mechanics 37: 1019-1030. Winkler E (1867), Die lehre von der elasticitaet and festigkeit, Prag. Dominicus. Yayli MÖ, Aras M, Aksoy S (2014), An efficient analytical method for vibration analysis of a beam on elastic foundation with elastically restrained ends. Shock and Vibration 2014: 1-7. Zheng XJ, Zhou YH (1988), Exact solution of nonlinear circular plate on elastic- foundation. Journal of Engineering Mechanics- ASCE 114: 1303-1316.