Harmonic Standing-Wave Excitations of Simply-Supported Isotropic Solid Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response.

Transcription

1 Haronic Standing-Wave Excitations of Siply-Supported Isotropic Solid Elastic Circular Cylinders: Exact 3D inear Elastodynaic Response Jaal Sakhr and Blaine A. Chronik Departent of Physics and Astronoy, The University of Western Ontario, arxiv: v1 [ath-ph] 3 Apr 019 ondon, Ontario, Canada N6A 3K7 Dated: April 4, 019 Abstract The vibration of a solid elastic cylinder is one of the classical applied probles of elastodynaics. Many fundaental forced-vibration probles involving solid elastic cylinders have not yet been studied or solved using the full three-diensional 3D theory of linear elasticity. One such proble is the steady-state forced-vibration response of a siply-supported isotropic solid elastic circular cylinder subjected to two-diensional haronic standing-wave excitations on its curved surface. In this paper, we exploit certain previously-obtained particular solutions to the Navier-aé equation of otion and exact atrix algebra to construct an exact closed-for 3D elastodynaic solution to the proble. The ethod of solution is direct and deonstrates a general approach that can be applied to solve other siilar forced-vibration probles involving elastic cylinders. Two coplete analytical solutions are in fact constructed corresponding to two different but closely-related failies of haronic standing-wave excitations. The second of these analytical solutions is evaluated nuerically in order to study the steady-state frequency response in soe exaple excitation cases. In each case, the solution generates a series of resonances that are in correspondence with a subset of the natural frequencies of the siply-supported cylinder. The considered proble is of general interest both as an exactly-solvable 3D elastodynaics proble and as a benchark forced-vibration proble involving a solid elastic cylinder. Keywords: Analytically-solvable 3D elastodynaics probles; Benchark forced-vibration probles; inear-elastic boundary-value probles with siply-supported boundary conditions; Isotropic solid elastic cylinders; haronic standing-wave boundary stresses 1

2 I. INTRODUCTION The vibration of a solid elastic cylinder is one of the classical applied probles of elastodynaics see Ref. [1] and references therein and a proble of fundaental interest to the study of sound and vibration []. The literature on the free vibration of finite-length solid circular cylinders is vast with any new contributions over the last few decades advocating various ethodological approaches to obtaining natural frequencies and ode shapes under a variety of end conditions. Key theoretical contributions to this topic include Refs. [3 18] for isotropic cylinders, Refs. [19 ] for transversely isotropic cylinders, and Refs. [3, 4] for anisotropic and inhoogeneous cylinders. See also Refs. [5, 6] for detailed reviews of work done prior to 000. There is, in coparison, a scant literature on the forced vibrational characteristics of solid elastic cylinders finite-length, isotropic, circular, or otherwise. The ost notable work on this subject is due to Ebenezer and co-workers [7, 8], who devised an exact series ethod to deterine the steady-state forced-vibration response of an isotropic solid elastic cylinder subjected to arbitrary axisyetric excitations on its surfaces. The work of Pan and Pan [9] is also noteworthy since it is a rare exaple of a study that provides an exact analytical solution to a forced-vibration proble involving elastic cylinders. In Ref. [9], Pan and Pan obtained analytically the exact forced response in fixed-free isotropic solid cylinders subjected to purely torsional oent excitations. As pointed out in Ref. [30] and elsewhere, any basic forced-vibration probles involving elastic cylinders have not yet been studied or solved using the full three-diensional 3D theory of linear elasticity. One such proble that has hitherto not been considered in the literature is the steady-state vibration response of a siply-supported isotropic solid elastic circular cylinder subjected to two-diensional D haronic standing-wave excitations on its curved surface. The posited proble is closely related to the proble of deterining the wave-propagation characteristics of free haronic standing waves in a siply-supported isotropic solid elastic circular cylinder; the forer is the forced analog of the latter. Explicit forulation and solution of the latter proble is not easy to find in the literature. It is however a well-known and often-cited fact that the characteristics of free haronic standing waves in a finite-length siply-supported elastic cylinder are forally equivalent to those of free haronic traveling waves in a corresponding cylinder of infinite length, the latter

3 of which are well-studied. Nuerical solutions to the free standing-wave proble are thus readily available as a corollary [31]. Certain analytical solutions will also be applicable, with certain odifications, in certain special cases e.g., the Pochhaer-Chree solution will be applicable, with certain odifications, to axisyetric vibrations [5]. A solution to the analogous forced-vibration proble does not however appear to be in the literature. Given the atheatical for of the excitations c.f., Eqs. 1 and, one ight expect that the proposed proble can be solved exactly and that the solution has a closed for. This is indeed the case, and in this paper, we shall exploit certain previously-obtained particular solutions to the Navier-aé equation [3] and exact atrix algebra to construct an exact closed-for 3D elastodynaic solution. The ethod of solution is direct and deonstrates a general approach that can be applied to solve other siilar forced-vibration probles involving elastic cylinders. The radial part of the solution, which involves Bessel functions of the first kind, can be expressed in several different but equivalent fors. We ake use of certain well-known Bessel function identities in order to cast the radial part of the solution in a syetric for that is free of derivatives thereby aking the obtained analytical solution apt for nuerical coputation. The analytical solution is later evaluated nuerically in order to study the steady-state frequency response of the syste in soe exaple cases. In each case, consistency with published natural frequency data is observed. The proposed proble is of general interest both as an exactly-solvable 3D elastodynaics proble and as a benchark forced-vibration proble involving a solid elastic cylinder. The obtained analytical solution is not only useful for revealing the ain physical features of the present proble, but can also serve as a benchark solution for assessent or validation of nuerical ethods and/or coputational software. II. MATHEMATICA DEFINITION OF THE PROBEM Consider the proble of a siply-supported isotropic solid elastic circular cylinder of finite but arbitrary length and radius R subjected to tie-haronic stresses on its curved surface. These stresses are spatially non-unifor and are such that the circuferential and longitudinal variations are also haronic. In this paper, we shall work in the circular cylindrical coordinate syste wherein all physical quantities depend on the spatial coordinates r, θ, z, which denote the radial, circuferential, and longitudinal coordinates, respectively, 3

4 and on the tie t. Using this notation, the boundary stresses on the curved surface are: σ rr R, θ, z, t = A sinθ sin 1a σ rθ R, θ, z, t = B cosθ sin 1b σ rz R, θ, z, t = C sinθ cos 1c or, σ rr R, θ, z, t = A cosθ sin z σ rθ R, θ, z, t = B sinθ sin sinωt, sinωt, z σ rz R, θ, z, t = C cosθ cos where σ rr r, θ, z, t is a noral coponent of stress, σ rθ r, θ, z, t and σ rz r, θ, z, t are shear coponents of stress, {A, B, C} are prescribed constant stresses, each having units of pressure, k Z + a b c and Z + {0} are prescribed diensionless constants, and ω is the prescribed angular frequency of excitation. Since the haronic teporal variation is separable fro the haronic spatial variations in each of the excitations 1a-1c and a-c, these represent haronic standing-wave excitations. The specific proble of interest here is to deterine the elastodynaic response of the cylinder when it is subjected to the non-unifor distribution of stress 1 or on its curved surface. We thus seek to deterine the displaceent at all points of the cylinder. The governing equation of otion for the displaceent is the Navier-aé N equation, which can be written in vector for as [1]: λ + µ u µ u + b = ρ u t, 3 where u ur, θ, z, t is the displaceent field, λ > 0 and µ > 0 are the first and second aé constants, respectively 1, and ρ > 0 is the constant density of the cylinder. Since there are only surface forces acting on the cylinder, the local body force is zero i.e., b = 0. The radial, circuferential, and longitudinal coponents of u shall here be denoted by u r r, θ, z, t, u θ r, θ, z, t, and u z r, θ, z, t, respectively. 1 Note that the first aé constant λ need not be positive, but we have assued it to be so for the purposes of this paper. 4

5 Although we have stated that the cylinder is siply supported, we have not yet specified the boundary conditions at the flat ends of the cylinder, which are situated at z = 0 and z =. The classical siply-supported boundary conditions for the stress and displaceent at the flat ends of the cylinder are: u r r, θ, 0, t = u r r, θ,, t = 0, u θ r, θ, 0, t = u θ r, θ,, t = 0, σ zz r, θ, 0, t = σ zz r, θ,, t = 0, 4a 4b 4c where σ zz r, θ, z, t is the noral coponent of stress along the axis of the cylinder. Conditions 1 and ust be satisfied for all θ [0, π], z 0,, and arbitrary t. Conditions 4a-4c ust be satisfied for all r [0, R, θ [0, π], and arbitrary t. The condition that the displaceent field is finite everywhere in the cylinder is iplicit. Note that we have not given any inforation about the displaceent field and its tie derivatives at soe initial tie t = t 0, and thus the proble as defined is not an initial-boundary-value proble. In the following, we shall refer to the linear-elastic boundary-value proble BVP defined by 1, 3, and 4 as BVP 1, and the boundary-value proble defined by, 3, and 4 as BVP. Note that, for forced otion, at least one of {A, B, C} ust be non-zero when 0. If = 0, then B 0 is required in boundary conditions 1 and at least one of {A, C} is required to be non-zero in boundary conditions. For future reference, we cite here the stress-displaceent relations fro the theory of linear elasticity [30]: σ rr = λ + µ u r r + λ uθ r θ + u r + λ u z z, 5a σ θθ = λ u r λ + µ uθ + r r θ + u r + λ u z z, 5b σ zz = λ u r r + λ uθ r θ + u r + λ + µ u z z, 5c 1 u r σ rθ = µ r θ + u θ r u θ = σ θr, 5d r ur σ rz = µ z + u z = σ zr, 5e r 5

6 uθ σ θz = µ z + 1 r u z = σ zθ. 5f θ Relations 5, which provide the general atheatical connection between the coponents of the displaceent and stress fields, will be used extensively in solving the above-defined BVPs. Since the excitations [1 or ] are tie-haronic, relations 5 and solution uniqueness together iply that the response of the cylinder ust necessarily be so as well. In other words, the displaceent field ur, θ, z, t = u s r, θ, z sinωt, where u s r, θ, z denotes the stationary or tie-independent part of the displaceent field. It is the latter object that we ultiately seek to deterine and to then study. III. SOME PARAMETRIC SOUTIONS TO THE NAVIER-AMÉ EQUATION In the absence of body forces, the following paraetric solutions to Eq. 3 can be obtained using a Buchwald decoposition of the displaceent field see Ref. [3] for details: J u r = nα a s r s I nα s r + b Y nα s r [ s c K nα s r s cosnθ + d s sinnθ] φ z zφ t t + n J n α r a 3 r I n α r + b Y n α r [ ] 3 c 3 sinnθ + d 3 cosnθ χ K n α r z zχ t t, 6 u θ = n J n α s r a s r I n α s r + b Y n α s r [ s c s sinnθ + d s cosnθ] φ z zφ t t K n α s r J nα a r 3 I nα r + b Y nα r [ ] 3 c K nα r 3 cosnθ + d 3 sinnθ χ z zχ t t, 7 and u z = γ s J n α s r a s I n α s r + b Y n α s r [ ] s c K n α s r s cosnθ + d s sinnθ dψ zz dz ψ t t, 8 where n is a non-negative integer and {a 1, a, a 3, b 1, b, b 3, c 1, c, c 3, d 1, d, d 3 } are arbitrary constants. The constituents of Eqs. 6-8 are as follows: 6

7 i The constants α 1 and α in the arguents of the Bessel functions are given by α 1 = κ ρτ λ + µ, α = κ ρτ µ, 9 where κ R\{0} and τ R\{0} are free paraeters. ii The correct linear cobination of Bessel functions is deterined by the relative values of the paraeters {λ, µ, ρ, κ, τ} as given in Table I. inear Cobination s = 1 ter s = ter {J n α s r, Y n α s r} κ > {I n α s r, K n α s r} κ < ρτ λ + µ ρτ λ + µ κ > ρτ µ κ < ρτ µ TABE I: Conditions on the radial part of each ter in Eqs iii In Eqs. 6-7, pries denote differentiation with respect to the radial coordinate r. iv The functions φ z z, φ t t, ψ z z, ψ t t, χ z z, and χ t t are given by κ z κ z E cos + F sin if κ < 0 φ z z = ψ z z = E exp κz + F exp κz if κ > 0, 10 τ t τ t G cos + H sin if τ < 0 φ t t = ψ t t = G exp τt + H exp τt if τ > 0, 11 κ z Ẽ cos + F κ z sin if κ < 0 χ z z = Ẽ exp κz + F exp κz if κ > 0 τ t G cos + H τ t sin if τ < 0 χ t t = G exp τt + H exp τt if τ > 0, 1, 13 where { E, F, G, H, Ẽ, F }, G, H are arbitrary constants. 7

8 v The constant γ s in Eq. 8 is given by 1 if s = 1 γ s = 1 κ ρτ if s =. 14 κ µ Note that Eqs. 6-8 are valid so long as λ + µκ ρτ i.e., α 1 0 and µκ ρτ i.e., α 0; otherwise the radial parts ust be odified as discussed in Ref. [3]. In the following, these conditions will be satisfied, by construction. IV. GENERA FORM OF THE DISPACEMENT FIED General solutions suited to the boundary-value probles defined in Sec. II can be easily constructed fro the faily of paraetric solutions given in Sec. III by identifying one or a cobination of the physical paraeters {, R, k, ω} with the free atheatical paraeters κ and τ. et κ = and τ = ω, and then choose particular solutions defined by taking E = 0 in Eq. 10, Ẽ = 0 in Eq. 1, G = 0 in Eq. 11, G = 0 in Eq. 13, and n = in Eqs Noting the fors of Eqs. 6-8 and coparing 1a/a with 5a, 1b/b with 5d, and 1c/c with 5e, we ay iediately deduce that the axial and teporal parts of the displaceent coponents are given by φ z z = ψ z z = F sin z, χ z z = F sin By defining a new set of arbitrary constants z, 15 φ t t = ψ t t = H sinωt, χ t t = H sinωt. 16 Ā s a s c s F H, Bs b s c s F H, s = 1, 17a à s a s d s F H, Bs b s d s F H, s = 1, 17b à 3 a 3 c 3 F H, B3 b 3 c 3 F H, 17c Ā 3 a 3 d 3 F H, B3 b 3 d 3 F H, 17d the following two independent particular solutions ay be extracted fro Eqs. 6-8: [ { u r = à s 1 } { Ã3 } ] sinθ sin s r 18a 8

9 and where { 1 } s = u θ = [ { Ã s } { r Ã3 1 } ] cosθ sin s u z = [ { u r = Ā s u θ = [ u z = r { Ã s γ s 1 } s { Ā s } s + Ā3 { Ā s γ s } s z sinωt, sinθ cos { } ] cosθ sin r ] + Ā3 { } s 1 } J α s r = J r α s r α s J +1 α s r I α s r = I r α s r + α s I +1 α s r z sinωt, sinθ sin cosθ cos {, } J α s r = s I α s r, 18b 18c 19a 19b 19c 0a { 1 } = J α r = J r α r α J +1 α r I α r = I r α r + α I +1 α r, { } = J α r I α r. 0b Since the Bessel functions of the second kind {Y p α s r, K p α s r} ± as r 0 p 0, the finiteness condition dictates that all such radial ters should be discarded leaving only Bessel functions of the first kind. Note that particular solutions 18 and 19 autoatically satisfy boundary conditions 4a and 4b. Note also that by virtue of 5c σ zz r, θ, z, t = F r, θ sin z sinωt, where the precise for of F r, θ is not relevant for our purposes, and thus displaceents 18 and 19 as well autoatically satisfy boundary condition 4c. It can be deduced fro inspection of 5, 18, and 19, that solution 18 is appropriate for BVP 1, whereas solution 19 is appropriate for BVP. The proper choices of Bessel functions in the radial parts of the displaceent coponents depend on the relative values of the aterial and excitation paraeters; three cases can be distinguished as listed in Table II. The proble should be solved separately for each of these three cases. Note that there ρω are two special cases not included in Table II: i = ; λ+µ and ii ρω =. µ These two singular cases require special treatent and shall not be considered here. As a final reark, note that solutions 18 and 19 are valid only when k Z +. 9

10 Case Paraetric Relationship k, ω Paraetric Relationship κ, τ 1 ρω λ + µ < ρω µ < κ < ρτ λ + µ and κ < ρτ µ 3 < ρω λ + µ < ρω µ ρω λ + µ < < ρω µ κ > κ < ρτ λ + µ and κ > ρτ µ ρτ λ + µ and κ > ρτ µ TABE II: Paraetric relationships defining three distinct sub-probles. In the second colun, the relationship is expressed in ters of the physical excitation paraeters k and ω, whereas in the third colun, the relationship is expressed in ters of the atheatical paraeters κ and τ. V. ANAYTICA SOUTION TO BVP 1 A. Case 1: ρω λ + µ < ρω µ < In this case, κ < ρτ/λ + µ and κ < ρτ/µ c.f., Table II, and thus, according to Table I, the odified Bessel functions I α s r and their derivatives should be eployed in the radial parts of Eqs. 18, where the constants α 1 and α, as deterined fro Eq. 9, are: α 1 = ρω λ + µ, α = ρω µ. 1 The constant γ s in Eq. 18c, as deterined fro Eq. 14, is given by 1 if s = 1 γ s = [ ] ρω 1 if s =. µ Inputting the above ingredients into 18, the displaceent coponents take the for: { [ ] } u r = Ã s r I α s r + α s I +1 α s r Ã3 r I α r sinθ sin u θ = 3a { [ ] } Ã s I α s r r Ã3 r I α r + α I +1 α r cosθ sin 3b 10

11 u z = { } à s γ s I α s r sinθ cos where the constants α s and γ s are given by Eqs. 1 and, respectively. To coplete the solution, we ust deterine the values of the constants 3c {Ã1, Ã, Ã3} in Eqs. 3a-3c that satisfy boundary conditions 1. Substituting Eqs. 3a-3c into Eqs. 5a, 5d, and 5e, and perforing the lengthy algebra yields the stress coponents: { [ 1 σ rr r, θ, z, t = à s β s + µ I r α s r µα ] s I +1 α s r r [ 1 µã3 I r α r + α ] } 4a r I +1α r sinθ sin where [ ] β s = λ αs γ s + µα s, s = 1, 4b and +Ã3 { [ 1 σ rθ r, θ, z, t = µ à s I r α s r + α ] s r I +1α s r [ α 1 + I r α r + α ] } r I +1α r cosθ sin σ rz r, θ, z, t = µ { à s 1 + γ s r I α s r + α s I +1 α s r } Ã3 r I α r sinθ cos z sinωt. 5 6 When 0, application of the boundary conditions proceeds by substituting Eqs. 4, 5, and 6 into the HSs of Eqs. 1a, 1b, and 1c, respectively, and then canceling identical sinusoidal ters on both sides of the resulting equations. This yields the following conditions: à s [ β s + µ µã3 1 R [ 1 R I α s R µα ] s R I +1α s R I α R + α R I +1α R ] = A, 7a 11

12 +Ã3 [ [ 1 à s I R α s R + α ] s R I +1α s R α + 1 R à s 1 + γ s R I α s R + α s I +1 α s R I α R + α R I +1α R Ã3 Conditions 7 can be written as the 3 3 linear syste: ] = Bµ, 7b R I α R = C µ. 7c f v +1 g w +1 1q + w +1 1p + v +1 1q + w +1 h w +1 p + v γ q + w +1 q à 1 à à 3 A = B, C 8a where [ β1 R f µ [ β R g µ [ α h R ] 1 + I α 1 R, R + 1 R + 1 R ] I α R, ] I α R, 8b 8c 8d p R I α 1 R, q R I α R, 8e v +1 α 1 I +1 α 1 R, w +1 α I +1 α R, 8f A B C A R, B R, C. 8g µ µ µ The general solution to syste 8 can be expressed in the for: à i = δ i C i A D A + Ci B B + Ci C, C i = 1,, 3, 9a where 1 if i = 1 δ i = +1 if i = +1 if i = 3, 9b and the values of the coefficients { C i A, Ci B, Ci C : i = 1,, 3} and deterinant D obtained using exact algebra are given by A1 in Appendix A. 1

13 1. Special Case: = 0 When = 0, σ rr = σ rz = 0 and u r = u z = 0. Substituting Eq. 5 into the HS of boundary condition 1b and then canceling identical sinusoidal ters on both sides of the resulting equation yields the coefficient Ã3 in the reaining displaceent coponent u θ, which fro Eq. 3b then reduces to B/µ u θ r, θ, z, t = α I 0 α R 1 R I 1α R I 1 α r sin z sinωt. 30. Exaple Case: = 1 When = 1, the coefficients { C i A, Ci B, Ci C : i = 1,, 3} reduce to: and the deterinant D reduces to: C 1 A = q 1w 1 + γ h 1 w q1 + w, 31a C 1 B = 1 + γ q 1 + w w g 1 w q1, 31b C 1 C = g 1 w h1 w w w, 31c C A = q 1v h 1 w p1 + v, 31d C B = p 1 + v w f 1 v q1, 31e C C = f 1 v h1 w v w, 31f C 3 A = 1 + γ q 1 + w v p 1 + v w, 31g C 3 B = g 1 w p1 + v 1 + γ f 1 v q1 + w, 31h C 3 C = f 1 v w g 1 w v, 31i D = [w w ] g 1 w h1 w p1 + v [ f1 ] q1 v h1 w v w + w γ + [ g1 w v f 1 v w ] q 1. 31j 13

14 B. Case : < ρω λ + µ < ρω µ According to Tables I and II, the Bessel functions J α s r and their derivatives should in this case be eployed in the radial parts of Eqs. 18. The displaceent coponents thus take the for: { [ ] } u r = à s r J α s r α s J +1 α s r Ã3 r J α r sinθ sin u θ = where 3a { [ ] } à s J α s r r Ã3 r J α r α J +1 α r cosθ sin 3b u z = α 1 = { } à s γ s J α s r sinθ cos 3c + ρω λ + µ, α = + ρω µ, 33 and γ s is again given by Eq.. } The constants {Ã1, Ã, Ã3 in Eqs. 3a-3c ust as before be chosen so as to satisfy boundary conditions 1. Proceeding as in the previous case, we first obtain general forulas for the radial coponents of the stress field. Substituting Eqs. 3a-3c into Eqs. 5a, 5d, and 5e, and perforing the lengthy algebra yields the required stress coponents: { [ 1 σ rr r, θ, z, t = à s η s + µ J r α s r + µα ] s J +1 α s r r +µã3 [ 1 J r α r + α r J +1α r ] } sinθ sin z sinωt, 34a where [ ] η s = λ αs + γ s + µα s, s = 1, 34b +Ã3 [ α σ rθ r, θ, z, t = µ { [ 1 à s J r α s r α ] s r J +1α s r 1 J r α r α ] } r J +1α r cosθ sin z sinωt, 35 14

15 and σ rz r, θ, z, t = µ { Ã s 1 + γ s r J α s r α s J +1 α s r } Ã3 r J α r sinθ cos z sinωt. 36 When 0, application of the boundary conditions 1 as described in Sec. V A yields three conditions involving the constants {Ã1, Ã, Ã3}. Collectively, these three conditions can be written as the 3 3 linear syste: F + V +1 G + W +1 1Q W +1 1P V +1 1Q W +1 H + W +1 P V γ Q W +1 Q Ã 1 Ã Ã 3 A = B, C 37a where [ η1 R F + µ [ η R G + µ [ α H R + ] 1 J α 1 R, R 1 R 1 R ] J α R, ] J α R, 37b 37c 37d P R J α 1 R, Q R J α R, 37e and {A, B, C} are as given by 8g. V +1 α 1 J +1 α 1 R, W +1 α J +1 α R, 37f The general solution to syste 37 can again be expressed in the for given by Eq. 9 with the exact values of the coefficients { C i A, Ci B, Ci C : i = 1,, 3} and deterinant D in this case given by A in Appendix A. 1. Special Case: = 0 When = 0, σ rr = σ rz = 0 and u r = u z = 0. Using Eq. 35 in boundary condition 1b yields the coefficient Ã3 in the reaining displaceent coponent u θ, which fro Eq. 3b then reduces to u θ r, θ, z, t = B/µ α J 0 α R 1 J R 1α R 15 J 1 α r sin z sinωt. 38

16 . Exaple Case: = 1 When = 1, the coefficients { C i A, Ci B, Ci C : i = 1,, 3} reduce to: C 1 A = Q 1W 1 + γ H 1 + W Q1 W, 39a C 1 B = 1 + γ Q 1 W W G 1 + W Q1, 39b C 1 C = G 1 + W H1 + W W W, 39c C A = Q 1V H 1 + W P1 V, 39d C B = P 1 V W F 1 + V Q1, 39e C C = F 1 + V H1 + W V W, 39f C 3 A = 1 + γ Q 1 W V + P 1 V W, 39g C 3 B = G 1 + W P1 V 1 + γ F 1 + V Q1 W, 39h and the deterinant D reduces to: C 3 C = F 1 + V W + G 1 + W V, 39i D = [W W ] G 1 + W H1 + W P1 V [ F1 ] Q1 + V H1 + W V W W γ + [ G 1 + W V + F 1 + V W ] Q 1. 39j C. Case 3: ρω λ + µ < < ρω µ According to Tables I and II, the s = 1 ter in each of the radial parts of Eqs.18a-18c should eploy the odified Bessel functions I α 1 r and their derivatives while the s = ters should eploy the Bessel functions J α r and their derivatives. The reaining ters are unodified fro those of Case. The displaceent coponents thus take the for: u r = { [ ] Ã 1 r I α 1 r + α 1 I +1 α 1 r + Ã } Ã3 r J α r sinθ sin 16 [ ] r J α r α J +1 α r 40a

17 where u θ = u z = { [ ] Ã1 I α 1 r + r ÃJ α r } Ã3 r J α r α J +1 α r cosθ sin { à 1 γ 1 I α 1 r + Ãγ J α r} sinθ cos α 1 = ρω λ + µ, α = 40b 40c + ρω µ, 41 and γ s is again given by Eq.. } The constants {Ã1, Ã, Ã3 in Eqs. 40a-40c ust as before be chosen so as to satisfy boundary conditions 1. Substituting Eqs. 40a-40c into Eqs. 5a, 5d, and 5e, and perforing the lengthy algebra yields the required stress coponents: [ 1 σ rr r, θ, z, t = {à 1 β 1 + µ I r α 1 r µα ] 1 I +1 α 1 r r [ 1 + à η + µ J r α r + µα ] J +1 α r r where and + µã3 [ β 1 = λ 1 J r α r + α r J +1α r [ σ rθ r, θ, z, t = µ + à + Ã3 α 1 γ 1 { ] [ 1 J α r α [ α r σ rz r, θ, z, t = µ 1 r + µα 1, η = λ [ 1 à 1 I r α 1 r + α 1 ] r J +1α r J α r α r J +1α r { à γ 1 + à 1 + γ r J α r α J +1 α r ] } sinθ sin [ α + γ ] r I +1α 1 r ] z ] } cosθ sin r I α 1 r + α 1 I +1 α 1 r Ã3 r J α r 17 sinωt, 4a + µα, 4b z sinωt, 43 } sinθ cos z sinωt. 44

18 When 0, application of the boundary conditions 1 as described in Sec. V A yields three conditions, which can be written as the 3 3 linear syste: f v +1 G + W +1 1Q W +1 1p + v +1 1Q W +1 H + W +1 p + v γ Q W +1 Q Ã 1 Ã Ã 3 A = B, C where all atrix eleents have been previously defined by Eqs. 8b-8g and Eqs. 37b- 37f. The general solution to syste 45 can once ore be expressed in the for given by Eq. 9 with the exact values of the coefficients { C i A, Ci B, Ci C : i = 1,, 3} and deterinant D in this case given by A3 in Appendix A Special Case: = 0 When = 0, σ rr = σ rz = 0 and u r = u z = 0. It can be shown that application of boundary condition 1b yields the sae result for u θ as that found for Case, naely, Eq Exaple Case: = 1 When = 1, the coefficients { C i A, Ci B, Ci C : i = 1,, 3} reduce to: C 1 A = Q 1W 1 + γ H 1 + W Q1 W, 46a C 1 B = 1 + γ Q 1 W W G 1 + W Q1, 46b C 1 C = G 1 + W H1 + W W W, 46c C A = Q 1v H 1 + W p1 + v, 46d C B = p 1 + v W f 1 v Q1, 46e C C = f 1 v H1 + W + v W, 46f C 3 A = 1 + γ Q 1 W v + p 1 + v W, 46g 18

19 C 3 B = G 1 + W p1 + v 1 + γ f 1 v Q1 W, 46h and the deterinant D reduces to: C 3 C = f 1 v W G 1 + W v, 46i D = [W W ] G 1 + W H1 + W p1 + v [ f1 ] Q1 v H1 + W + v W W γ + [ G1 + W v + f 1 v W ] Q 1. 46j VI. ANAYTICA SOUTION TO BVP A. Case 1: ρω λ + µ < ρω µ < Eploying 19 and following the logic at the beginning of Sec. V A, the displaceent coponents take the for: { [ ] } u r = Ā s r I α s r + α s I +1 α s r + Ā3 r I α r cosθ sin u θ = { r Ā s I α s r +Ā3 u z = 47a [ ] } r I α r+α I +1 α r sinθ sin { } Ā s γ s I α s r cosθ cos where the constants α s and γ s are given by Eqs. 1 and, respectively. 47b 47c To deterine the values of the constants { Ā 1, Ā, Ā3} in Eqs. 47a-47c that satisfy boundary conditions, we follow the sae procedure outlined in Sec. V A. This leads to the 3 3 linear syste: f v +1 g w +1 1q + w +1 1p + v +1 1q + w +1 h w +1 p + v γ q + w +1 q Ā 1 Ā Ā 3 A = B C 48, 19

20 where the entries in the above 3 3 coefficient atrix are given by Eqs. 8b-8f, and {A, B, C} are given by 8g. The general solution to syste 48 is: Ā i = δ i C i A D A Ci B B + Ci C, C i = 1,, 3, 49a where +1 even i δ i = 1 odd i, 49b the coefficients { C i A, Ci B, Ci C : i = 1,, 3} are given by Eqs. A1a-A1i, and D is given by Eq. A1j. 1. Special Case: = 0 When = 0, u θ = 0, σ rθ = 0, and boundary condition b is identically satisfied. Application of boundary conditions a and c yields the linear syste: f 0 v 1 g 0 w 1 v γ w1 Ā1 Ā = A. 50 C The general solution to syste 50 is: 1 + γ w1 A g 0 w 1 C Ā 1 =, 51a 1 + γ w1 f0 v 1 v1 g0 w 1 Ā = f0 v 1 C v1 A 1 + γ w1 f0 v 1 v1 g0 w 1, 51b where {f 0, g 0 } and {v 1, w 1 } are the evaluated zero- and first-order Bessel functions respectively obtained fro substituting = 0 in Eqs. 8b, 8c, and 8f. Thus, in this special case, the non-zero coponents of the displaceent field reduce to: [ ] u r r, z, t = Ā s α s I 1 α s r sin 5a u z r, z, t = [ Ā s γ s I 0 α s r ] cos 5b where constants α s and γ s are given by Eqs. 1 and, respectively, and constants } {Ā1, Ā given by Eq

21 B. Case : < ρω λ + µ < ρω µ Eploying 19 and following the logic at the beginning of Sec. V B, the displaceent coponents take the for: { [ ] } u r = Ā s r J α s r α s J +1 α s r + Ā3 r J α r cosθ sin u θ = { r [ ] } Ā s J α s r +Ā3 r J α r α J +1 α r sinθ sin z u z = { } Ā s γ s J α s r cosθ cos where constants α s and γ s are given by Eqs. 33 and, respectively. 53a sinωt, 53b 53c The values of the constants { Ā 1, Ā, Ā3} in Eqs. 53a-53c are again obtained by foral application of boundary conditions, which in the present case leads to the 3 3 linear syste: F + V +1 G + W +1 1Q W +1 1P V +1 1Q W +1 H + W +1 P V γ Q W +1 Q Ā 1 Ā Ā 3 A = B C where the entries in the above 3 3 coefficient atrix are given by Eqs. 37b-37f, and {A, B, C} are given by 8g. The general solution to syste 54 is given by Eqs. 49, where the coefficients { C i A, Ci B, Ci C : i = 1,, 3} are given by Eqs. Aa-Ai, and D is given by Eq. Aj. 54, 1. Special Case: = 0 When = 0, u θ = 0, σ rθ = 0, and boundary condition b is identically satisfied. Application of boundary conditions a and c yields the linear syste: F 0 + V 1 G 0 + W 1 V γ W1 Ā1 Ā = A. 55 C 1

22 The general solution to syste 55 is: 1 + γ W1 A + G 0 + W 1 C Ā 1 =, 56a 1 + γ W1 F0 + V 1 V1 G0 + W 1 V 1 A + F 0 + V 1 C Ā =, 56b 1 + γ W1 F0 + V 1 V1 G0 + W 1 where {F 0, G 0 } and {V 1, W 1 } are the evaluated zero- and first-order Bessel functions respectively obtained fro substituting = 0 in Eqs. 37b, 37c, and 37f. Thus, in this special case, the non-zero coponents of the displaceent field reduce to: [ ] u r r, z, t = Ā s α s J 1 α s r sin 57a u z r, z, t = [ Ā s γ s J 0 α s r ] cos 57b where constants α s and γ s are given by Eqs. 33 and, respectively, and constants } {Ā1, Ā given by Eq. 56. C. Case 3: ρω λ + µ < < ρω µ Eploying 19 and following the logic at the beginning of Sec. V C, the displaceent coponents take the for: { [ ] u r = Ā 1 r I α 1 r + α 1 I +1 α 1 r + Ā } + Ā3 r J α r cosθ sin z [ ] r J α r α J +1 α r sinωt, 58a u θ = { [ ] Ā1 I α 1 r + r ĀJ α r } + Ā3 r J α r α J +1 α r sinθ sin 58b u z = { Ā 1 γ 1 I α 1 r + Āγ J α r} cosθ cos 58c where constants α s and γ s are given by Eqs. 41 and, respectively.

23 The values of the constants { Ā 1, Ā, Ā3} in Eqs. 58a-58c are again obtained by foral application of boundary conditions, which in the present case leads to the 3 3 linear syste: f v +1 G + W +1 1Q W +1 1p + v +1 1Q W +1 H + W +1 p + v γ Q W +1 Q Ā 1 Ā Ā 3 A = B C where the entries in the above 3 3 coefficient atrix are given by Eqs. 8b-8f and Eqs. 37b-37f, and {A, B, C} are as before given by 8g. The general solution to syste 59 is given by Eqs. 49, where the coefficients { C i A, Ci B, Ci C : i = 1,, 3} are given by Eqs. A3a-A3i, and D is given by Eq. A3j. 59, 1. Special Case: = 0 When = 0, u θ = 0, σ rθ = 0, and boundary condition b is identically satisfied. Application of boundary conditions a and c yields the linear syste: f 0 v 1 G 0 + W 1 v γ W1 Ā1 Ā = A. 60 C The general solution to syste 60 is: 1 + γ W1 A + G 0 + W 1 C Ā 1 =, 61a 1 + γ W1 f0 v 1 + v1 G0 + W 1 Ā = v 1 A f 0 v 1 C 1 + γ W1 f0 v 1 + v1 G0 + W 1, 61b where {f 0, G 0 } and {v 1, W 1 } are the evaluated zero- and first-order Bessel functions respectively obtained fro substituting = 0 in Eqs. 8b, 37c, 8f, and 37f. Thus, in this special case, the non-zero coponents of the displaceent field reduce to: [ ] u r r, z, t = Ā 1 α 1 I 1 α 1 r Āα J 1 α r sin 6a u z r, z, t = [ Ā 1 γ 1 I 0 α 1 r + Āγ J 0 α r] cos 6b where constants α s and γ s are given by Eqs. 41 and, respectively, and constants } {Ā1, Ā given by Eq

24 D. Consistency with the ENBKS Field Equations As an analytical check, we have verified that, in the special = 0 case, the general stress and displaceent fields obtained fro applying our ethod of solution to BVP are consistent with the general axisyetric field equations that would be obtained fro applying the foralis of Ebenezer et al. [8] henceforth referred to as the ENBKS ethod to BVP. Deonstration of this consistency is soewhat intricate; the details are therefore consigned to Appendix C. Equivalency of our = 0 solution with the axisyetric solution that would be obtained fro the ENBKS ethod then directly follows fro application of the boundary conditions. VII. NUMERICA EXAMPE Paraeter ength Radius R Nuerical Value Mass density ρ 8000 kg/ 3 Young odulus E 190 GPa Poisson ratio ν 0.30 First aé constant λ Second aé constant µ GPa GPa TABE III: Geoetric and aterial paraeter values used in the nuerical exaple. As an exaple, we exaine the steady-state forced-vibration response of a solid steel cylinder subjected to haronic standing-wave excitations of type on its curved surface. The geoetric and aterial properties of the cylinder are specified in Table III. While there are several possible types of analyses that could be considered in studying the steady-state forced response, the ost coon is to study the behavior of the stationary displaceent as a function of the excitation frequency. We shall here restrict attention to this type of analysis. Recall that the stationary displaceent in the context of steady-state i.e., tieharonic vibrations is erely the tie-independent part of the displaceent field. While 4

25 Mode nuber Circuferential wave nuber = 0 = 1 = = TABE IV: Natural frequencies of a siply-supported isotropic solid elastic circular cylinder having the geoetrical and aterial properties given in Table III. All values are in units of khz. The above data was coputed using the free-vibration frequency data given in Table 7 of Ref. [18]. this is a 3D function of the spatial coordinates and thus varies fro point to point in the cylinder, its qualitative global behavior as a function of the excitation frequency should generally be independent of location in the cylinder. This will certainly be true under general variations in the circuferential and longitudinal coordinates since the corresponding parts of the displaceent field are independent of excitation frequency. Variations in the radial coordinate are effectively scale transforations of the otherwise frequency-dependent Bessel functions. In frequency space, the effects of such transforations can be significant locally but not so globally for the siple reason that all physically relevant features in frequency space are due to the boundary conditions and are thereby ainly deterined by the frequency-dependent aplitudes { Ā 1, Ā, Ā3}. As is usual for this type of analysis, we shall thus study the stationary part of the displaceent field, at a few suitably-chosen representative points in the cylinder, as a function of the excitation frequency. To do so, we shall nuerically evaluate the analytical solution to BVP obtained in Sec. VI. Note that, aside fro nodal points, we are free to choose any point in the cylinder as a representative point. 5

26 Stationary Radial Displaceent 1 # k = # k = # k = # k = # k = Excitation Frequency khz FIG. 1: Stationary radial displaceent at point r, θ, z = R/, 0, /7 for various values of the longitudinal wave nuber k and circuferential wave nuber = 0. For reference, the natural frequencies listed in Table IV are arked by X s on the frequency axis of each subplot. Using paraeter values as specified in Table III, the coponents of the displaceent field were coputed at excitation frequencies that are integer ultiples of 10 Hz with lower and upper bounds of 10 Hz and 100 khz, respectively. In all calculations, the excitation aplitudes were set as follows: A = B = C = 10 5 Pa. Given that the cylinder is being forced to vibrate, we expect to observe unusually large displaceents i.e., resonances when the excitation frequency is close to one of the natural frequencies of the siply-supported cylinder. The first nine of these frequencies, coputed using free-vibration frequency data 6

27 Stationary Radial Displaceent 1 # k = # k = # k = # k = # k = Excitation Frequency khz FIG. : Sae as Figure 1 except the circuferential wave nuber = 1. obtained fro Ref. [18], are given in Table IV. The stationary radial displaceent at the representative interior point r = R/, θ = 0, z = /7 is shown in Figs. 1-3 for various values of the longitudinal wave nuber k and circuferential wave nubers = 0, 1, and, respectively. For visual reference, the natural frequencies listed in Table IV are arked by X s on the frequency axis of each subplot. As seen in Figs. 1-3 and as well observed in other ore general forced-vibration analyses of solid elastic cylinders [7, 8], the response at a particular frequency depends on the spatial distribution of the excitation, which is here deterined by the specific values of the wave nubers and k. In each case, we observe unistakable resonances close to a subset 7

28 Stationary Radial Displaceent 1 # k = # k = # k = # k = # k = Excitation Frequency khz FIG. 3: Sae as Figure 1 except the circuferential wave nuber =. of the natural frequencies of the siply-supported cylinder. Note that each individual excitation obtained by specifying a single pair of, k values generates a unique series of resonances, as opposed to producing just a single resonance. In other words, a single haronic excitation excites a set of resonant odes instead of exciting only one resonant ode. Unfortunately, there does not appear to be any siple rule for predicting which resonant odes will be excited by a particular excitation. In Figs. 1-3, we used coon vertical and horizontal scales in all subplots in order to ake it easier to copare the different cases. We should however ention that the aplitudes of the resonances are not all equal, and this is evident when one views the 8

29 Stationary Radial Displaceent #10-4 # k = 1 k = k = 3 k = 4 k = 5 Natural Frequencies Excitation Frequency khz FIG. 4: Responses at point r, θ, z = R/, 0, /7 to various standing-wave excitations with longitudinal wave nubers k as indicated and circuferential wave nuber = 3. As discussed in the text, the inset shows a closer view of the neighborhood around the third resonance. A near-degeneracy in the natural frequency spectru occurs inside the interval 40, 41 khz. response outside the displaceent range shown in the figures. This feature is illustrated in Fig. 4, which shows the responses to the five lowest k-haronic standing-wave excitations [at the representative point r = R/, θ = 0, z = /7] when = 3. Higher k-haronics excite resonant odes at higher excitation frequencies than those considered here. Note that the displaceents generated by these five different standing-wave excitations {, k : = 3, k = 1,..., 5} are overlaid on the sae plot with the understanding that they actually correspond to different excitation cases. So, it should be understood that, for instance, the red curve is the response to a standing-wave excitation with longitudinal wave nuber k = 1, whereas the green curve is the response to a standing-wave excitation with longitudinal wave nuber k = 3. The natural frequencies listed in the last colun of Table IV are again arked by X s on the frequency axis. No resonances occur in the frequency interval 0, 5 khz. For this reason, we only display the result on the frequency interval [5, 55] khz. It is clear fro this plot, which is representative, that the aplitudes of the resonances generally differ. Note that differences in aplitude not only occur between the 9

30 Stationary Axial Displaceent k = 0 1 # = # = # = #10-6 = 3 5 # Excitation Frequency khz FIG. 5: Stationary axial displaceent at point r, θ, z = R/, 0, /7 for various values of the circuferential wave nuber and longitudinal wave nuber k = 0. For reference, the natural frequencies listed in Table IV are arked by X s on the frequency axis of each subplot. different excitation cases; the aplitudes of the resonances generated by each individual excitation also vary. While it ay be obvious to soe readers, it is worth ephasizing that the aplitudes, which inherently depend on the frequency resolution 3 and on where in the 3 In the ain plot of Fig. 4 i.e., not the inset, the coponents of the displaceent field were coputed at excitation frequencies that are integer ultiples of 1 Hz with lower and upper bounds of 1 Hz and 100 khz, respectively. When a different frequency increent is used e.g., 0.5 Hz instead of 1 Hz, the aplitudes change. 30

31 cylinder the response is deterined, should not be interpreted as resonance intensities. In the present context of a lossless i.e., undaped cylinder, the aplitudes are insignificant since, in theory, the aplitude of any resonance asyptotically approaches infinity as the excitation frequency approaches the associated natural frequency. Resonance intensities are actually related non-trivially to the shapes and widths of the resonances, and thus eaningful coparisons of resonance intensities can be ade by careful exaination of the resonance widths. Accurately calculating resonance intensities is a specialized subject in its own right and one that is ore appropriately discussed when there are aterial daping effects. We shall thus not consider this topic any further here. With respect to studying the elastodynaic response as a function of excitation frequency, our intent here is only to illustrate and discuss the results of nuerically evaluating our analytical solution. Near-degeneracies in the natural frequency spectru are not uncoon. Note that we are here specifically referring to the situation in which two or ore consecutive natural frequencies have nearly equal values. This occurs, for exaple, in the low end of the frequency spectru when = 3. As can be seen fro the last colun of Table IV, the third and fourth natural frequencies differ by less than 10 Hz, which is a sall difference on a khz scale. Such situations are interesting in the context of forced vibrations since it is not generally obvious how any individual resonances will occur or can be resolved when the excitation frequency is in the vicinity of several closely-spaced natural frequencies. In theory, one generally expects to observe one resonance per natural frequency, but for a variety of reasons, this does not always occur in practice. If fewer resonances occur than expected, then it is inforative to deterine their relative intensities. Again, it does not suit our purpose to delve into this topic here. We erely want to point out that no further individually-resolved resonances are revealed when zooing in on the third resonance in Fig. 4. To be certain, we re-coputed the coponents of the displaceent field at excitation frequencies in the interval [40, 41] khz in increents of 0.1 Hz. A frequency increent of 1 Hz was eployed in obtaining the ain plot of Fig. 4. A zooed view of the result is shown in the inset of Fig. 4 fro which it is clear that there is no resonance associated with the third natural frequency. In Figs. 1-4, the absence of resonances in the neighborhoods of certain natural frequencies, in particular, the fifth natural frequency when = 0 and the third natural frequency when = {1,, 3} is noteworthy and interesting. As it turns out, resonant odes at these 31

32 frequencies are excited by haronic boundary stresses of type with longitudinal wave nuber k = 0, as shown in Fig. 5. The analytical solutions obtained in Secs. V and VI are valid only when k is a positive integer. Analytical solutions to BVP 1 and BVP can however also be obtained when k = 0. The solution to BVP is given in Appendix B. The results of Figs. 1-5 therefore iply that boundary stresses of the haronic standing-wave type excite resonant odes at all natural frequencies. Siilar results are obtained for all coponents of the displaceent field evaluated at any suitably-chosen representative point. Nuerical experients bear out that, regardless of displaceent coponent or location in the cylinder, the frequency response is qualitatively always the sae; each standing-wave excitation generates a series of resonances that are in correspondence with a unique subset of the natural frequency spectru of the siplysupported cylinder. Quantitative differences in the detailed features of the resonances e.g., their shapes and widths are observed when results for the different displaceent coponents are overlaid or when the representative point is varied, but these differences are not usually of interest in steady-state vibration analyses of undaped solid elastic cylinders [7, 8]. VIII. CONCUSION In this paper, we considered the proble of deterining the steady-state forced-vibration response of a siply-supported isotropic solid elastic circular cylinder subjected to D haronic standing-wave excitations on its curved surface. Exploiting previously-obtained particular solutions to the Navier-aé equation [3] and exact atrix algebra, we constructed an exact closed-for 3D elastodynaic solution. In constructing the solution, we ade use of standard Bessel function identities in order to cast the radial part of the solution, which involves Bessel functions of the first kind, in a syetric for that is free of derivatives thereby aking the obtained analytical solution apt for nuerical coputation. Two coplete analytical solutions were in fact constructed corresponding to two different but closely-related failies of haronic standing-wave excitations. The second of these analytical solutions i.e., the solution to BVP was evaluated nuerically in order to study the frequency response in soe exaple cases. In each case, the solution generates a series of resonances in correspondence with a subset of the natural frequencies of the siplysupported cylinder. It is worth ephasizing that each individual standing-wave excitation 3

33 generates a unique series of resonances, as opposed to producing just a single resonance. In general, the response at a particular frequency depends on the spatial distribution of the excitation, which is, in the present context, deterined by the specific values of the wave nubers and k. This is consistent with what is observed in other forced-response studies of solid elastic cylinders [7, 8]. Finally, we note that while haronic standing-wave excitations of type 1 and excite resonant odes at all natural frequencies, there does not appear to be any siple rule for predicting which resonant odes will be excited by a particular excitation. The proble considered in this paper is of general interest both as an exactly-solvable 3D elastodynaics proble and as a benchark forced-vibration proble involving a solid elastic cylinder. The obtained analytical solution is not only useful for revealing the ain physical features of the considered proble, but can also serve as a benchark solution for assessent and/or validation purposes. The ethod of solution eployed in this paper deonstrates a general approach that can be applied to solve other elastodynaic forcedresponse probles involving isotropic, open or closed, solid or hollow, elastic cylinders with siply-supported or other boundary conditions. Acknowledgents The authors acknowledge financial support fro the Natural Sciences and Engineering Research Council NSERC of Canada and the Ontario Research Foundation ORF. Appendix A: Exact Values of the Solution Coefficients { C i A, Ci B, Ci C : i = 1,, 3} 1. Case 1 C 1 A = 1q + w +1 q 1 + γ h w +1 q + w +1, A1a C 1 B = 1 + γ 1q + w +1 q + w +1 g w +1 q, A1b C 1 C = g w +1 h w +1 1q + w +1 1q + w +1, A1c C A = 1p + v +1 q h w +1 p + v +1, A1d C B = 1q + w +1 p + v +1 f v +1 q, A1e 33

34 C C = f v +1 h w +1 1q + w +1 1p + v +1, A1f C 3 A and = 1+γ 1p +v +1 q +w +1 1q +w +1 p +v +1, A1g C 3 B = g w +1 p + v γ f v +1 q + w +1, A1h C 3 C = f v +1 1q + w +1 g w +1 1p + v +1, A1i [ ] D = 1q + w +1 g w +1 h w +1 p + v +1 [ f ] γ v +1 h w +1 1q + w +1 1p + v +1 q + w +1 [ g ] + w +1 1p + v +1 f v +1 1q + w +1 q. A1j. Case C 1 C C C C 1 A = 1Q W +1 Q 1 + γ H + W +1 Q W +1, Aa C 1 B = 1 + γ 1Q W +1 Q W +1 G + W +1 Q, Ab = G +W +1 H +W +1 1Q W +1 1Q W +1, Ac C A = 1P V +1 Q H + W +1 P V +1, Ad C B = 1Q W +1 P V +1 F + V +1 Q, Ae = F + V +1 H + W +1 1Q W +1 1P V +1, Af C 3 A = 1 + γ 1P V +1 Q W +1 1Q W +1 P V +1, C 3 C and Ag C 3 B = G + W +1 P V γ F + V +1 Q W +1, Ah = F + V +1 1Q W +1 G + W +1 1P V +1, Ai [ ] D = 1Q W +1 G + W +1 H + W +1 P V +1 [ F ] γ + V +1 H + W +1 1Q W +1 1P V +1 Q W +1 [ G ] + + W +1 1P V +1 F + V +1 1Q W +1 Q. Aj 34

35 3. Case 3 C 1 C C C C 1 A = 1Q W +1 Q 1 + γ H + W +1 Q W +1, A3a C 1 B = 1 + γ 1Q W +1 Q W +1 G + W +1 Q, A3b = G +W +1 H +W +1 1Q W +1 1Q W +1, A3c C A = 1p + v +1 Q H + W +1 p + v +1, A3d C B = 1Q W +1 p + v +1 f v +1 Q, A3e = f v +1 H + W +1 1Q W +1 1p + v +1, A3f C 3 A = 1 + γ 1p + v +1 Q W +1 1Q W +1 p + v +1, C 3 C and A3g C 3 B = G + W +1 p + v γ f v +1 Q W +1, A3h = f v +1 1Q W +1 G + W +1 1p + v +1, A3i [ ] D = 1Q W +1 G + W +1 H + W +1 p + v +1 [ f ] γ v +1 H + W +1 1Q W +1 1p + v +1 Q W +1 [ G ] + + W +1 1p + v +1 f v +1 1Q W +1 Q. A3j Appendix B: Solution to BVP in the Special Case k = 0 It can be established eploying results fro Ref. [3] or otherwise that u r = 0, u θ = 0, u z = AJ αr cosθ sinωt, B1 where α = ρω /µ, is a particular solution to Eq. 3 copatible with boundary conditions a-c when k = 0. Solution B1 therefore furnishes the general for of the displaceent field appropriate to the boundary-value proble defined in Sec. II in the special case k = 0. We need now only to deterine the value of the constant A in B1 that satisfies boundary 35

36 conditions a-c. Substituting the coponents of solution B1 into Eqs. 5a, 5d, and 5e iediately yields the stress coponents: σ rr r, θ, z, t = σ rθ r, θ, z, t = 0, Ba and [ ] σ rz r, θ, t = µa r J αr αj +1 αr cosθ sinωt. Bb Boundary conditions a and b are therefore satisfied identically. Application of boundary condition c with k = 0 yields a condition involving the constant A that can be subsequently solved to give A = C µ [ J R αr αj +1 αr ]. B3 Appendix C: Axisyetric Solution to BVP Using the ENBKS Method In this appendix, we show how the axisyetric solution to BVP ay be reproduced using the ENBKS ethod. We shall here only consider Case 1. The solution in the other two cases ay be siilarly reproduced. 1. General ENBKS Solution Quoting fro Ref. [8], the following is an exact axisyetric solution to Eq. 3, for arbitrary values of k rn n = 1,, 3,..., N r and k zn n = 1,, 3,..., N z : where u1 z u 1 r u z u r uenbks z u ENBKS r = u1 z u 1 r + u z u r N r P cosk 1 z + P ns J 0 k rn r cosk zns z = n=1 N r, P ns ψ ns J 1 k rn r sink zns z n=1 N z QJ 0 K r + Q ns J 0 k rns r cosk zn z = n=1 N z, Q ns χ ns J 1 k rns r sink zn z n=1 36, C1a C1b C1c