Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 1, pp. 87-101, Warsaw 2017 DOI: 10.15632/jtam-pl.55.1.87 QUASI-GREEN’S FUNCTION APPROACH TO FUNDAMENTAL FREQUENCY ANALYSIS OF ELASTICALLY SUPPORTED THIN CIRCULAR AND ANNULAR PLATES WITH ELASTIC CONSTRAINTS Krzysztof K. Żur Bialystok University of Technology, Faculty of Management, Kleosin, Poland e-mail: k.zur@pb.edu.pl Free vibration analysis of homogeneous and isotropic thin circular and annular plates with discrete elements such as elastic ring supports is considered. The general form of quasi- -Green’s function for thin circular and annular plates is obtained. The nonlinear characteri- stic equations are defined for thin circular and annular plateswith different boundary condi- tions anddifferent combinations of the core and support radius.The continuity conditions at the ring supports are omitted based on the properties of Green’s function. The fundamental frequency of axisymmetric vibration has been calculatedusing theNewton-Raphsonmethod and calculation software. The obtained results are comparedwith selected results presented in literature. The exact frequencies of vibration presented in a non-dimensional form can serve as benchmark values for researchers to validate their numericalmethods when applied for uniform thin circular and annular plate problems. Keywords: quasi-Green’s function, ring supports, movable edges, elastic constraints 1. Introduction The study of vibration of a thin circular and annular plate is basic in structural mechanics. Components of circular and annular plates are commonly used in the aerospace industry and aviation as well as in marine and civil engineering applications. Circular and annular plates are the most critical structural elements in high speed rotating engineering systems. The natural frequencies of circular and annular plates have been studied extensively for more than a centu- ry, because if only the frequency of external load matches the natural frequency of the plate, destruction may occur. Additionally, the influence of elastic or rigid ring supports on dynamic behavior of plates have been studied in a lot of works, because it used to stabilize or to increase the frequency of plates. Knowledge about distribution of ring supports of variable stiffness can allow one to predict dynamic behavior of structural elements suchus circular and annular plates. The free vibration of circular and annular plates with concentric ring supports have been studied in a lot of works. Bodine (1967) studied the influence of rigid supports on the funda- mental frequency of circular plates in which radius of the supports was small. Kunukkasseril and Swamidas (1974) formulated equations for circular plates with elastic supports, but they solved the free vibration problem for a free circular plate. Singh and Mirza (1976) studied free axisymmetric vibration of circular plates elastically supported along two concentric circles. Azi- mi (1988) studied natural vibration of circular plates with elastic and rigid supports using the receptancemethod.Wang andThevendran (1993) analyzed free vibration of annular plates with concentric supports using by the Rayleigh-Ritz method. Ding (1994) solved the free vibration problem for arbitrarily shaped plates with concentric elastic and rigid ring supports. Liu and Chen (1995) studied axisymmetric vibration of annular and circular plates using simple finite analysis. Inworks byVega et al. (1999) free vibration analysis was presented for a concentrically 88 K.K. Żur supported annular plate with a free edge using the optimized Rayleigh-Ritz method. Laura et al. (1999) analyzed transverse vibration of a circular plate with a free edge and concentric ring supports. Vega et al. (2000) analyzed free vibration of concentrically supported annular plates with one edge clamped or simply supported. The fundamental frequency of a free thin circular plate supported on a ring was analyzed by Wang (2001). Influence of the stiffness and location of elastic ring supports on the fundamental frequency of circular plates were analyzed byWang and Wang (2003). Wang (2006, 2014) studied vibration modes of concentrically supported free circular and annular plates withmovable edges. Rao andRao (2014a) analyzed free vibration of annular plates with both edges elastically restrained and resting on theWinkler foundation.Ad- ditionally, Rao and Rao (2014b) analyzed free vibration of a thin circular plate with concentric ring and elastic edge support. In the works presented above, the analyzed plates were separated into two regions for one ring supports. The number of separated regions increases if the number of considered elastic ring supports increases. In this approach, the solution to boundary value problem is complica- ted. Additionally, continuity conditions between the support and plate must be used to obtain characteristic equations. Solution to the boundary value problem is very tedious andmore com- plicated based on continuity conditions, because characteristic matrices have a large dimension. Application of Green’s function to the solution to the boundary value problem of free vibra- tion of plates allow one to neglect the continuity condition. In theworks of Kukla and Szewczyk (2004, 2005, 2007) Green’s function approach to frequency analysis is presented for circular and annular thin plates with elastic supports. The authors calculated nontrivial constants of general solutions to the differential equation to obtain a full form of Green’s function for free, simply-supported and clamped plates. The nontrivial constants have a very complicated form, and calculating them is very tedious for different boundary conditions such as sliding supports or elastic constraints. The novelty of the paper is quasi-Green’s function (not full form) approach to obtain cha- racteristic equations of concentrically supported circular and annular plates with clamped, free, simply-supported and sliding (movable) edges or elastic constraints. The quasi-Green function is obtained by the method presented in the previous works (Żur, 2015, 2016a). Nonlinear charac- teristic equations of plates are obtained without calculating nontrivial constants of the general solution to the differential equation. The numerical results of investigation are compared with selected results presented in literature. The exact fundamental frequencies of axisymmetric vi- bration are presented in a non-dimensional form for different combinations of the core and support radius as well as selected values of parameters of elastic constraints. 2. Statement of the problem Consider an isotropic, homogeneous annular (circular) thin plate of constant thickness h in cylindrical coordinates (r,θ,z) with the z-axis along the longitudinal direction. The geometry and coordinate systemof the consideredplate is shown inFig. 1.Thepartial differential equation for free vibration of thin uniform annular (circular) plates has the following form ∇4W(r,t)+ ρh D ∂2W(r,t) ∂t2 =− χ ∑ j=1 KjW(r,t)δ(r −rj) (2.1) where ρ is mass density, D = Eh3/[12(1−ν2)] is flexural rigidity, E is Young’s modulus, ν is Poisson’s ratio, ∇2 = (∂2/∂r2) + (1/r)(∂/∂r) is Laplacian, Kj is a coefficient of normalized stiffness of the supports, δ is Dirac’s delta function, rj is the position of elastic ring supports, χ is thenumber of elastic ring supports andW(r,t) is small deflection comparedwith thicknessh of the plate. Quasi-Green’s function approach to fundamental frequency analysis... 89 Fig. 1. The geometry and coordinate system of the annular plate with radius of the hole R1 The axisymmetric deflection of an annular (circular) plate may be expressed as follows W(r,t) =w(r)eiωt (2.2) where w(r) is the radial mode function, ω is the natural frequency, and i2 = −1. Substituting Eq. (2.2) into Eq. (2.1) and using the dimensionless coordinates ξ = r/R and κj = rj/R, the governing differential equation of the annular (circular) plate is obtained L(w)−λ2w =− χ ∑ j=1 Kjw(κj)δ(ξ −κj) (2.3) where L(w)≡ d4w dξ4 + 2 ξ d3w dξ3 − 1 ξ2 d2w dξ2 + 1 ξ3 dw dξ (2.4) is the differential operator and λ = ωR2 √ ρh/D (2.5) is the dimensionless frequency of vibration. Theboundary conditions at the outer edge (ξ =1) of the annular (circular) platemaybeone of the following: clamped, simply supported, free, sliding supports and elastic supports. These conditions may be written in terms of the radial mode function w(ξ) in the following form: — clamped w(ξ)|ξ=1 =0 dw dξ ∣ ∣ ∣ ξ=1 =0 (2.6) — simply supported w(ξ)|ξ=1 =0 M(w)|ξ=1 = (d2w dξ2 + ν ξ dw dξ ) ξ=1 =0 (2.7) — free M(w)|ξ=1 =0 V (w)|ξ=1 = (d3w dξ3 + 1 ξ d2w dξ2 − 1 ξ2 dw dξ ) ξ=1 =0 (2.8) 90 K.K. Żur —movable edges (sliding) dw dξ ∣ ∣ ∣ ξ=1 =0 V (w)|ξ=1 =0 (2.9) — elastic supports Φ(w)|ξ=1 = [(d2w dξ2 +ν dw dξ ) +φ dw dξ ] ξ=1 =0 Ψ(w)|ξ=1 = [(d3w dξ3 + d2w dξ2 − dw dξ ) −ψw ] ξ=1 =0 (2.10) M(w) and V (w) are the normalized radial bendingmoment and the normalized effective shear force, respectively. φ = KφR/DR and ψ = KψR 3/DR are the parameters of elastic constraints. Kφ and Kψ are the rotational and translational spring constants, respectively. Similar boundary conditions may be defined at the inner edge (ξ =R1/R = ξ1), depending on considered annular plates. 3. Finding quasi-Green’s function The general solution to the homogeneous differential equation for thin annular (circular) plates L(w)−λ2w =0 (3.1) is a linear combination of theBessel functions presented in the following form (McLachlan, 1955) w(ξ) = C1J0(λξ)+C2I0(λξ)+C3Y0(λξ)+C4K0(λξ) (3.2) where J0(λξ), Y0(λξ) are the Bessel functions of the first and second kind, I0(λξ), K0(λξ) are themodifiedBessel functions of the first and second kind.The quasi-Green function K(ξ,α) is a particular solution to Eq. (3.1) andmay be received from the formula presented in the following form (Jaroszewicz and Zoryj, 2005; Żur, 2015) K(ξ,α) = D(ξ,α) W(α)p0(α) (3.3) where p0(α) = 1 is a coefficient placed in front of the highest order of derivative of differential equation (3.1), and D(ξ,α) = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ J0(λα) I0(λα) Y0(λα) K0(λα) dJ0(λα) dα dI0(λα) dα dY0(λα) dα dK0(λα) dα d2J0(λα) dα2 d2I0(λα) dα2 d2Y0(λα) dα2 d2K0(λα) dα2 J0(λξ) I0(λξ) Y0(λξ) K0(λξ) ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ W(α)= ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ J0(λα) I0(λα) Y0(λα) K0(λα) dJ0(λα) dα dI0(λα) dα dY0(λα) dα dK0(λα) dα d2J0(λα) dα2 d2I0(λα) dα2 d2Y0(λα) dα2 d2K0(λα) dα2 d3J0(λα) dα3 d3I0(λα) dα3 d3Y0(λα) dα3 d3K0(λα) dα3 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (3.4) Quasi-Green’s function approach to fundamental frequency analysis... 91 The elements of the matrixD andW have the following form dJ0(λα) dα =−λJ1(λα) dI0(λα) dα = λI1(λα) dY0(λα) dα =−λY1(λα) dK0(λα) dα =−λK1(λα) (3.5) d2J0(λα) dα2 = λ2 2 [J0(λα)+J2(λα)] d2I0(λα) dα2 = λ2 2 [I0(λα)+ I2(λα)] d2Y0(λα) dα2 = λ2 2 [Y0(λα)+Y2(λα)] d2K0(λα) dα2 = λ2 2 [K0(λα)+K2(λα)] (3.6) d3J0(λα) dα3 = λ3 4 [3J1(λα)+J3(λα)] d3I0(λα) dα3 = λ3 4 [3I1(λα)+ I3(λα)] d3Y0(λα) dα3 = λ3 4 [3Y1(λα)−Y3(λα)] d3K0(λα) dα3 =− λ3 4 [3K1(λα)+K3(λα)] (3.7) After calculations, the function D(ξ,α) has the form D(ξ,α) = 2λ2 πα [2I0(λξ)K0(λα)−2I0(λα)K0(λξ)+πJ0(λξ)Y0(λα)−πJ0(λα)Y0(λξ)] (3.8) Bessel function (3.2) expresses linear independent solutions, thus the Wronskian must satisfy the condition (Stakgold and Holst, 2011) W(α)= 8λ4 πα2 6=0 (3.9) Condition (3.9) is satisfied for a circularplate (0 < α ¬ 1) andanannularplate (0 < ξ1 ¬ α ¬ 1). After calculations, the quasi-Green function has the form K(ξ,α) = α 4λ2 [2I0(λξ)K0(λα)−2I0(λα)K0(λξ)−πJ0(λα)Y0(λξ)+πJ0(λξ)Y0(λα)] (3.10) and satisfies the conditions K(a,a) = ∂K(ξ,α) ∂ξ ∣ ∣ ∣ ξ=a = ∂2K(ξ,α) ∂ξ2 ∣ ∣ ∣ ξ=a =0 ∂3K(ξ,α) ∂ξ3 ∣ ∣ ∣ ξ=a =1 (3.11) according to properties of the influence functions (Stakgold and Holst, 2011). 4. Solution of the problem for the circular plate In the previous paper (Żur, 2016b), the possibility of solving the similar boundaryvalue problem was proposed for non-uniform annular plates without calculations. Based on the paper of Żur (2016b), the limit limξ→0Y0(λξ) = ∞, limξ→0K0(λξ) = ∞ of linear independent solutions to Eq. (2.3) for the circular plate can be presented in the following form K(ξ,λ,κ,K)a = J0(λξ)− χ ∑ j=1 KjJ0(λκj)G(ξ,κj) K(ξ,λ,κ,K)b = I0(λξ)− χ ∑ j=1 KjI0(λκj)G(ξ,κj) (4.1) 92 K.K. Żur where G(ξ,κj)= K(ξ,κj)H(ξ −κj) K(ξ,κj)= κj 4λ2 [2I0(λξ)K0(λκj)−2I0(λκj)K0(λξ)−πJ0(λκj)Y0(λξ)+πJ0(λξ)Y0(λκj)] (4.2) and κ= [κ1, . . . ,κχ] K= [K1, . . . ,Kχ] (4.3) and H(ξ−κj) is the Heaviside function. The characteristic equations ∆ = 0 of the circular plate for different boundary conditions and different values of the parameters κj and Kj are obtained from well known characteristic determinants given by: — clamped ∆(λ,κ,K)≡ ∣ ∣ ∣ ∣ ∣ ∣ K(ξ,λ,κ,K)a K(ξ,λ,κ,K)b ∂K(ξ,λ,κ,K)a ∂ξ ∂K(ξ,λ,κ,K)b ∂ξ ∣ ∣ ∣ ∣ ∣ ∣ |ξ=1 (4.4) — simply supported ∆(λ,κ,K)≡ ∣ ∣ ∣ ∣ ∣ K(ξ,λ,κ,K)a K(ξ,λ,κ,K)b M[K(ξ,λ,κ,K)a] M[K(ξ,λ,κ,K)b] ∣ ∣ ∣ ∣ ∣ ξ=1 (4.5) — free ∆(λ,κ,K)≡ ∣ ∣ ∣ ∣ ∣ M[K(ξ,λ,κ,K)a] M[K(ξ,λ,κ,K)b] V [K(ξ,λ,κ,K)a] V [K(ξ,λ,κ,K)b] ∣ ∣ ∣ ∣ ∣ ξ=1 (4.6) — sliding ∆(λ,κ,K)≡ ∣ ∣ ∣ ∣ ∣ ∣ ∂K(ξ,λ,κ,K)a ∂ξ ∂K(ξ,λ,κ,K)b ∂ξ V [K(ξ,λ,κ,K)a] V [K(ξ,λ,κ,K)b] ∣ ∣ ∣ ∣ ∣ ∣ ξ=1 (4.7) — elastic supports ∆(λ,κ,K,φ,ψ)≡ ∣ ∣ ∣ ∣ ∣ Φ[K(ξ,λ,κ,K)a] Φ[K(ξ,λ,κ,K)b] Ψ[K(ξ,λ,κ,K)a] Ψ[K(ξ,λ,κ,K)b] ∣ ∣ ∣ ∣ ∣ ξ=1 (4.8) 5. Solution of the problem for the annular plate The linear independent solutions to Eq. (2.3) for the annular plate can be presented in the following form Ba ≡ K(ξ,λ,κ,K)a = J0(λξ)− χ ∑ j=1 KjJ0(λκj)G(ξ,κj) Bb ≡ K(ξ,λ,κ,K)b = I0(λξ)− χ ∑ j=1 KjI0(λκj)G(ξ,κj) Bc ≡ K(ξ,λ,κ,K)c = Y0(λξ)− χ ∑ j=1 KjY0(λκj)G(ξ,κj) Bd ≡ K(ξ,λ,κ,K)d = K0(λξ)− χ ∑ j=1 KjK0(λκj)G(ξ,κj) (5.1) Quasi-Green’s function approach to fundamental frequency analysis... 93 The characteristic equations ∆ = 0 of the annular plate for different boundary conditions and different values of the parameters κj and Kj are obtained fromwell known characteristic deter- minants given by: — free outer edge and clamped inner edge ∆(λ,κ,K)≡ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ M[Ba]|ξ=1 M[Bb]|ξ=1 M[Bc]|ξ=1 M[Bd]|ξ=1 V [Ba]|ξ=1 V [Bb]|ξ=1 V [Bc]|ξ=1 V [Bd]|ξ=1 Ba|ξ=ξ1 Bb|ξ=ξ1 Bc|ξ=ξ1 Bd|ξ=ξ1 ∂Ba ∂ξ ∣ ∣ ∣ ξ=ξ1 ∂Bb ∂ξ ∣ ∣ ∣ ξ=ξ1 ∂Bc ∂ξ ∣ ∣ ∣ ξ=ξ1 ∂Bd ∂ξ ∣ ∣ ∣ ξ=ξ1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (5.2) — free outer edge and simply supported inner edge ∆(λ,κ,K)≡ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (M[Ba]|ξ=1 M[Bb]|ξ=1 M[Bc]|ξ=1 M[Bd]|ξ=1 V [Ba]|ξ=1 V [Bb]|ξ=1 V [Bc]|ξ=1 V [Bd]|ξ=1 Ba|ξ=ξ1 Bb|ξ=ξ1 Bc|ξ=ξ1 Bd|ξ=ξ1 M[Ba]|ξ=ξ1 M[Bb]|ξ=ξ1 M[Bc]|ξ=ξ1 M[Bd]|ξ=ξ1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (5.3) — free both edges ∆(λ,κ,K)≡ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ M[Ba]|ξ=1 M[Bb]|ξ=1 M[Bc]|ξ=1 M[Bd]|ξ=1 V [Ba]|ξ=1 V [Bb]|ξ=1 V [Bc]|ξ=1 V [Bd]|ξ=1 M[Ba]|ξ=ξ1 M[Bb]|ξ=ξ1 M[Bc]|ξ=ξ1 M[Bd]|ξ=ξ1 V [Ba]|ξ=ξ1 V [Bb]|ξ=ξ1 V [Bc]|ξ=ξ1 V [Bd]|ξ=ξ1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (5.4) — free outer edge and sliding inner edge ∆(λ,κ,K)≡ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ M[Ba]|ξ=1 M[Bb]|ξ=1 M[Bc]|ξ=1 M[Bd]|ξ=1 V [Ba]|ξ=1 V [Bb]|ξ=1 V [Bc]|ξ=1 V [Bd]|ξ=1 ∂Ba ∂ξ ∣ ∣ ∣ ξ=ξ1 ∂Bb ∂ξ ∣ ∣ ∣ ξ=ξ1 ∂Bc ∂ξ ∣ ∣ ∣ ξ=ξ1 ∂Bd ∂ξ ∣ ∣ ∣ ξ=ξ1 V [Ba]|ξ=ξ1 V [Bb]|ξ=ξ1 V [Bc]|ξ=ξ1 V [Bd]|ξ=ξ1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (5.5) — free inner edge and clamped outer edge ∆(λ,κ,K)≡ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ M[Ba]|ξ=ξ1 M[Bb]|ξ=ξ1 M[Bc]|ξ=ξ1 M[Bd]|ξ=ξ1 V [Ba]|ξ=ξ1 V [Bb]|ξ=ξ1 V [Bc]|ξ=ξ1 V [Bd]|ξ=ξ1 Ba|ξ=1 Bb|ξ=1 Bc|ξ=1 Bd|ξ=1 ∂Ba ∂ξ ∣ ∣ ∣ ξ=1 ∂Bb ∂ξ ∣ ∣ ∣ ξ=1 ∂Bc ∂ξ ∣ ∣ ∣ ξ=1 ∂Bd ∂ξ ∣ ∣ ∣ ξ=1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (5.6) — elastic constraints at the inner edge and free outer edge ∆(λ,κ,K,φ,ψ)≡ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ M[Ba]|ξ=1 M[Bb]|ξ=1 M[Bc]|ξ=1 M[Bd]|ξ=1 V [Ba]|ξ=1 V [Bb]|ξ=1 V [Bc]|ξ=1 V [Bd]|ξ=1 Φ[Ba]|ξ=ξ1 Φ[Bb]|ξ=ξ1 Φ[Bc]|ξ=ξ1 Φ[Bd]|ξ=ξ1 Ψ[Ba]|ξ=ξ1 Ψ[Bb]|ξ=ξ1 Ψ[Bc]|ξ=ξ1 Ψ[Bd]|ξ=ξ1 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ (5.7) 94 K.K. Żur 6. Results and discussion The numerical results for fundamental frequencies of elastically supported circular plates are presented in Tables 1 and 2 with comparison to the results by Azimi (1988), Ding (1994), Wang and Wang. (2003). The numerical results for fundamental frequencies of free vibration of free circular plates with rigid ring supports are presented in Table 3 with comparison to the results by Wang (2014). The numerical results for fundamental frequencies of free vibration of free elastically supported annular plates with different boundary condition at the inner edge are presented inTables 4 and5 for different combinations of the radius of the core and supports.The fundamental frequencies of free vibration of circular plates with elastic constraints and interior ring supports of variable stiffness are presented in Table 6. Additionally, the eigenvalues of circular plates with elastic constraints depending on radius and stiffness of interior ring supports are shown in Figs. 2 and 3. Table 1.The fundamental frequency λ0 of free vibration of circular plates with the elastic ring support K1 κ1 Dimensionless frequency λ0 Boundary conditions Clamped Simply Free Sliding supported 10 0 GF 3.196 2.221 0.211 0.212 0.1 GF 3.272 2.360 1.115 1.171 0.2 GF 3.326 2.460 1.357 1.383 Wang andWang (2003) 3.325 2.460 – – Azimi (1988) 3.326 2.461 – – Ding (1994) 3.322 – – – 0.3 GF 3.348 2.523 1.497 1.532 0.4 GF 3.338 2.547 1.620 1.656 Wang andWang (2003) 3.338 2.547 – – Azimi (1988) 3.338 2.547 – – Ding (1994) 3.334 – – – 0.5 GF 3.304 2.530 1.736 1.765 0.6 GF 3.262 2.478 1.844 1.856 Wang andWang (2003) 3.262 2.478 – – Azimi (1988) 3.262 2.479 – – Ding (1994) 3.262 – – – 0.7 GF 3.225 2.403 1.928 1.928 0.8 GF 3.204 2.321 1.960 1.980 Wang andWang (2003) 3.204 2.321 1.961 – Azimi (1988) 3.199 2.321 – – Ding (1994) 3.204 – – – The fundamental frequencies of free vibration of annular plates with the clamped outer edge and the free inner edge (rigid interior support) are presented in Table 7 with comparison to the results by Vega (2000). The eigenvalues of free annular plates with elastic constraints at the inner edge and interior ring supports are presented in Table 8 for different combinations of the radius of the core and supports. Quasi-Green’s function approach to fundamental frequency analysis... 95 Table 2.The fundamental frequency λ0 of free vibration of circular plates with the elastic ring support K1 κ1 Dimensionless frequency λ0 Boundary conditions Clamped Simply Free Sliding supported 1000 0 GF 3.204 2.223 0.666 0.667 0.1 GF 4.677 3.805 1.946 2.238 0.2 GF 5.175 4.202 2.049 2.418 Wang andWang (2003) 5.175 4.202 – – Azimi (1988) 5.187 4.210 – – Ding (1994) 4.929 – – – 0.3 GF 5.763 4.682 2.187 2.656 0.4 GF 6.110 5.276 2.374 2.979 Wang andWang (2003) 6.110 5.276 – – Azimi (1988) 6.129 5.282 – – Ding (1994) 6.114 – – – 0.5 GF 5.195 5.136 2.619 3.403 0.6 GF 4.503 4.479 2.891 3.803 Wang andWang (2003) 4.503 4.479 – – Azimi (1988) 4.512 4.486 – – Ding (1994) 4.492 – – – 0.7 GF 3.967 3.962 2.992 3.707 0.8 GF 3.539 3.532 2.787 3.438 Wang andWang (2003) 3.539 3.532 – – Azimi (1988) 3.547 3.537 – – Ding (1994) 3.547 – – – Table 3. The fundamental frequency λ0 of free vibration of free circular plates with the rigid ring support K1 κ1 Dimensionless frequency λ0 GF Wang (2014) ∞ 0 3.751 3.752 0.1 3.909 3.909 0.2 4.275 4.275 0.3 4.851 4.851 0.4 5.706 5.707 0.5 6.929 6.929 0.6 8.396 8.390 0.7 8.960 8.959 0.8 7.809 7.809 0.9 6.235 6.235 1.0 4.935 4.935 96 K.K. Żur Table 4. The fundamental frequency λ0 of free vibration of free annular plates with different boundary conditions at the inner edge and interior elastic support K1 ξ1 κ1 Dimensionless frequency Boundary conditions at the inner edge Clamped Simply Free Sliding supported 10 0.1 0.2 λ0 2.043 1.826 1.350 1.364 0.1 0.4 2.050 1.886 1.617 1.624 0.1 0.6 2.207 2.091 1.848 1.848 0.1 0.9 2.658 2.537 1.931 1.946 0.3 0.5 2.569 1.918 1.730 1.778 0.3 0.7 2.697 2.209 1.975 1.975 0.3 0.9 3.003 2.587 1.959 2.041 0.5 0.7 3.616 2.235 2.039 2.068 0.5 0.9 3.812 2.736 2.086 2.193 0.7 0.8 6.077 2.634 2.311 2.363 0.7 0.9 6.119 3.008 2.401 2.436 0.8 0.9 9.197 3.275 2.658 2.658 Table 5. The fundamental frequency λ0 of free vibration of free annular plates with different boundary conditions at the inner edge and interior elastic support K1 ξ1 κ1 Dimensionless frequency Boundary conditions at the inner edge Clamped Simply Free Sliding supported 1000 0.1 0.2 λ0 2.087 1.925 1.979 2.091 0.1 0.4 2.117 1.105 2.331 2.400 0.1 0.6 9.335 3.843 2.852 2.934 0.1 0.9 4.685 4.215 2.467 2.545 0.3 0.5 2.684 7.401 2.352 2.859 0.3 0.7 12.106 4.769 2.887 3.561 0.3 0.9 6.175 5.214 2.439 2.987 0.5 0.7 13.526 13.323 2.859 4.266 0.5 0.9 9.260 7.249 2.648 4.106 0.7 0.8 5.526 6.437 3.293 6.321 0.7 0.9 8.396 8.054 3.432 7.049 0.8 0.9 9.697 7.774 4.294 8.334 Quasi-Green’s function approach to fundamental frequency analysis... 97 Fig. 2. The fundamental frequency of the circular plate with elastic constraints (φ =100, ψ =10) depending on the radius and stiffness of interior ring supports Fig. 3. The fundamental frequency of the circular plate with elastic constraints (φ =0.1, ψ =100) depending on the radius and stiffness of interior ring supports Table 7. The fundamental frequency λ0 of free vibration of annular plates with the clamped outer edge and free inner edge and the rigid interior support (K1 →∞) κ1 ξ1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Dimensionless frequency λ0 0.1 GF 5.335 5.946 6.262 5.537 4.782 4.215 3.786 3.453 Vega et al. (2000) 5.335 5.946 6.262 5.537 4.782 4.215 3.786 3.453 0.2 GF 5.890 6.530 5.996 5.051 4.370 3.880 3.512 Vega et al. (2000) 5.890 6.530 5.996 5.051 4.370 3.880 3.512 0.3 GF 6.723 7.100 5.853 4.856 4.194 3.727 Vega et al. (2000) 6.723 7.100 5.853 4.856 4.194 3.727 0.4 GF 7.912 7.527 5.856 4.821 4.154 Vega et al. (2000) 7.912 7.527 5.856 4.821 4.154 0.5 GF 9.612 7.874 5.969 4.894 Vega et al. (2000) 9.612 7.874 5.969 4.894 0.6 GF 11.912 8.256 6.195 Vega et al. (2000) 11.912 8.256 6.196 0.7 GF 14.146 8.785 Vega et al. (2000) 14.147 8.785 0.8 GF 15.991 98 K.K. Żur Table 6.The fundamental frequency λ0 of free vibration of circular plates with elastic constra- ints and the interior ring support K1 κ1 Dimensionless Elastic parameters at the outer edge frequency φ =100, ψ =10 φ =0.1, ψ =100 0 2.056 2.203 0.1 2.129 2.337 0.2 2.190 2.434 0.3 2.241 2.497 10 0.4 λ0 2.285 2.524 0.5 2.320 2.513 0.6 2.344 2.469 0.7 2.357 2.400 0.8 2.361 2.321 0 2.056 2.203 0.1 2.441 2.928 0.2 2.610 3.247 0.3 2.770 3.486 100 0.4 λ0 2.960 3.650 0.5 3.188 3.648 0.6 3.387 3.445 0.7 3.318 3.145 0.8 3.142 2.809 0 2.056 3.563 0.1 2.715 3.600 0.2 2.850 3.878 0.3 3.032 4.176 1000 0.4 λ0 3.287 4.490 0.5 3.629 4.700 0.6 3.907 4.321 0.7 3.719 3.786 0.8 3.429 3.263 0 2.714 3.624 0.1 2.769 3.746 0.2 2.888 3.973 0.3 3.068 4.250 ∞ 0.4 λ0 3.327 4.536 0.5 3.664 4.700 0.6 3.908 4.410 0.7 3.743 3.892 0.8 3.458 3.354 The Poisson ratio is taken as ν = 0.3 for all considered cases. The numerical results are obtained by using the Newton-Raphson method and Mathematica v10 software. The obtained results are in good agreement with the results obtained by othermethods presented in literature and can be used to validate the accuracy of other numerical methods as benchmark values. Quasi-Green’s function approach to fundamental frequency analysis... 99 Table 8. The fundamental frequency λ0 of free vibration of free annular plates with elastic constraints at the inner edge and interior ring support K1 ξ1 κ1 Dimensionless Elastic parameters at the inner edge frequency φ =100, ψ =10 φ =0.1, ψ =100 10 0.1 0.2 λ0 1.146 1.556 0.1 0.4 1.530 1.740 0.1 0.6 1.814 2.001 0.1 0.9 2.031 2.439 0.3 0.5 1.537 1.956 0.3 0.7 1.839 2.238 0.3 0.9 1.974 2.625 0.5 0.7 1.648 2.296 0.5 0.9 1.894 2.788 0.7 0.8 1.510 2.680 0.7 0.9 1.759 3.050 0.8 0.9 1.603 3.237 1000 0.1 0.2 λ0 2.101 1.941 0.1 0.4 2.460 6.152 0.1 0.6 3.091 3.663 0.1 0.9 2.855 4.239 0.3 0.5 2.906 0.852 0.3 0.7 3.739 4.951 0.3 0.9 3.212 6.150 0.5 0.7 4.332 6.707 0.5 0.9 4.267 7.911 0.7 0.8 6.374 7.977 0.7 0.9 7.185 8.053 0.8 0.9 8.319 8.196 ∞ 0.1 0.2 λ0 3.308 1.296 0.1 0.4 3.687 1.463 0.1 0.6 3.972 2.000 0.1 0.9 3.022 1.351 0.3 0.5 1.175 1.937 0.3 0.7 2.071 2.120 0.3 0.9 3.834 1.271 0.5 0.7 2.995 2.030 0.5 0.9 2.644 1.335 0.7 0.8 1.998 0.999 0.7 0.9 1.683 1.385 0.8 0.9 1.046 1.293 7. Conclusions In thispaper, thequasi-Green functionhasbeenemployed to solvenatural vibrationof elastically supported thin circular and annular plates with different boundary conditions. The advantage of quasi-Green’s function is the obtaining of characteristic equations without calculating nontrivial constants in complicated forms. Additionally, the number of supports of circular and annular 100 K.K. Żur plates does not influence the dimension of characteristic matrices, because the continuity condi- tions can beneglected. In the presented approach, the solution to the boundaryvalue problem is much simpler.Thequasi-Green function approach canbeused to the frequency analysis of plates and beams with other discrete elements such as an additional mass or a mass on the spring. The exact frequencies of vibration presented in a non-dimensional form can serve as benchmark values for researchers to validate their numerical methods applied in similar problems presented in the paper. Acknowledgements The research was conducted within S/WZ/1/2014 project and was financed by the funds of the Ministry of Science and Higher Education, Poland. References 1. Azimi S., 1988, Free vibration of circular plates with elastic or rigid interior support, Journal of Sound and Vibration, 120, 37-53 2. BodineR.Y., 1967,Vibrationof a circular plate supportedbya concentric ringof arbitrary radius, Journal of the Acoustical Society of America, 41, 1551 3. Ding Z., 1994, Free vibration of arbitrarily shaped plates with concentric ring elastic and rigid supports,Computers and Structures, 50, 685-692 4. Jaroszewicz J., Zoryj L., 2005,Methods of Free Axisymmetric Vibration Analysis of Circular Plates Using by Influence Functions, Bialystok University of Technology, Poland 5. Kukla S., Szewczyk M., 2004, The Green’s functions for vibration problems of circular pla- tes with elastic ring supports, Scientific Research of the Institute of Mathematics and Computer Science, 4, 1, 79-86 6. Kukla S., Szewczyk M., 2005, Application of Green’s function method in frequency analysis of axisymmetric vibration of annular plates with elastic ring supports, Scientific Research of the Institute of Mathematics and Computer Science, 3, 1, 67-72 7. Kukla S., Szewczyk M., 2007, Frequency analysis of annular plates with elastic concentric supports by Green’s function method, Journal of Sound and Vibration, 300, 387-393 8. Kunukkasseril V.X., Swamidas A.S.J., 1974,Vibration of continuous circular plates, Interna- tional Journal of Solids and Structures, 10, 603-619 9. Laura P.A.A., Gutierrez R.H., Vera S.A., Vega D.A., 1999, Transverse vibrations of cir- cular plate with a free edge and a concentric circular support, Journal of Sound and Vibration, 223, 5, 843-845 10. LiuC.F., ChenG.T., 1995,A simple finite element analysis of axisymmetric vibration of annular and circular plates, International Journal of Mechanical Sciences, 37, 8, 861-871 11. McLachlan N.W., 1955,Bessel Functions for Engineers, Clarendon Press, Oxford 12. Rao L.B., Rao C.K., 2014a, Frequencies of circular plate with concentric ring and elastic edge support, Frontiers on Mechanical Engineering, 9, 2, 168-176 13. Rao L.B., Rao C.K., 2014b, Frequency analysis of annular plates with inner and outer ed- ges elastically restrained and resting on Winkler foundation, International Journal of Mechanical Sciences, 81, 184-194 14. SinghA.V.,Mirza S., 1976, Free axisymmetric vibration of a circular plate elastically supported along two concentric circles, Journal of Sound and Vibration, 48, 425-429 15. Vega D.A., Vera S.A., Laura P.A.A.,Gutierrez R.H., Pronsato M.E., 1999, Transverse vibrations of an annular circular plate with free edges and an intermediate concentric circular support, Journal of Sound and Vibration, 223, 493-496 Quasi-Green’s function approach to fundamental frequency analysis... 101 16. Vega D.A., Laura P.A.A., Vera S.A., 2000, Vibrations of an annular isotropic plate with one edge clamped or simply supported and an intermediate concentric circular supports, Journal of Sound and Vibration, 233, 1, 171-174 17. WangC.M.,ThevendranV., 1993,Vibrationanalysis of annularplateswith concentric supports using a variant of Rayleigh-Ritzmethod, Journal of Sound and Vibration, 163, 137-149 18. Wang C.Y., 2001,On the fundamental frequency of a circular plate supported on a ring, Journal of Sound and Vibration, 243, 5, 945-946 19. Wang C.Y., Wang C.M., 2003, Fundamental frequencies of circular plates with internal elastic ring support, Journal of Sound and Vibration, 263, 1071-1078 20. Wang C.Y., 2006, Fundamental frequencies of annular plates with movable edges, Journal of Sound and Vibration, 290, 524-528 21. Wang C.Y., 2014, The vibration modes of concentrically supported free circular plates, Journal of Sound and Vibration, 333, 835-847 22. ŻurK.K., 2015,Green’s function in frequency analysis of circular thin plates of variable thickness, Journal of Theoretical and Applied Mechanics, 53, 4, 873-884 23. Żur K.K., 2016a,Green’s function approach to frequency analysis of thin circular plates,Bulletin of the Polish Academy of Sciences – Technical Sciences, 64, 1, 181-188 24. Zur K.K., 2016b, Green’s function for frequency analysis of thin annular plates with nonlinear variable thickness,Applied Mathematical Modelling, 40, 5-6, 3601-3619 Manuscript received November 2, 2015; accepted for print May 6, 2016