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Frequency Equations for the In-Plane Vibration of Circular Annular Disks

Frequency Equations for the In-Plane Vibration of Circular Annular Disks Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2010, Article ID 501902, 8 pages doi:10.1155/2010/501902 Research Article Frequency Equations for the In-Plane Vibration of Circular Annular Disks S. Bashmal, R. Bhat, and S. Rakheja Department of Mechanical and Industrial Engineering, Concordia University, 1455 De Maisonneuve Bloulevard W., Montreal, QC, Canada H3G 1M8 Correspondence should be addressed to S. Bashmal, bashmal@gmail.com Received 23 October 2009; Revised 23 June 2010; Accepted 25 June 2010 Academic Editor: Miguel Ayala Botto Copyright © 2010 S. Bashmal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with the in-plane vibration of circular annular disks under combinations of different boundary conditions at the inner and outer edges. The in-plane free vibration of an elastic and isotropic disk is studied on the basis of the two-dimensional linear plane stress theory of elasticity. The exact solution of the in-plane equation of equilibrium of annular disk is attainable, in terms of Bessel functions, for uniform boundary conditions. The frequency equations for different modes can be obtained from the general solutions by applying the appropriate boundary conditions at the inner and outer edges. The presented frequency equations provide the frequency parameters for the required number of modes for a wide range of radius ratios and Poisson’s ratios of annular disks under clamped, free, or flexible boundary conditions. Simplified forms of frequency equations are presented for solid disks and axisymmetric modes of annular disks. Frequency parameters are computed and compared with those available in literature. The frequency equations can be used as a reference to assess the accuracy of approximate methods. 1. Introduction solid disks clamped at the outer edge have been investigated in a few recent studies. Farag and Pan [11] evaluated the frequency parameters and the mode shapes of in-plane The out-of-plane vibration properties of circular disks sub- jected to a variety of boundary conditions have been exte- vibration of solid disks clamped at the outer edge using nsively investigated (e.g., [1–4]). The in-plane vibration assumed deflection modes in terms of trigonometric and analyses of circular disk, however, have been gaining increas- Bessel functions. Park [12] studied the exact frequency ing attention only in the recent years. Much of the interest equation for the solid disk clamped at the outer edge. could be attributed to important significance of the in- The in-plane vibration analyses in the above reported plane vibration in various practical problems such as the studies were limited to solid disks with either free or clamped vibration of railway wheels, disk brakes, and hard disk drives outer edge. The in-plane free vibration of annular disks contributing to noise and structural vibration [5–7]. with different boundary conditions has also been addressed The in-plane vibration of circular disks was first in a few studies. The variations in the in-plane vibration attempted by Love [8] who formulated the equations of frequency parameters of annular disks with free edges were investigated as function of the size of the opening by Ambati motion for a thin solid circular disk with free outer edge together with the general solution. The equations of motion et al. [13]. The variation ranged from a solid disk to a were subsequently solved by Onoe [9] to obtain the exact thin ring, while the validity of the analytical results was frequency equations corresponding to different modes of a demonstrated using the experimental data. Another study solid disk with free outer edge. Holland [10] evaluated the investigated the free vibration and dynamic response char- frequency parameters and the corresponding mode shapes acteristics of an annular disk with clamped inner boundary for a wide range of Poisson’s ratios and the vibration response and a concentrated radial force applied at the outer boundary to an in-plane force. The in-plane vibration characteristics of [14]. Irie et al. [15] investigated the modal characteristics 2 Advances in Acoustics and Vibration of in-plane vibration of annular disks using transfer matrix formulation while considering free and clamped inner and outer edges. The above reported studies on in-plane vibration of solid and annular disks have employed different methods of analyses. The finite-element technique has also been used to examine the validity of analytical methods (e.g., [11, 14]). The exact frequency equations of in-plane vibration, how- ever, have been limited only to solid disks. Such analyses for the annular disks pose more complexities due to presence of different combinations of boundary conditions at the inner b and outer edges. This study aims at generalized formulation for in-plane vibration analyses of circular annular disks under different combinations of clamped, free, or flexible Figure 1: Geometry and coordinate system used for in-plane boundary conditions at the inner and outer edges. The vibration analysis of an annular disk. equations of motion are solved for the general case of annular disks. The exact frequency equations are presented for different combinations of boundary conditions, including Assuming harmonic oscillations corresponding to a the flexible boundaries, for various radius ratios, while the natural frequency ω, the potential functions φ and ψ can be solid disk is considered as special cases of the generalized represented by formulation. φ(ξ, θ, t) = Φ(ξ) cos nθ sin ωt, (4) 2. Theory ψ(ξ, θ, t) = Ψ(ξ) sin nθ sin ωt, (5) The equations of the in-plane vibration of a circular disk are formulated for an annular disk shown in Figure 1. The disk is considered to be elastic with thickness h,outer radius b where n is the circumferential wave number or nodal and inner radius a. The material is assumed to be isotropic diameter number. Upon substituting for u and u in terms r θ with mass density ρ,Young’s modulus E, and Poisson ratio v. of ξ, Φ,and Ψ from (2)to(5), in (1), the equations of motion The equations of dynamic equilibrium in terms of in-plane reduce to the following uncoupled form: displacements along the radial and circumferential directions can be found in many reported studies (e.g., [11, 16]). These 2 2 ∇ Φ =−λ Φ, (6) equations of motion in the polar coordinate system (r, θ)can be written as 2 2 ∇ Ψ =−λ Ψ, (7) 2 2 2 2 ∂ u ∂ u 1 ∂u u C ∂ u r r r r r 2 T − C + − − 2 2 2 2 2 ∂t ∂r r ∂r r r ∂θ where λ and λ are nondimensional frequency parameters 1 2 defined as 1 1+ v ∂ u 1 3 − v ∂u θ θ 2 2 − C + C = 0, T T r 1 − v ∂r∂θ r 1 − v ∂θ 2 2 2 2 2 (1) ω 1 − v ρb 2ω (1+ v)ρb 2 2 2 2 2 2 λ = , λ = , 1 2 ∂ u ∂ u 1 ∂u u C ∂ u θ θ θ θ θ 2 L E E − C + − − 2 2 2 2 2 (8) ∂t ∂r r ∂r r r ∂θ 2 2 ∂ 1 ∂ n ∇ = + − . 2 2 1 1+ v ∂ u 1 3 − v ∂u ∂ξ ξ ∂ξ ξ r r 2 2 − C + C = 0, T T r 1 − v ∂r∂θ r 1 − v ∂θ Equations (6)and (7) are the parametric Bessel equations where u and u are the radial and circumferential dis- r θ and their general solutions are attainable in terms of the placements, respectively, along the r and θ directions, C = Bessel functions as [18] E/ρ(1 − v )and C = E/2ρ(1 + v). Following Love’s theory [8], the radial and circumfer- ential displacements can be expressed in terms of the Lame´ Φ = A J (λ ξ) + B Y (λ ξ) , (9) n n 1 n n 1 Potentials φ and ψ [17], as Ψ = C J (λ ξ) + D Y (λ ξ) , (10) n n 2 n n 2 1 ∂φ 1 ∂ψ u = + ,(2) b ∂ξ ξ ∂θ where J and Y are the Bessel functions of the first and n n second kind of order n,respectively, and A , B , C ,and D n n n n 1 1 ∂φ ∂ψ u = − ,(3) are the deflection coefficients. b ξ ∂θ ∂ξ The radial and circumferential displacements can then be where ξ = r/b. expressed in terms of the Bessel functions by substituting for Advances in Acoustics and Vibration 3 Φ and Ψ in (2)and (3). The resulting expressions for the A direct substitution of u and u from (11) in the above r θ radial and circumferential displacements can be expressed as: equations would result in second derivatives of the Bessel functions. Alternatively, the above equation for the boundary conditions may be expressed in terms of Φ and Ψ through u = A X (λ ξ) + B Z (λ ξ) r n n 1 n n 1 direct substitution of u and u from (2)and (3), respectively. r θ The boundary conditions in terms of N canthusbeobtained as + [C J (λ ξ) + D Y (λ ξ)] cos nθ sin ωt, n n 2 n n 2 (11) 2 2 ∂ Φ n ∂Ψ n n v nv ∂Ψ + − Ψ − Φ − ( ) ( ) u =− C X λ ξ + D Z λ ξ θ n n 2 n n 2 2 2 2 ξ ∂ξ ξ ξ ξ ∂ξ b ∂ξ (19) v ∂Φ n + + Ψ = 0. + [A J (λ ξ) + B Y (λ ξ)] sin nθ sin ωt, n n 1 n n 1 ξ ∂ξ ξ where Rearranging (19) results in ∂ n X (λ ξ) = J (λ ξ) =− J (λ ξ) + λ J (λ ξ), n i n i n i i n−1 i ∂ξ ξ 2 2 ∂ v ∂ n v n ∂ n + − Φ + (1 − v) − Ψ = 0. (20) ∂ n 2 2 ξ ∂ξ ξ ξ ∂ξ ξ ∂ξ Z (λ ξ) = Y (λ ξ) = − Y (λ ξ) + λ Y (λ ξ) , n i n i n i i n−1 i ∂ξ ξ i = 1, 2. The second order derivative term (∂ Φ/∂ξ )in(20)can be (12) eliminated by adding and subtracting the term ((1/ξ)(∂/∂ξ)− 2 2 2 (n /ξ )+ λ )Φ,which yields 2.1. Free and Clamped Boundary Conditions. Equations (11) represent the solutions for distributions of the radial and 2 2 2 ∂ 1 ∂ n 1 ∂ n 2 2 circumferential displacements for the general case of an + − + λ − − + λ 1 1 2 2 ξ ∂ξ ξ ξ ∂ξ ξ ∂ξ annular disk. The evaluations of the natural frequencies (21) and arbitrary deflection coefficients (A , B , C and D ), n n n n v ∂ n v n ∂ n however, necessitate the consideration of the in-plane free ( ) + − Φ + 1 − v − Ψ = 0. 2 2 ξ ∂ξ ξ ξ ∂ξ ξ vibration response under different combinations of bound- ary conditions at the inner and the outer edges. For the annular disk clamped at the outer edge (ξ = 1), the From (6), it can be seen that the terms within the first application of boundary conditions (u = 0and u = 0) r θ parenthesis are identically equal to zero. Equation (21) must satisfy the following for the general solutions (11): describing the boundary condition associated with N can be 2 2 further simplified upon substitutions for λ = λ (1 − v)/2, A X (λ ) + B Z (λ ) + n[C J (λ ) + D Y (λ )] = 0, 1 2 (13) n n 1 n n 1 n n 2 n n 2 which yields C X (λ ) + D Z (λ ) + n[A J (λ ) + B Y (λ )] = 0. (14) n n 2 n n 2 n n 1 n n 1 In a similar manner, the solution must satisfy the n 1 ∂ 1 1 ∂ 1 − − λ Φ + n − Ψ = 0. (22) 2 2 following for the clamped inner edge (ξ = β), where β = a/b ξ ξ ∂ξ 2 ξ ∂ξ ξ is the radius ratio between inner and outer radii of the disk: A X λ β + B Z λ β + C J λ β + D Y λ β = 0, Similarly, the boundary condition equation associated with n n 1 n n 1 n n 2 n n 2 N (18) can be simplified as rθ (15) 2n ∂ 1 2 ∂ 2n C X λ β + D Z λ β + A J λ β + B Y λ β = 0. n n 2 n n 2 n n 1 n n 1 2 − + Φ + − + λ Ψ = 0. (23) ξ ∂ξ ξ ξ ∂ξ ξ (16) The conditions involving at the free edges are satisfied when Upon substitutions for Φ and Ψ from (16)in(26)and (27), the radial (N ) and circumferential (N ) in-plane forces at r rθ the boundary condition equations for the free edges are the edge are zero [11], such that obtained, which involve only first derivatives of the Bessel functions. For an annular disk with free inner and outer −Eh ∂u v ∂u v r θ N = + + u = 0, (17) r ξ edges, (22)and (23) represent the conditions at both the (1 − v )b ∂ξ ξ ∂θ ξ inner and the outer boundaries (ξ = 1and ξ = β). The equations for the free edge boundary conditions can −Eh 1 ∂u ∂u u r θ θ N = + − = 0. (18) be expressed in the matrix form in the four deflection rθ ( ) 2 1+ v b ξ ∂θ ∂ξ ξ coefficients, as 4 Advances in Acoustics and Vibration Table 1: Frequency equations for the solid disks corresponding to free and clamped edge conditions. Boundary conditions at ξ = 1 Clamped Free Radial J (λ ) = 0 λ J (λ ) = (1 − v)J (λ ) 1 1 1 0 1 1 1 n = 0 Circumferential J (λ ) = 0 λ J (λ ) = 2J (λ ) 1 2 2 0 2 1 2 [X (λ )J (λ )] + [X (λ )J (λ )] = 1 2 1 1 1 1 1 2 n = 1 X (λ )X (λ ) = n J (λ )J n 1 n 2 n 1 n 2 (2 − (λ /2))J (λ )J (λ ) 2 1 2 1 1 (λ ) [X (λ )−QJ (λ )][X (λ )−QJ (λ )] = n (Q − 1) n 2 n 2 n 1 n 1 n> 1 where Q = λ /2(n − 1) ⎡ ⎤ 2 2 λ λ 2 2 2 2 n − J (λ ) − X (λ ) n − Y (λ ) − Z (λ ) nX (λ ) − nJ (λ ) nZ (λ ) − nY (λ ) ⎢ n 1 n 1 n 1 n 1 n 2 n 2 n 2 n 2 ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ 2 2 2 2 ⎥ 2nJ (λ ) − 2nX (λ ) 2nY (λ ) − 2nZ (λ ) 2X (λ ) − 2n − λ J (λ ) 2Z (λ ) − 2n − λ Y (λ ) ⎢ n 1 n 1 n 1 n 1 n 2 n 2 n 2 n 2 ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ 2 2 2 2 ⎢                 ⎥ n λ 1 n λ 1 n n n n 2 2 ⎢ ⎥ − J λ β − X λ β − Y λ β − Z λ β X λ β − J λ β Z λ β − Y λ β n 1 n 1 n 1 n 1 n 2 n 2 n 2 n 2 ⎢ 2 2 2 2 ⎥ β 2 β β 2 β β β β β ⎢ ⎥ ⎢ ⎥ 2 2 (24) ⎣ 2n 2n 2n 2n 2 2n 2 2n ⎦ 2 2 J λ β − X λ β Y λ β − Z λ β X λ β − − λ J λ β Z λ β − − λ Y λ β n 1 n 1 n 1 n 1 n 2 n 2 n 2 n 2 2 2 2 2 2 2 β β β β β β β β ⎧ ⎫ ⎪ n⎪ ⎪ ⎪ ⎨ ⎬ × ={0}. ⎪ C ⎪ ⎪ ⎪ ⎩ ⎭ The determinant of the above matrix yields the frequency involving the two boundary conditions are summarized in equation for the annular disk with free inner and outer edge Table 1,where X (λ ) is the derivative of the Bessel function n 1 conditions. J evaluated at the outer edge (ξ = 1). For annular disks, For the clamped inner and outer edges, the equations for simplified frequency equations can be obtained for the the boundary conditions can be obtained directly from (13) axisymmetric modes. These equations where expressed in to (16), such that Table 2 for the four combinations of boundary conditions. ⎡ ⎤ X (λ ) Z (λ ) nJ (λ ) nY (λ ) n 1 n 1 n 2 n 2 ⎧ ⎫ ⎢ ⎥ ⎪ A ⎪ nJ (λ ) nY (λ ) X (λ ) Z (λ ) ⎪ ⎪ ⎢ n 1 n 1 n 2 n 2 ⎥ ⎪ ⎪ 2.2. Flexible Boundary Conditions. In the above analysis, ⎨ ⎬ ⎢        ⎥ n n ⎢ ⎥ X λ β Z λ β J λ β Y λ β ={0}. the boundary conditions considered are either clamped n 1 n 1 n 2 n 2 ⎢ ⎥ ⎪ C ⎪ β β ⎪ n⎪ ⎢ ⎥ ⎪ ⎪ ⎩ ⎭ or free. However, Flexible boundary conditions may be ⎣ n n ⎦ J λ β Y λ β X λ β Z λ β n 1 n 1 n 2 n 2 considered more representative of many practical situations. β β The proposed formulations can be further employed to (25) study the in-plane vibration of solid as well as annular In the above equations, (24)and (25), the top two rows disks with flexible boundary conditions. Artificial springs describe the boundary condition at the outer edge, while may be applied to describe the flexible boundary conditions the bottom two rows are associated with those at the at the inner or the outer edge of an annular disk. A inner edge. The equations for the boundary conditions number of studies on the analysis of out-of-plane vibration involving combinations of free and clamped edges can thus characteristics of circular plates and cylindrical shells have be directly obtained from the above two equations. For the employed uniformly distributed artificial springs around the free inner edge and clamped outer edge, denoted as “free- edge to represent a flexible boundary conditions or a flexible clamped” condition, the matrix equation comprises the tip joint [19–22]. two rows of the matrix in (25) and the lower two rows The effects of flexible boundary conditions on the in- from (24). For the clamped inner edge and free outer edge, plane free vibration of circular disks have been considered denoted as “Clamped-Free” condition, the matrix equation is in a recent study by the authors [23] using the Rayleigh- formulated in the similar manner using the lower and upper Ritz approach. Artificial springs, distributed along the radial two rows from (25)and (24), respectively. and circumferential directions at the free outer and/or inner The in-plane vibration analysis of a solid disk can be edges, were considered to simulate for flexible boundary shown as a special case of the above generalized formulations. conditions. The exact solution of the frequency equations Upon eliminating the coefficients associated with Bessel for the disk with flexible supports can be attained from function of the second kind, Equations (24)and (25)reduce (11) together with the consideration of the flexible boundary to those reported by Onoe [9] for free solid disk and by conditions. The conditions involving flexible edge supports Park [12] for the clamped solid disk. The frequency equation at the inner and outer edges are satisfied when the radial corresponding to different values of n for the solid disks (N ) and circumferential (N ) in-plane forces at the edges r rθ Advances in Acoustics and Vibration 5 Table 2: Frequency equations of axisymmetric modes for annular disks. Boundary conditions radial Circumferential inner outer Clamped Clamped J (λ )Y (λ β) − J (λ β)Y (λ ) = 0 J (λ )Y (λ β) − J (λ β)Y (λ ) = 0 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 [(−λ /(1 − v))J (λ )+ J (λ )][(−λ /(1 − 1 0 1 1 1 1 v))Y (λ β)+(1/β)Y (λ β)] − λ /(1 − [−2J (λ )+ λ J (λ )][−(2/β)Y (λ β)+ λ Y (λ β)] − 0 1 1 1 1 1 2 2 0 2 1 2 2 0 2 Free Free v)Y (λ )+ Y (λ )[(−λ /(1 − v))J (λ β)+ [−(2/β)J (λ β)+ λ J (λ β)][−2Y (λ )+ λ Y (λ )] = 0 0 1 1 1 1 0 1 1 2 2 0 2 1 2 2 0 2 (1/β)J (λ β)] = 0 1 1 [(λ /(1 − v))J (λ ) − J (λ )]Y (λ β) − [2J (λ ) − λ J (λ )]Y (λ β) − J (λ β)[2Y (λ ) − 1 0 1 1 1 1 1 1 2 2 0 2 1 2 1 2 1 2 Clamped Free J (λ β)[(λ /(1 − v))Y (λ ) − Y (λ )] = 0 λ Y (λ )] = 0 1 1 1 0 1 1 1 2 0 2 J (λ )[(−λ /(1 − v))Y (λ β)+ 1 1 1 0 1 J (λ )[− (2/β)Y (λ β)+ λ Y (λ β)] − [−(2/β)J (λ β)+ 1 2 1 2 2 0 2 1 2 Free Clamped (1/β)Y (λ β)] + [(−λ /(1 − v))J (λ β)+ 1 1 1 0 1 λ J (λ β)]Y (λ ) = 0 2 0 2 1 2 (1/β)J (λ β)]Y (λ ) = 0 1 1 1 1 Table 3: Exact frequency parameters of in-plane vibration of a solid Table 4: Exact frequency parameters of in-plane vibration of a solid disk with free edge (v = 0.3). disk with clamped edge (v = 0.33). Mode n = 1 n = 2 n = 3 n = 4 Mode n = 1 n = 2 n = 3 n = 4 1 1.6176 1.3877 2.1304 2.7740 1 1.9441 3.0185 3.0185 4.7021 2 3.5291 2.5112 3.4517 4.4008 2 3.1126 4.0127 4.0127 5.8985 3 4.0474 4.5208 5.3492 6.1396 3 4.9104 5.7398 5.7398 7.3648 4 5.8861 5.2029 6.3695 7.4633 4 5.3570 6.7079 6.7079 8.9816 5 6.9113 6.7549 7.6186 8.5007 5 6.7763 7.6442 7.6442 9.5296 6 7.7980 8.2639 9.3470 10.2350 6 8.4938 9.4356 9.4356 11.1087 7 9.6594 8.7342 9.8366 11.0551 7 8.6458 9.9894 9.9894 12.5940 are equal to the respective radial and circumferential spring equations. The above boundary equations reduce to those in forces, such that (22)and (23) for the free edge conditions by letting K = 0 and K = 0. Furthermore, the clamped edge condition can −Eh ∂u v ∂u v r θ be represented by considering infinite values of K and K . r θ N = + + u = K u (ξ, θ), r r r r (1 − v )b ∂ξ ξ ∂θ ξ Equations (27) further show that the flexible edge support (26) conditions involve combinations of the free and clamped −Eh 1 ∂u ∂u u r θ θ edge conditions. ( ) N = + − = K u ξ, θ , rθ θ θ 2(1+ v)b ξ ∂θ ∂ξ ξ 3. Results where K and K are the radial and circumferential stiffness r θ coefficients, respectively. Introducing the nondimensional The frequency parameters (λ )derived fordifferent com- 2 1 stiffness parameters, K = K b(1− v )/Eh and K = K b(1 + r r θ θ binations of boundary conditions are compared with those v)/Eh,(26)can be writtenas reported in different studies to demonstrate the validity of 2 the proposed formulation. For this purpose, the frequency n 1 ∂ 1 ∂ − − λ + K Φ 2 parameters of a solid disk with free and clamped outer edge ξ ξ ∂ξ 2 ∂ξ are initially evaluated and compared with those reported by Holland [10]and Park [12], respectively. The results 1 ∂ 1 1 + n − + K Ψ = 0, presentedinTables 3 and 4 for the free and clamped outer ξ ∂ξ ξ ξ edge conditions, respectively, were found to be identical to (27) those reported in [10, 12] for solid disks with clamped edge. 1 ∂ 1 1 n − − K Φ 2 The exact frequency parameters for the annular disks ξ ∂ξ ξ ξ were subsequently obtained under different combinations of boundary conditions at the inner and outer edges. The edge n 1 ∂ 1 ∂ + − − λ − K Ψ = 0. 2 conditions are presented for the inner followed by that of ξ ξ ∂ξ 2 ∂ξ the outer edge. For instance, a “Free-Clamped” condition The application of the above conditions yields the matrix refers to free inner edge and clamped outer edge. The equations for the disk with flexible supports at the inner solutions obtained for conditions involving free and clamped and outer edges, similar to (24). The frequency parameters edges (“Free-Free”, “Free-Clamped”, “Clamped-Clamped”, are subsequently obtained through solution of the matrix and “Clamped-Free”) are compared with those reported 6 Advances in Acoustics and Vibration Table 5: Frequency parameters of in-plane vibration of an annular disk with “Free-Free” conditions. n = 1 n = 2 n = 3 n = 4 Radius ratio (β) Reference present Reference Present Reference Present Reference Present [15] study [15] study [15] study [15] study 1.652 1.651 1.110 1.110 2.071 2.071 2.767 2.766 0.2 3.842 3.842 2.403 2.402 3.401 3.400 4.389 4.388 1.683 1.682 0.721 0.721 1.618 1.619 2.482 2.482 0.4 4.044 4.044 2.451 2.450 3.346 3.346 4.227 4.226 Table 6: Frequency parameters of in-plane vibration of an annular disk with “Free-Clamped” conditions. n = 1 n = 2 n = 3 n = 4 Radius ratio (β) Reference Present Reference Present Reference Present Reference Present [15] study [15] study [15] study [15] study 2.104 2.103 2.553 2.553 3.688 3.688 4.712 4.711 0.2 3.303 3.302 3.948 3.948 4.859 4.858 5.894 5.893 2.517 2.517 2.721 2.721 3.214 3.214 3.955 3.956 0.4 3.508 3.508 4.147 4.147 4.998 4.998 5.874 5.873 Table 7: Frequency parameters of in-plane vibration of an annular disk with “Clamped-Clamped” conditions. n = 1 n = 2 n = 3 n = 4 Radius ratio (β) Reference present Reference present Reference present Reference present [15] study [15] study [15] study [15] study 2.783 2.783 3.378 3.378 4.066 4.065 4.802 4.800 0.2 4.060 4.060 4.360 4.359 5.104 5.103 6.003 6.001 3.429 3.429 4.023 4.022 4.707 4.707 5.287 5.286 0.4 5.306 5.306 5.311 5.311 5.619 5.619 6.289 6.288 Table 8: Frequency parameters of in-plane vibration of an annular disk with “Clamped-Free” conditions. n = 1 n = 2 n = 3 n = 4 Radius ratio (β) Reference present Reference present Reference present Reference present [15] study [15] study [15] study [15] study 0.919 0.919 1.542 1.541 2.157 2.157 2.778 2.777 0.2 2.121 2.121 2.605 2.604 3.473 3.472 4.408 4.406 1.281 1.281 1.965 1.964 2.445 2.445 2.911 2.911 0.4 2.691 2.691 2.908 2.907 3.604 3.603 4.492 4.491 Table 9: Frequency parameters of in-plane vibration of an annular disk with flexible boundary conditions. n = 1 n = 2 n = 3 n = 4 Boundary conditions Reference present Reference present Reference present Reference present [19] study [19] study [19] study [19] study 0.771 0.771 1.408 1.408 2.121 2.121 2.772 2.772 Elastic-Free 1.906 1.906 2.524 2.524 3.444 3.444 4.397 4.397 2.590 2.590 3.117 3.116 3.928 3.926 4.759 4.760 Elastic-Clamped 3.625 3.625 4.134 4.136 5.000 5.001 5.957 5.957 1.494 1.492 1.815 1.813 2.474 2.472 3.123 3.120 Elastic-Elastic 2.603 2.601 3.209 3.207 4.004 4.002 4.859 4.857 1.040 1.046 1.505 1.504 2.414 2.416 3.114 3.116 Free-Elastic 2.432 2.432 3.018 3.018 3.895 3.896 4.833 4.834 1.686 1.686 1.960 1.959 2.514 2.512 3.129 3.127 Clamped-Elastic 2.764 2.765 3.319 3.317 4.061 4.060 4.879 4.877 Advances in Acoustics and Vibration 7 by Irie et al. [15] in Tables 5, 7, 8, 9,respectively. The r: Radial coordinate u , u , u : Radial, circumferential, and normal simulations results were obtained for two different values r θ z of the radial ratios (β = 0.2 and 0.4), and v = 0.3. The displacements of the disk Y (ξ): Bessel function of the second kind of results show excellent agreements of the values obtained in n order n the present study with those reported in [15], irrespective of the boundary condition and radius ratio considered. z: Normal coordinate β:Radiusratio The exact frequency parameters of the annular disk are further investigated for boundary conditions involving dif- θ: Circumferential coordinate ferent combinations of free, clamped, and elastic edges. The λ: Nondimensional frequency parameters v: Poisson’s ratio solutions corresponding to selected modes are obtained for (“Elastic-Free”, “Elastic-Clamped”, “Elastic-Elastic”, “Free- ξ: Nondimensional radial coordinate of the disk Elastic”, and “Clamped-Elastic”) conditions are presented in ρ: Mass density of the disk Table 9. The results were attained for β = 0.2and v = 0.3. The nondimensional radial and circumferential stiffness φ, ψ:LameP ´ otentials Φ, Ψ: Radial variations of LameP ´ otentials parameters were chosen as K = 1and K = 1. The results r θ are also compared with those obtained using the Raleigh-Ritz ω: Radian natural frequency. methods, as reported in [23]. The comparisons reveal very good agreements between the analytical and the reported References results irrespective of the boundary condition considered. The results suggest that the proposed frequency equations [1] A. Leissa, Vibration of Plates, Nasa SP-160, 1969. could serve as the reference for approximate methods on [2] J. Rao, Dynamics of Plates, Narosa Publishing House, 1999. in-plane vibration characteristics of the annular disks with ¨ [3] O. Civalek, “Discrete singular convolution method and appli- different combinations of edge conditions. cations to free vibration analysis of circular and annular plates,” Structural Engineering and Mechanics, vol. 29, no. 2, pp. 237–240, 2008. 4. Conclusions [4] O. Civalek and H. Ersoy, “Free vibration and bending analysis of circular Mindlin plates using singular convolution method,” The characteristics of in-plane vibration for circular disks Communications in Numerical Methods in Engineering, vol. 25, are investigated under different combinations of boundary no. 8, pp. 907–922, 2009. conditions. The governing equations are solved to obtain the [5] K. I. Tzou, J. A. Wickert, and A. Akay, “In-plane vibration exact frequency equation of solid and annular disks. Fre- modes of arbitrarily thick disks,” Journal of Vibration and quency equations are presented for different combinations Acoustics, vol. 120, no. 2, pp. 384–391, 1998. of boundary conditions, including flexible boundaries, at [6] H. Lee and R. Singh, “Self and mutual radiation from flexural the inner and outer edges. The nondimensional frequency and radial modes of a thick annular disk,” Journal of Sound and parameters obtained by the present approach compare very Vibration, vol. 286, no. 4-5, pp. 1032–1040, 2005. well with those available in literature, irrespective of the [7] D. J. Thompson and C. J.C. Jones, “A review of the modelling boundary condition and radius ratio considered. The exact of wheel/rail noise generation,” Journal of Sound and Vibra- frequency parameters can serve as a reference to assess the tion, vol. 231, no. 3, pp. 519–536, 2000. accuracy of approximate methods. The presented frequency [8] A. Love, A Treatise on the Mathematical Theory of Elasticity, equations can be numerically evaluated to obtain the in- Dover Publications, New York, NY, USA, 1944. plane modal characteristics of circular disk for a wide range [9] M. Onoe, “Contour vibrations of isotropic circular plates,” of constraints conditions and geometric parameters. Journal of the Acoustical Society of America, vol. 28, pp. 1158– 1162, 1956. Nomenclature [10] R. Holland, “Numerical studies of elastic-disk contour modes lacking axial symmetry,” Journal of Acoustical Society of A , B , C , D :Deflection coefficients of the exact n n n n America, vol. 40, pp. 1051–1057, 1966. solution [11] N. H. Farag and J. Pan, “Modal characteristics of in-plane a: Inner radius of the annular disk vibration of circular plates clamped at the outer edge,” Journal b: Outer radius of the annular disk of the Acoustical Society of America, vol. 113, no. 4, pp. 1935– C : E/ρ(1 − v ) 1946, 2003. C : E/2ρ(1 + v) [12] C. I. Park, “Frequency equation for the in-plane vibration of E: Young’s modulus of disk a clamped circular plate,” Journal of Sound and Vibration, vol. h: Thickness of the annular disk 313, no. 1-2, pp. 325–333, 2008. J (ξ): Bessel function of the first kind of order n [13] G. Ambati,J.F.W.Bell, andJ.C.K.Sharp,“In-plane vibrations K , K : Radial and circumferential stiffness r θ of annular rings,” Journal of Sound and Vibration, vol. 47, no. coefficients 3, pp. 415–432, 1976. K , K : Nondimensional radial and r θ [14] V. Srinivasan and V. Ramamurti, “Dynamic response of an circumferential stiffness parameters annular disk to a moving concentrated, in-plane edge load,” n: Nodal diameter number Journal of Sound and Vibration, vol. 72, no. 2, pp. 251–262, N , N : Radial and circumferential in-plane forces r rθ 8 Advances in Acoustics and Vibration [15] T. Irie, G. Yamada, and Y. Muramoto, “Natural frequencies of in-plane vibration of annular plates,” Journal of Sound and Vibration, vol. 97, no. 1, pp. 171–175, 1984. [16] J.-S. Chen and J.-L. Jhu, “On the in-plane vibration and stability of a spinning annular disk,” Journal of Sound and Vibration, vol. 195, no. 4, pp. 585–593, 1996. [17] A. C. Eringen and E. S. Suhubi, Elastodynamics, vol. 2, Academic Press, New York, NY, USA, 1975. [18] D. G. Zill and M. R. Cullen, Advanced Engineering Mathemat- ics, Jones and Bartlett Publishers, 2006. [19] P. A. A. Laura, L. E. Luisoni, and J. J. Lopez, “A note on free andforcedvibrationsofcircularplates: theeffect of support flexibility,” Journal of Sound and Vibration,vol. 47, no.2,pp. 287–291, 1976. [20] K. S. Rao and C. L. Amba-Rao, “Lateral vibration and stability relationship of elastically restrained circular plates,” AIAA Journal, vol. 10, no. 12, pp. 1689–1690, 1972. [21] P. A. A. Laura, J. C. Paloto, and R. D. Santos, “A note on the vibration and stability of a circular plate elastically restrained against rotation,” Journal of Sound and Vibration, vol. 41, no. 2, pp. 177–180, 1975. [22] C. S. Kim and S. M. Dickinson, “The flexural vibration of thin isotropic and polar orthotropic annular and circular plates with elastically restrained peripheries,” Journal of Sound and Vibration, vol. 143, no. 1, pp. 171–179, 1990. [23] S. Bashmal, R. Bhat, and S. Rakheja, “In-plane free vibration of circular annular disks,” Journal of Sound and Vibration, vol. 322, no. 1-2, pp. 216–226, 2009. 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Frequency Equations for the In-Plane Vibration of Circular Annular Disks

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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2010, Article ID 501902, 8 pages doi:10.1155/2010/501902 Research Article Frequency Equations for the In-Plane Vibration of Circular Annular Disks S. Bashmal, R. Bhat, and S. Rakheja Department of Mechanical and Industrial Engineering, Concordia University, 1455 De Maisonneuve Bloulevard W., Montreal, QC, Canada H3G 1M8 Correspondence should be addressed to S. Bashmal, bashmal@gmail.com Received 23 October 2009; Revised 23 June 2010; Accepted 25 June 2010 Academic Editor: Miguel Ayala Botto Copyright © 2010 S. Bashmal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with the in-plane vibration of circular annular disks under combinations of different boundary conditions at the inner and outer edges. The in-plane free vibration of an elastic and isotropic disk is studied on the basis of the two-dimensional linear plane stress theory of elasticity. The exact solution of the in-plane equation of equilibrium of annular disk is attainable, in terms of Bessel functions, for uniform boundary conditions. The frequency equations for different modes can be obtained from the general solutions by applying the appropriate boundary conditions at the inner and outer edges. The presented frequency equations provide the frequency parameters for the required number of modes for a wide range of radius ratios and Poisson’s ratios of annular disks under clamped, free, or flexible boundary conditions. Simplified forms of frequency equations are presented for solid disks and axisymmetric modes of annular disks. Frequency parameters are computed and compared with those available in literature. The frequency equations can be used as a reference to assess the accuracy of approximate methods. 1. Introduction solid disks clamped at the outer edge have been investigated in a few recent studies. Farag and Pan [11] evaluated the frequency parameters and the mode shapes of in-plane The out-of-plane vibration properties of circular disks sub- jected to a variety of boundary conditions have been exte- vibration of solid disks clamped at the outer edge using nsively investigated (e.g., [1–4]). The in-plane vibration assumed deflection modes in terms of trigonometric and analyses of circular disk, however, have been gaining increas- Bessel functions. Park [12] studied the exact frequency ing attention only in the recent years. Much of the interest equation for the solid disk clamped at the outer edge. could be attributed to important significance of the in- The in-plane vibration analyses in the above reported plane vibration in various practical problems such as the studies were limited to solid disks with either free or clamped vibration of railway wheels, disk brakes, and hard disk drives outer edge. The in-plane free vibration of annular disks contributing to noise and structural vibration [5–7]. with different boundary conditions has also been addressed The in-plane vibration of circular disks was first in a few studies. The variations in the in-plane vibration attempted by Love [8] who formulated the equations of frequency parameters of annular disks with free edges were investigated as function of the size of the opening by Ambati motion for a thin solid circular disk with free outer edge together with the general solution. The equations of motion et al. [13]. The variation ranged from a solid disk to a were subsequently solved by Onoe [9] to obtain the exact thin ring, while the validity of the analytical results was frequency equations corresponding to different modes of a demonstrated using the experimental data. Another study solid disk with free outer edge. Holland [10] evaluated the investigated the free vibration and dynamic response char- frequency parameters and the corresponding mode shapes acteristics of an annular disk with clamped inner boundary for a wide range of Poisson’s ratios and the vibration response and a concentrated radial force applied at the outer boundary to an in-plane force. The in-plane vibration characteristics of [14]. Irie et al. [15] investigated the modal characteristics 2 Advances in Acoustics and Vibration of in-plane vibration of annular disks using transfer matrix formulation while considering free and clamped inner and outer edges. The above reported studies on in-plane vibration of solid and annular disks have employed different methods of analyses. The finite-element technique has also been used to examine the validity of analytical methods (e.g., [11, 14]). The exact frequency equations of in-plane vibration, how- ever, have been limited only to solid disks. Such analyses for the annular disks pose more complexities due to presence of different combinations of boundary conditions at the inner b and outer edges. This study aims at generalized formulation for in-plane vibration analyses of circular annular disks under different combinations of clamped, free, or flexible Figure 1: Geometry and coordinate system used for in-plane boundary conditions at the inner and outer edges. The vibration analysis of an annular disk. equations of motion are solved for the general case of annular disks. The exact frequency equations are presented for different combinations of boundary conditions, including Assuming harmonic oscillations corresponding to a the flexible boundaries, for various radius ratios, while the natural frequency ω, the potential functions φ and ψ can be solid disk is considered as special cases of the generalized represented by formulation. φ(ξ, θ, t) = Φ(ξ) cos nθ sin ωt, (4) 2. Theory ψ(ξ, θ, t) = Ψ(ξ) sin nθ sin ωt, (5) The equations of the in-plane vibration of a circular disk are formulated for an annular disk shown in Figure 1. The disk is considered to be elastic with thickness h,outer radius b where n is the circumferential wave number or nodal and inner radius a. The material is assumed to be isotropic diameter number. Upon substituting for u and u in terms r θ with mass density ρ,Young’s modulus E, and Poisson ratio v. of ξ, Φ,and Ψ from (2)to(5), in (1), the equations of motion The equations of dynamic equilibrium in terms of in-plane reduce to the following uncoupled form: displacements along the radial and circumferential directions can be found in many reported studies (e.g., [11, 16]). These 2 2 ∇ Φ =−λ Φ, (6) equations of motion in the polar coordinate system (r, θ)can be written as 2 2 ∇ Ψ =−λ Ψ, (7) 2 2 2 2 ∂ u ∂ u 1 ∂u u C ∂ u r r r r r 2 T − C + − − 2 2 2 2 2 ∂t ∂r r ∂r r r ∂θ where λ and λ are nondimensional frequency parameters 1 2 defined as 1 1+ v ∂ u 1 3 − v ∂u θ θ 2 2 − C + C = 0, T T r 1 − v ∂r∂θ r 1 − v ∂θ 2 2 2 2 2 (1) ω 1 − v ρb 2ω (1+ v)ρb 2 2 2 2 2 2 λ = , λ = , 1 2 ∂ u ∂ u 1 ∂u u C ∂ u θ θ θ θ θ 2 L E E − C + − − 2 2 2 2 2 (8) ∂t ∂r r ∂r r r ∂θ 2 2 ∂ 1 ∂ n ∇ = + − . 2 2 1 1+ v ∂ u 1 3 − v ∂u ∂ξ ξ ∂ξ ξ r r 2 2 − C + C = 0, T T r 1 − v ∂r∂θ r 1 − v ∂θ Equations (6)and (7) are the parametric Bessel equations where u and u are the radial and circumferential dis- r θ and their general solutions are attainable in terms of the placements, respectively, along the r and θ directions, C = Bessel functions as [18] E/ρ(1 − v )and C = E/2ρ(1 + v). Following Love’s theory [8], the radial and circumfer- ential displacements can be expressed in terms of the Lame´ Φ = A J (λ ξ) + B Y (λ ξ) , (9) n n 1 n n 1 Potentials φ and ψ [17], as Ψ = C J (λ ξ) + D Y (λ ξ) , (10) n n 2 n n 2 1 ∂φ 1 ∂ψ u = + ,(2) b ∂ξ ξ ∂θ where J and Y are the Bessel functions of the first and n n second kind of order n,respectively, and A , B , C ,and D n n n n 1 1 ∂φ ∂ψ u = − ,(3) are the deflection coefficients. b ξ ∂θ ∂ξ The radial and circumferential displacements can then be where ξ = r/b. expressed in terms of the Bessel functions by substituting for Advances in Acoustics and Vibration 3 Φ and Ψ in (2)and (3). The resulting expressions for the A direct substitution of u and u from (11) in the above r θ radial and circumferential displacements can be expressed as: equations would result in second derivatives of the Bessel functions. Alternatively, the above equation for the boundary conditions may be expressed in terms of Φ and Ψ through u = A X (λ ξ) + B Z (λ ξ) r n n 1 n n 1 direct substitution of u and u from (2)and (3), respectively. r θ The boundary conditions in terms of N canthusbeobtained as + [C J (λ ξ) + D Y (λ ξ)] cos nθ sin ωt, n n 2 n n 2 (11) 2 2 ∂ Φ n ∂Ψ n n v nv ∂Ψ + − Ψ − Φ − ( ) ( ) u =− C X λ ξ + D Z λ ξ θ n n 2 n n 2 2 2 2 ξ ∂ξ ξ ξ ξ ∂ξ b ∂ξ (19) v ∂Φ n + + Ψ = 0. + [A J (λ ξ) + B Y (λ ξ)] sin nθ sin ωt, n n 1 n n 1 ξ ∂ξ ξ where Rearranging (19) results in ∂ n X (λ ξ) = J (λ ξ) =− J (λ ξ) + λ J (λ ξ), n i n i n i i n−1 i ∂ξ ξ 2 2 ∂ v ∂ n v n ∂ n + − Φ + (1 − v) − Ψ = 0. (20) ∂ n 2 2 ξ ∂ξ ξ ξ ∂ξ ξ ∂ξ Z (λ ξ) = Y (λ ξ) = − Y (λ ξ) + λ Y (λ ξ) , n i n i n i i n−1 i ∂ξ ξ i = 1, 2. The second order derivative term (∂ Φ/∂ξ )in(20)can be (12) eliminated by adding and subtracting the term ((1/ξ)(∂/∂ξ)− 2 2 2 (n /ξ )+ λ )Φ,which yields 2.1. Free and Clamped Boundary Conditions. Equations (11) represent the solutions for distributions of the radial and 2 2 2 ∂ 1 ∂ n 1 ∂ n 2 2 circumferential displacements for the general case of an + − + λ − − + λ 1 1 2 2 ξ ∂ξ ξ ξ ∂ξ ξ ∂ξ annular disk. The evaluations of the natural frequencies (21) and arbitrary deflection coefficients (A , B , C and D ), n n n n v ∂ n v n ∂ n however, necessitate the consideration of the in-plane free ( ) + − Φ + 1 − v − Ψ = 0. 2 2 ξ ∂ξ ξ ξ ∂ξ ξ vibration response under different combinations of bound- ary conditions at the inner and the outer edges. For the annular disk clamped at the outer edge (ξ = 1), the From (6), it can be seen that the terms within the first application of boundary conditions (u = 0and u = 0) r θ parenthesis are identically equal to zero. Equation (21) must satisfy the following for the general solutions (11): describing the boundary condition associated with N can be 2 2 further simplified upon substitutions for λ = λ (1 − v)/2, A X (λ ) + B Z (λ ) + n[C J (λ ) + D Y (λ )] = 0, 1 2 (13) n n 1 n n 1 n n 2 n n 2 which yields C X (λ ) + D Z (λ ) + n[A J (λ ) + B Y (λ )] = 0. (14) n n 2 n n 2 n n 1 n n 1 In a similar manner, the solution must satisfy the n 1 ∂ 1 1 ∂ 1 − − λ Φ + n − Ψ = 0. (22) 2 2 following for the clamped inner edge (ξ = β), where β = a/b ξ ξ ∂ξ 2 ξ ∂ξ ξ is the radius ratio between inner and outer radii of the disk: A X λ β + B Z λ β + C J λ β + D Y λ β = 0, Similarly, the boundary condition equation associated with n n 1 n n 1 n n 2 n n 2 N (18) can be simplified as rθ (15) 2n ∂ 1 2 ∂ 2n C X λ β + D Z λ β + A J λ β + B Y λ β = 0. n n 2 n n 2 n n 1 n n 1 2 − + Φ + − + λ Ψ = 0. (23) ξ ∂ξ ξ ξ ∂ξ ξ (16) The conditions involving at the free edges are satisfied when Upon substitutions for Φ and Ψ from (16)in(26)and (27), the radial (N ) and circumferential (N ) in-plane forces at r rθ the boundary condition equations for the free edges are the edge are zero [11], such that obtained, which involve only first derivatives of the Bessel functions. For an annular disk with free inner and outer −Eh ∂u v ∂u v r θ N = + + u = 0, (17) r ξ edges, (22)and (23) represent the conditions at both the (1 − v )b ∂ξ ξ ∂θ ξ inner and the outer boundaries (ξ = 1and ξ = β). The equations for the free edge boundary conditions can −Eh 1 ∂u ∂u u r θ θ N = + − = 0. (18) be expressed in the matrix form in the four deflection rθ ( ) 2 1+ v b ξ ∂θ ∂ξ ξ coefficients, as 4 Advances in Acoustics and Vibration Table 1: Frequency equations for the solid disks corresponding to free and clamped edge conditions. Boundary conditions at ξ = 1 Clamped Free Radial J (λ ) = 0 λ J (λ ) = (1 − v)J (λ ) 1 1 1 0 1 1 1 n = 0 Circumferential J (λ ) = 0 λ J (λ ) = 2J (λ ) 1 2 2 0 2 1 2 [X (λ )J (λ )] + [X (λ )J (λ )] = 1 2 1 1 1 1 1 2 n = 1 X (λ )X (λ ) = n J (λ )J n 1 n 2 n 1 n 2 (2 − (λ /2))J (λ )J (λ ) 2 1 2 1 1 (λ ) [X (λ )−QJ (λ )][X (λ )−QJ (λ )] = n (Q − 1) n 2 n 2 n 1 n 1 n> 1 where Q = λ /2(n − 1) ⎡ ⎤ 2 2 λ λ 2 2 2 2 n − J (λ ) − X (λ ) n − Y (λ ) − Z (λ ) nX (λ ) − nJ (λ ) nZ (λ ) − nY (λ ) ⎢ n 1 n 1 n 1 n 1 n 2 n 2 n 2 n 2 ⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ 2 2 2 2 ⎥ 2nJ (λ ) − 2nX (λ ) 2nY (λ ) − 2nZ (λ ) 2X (λ ) − 2n − λ J (λ ) 2Z (λ ) − 2n − λ Y (λ ) ⎢ n 1 n 1 n 1 n 1 n 2 n 2 n 2 n 2 ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ 2 2 2 2 ⎢                 ⎥ n λ 1 n λ 1 n n n n 2 2 ⎢ ⎥ − J λ β − X λ β − Y λ β − Z λ β X λ β − J λ β Z λ β − Y λ β n 1 n 1 n 1 n 1 n 2 n 2 n 2 n 2 ⎢ 2 2 2 2 ⎥ β 2 β β 2 β β β β β ⎢ ⎥ ⎢ ⎥ 2 2 (24) ⎣ 2n 2n 2n 2n 2 2n 2 2n ⎦ 2 2 J λ β − X λ β Y λ β − Z λ β X λ β − − λ J λ β Z λ β − − λ Y λ β n 1 n 1 n 1 n 1 n 2 n 2 n 2 n 2 2 2 2 2 2 2 β β β β β β β β ⎧ ⎫ ⎪ n⎪ ⎪ ⎪ ⎨ ⎬ × ={0}. ⎪ C ⎪ ⎪ ⎪ ⎩ ⎭ The determinant of the above matrix yields the frequency involving the two boundary conditions are summarized in equation for the annular disk with free inner and outer edge Table 1,where X (λ ) is the derivative of the Bessel function n 1 conditions. J evaluated at the outer edge (ξ = 1). For annular disks, For the clamped inner and outer edges, the equations for simplified frequency equations can be obtained for the the boundary conditions can be obtained directly from (13) axisymmetric modes. These equations where expressed in to (16), such that Table 2 for the four combinations of boundary conditions. ⎡ ⎤ X (λ ) Z (λ ) nJ (λ ) nY (λ ) n 1 n 1 n 2 n 2 ⎧ ⎫ ⎢ ⎥ ⎪ A ⎪ nJ (λ ) nY (λ ) X (λ ) Z (λ ) ⎪ ⎪ ⎢ n 1 n 1 n 2 n 2 ⎥ ⎪ ⎪ 2.2. Flexible Boundary Conditions. In the above analysis, ⎨ ⎬ ⎢        ⎥ n n ⎢ ⎥ X λ β Z λ β J λ β Y λ β ={0}. the boundary conditions considered are either clamped n 1 n 1 n 2 n 2 ⎢ ⎥ ⎪ C ⎪ β β ⎪ n⎪ ⎢ ⎥ ⎪ ⎪ ⎩ ⎭ or free. However, Flexible boundary conditions may be ⎣ n n ⎦ J λ β Y λ β X λ β Z λ β n 1 n 1 n 2 n 2 considered more representative of many practical situations. β β The proposed formulations can be further employed to (25) study the in-plane vibration of solid as well as annular In the above equations, (24)and (25), the top two rows disks with flexible boundary conditions. Artificial springs describe the boundary condition at the outer edge, while may be applied to describe the flexible boundary conditions the bottom two rows are associated with those at the at the inner or the outer edge of an annular disk. A inner edge. The equations for the boundary conditions number of studies on the analysis of out-of-plane vibration involving combinations of free and clamped edges can thus characteristics of circular plates and cylindrical shells have be directly obtained from the above two equations. For the employed uniformly distributed artificial springs around the free inner edge and clamped outer edge, denoted as “free- edge to represent a flexible boundary conditions or a flexible clamped” condition, the matrix equation comprises the tip joint [19–22]. two rows of the matrix in (25) and the lower two rows The effects of flexible boundary conditions on the in- from (24). For the clamped inner edge and free outer edge, plane free vibration of circular disks have been considered denoted as “Clamped-Free” condition, the matrix equation is in a recent study by the authors [23] using the Rayleigh- formulated in the similar manner using the lower and upper Ritz approach. Artificial springs, distributed along the radial two rows from (25)and (24), respectively. and circumferential directions at the free outer and/or inner The in-plane vibration analysis of a solid disk can be edges, were considered to simulate for flexible boundary shown as a special case of the above generalized formulations. conditions. The exact solution of the frequency equations Upon eliminating the coefficients associated with Bessel for the disk with flexible supports can be attained from function of the second kind, Equations (24)and (25)reduce (11) together with the consideration of the flexible boundary to those reported by Onoe [9] for free solid disk and by conditions. The conditions involving flexible edge supports Park [12] for the clamped solid disk. The frequency equation at the inner and outer edges are satisfied when the radial corresponding to different values of n for the solid disks (N ) and circumferential (N ) in-plane forces at the edges r rθ Advances in Acoustics and Vibration 5 Table 2: Frequency equations of axisymmetric modes for annular disks. Boundary conditions radial Circumferential inner outer Clamped Clamped J (λ )Y (λ β) − J (λ β)Y (λ ) = 0 J (λ )Y (λ β) − J (λ β)Y (λ ) = 0 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 [(−λ /(1 − v))J (λ )+ J (λ )][(−λ /(1 − 1 0 1 1 1 1 v))Y (λ β)+(1/β)Y (λ β)] − λ /(1 − [−2J (λ )+ λ J (λ )][−(2/β)Y (λ β)+ λ Y (λ β)] − 0 1 1 1 1 1 2 2 0 2 1 2 2 0 2 Free Free v)Y (λ )+ Y (λ )[(−λ /(1 − v))J (λ β)+ [−(2/β)J (λ β)+ λ J (λ β)][−2Y (λ )+ λ Y (λ )] = 0 0 1 1 1 1 0 1 1 2 2 0 2 1 2 2 0 2 (1/β)J (λ β)] = 0 1 1 [(λ /(1 − v))J (λ ) − J (λ )]Y (λ β) − [2J (λ ) − λ J (λ )]Y (λ β) − J (λ β)[2Y (λ ) − 1 0 1 1 1 1 1 1 2 2 0 2 1 2 1 2 1 2 Clamped Free J (λ β)[(λ /(1 − v))Y (λ ) − Y (λ )] = 0 λ Y (λ )] = 0 1 1 1 0 1 1 1 2 0 2 J (λ )[(−λ /(1 − v))Y (λ β)+ 1 1 1 0 1 J (λ )[− (2/β)Y (λ β)+ λ Y (λ β)] − [−(2/β)J (λ β)+ 1 2 1 2 2 0 2 1 2 Free Clamped (1/β)Y (λ β)] + [(−λ /(1 − v))J (λ β)+ 1 1 1 0 1 λ J (λ β)]Y (λ ) = 0 2 0 2 1 2 (1/β)J (λ β)]Y (λ ) = 0 1 1 1 1 Table 3: Exact frequency parameters of in-plane vibration of a solid Table 4: Exact frequency parameters of in-plane vibration of a solid disk with free edge (v = 0.3). disk with clamped edge (v = 0.33). Mode n = 1 n = 2 n = 3 n = 4 Mode n = 1 n = 2 n = 3 n = 4 1 1.6176 1.3877 2.1304 2.7740 1 1.9441 3.0185 3.0185 4.7021 2 3.5291 2.5112 3.4517 4.4008 2 3.1126 4.0127 4.0127 5.8985 3 4.0474 4.5208 5.3492 6.1396 3 4.9104 5.7398 5.7398 7.3648 4 5.8861 5.2029 6.3695 7.4633 4 5.3570 6.7079 6.7079 8.9816 5 6.9113 6.7549 7.6186 8.5007 5 6.7763 7.6442 7.6442 9.5296 6 7.7980 8.2639 9.3470 10.2350 6 8.4938 9.4356 9.4356 11.1087 7 9.6594 8.7342 9.8366 11.0551 7 8.6458 9.9894 9.9894 12.5940 are equal to the respective radial and circumferential spring equations. The above boundary equations reduce to those in forces, such that (22)and (23) for the free edge conditions by letting K = 0 and K = 0. Furthermore, the clamped edge condition can −Eh ∂u v ∂u v r θ be represented by considering infinite values of K and K . r θ N = + + u = K u (ξ, θ), r r r r (1 − v )b ∂ξ ξ ∂θ ξ Equations (27) further show that the flexible edge support (26) conditions involve combinations of the free and clamped −Eh 1 ∂u ∂u u r θ θ edge conditions. ( ) N = + − = K u ξ, θ , rθ θ θ 2(1+ v)b ξ ∂θ ∂ξ ξ 3. Results where K and K are the radial and circumferential stiffness r θ coefficients, respectively. Introducing the nondimensional The frequency parameters (λ )derived fordifferent com- 2 1 stiffness parameters, K = K b(1− v )/Eh and K = K b(1 + r r θ θ binations of boundary conditions are compared with those v)/Eh,(26)can be writtenas reported in different studies to demonstrate the validity of 2 the proposed formulation. For this purpose, the frequency n 1 ∂ 1 ∂ − − λ + K Φ 2 parameters of a solid disk with free and clamped outer edge ξ ξ ∂ξ 2 ∂ξ are initially evaluated and compared with those reported by Holland [10]and Park [12], respectively. The results 1 ∂ 1 1 + n − + K Ψ = 0, presentedinTables 3 and 4 for the free and clamped outer ξ ∂ξ ξ ξ edge conditions, respectively, were found to be identical to (27) those reported in [10, 12] for solid disks with clamped edge. 1 ∂ 1 1 n − − K Φ 2 The exact frequency parameters for the annular disks ξ ∂ξ ξ ξ were subsequently obtained under different combinations of boundary conditions at the inner and outer edges. The edge n 1 ∂ 1 ∂ + − − λ − K Ψ = 0. 2 conditions are presented for the inner followed by that of ξ ξ ∂ξ 2 ∂ξ the outer edge. For instance, a “Free-Clamped” condition The application of the above conditions yields the matrix refers to free inner edge and clamped outer edge. The equations for the disk with flexible supports at the inner solutions obtained for conditions involving free and clamped and outer edges, similar to (24). The frequency parameters edges (“Free-Free”, “Free-Clamped”, “Clamped-Clamped”, are subsequently obtained through solution of the matrix and “Clamped-Free”) are compared with those reported 6 Advances in Acoustics and Vibration Table 5: Frequency parameters of in-plane vibration of an annular disk with “Free-Free” conditions. n = 1 n = 2 n = 3 n = 4 Radius ratio (β) Reference present Reference Present Reference Present Reference Present [15] study [15] study [15] study [15] study 1.652 1.651 1.110 1.110 2.071 2.071 2.767 2.766 0.2 3.842 3.842 2.403 2.402 3.401 3.400 4.389 4.388 1.683 1.682 0.721 0.721 1.618 1.619 2.482 2.482 0.4 4.044 4.044 2.451 2.450 3.346 3.346 4.227 4.226 Table 6: Frequency parameters of in-plane vibration of an annular disk with “Free-Clamped” conditions. n = 1 n = 2 n = 3 n = 4 Radius ratio (β) Reference Present Reference Present Reference Present Reference Present [15] study [15] study [15] study [15] study 2.104 2.103 2.553 2.553 3.688 3.688 4.712 4.711 0.2 3.303 3.302 3.948 3.948 4.859 4.858 5.894 5.893 2.517 2.517 2.721 2.721 3.214 3.214 3.955 3.956 0.4 3.508 3.508 4.147 4.147 4.998 4.998 5.874 5.873 Table 7: Frequency parameters of in-plane vibration of an annular disk with “Clamped-Clamped” conditions. n = 1 n = 2 n = 3 n = 4 Radius ratio (β) Reference present Reference present Reference present Reference present [15] study [15] study [15] study [15] study 2.783 2.783 3.378 3.378 4.066 4.065 4.802 4.800 0.2 4.060 4.060 4.360 4.359 5.104 5.103 6.003 6.001 3.429 3.429 4.023 4.022 4.707 4.707 5.287 5.286 0.4 5.306 5.306 5.311 5.311 5.619 5.619 6.289 6.288 Table 8: Frequency parameters of in-plane vibration of an annular disk with “Clamped-Free” conditions. n = 1 n = 2 n = 3 n = 4 Radius ratio (β) Reference present Reference present Reference present Reference present [15] study [15] study [15] study [15] study 0.919 0.919 1.542 1.541 2.157 2.157 2.778 2.777 0.2 2.121 2.121 2.605 2.604 3.473 3.472 4.408 4.406 1.281 1.281 1.965 1.964 2.445 2.445 2.911 2.911 0.4 2.691 2.691 2.908 2.907 3.604 3.603 4.492 4.491 Table 9: Frequency parameters of in-plane vibration of an annular disk with flexible boundary conditions. n = 1 n = 2 n = 3 n = 4 Boundary conditions Reference present Reference present Reference present Reference present [19] study [19] study [19] study [19] study 0.771 0.771 1.408 1.408 2.121 2.121 2.772 2.772 Elastic-Free 1.906 1.906 2.524 2.524 3.444 3.444 4.397 4.397 2.590 2.590 3.117 3.116 3.928 3.926 4.759 4.760 Elastic-Clamped 3.625 3.625 4.134 4.136 5.000 5.001 5.957 5.957 1.494 1.492 1.815 1.813 2.474 2.472 3.123 3.120 Elastic-Elastic 2.603 2.601 3.209 3.207 4.004 4.002 4.859 4.857 1.040 1.046 1.505 1.504 2.414 2.416 3.114 3.116 Free-Elastic 2.432 2.432 3.018 3.018 3.895 3.896 4.833 4.834 1.686 1.686 1.960 1.959 2.514 2.512 3.129 3.127 Clamped-Elastic 2.764 2.765 3.319 3.317 4.061 4.060 4.879 4.877 Advances in Acoustics and Vibration 7 by Irie et al. [15] in Tables 5, 7, 8, 9,respectively. The r: Radial coordinate u , u , u : Radial, circumferential, and normal simulations results were obtained for two different values r θ z of the radial ratios (β = 0.2 and 0.4), and v = 0.3. The displacements of the disk Y (ξ): Bessel function of the second kind of results show excellent agreements of the values obtained in n order n the present study with those reported in [15], irrespective of the boundary condition and radius ratio considered. z: Normal coordinate β:Radiusratio The exact frequency parameters of the annular disk are further investigated for boundary conditions involving dif- θ: Circumferential coordinate ferent combinations of free, clamped, and elastic edges. The λ: Nondimensional frequency parameters v: Poisson’s ratio solutions corresponding to selected modes are obtained for (“Elastic-Free”, “Elastic-Clamped”, “Elastic-Elastic”, “Free- ξ: Nondimensional radial coordinate of the disk Elastic”, and “Clamped-Elastic”) conditions are presented in ρ: Mass density of the disk Table 9. The results were attained for β = 0.2and v = 0.3. The nondimensional radial and circumferential stiffness φ, ψ:LameP ´ otentials Φ, Ψ: Radial variations of LameP ´ otentials parameters were chosen as K = 1and K = 1. The results r θ are also compared with those obtained using the Raleigh-Ritz ω: Radian natural frequency. methods, as reported in [23]. The comparisons reveal very good agreements between the analytical and the reported References results irrespective of the boundary condition considered. The results suggest that the proposed frequency equations [1] A. Leissa, Vibration of Plates, Nasa SP-160, 1969. could serve as the reference for approximate methods on [2] J. Rao, Dynamics of Plates, Narosa Publishing House, 1999. in-plane vibration characteristics of the annular disks with ¨ [3] O. Civalek, “Discrete singular convolution method and appli- different combinations of edge conditions. cations to free vibration analysis of circular and annular plates,” Structural Engineering and Mechanics, vol. 29, no. 2, pp. 237–240, 2008. 4. Conclusions [4] O. Civalek and H. Ersoy, “Free vibration and bending analysis of circular Mindlin plates using singular convolution method,” The characteristics of in-plane vibration for circular disks Communications in Numerical Methods in Engineering, vol. 25, are investigated under different combinations of boundary no. 8, pp. 907–922, 2009. conditions. The governing equations are solved to obtain the [5] K. I. Tzou, J. A. Wickert, and A. Akay, “In-plane vibration exact frequency equation of solid and annular disks. Fre- modes of arbitrarily thick disks,” Journal of Vibration and quency equations are presented for different combinations Acoustics, vol. 120, no. 2, pp. 384–391, 1998. of boundary conditions, including flexible boundaries, at [6] H. Lee and R. Singh, “Self and mutual radiation from flexural the inner and outer edges. The nondimensional frequency and radial modes of a thick annular disk,” Journal of Sound and parameters obtained by the present approach compare very Vibration, vol. 286, no. 4-5, pp. 1032–1040, 2005. well with those available in literature, irrespective of the [7] D. J. Thompson and C. J.C. Jones, “A review of the modelling boundary condition and radius ratio considered. The exact of wheel/rail noise generation,” Journal of Sound and Vibra- frequency parameters can serve as a reference to assess the tion, vol. 231, no. 3, pp. 519–536, 2000. accuracy of approximate methods. The presented frequency [8] A. Love, A Treatise on the Mathematical Theory of Elasticity, equations can be numerically evaluated to obtain the in- Dover Publications, New York, NY, USA, 1944. plane modal characteristics of circular disk for a wide range [9] M. Onoe, “Contour vibrations of isotropic circular plates,” of constraints conditions and geometric parameters. Journal of the Acoustical Society of America, vol. 28, pp. 1158– 1162, 1956. Nomenclature [10] R. Holland, “Numerical studies of elastic-disk contour modes lacking axial symmetry,” Journal of Acoustical Society of A , B , C , D :Deflection coefficients of the exact n n n n America, vol. 40, pp. 1051–1057, 1966. solution [11] N. H. Farag and J. Pan, “Modal characteristics of in-plane a: Inner radius of the annular disk vibration of circular plates clamped at the outer edge,” Journal b: Outer radius of the annular disk of the Acoustical Society of America, vol. 113, no. 4, pp. 1935– C : E/ρ(1 − v ) 1946, 2003. C : E/2ρ(1 + v) [12] C. I. Park, “Frequency equation for the in-plane vibration of E: Young’s modulus of disk a clamped circular plate,” Journal of Sound and Vibration, vol. h: Thickness of the annular disk 313, no. 1-2, pp. 325–333, 2008. J (ξ): Bessel function of the first kind of order n [13] G. Ambati,J.F.W.Bell, andJ.C.K.Sharp,“In-plane vibrations K , K : Radial and circumferential stiffness r θ of annular rings,” Journal of Sound and Vibration, vol. 47, no. coefficients 3, pp. 415–432, 1976. K , K : Nondimensional radial and r θ [14] V. Srinivasan and V. Ramamurti, “Dynamic response of an circumferential stiffness parameters annular disk to a moving concentrated, in-plane edge load,” n: Nodal diameter number Journal of Sound and Vibration, vol. 72, no. 2, pp. 251–262, N , N : Radial and circumferential in-plane forces r rθ 8 Advances in Acoustics and Vibration [15] T. Irie, G. Yamada, and Y. Muramoto, “Natural frequencies of in-plane vibration of annular plates,” Journal of Sound and Vibration, vol. 97, no. 1, pp. 171–175, 1984. [16] J.-S. Chen and J.-L. Jhu, “On the in-plane vibration and stability of a spinning annular disk,” Journal of Sound and Vibration, vol. 195, no. 4, pp. 585–593, 1996. [17] A. C. Eringen and E. S. Suhubi, Elastodynamics, vol. 2, Academic Press, New York, NY, USA, 1975. [18] D. G. Zill and M. R. Cullen, Advanced Engineering Mathemat- ics, Jones and Bartlett Publishers, 2006. [19] P. A. A. Laura, L. E. Luisoni, and J. J. Lopez, “A note on free andforcedvibrationsofcircularplates: theeffect of support flexibility,” Journal of Sound and Vibration,vol. 47, no.2,pp. 287–291, 1976. [20] K. S. Rao and C. L. Amba-Rao, “Lateral vibration and stability relationship of elastically restrained circular plates,” AIAA Journal, vol. 10, no. 12, pp. 1689–1690, 1972. [21] P. A. A. Laura, J. C. Paloto, and R. D. Santos, “A note on the vibration and stability of a circular plate elastically restrained against rotation,” Journal of Sound and Vibration, vol. 41, no. 2, pp. 177–180, 1975. [22] C. S. Kim and S. M. Dickinson, “The flexural vibration of thin isotropic and polar orthotropic annular and circular plates with elastically restrained peripheries,” Journal of Sound and Vibration, vol. 143, no. 1, pp. 171–179, 1990. [23] S. Bashmal, R. Bhat, and S. Rakheja, “In-plane free vibration of circular annular disks,” Journal of Sound and Vibration, vol. 322, no. 1-2, pp. 216–226, 2009. 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