- Email: [email protected]
Dynamic analysis of a hollow cylinder subject to a dual traveling force imposed on its inner surface
Journal of Sound and Vibration 340 (2015) 283–302
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Dynamic analysis of a hollow cylinder subject to a dual traveling force imposed on its inner surface Sooyoung Lee, Jongwon Seok n School of Mechanical Engineering, College of Engineering, Chung-Ang University, 84 HeukSeok-Ro, DongJak-Gu, Seoul 156-756, Republic of Korea
a r t i c l e i n f o
abstract
Article history: Received 31 May 2014 Received in revised form 24 November 2014 Accepted 25 November 2014 Handling Editor: S. Ilanko Available online 23 December 2014
The dynamic behavior of a hollow cylinder under a dual traveling force applied to the inner surface is investigated in this study. The cylinder is constrained at both the top and bottom surfaces not to move in the length direction but free in other directions. And a dual force travels at a constant velocity along the length direction on the inner surface of the hollow cylinder. The resulting governing field equations and the associated boundary conditions are ruled by the general Hooke's law. Due to the nature of the field equations, proper adjoint system of equations and biorthogonality conditions were derived in a precise and detailed manner. To solve these field equations in this study, the method of € separation of variable is used and the method of Frobenius is employed for the differential equations in the radial direction. Using the field equations, the eigenanalyses on both the original and its adjoint system were performed with great care, which results in the eigenfunction sets of both systems. The biorthogonality conditions were applied to the field equations to obtain the discretized equation for each mode. Using the solutions of the discretized equations that account for the boundary forcing terms, the critical speed for a dual traveling force for each mode could be computed. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction The dynamic behaviors of the structures subjected to a variety types of traveling forces have been studied for a long time in various engineering fields [1–8]. In particular, hollow cylindrical structure is used in many machines, which motivates the researches on the investigation of its dynamics under a traveling force. Single or dual slotting of a hollow cylinder is an illustrative example [9–11]. Although few in number, several studies dealing with the three-dimensional eigenanalysis of thick hollow cylinders have been conducted. Greenspon [12,13] addressed the flexural vibrations of a thick-walled circular cylinder of finite length using the work of Pochhammer [14]. In his study, he obtained the curves of the frequency parameter as a function of the length and thickness ratios. Similarly, Gazis [15,16] studied the analytical foundation for an investigation of the harmonic waves in a hollow circular cylinder of infinite extent using the work described in [14]. Mirsky [17] obtained the most general type of harmonic waves in both hollow and solid circular cylinders of transversely isotropic material. Ye and Soldatos [18] analyzed the free vibration of cross-ply laminated hollow cylinders based on a recursive solution. So and Leissa [19] performed an analysis of the free vibrations of a thick cylinder using a Ritz analysis. Brischetto [20] obtained an
n
Corresponding author. Tel.: þ82 2 820 5729; fax: þ 82 2 3280 9982. E-mail address: [email protected] (J. Seok).
http://dx.doi.org/10.1016/j.jsv.2014.11.032 0022-460X/& 2014 Elsevier Ltd. All rights reserved.
284
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
exact solution to a free vibration problem of single-layer spherical, cylindrical, and flat panels. Further, Soldatos and Hadjigeorgiou [21] solved a free vibration problem of a homogeneous isotropic thick cylindrical shell using a new iterative approach. This new approach allows the accurate prediction of the frequencies of vibration. Loy and Lam [22] performed an approximate analysis of the vibration of thick circular cylindrical shells using a concept that is similar to the finite-strip method. Owing to the difficulties faced in solving a three-dimensional problem, the hollow cylinder is often simplified to a twodimensional cylindrical shell equivalent. And the dynamic problem of the cylindrical shell under a traveling force has long been treated by many researchers [23–25] due to the relative simplicity compared to that of its three-dimensional counterpart. The dynamic response of shells based on the classical shell theory was studied by Bhuta [26], using the theory of a thin cylindrical shell. And the response of an infinitely long, circular, cylindrical shell axially propagating step wave was investigated by Forrestal and Hermann [27]. Later, Reismann and Padlog [28] presented a formal solution of the dynamic response of a cylindrical shell of finite length by the use of the method of eigenfunction expansion. Raymond Parnes [29] obtained the response of an infinite elastic cylindrical bore to a traveling load. Huang [30] investigated the dynamic response of the shell of infinite length acted upon by a ring load traveling at a constant velocity using the Fourier transform method in conjunction with the contour integral method. Also, Mangrum and Burns [31] studied the dynamic response of orthotropic cylindrical shells under an axial moving load using small deflection shell theory. Chonan [32] studied the dynamic response of a cylindrical shell imperfectly bonded to a surrounding elastic continuum of infinite extent. In his study [32], an axisymmetric ring pressure moving in the axial direction along the interior of the shell was treated as the applied load. Datta et al. [33] obtained the dynamic response of a cylindrical shell to moving disturbances. Bert and Birman [34] performed the dynamic stability analyses of a simply supported, finite-length, circular cylindrical thick shell subjected to a parametrically excited periodic axial loading. Panneton et al. [35] investigated the vibrations and sound radiation by the finite cylindrical shells subjected to a circumferentially moving force. Singh et al. [36] studied the nonaxisymmetric dynamic behavior of the orthotropic cylindrical shells subjected to a load moving along the axis of the shell. They used a thick shell model that includes the effects of shear deformation and rotary inertia. Liew et al. [37] studied the dynamic stability of the rotating cylindrical shells under static and periodic axial forces using the energy formulation. In their study [37], they used a combination of the Ritz method and Bolotin's first approximation. Wang et al. [38] obtained the nonlinear traveling wave response of a cantilever circular cylindrical shell subjected to a concentrated harmonic force moving in a concentric circular path. Donnell's nonlinear shallow-shell theory, together with the consideration of geometric nonlinearities, were used in their study [38]. Huang et al. [39] investigated the resonance behavior of a rotating cylindrical shell caused by the action of harmonic moving loads. They used the shell equations derived from the Love–Timoshenko theory for a nonrotating thin cylindrical shell. On the other hand, the Laplace transform technique was used to solve axially symmetrical thermoelastic problems. Sherief and Anwar [44] studied the heat conduction of an infinitely long thermoelastic cylindrical medium, for which a onedimensional solution was sought in the Laplace transform domain. Lee et al. [45] obtained a transient response for a onedimensional axisymmetric quasistatic thermoelastic problem. Cho and Kardomateas [46] investigated thermal shock stress caused by heat transfer into a medium at a constant temperature in a thick orthotropic cylindrical shell. Sherief et al. [47] studied a one-dimensional problem for an infinitely long hollow cylinder with one relaxation time. Lee [48] investigated one-dimensional axisymmetric quasi-state coupled magnetothermoelastic problems. Bai [49] obtained an analytical solution for a hollow cylinder of porothermoelastic media of infinite length. Elhagary [50] considered a homogeneous isotropic thermoelastic hollow cylinder with one relaxation time. Most of the abovementioned analytic approaches on the cylindrical shell problem subjected to a traveling load can provide valuable insight into the dynamic response of thin shell-like cylindrical structures caused by a traveling force. However, in the cases that the hollow cylinder is thick enough not to be considered as a thin shell, the radial mode can appear in a relatively low frequency range, so that the thin shell theory cannot provide a useful way to accurately compute its dynamic behaviors. In the present study, we treated a general hollow cylinder and solved its dynamic problems in an analytic way. As an illustrative example in this work, the hollow cylinder subjected to a dual traveling force on its inner surface is treated. For the analysis of this study, proper three-dimensional nonlinear governing field equations were derived. Using these equations, an eigenanalysis was performed first with great care. Due to the nonsymmetric nature of the field equations, the present eigensystem is non-self-adjoint, so that the biorthogonality conditions, along with the adjoint system of equations, were obtained. During this process, series form of eigensolutions, with respect to the radial coordinate, was attempted to € obtain using the Frobenius method [40,41], which results in the dispersion relations. It is well known that a threedimensional linear elasticity problem can be solved in terms of Bessel functions [15,16], after decomposing the problem into irrotational and solenoidal parts by introducing two types of potential functions. However, the decomposition of the adjoint system cannot be performed easily because of the additional terms originating from the boundary conditions, and therefore, we opted to derive and use the Fröbenius-type power series solutions in this study. Using the eigensolutions, we obtained the mode shape for each mode. Then, the same approach is applied to the adjoint system to solve its eigenproblem. Using the biorthogonality conditions between the eigenfunctions of the original and its adjoint systems, the field equations that account for the inhomogeneous boundary forcing terms could be successfully discretized. The critical speed was obtained for a set of several representative modes.
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
285
Fig. 1. Schematics of the hollow cylinder.
2. Mathematical model of a hollow cylinder In this study, the hollow cylinder subjected to a dual traveling force imposed on its inner surface is investigated. Fig. 1 shows a schematic of the hollow cylinder, where the inner radius is denoted as Ri , the outer radius as Ro and the height as H. Here, the cylindrical coordinate system (r, θ, z) with unit vectors (e^ r , e^ θ , e^ z ) respectively, are used as shown in Fig. 1. 2.1. Derivation of the governing field equations and boundary conditions in cylindrical coordinates The cylinder is fixed only in z-direction (uz ¼ 0) at z ¼ 0 and H and the forces travels at a constant velocity v, along the z-direction on the inner surface of the hollow cylinder (r ¼ Ri ) at θ ¼ θ1 and θ2 . Note that subscripts or superscripts i, j and k are used to denotes the coordinates, and this symbolism will be understood hereafter. The equations for the stress components in terms of the strain components in the cylindrical coordinate system can be expressed with Einstein's index notation as below:
σ^ ij ¼ Eε^ ij þ Eνδij ε^ kk ;
(1)
where E ¼ E=ð1 þ νÞ, ν ¼ ν=ð1 2νÞ, δij is the Kronecker delta function, E and ν are Young's modulus and Poisson's ratio, respectively. The stress equations of motion of the problem at hand in cylindrical coordinates can be obtained as the following usual forms: σ^ rr σ^ θθ σ^ ¼ ρu€^ r : Radial direction: σ^ rr;r þ rθ;θ þ σ^ rz;z þ (2a) r^ r^
σ^ 2σ^ Circumferential direction: σ^ rθ;r þ θθ;θ þ σ^ θz;z þ rθ ¼ ρu€^ θ : r^ r^
(2b)
σ^ σ^ rz ¼ ρu€^ z : Length direction: σ^ rz;r þ θz;θ þ σ^ zz;z þ r^ r^
(2c)
where ρ is the mass density of the cylinder material, ui means the i directional displacement component, dot is used over a symbol to represent the time derivative and a comma followed by subscript(s) denotes partial differentiation with respect to the spatial coordinate indicated by the subscript(s). The following strain–displacement relations are employed in this study:
ε^ rr ¼ u^ r;r ;
(3a)
u^ þ u^ θ;θ ^εθθ ¼ r ; r^
(3b)
ε^ zz ¼ u^ z;z ;
(3c)
1 u^ r;θ u^ θ þ u^ θ;r ; ε^ rθ ¼ ^ 2 r
(3d)
ε^ θz ¼
u^ 1 u^ θ;z þ z;θ ; ^ 2 r
(3e)
286
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
ε^ rz ¼
1 u^ r;z þ u^ z;r : 2
(3f)
For the convenience of the subsequent analyses, the following dimensionless parameters are introduced: ur ¼
z^ t^ 2ρR2 H2 u^ r u^ u^ z r^ ; uθ ¼ θ ; uz ¼ ; r ¼ ; z ¼ ; t ¼ ; ρ ¼ ; η ¼ 2; 2 H R R R H t R Et
where R ¼ Ro Ri . Using Eqs. (1)–(4), the displacement equations of motion can be derived in dimensionless forms u ηð3 þ 2νÞuθ;θ Radial direction: 2ηð1 þ νÞur;rr þ ur;zz þ ηr r;rθθ þ 2ð1 þ νÞ ur;r urr r2
uθ;rθ ¼ ρu€ r ; þ ηð1 þ 2νÞ uz;rz þ r Circumferential direction:
2ηð1 þ νÞuθ;θθ þ r2
þ
(4)
(5a)
ηuθ;rr þ uθ;zz þ ηr uθ;r ur2θ þ ηð3 þr22νÞur;θ
ηð1 þ 2νÞ uz;θz þ ur;rθ r
¼ ρu€ θ :
(5b)
ðu þ u Þ Length direction: 2ð1 þ νÞuz;zz þ η uz;rr þ urz;r þ urz:2θθ þur;rz þ r;z r θ;θz ur;z þuθ;θz ¼ ρu€ z : þ 2ν ur;rz þ r
(5c)
The number of the boundary/continuity conditions of the present model are 18 in total and they can be expressed as follows: Radial directional boundary conditions: σ rr Ro ; θ; z ¼ 0; σ rθ Ro ; θ; z ¼ 0; σ rz Ro ; θ; z ¼ 0; (6a)
σ rr Ri ; θ; z ¼ 0;
σ rθ Ri ; θ; z ¼ 0;
σ rz Ri ; θ; z ¼ 0:
(6b)
σ rθ ðr; 2π ; zÞ ¼ σ rθ ðr; 0; zÞ; σ θθ ðr; 2π ; zÞ ¼ σ θθ ðr; 0; zÞ; σ θz ðr; 2π ; zÞ ¼ σ θz ðr; 0; zÞ;
(6c)
Circumferential directional continuity conditions:
ur ðr; 2π ; zÞ ¼ ur ðr; 0; zÞ;
uθ ðr; 2π ; zÞ ¼ uθ ðr; 0; zÞ;
uz ðr; 2π ; zÞ ¼ uz ðr; 0; zÞ: Length directional boundary conditions: σ rz r; θ; H ¼ 0;
uz r; θ; H ¼ 0;
σ θz r; θ; H ¼ 0;
σ rz r; θ; 0 ¼ 0; σ θz r; θ; 0 ¼ 0; uz r; θ; 0 ¼ 0:
(6d)
(6e) (6f)
Eq. (6a)–(f) are the boundary/continuity conditions on the hollow cylinder in its natural state. In case that the traveling forces exist, the following boundary forcing term σ~ rr should be superposed to the σ rr Ri ; θ; z condition. σ~ rr Ri ; θ; z ¼ f 1 δ Ri θ Ri θ1 þ f 2 δ Ri θ Ri θ2 δðz Vt Þ; (7) where, δðÞ denotes the Dirac delta function and V is the traveling velocity of the forces in the z-direction. Substituting Eqs. (1) and (3) into Eqs. (6a)–(6f), the boundary/continuity conditions can be expressed in terms of displacements and their gradients as follows: Radial directional boundary conditions: uθ;θ Ro ; θ; z þur Ro ; θ; z 2ηð1 þ νÞur;r Ro ; θ; z þ 2ην uz;z Ro ; θ; z þ ¼ 0; (8a) Ro ur;θ Ro ; θ; z uθ Ro ; θ; z þ 2ηuθ;r Ro ; θ; z ¼ 0; 2η Ro
(8b)
ur;z Ro ; θ; z þ ηuz;r Ro ; θ; z ¼ 0;
(8c)
uθ;θ Ri ; θ; z þ ur Ri ; θ; z 2ηð1 þ νÞur;r Ri ; θ; z þ2ην uz;z Ri ; θ; z þ ¼ 0; Ri
(8d)
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
287
ur;θ Ri ; θ; z uθ Ri ; θ; z 2η þ 2ηuθ;r Ri ; θ; z ¼ 0; Ri
(8e)
ur;z Ri ; θ; z þ ηuz;r Ri ; θ; z ¼ 0:
(8f)
Circumferential directional continuity conditions: ur;θ ðr;2π ;zÞ uθ ðr;2π ;zÞ 2ηuθ;r ðr;2π ;zÞ þ r r2
2η
u ðr; 0; zÞ uθ ðr; 0; zÞ 2ηuθ;r ðr; 0; zÞ ; ¼ 2η r;θ þ r r2
(8g)
2ηð1 þ νÞðuθ;θ ðr;2π ;zÞ þ ur ðr;2π ;zÞÞ 2ηνfur;r ðr;2π ;zÞ þ uz;z ðr;2π ;zÞg þ r r2
¼
2ηð1 þ νÞ uθ;θ ðr; 0; zÞ þ ur ðr; 0; zÞ 2ην ur;r ðr; 0; zÞ þuz;z ðr; 0; zÞ ; þ r r2
(8h)
uθ;z ðr; 2π ; zÞ 2ηuz;θ ðr; 2π ; zÞ uθ;z ðr; 0; zÞ 2ηuz;θ ðr; 0; zÞ þ þ ¼ ; r r r2 r2
(8i)
ur ðr; 2π ; zÞ ¼ ur ðr; 0; zÞ;
(8j)
uθ ðr; 2π ; zÞ ¼ uθ ðr; 0; zÞ;
(8k)
uz ðr; 2π ; zÞ ¼ uz ðr; 0; zÞ:
(8l)
Length directional boundary conditions:
ur;z r; θ; H þ ηuz;r r; θ; H ¼ 0;
(8m)
2ηuz;θ r; θ; H ¼ 0; uθ;z r; θ; H þ r uz r; θ; H ¼ 0; ur;z r; θ; 0 þ ηuz;r r; θ; 0 ¼ 0;
(8n) (8o) (8p)
2ηuz;θ r; θ; 0 ¼ 0; uθ;z r; θ; 0 þ r uz r; θ; 0 ¼ 0:
(8q) (8r)
2.2. Eigenanalysis and derivation of the biorthogonality conditions In this section, an eigenanalysis is performed using the governing field equations together with the homogeneous boundary conditions. Assuming the displacement functions in nth mode eigenstate as ðr Þ
ðr Þ
ur ¼ ϕ eλt þ ϕ
eλt ;
ðθ Þ θ uθ ¼ ϕð Þ eλt þ ϕ eλt ; ðzÞ
uz ¼ ϕ eλt þ ϕ eλt ; ðzÞ
ðjÞ
(9)
ðjÞ
where ϕ is the eigenfunction of the jth mode and ϕ is its complex conjugate. Substituting Eqs. (9) into Eqs. (5) and (6), ðjÞ ðjÞ followed by replacing ϕ and λ with ϕn and λn , respectively, the following equations for the eigenvalue problem of the original system can be obtained as LΦD ¼ λn MΦD ; n
2
n
(10a)
n n Br ΦB1 r ¼ Ro ¼ Br ΦB2 r ¼ R ¼ 0;
(10b)
Bθ ΦB3 ¼ Bθ ΦB4 ;
(10c)
Bz ΦB5 ¼ Bz ΦB6 ¼ 0;
(10d)
i
n
n
n
n
288
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
where 2
L1;1 ðÞ 6 L ¼ 4 L2;1 ðÞ L3;1 ðÞ
L1;2 ðÞ L3;2 ðÞ
ρðÞ 6 M¼4 0 0 2 6 Br ¼ 6 4
L2;3 ðÞ 7 5;
L2;2 ðÞ
2
3
0
0
0 7 5; ρðÞ
2ηð1 þ νÞðÞ;r þ 2ην r ðÞ
2ηνðÞ;θ
η
ðÞ;z
ðÞ;r 2η r ðÞ;θ
0
ðÞ
0
ϕðθÞ r; θ; z
h
ϕðθÞ Ro ; θ; z
h
ϕðθÞ Ri ; θ; z
ΦB1 ¼ ϕðrÞ Ro ; θ; z
ΦB2 ¼ ϕðrÞ Ri ; θ; z
ηðÞ;r
(13a)
3 7
(13b)
3 7 5;
(13c)
7 7; 5
7 2ην 7; r ðÞ;z 5 2η 2η ðÞ þ r ðÞ r 2 ;θ
ϕðzÞ r; θ; z
3
0
2ηð1 þ νÞ ðÞ;θ þ 2rηðÞ r2 1 r ðÞ;z
ðÞ;z 6 Bz ¼ 4 0 0
0
2η 2η r ðÞ;r r2 ðÞ
2
h
2ηνðÞ;z
0
2η ðÞ þ 2rηðÞ r 2 ;θ 6 6 2ηð1 þ νÞ Bθ ¼ 6 r2 ðÞ þ 2ην r ðÞ;r 4 0
ΦD ¼ ϕðrÞ r; θ; z
(12)
2rηðÞ
ðÞ;z
2
(11)
L3;3 ðÞ
0 ρðÞ
2η r ðÞ;θ
3
L1;3 ðÞ
iT
;
(14a)
iT
iT
ϕðzÞ Ro ; θ; z ϕðzÞ Ri ; θ; z
;
(14b)
;
(14c)
ΦB3 ¼ ϕðrÞ ðr; 2π ; zÞ ϕðθÞ ðr; 2π ; zÞ ϕðzÞ ðr; 2π ; zÞ ;
(14d)
h
iT
h
iT
ΦB4 ¼ ϕðrÞ ðr; 0; zÞ ϕðθÞ ðr; 0; zÞ ϕðzÞ ðr; 0; zÞ ; h
ϕðθÞ r; θ; H
h
ϕðθÞ r; θ; 0
ΦB5 ¼ ϕðrÞ r; θ; H
ΦB6 ¼ ϕðrÞ r; θ; 0
(14e)
iT
iT
ϕðzÞ r; θ; H ϕðzÞ r; θ; 0
;
(14f)
;
(14g)
andthe linear operators associated with the governing field equations are defined as L1;1 ðÞ ¼ 2ηð1 þ νÞðÞ;rr L1;2 ðÞ ¼
2ηð1 þ νÞ η 2ηð1 þ νÞ ðÞ;r þ 2 ðÞ;θθ þðÞ;zz ðÞ; r r2 r2
ηð1 þ 2νÞ r
ðÞ;rθ
ηð3 þ2νÞ r2
ðÞ;θ ;
L1;3 ðÞ ¼ ηð1 þ 2νÞðÞ;rz ; L2;1 ðÞ ¼
ηð1 þ 2νÞ r
ðÞ;rθ þ
ηð3 þ2νÞ r2
ηð1 þ2νÞ r
L3;1 ðÞ ¼ ð1 þ2νÞðÞ;rz þ
(15b) (15c)
ðÞ;θ ;
η 2η η L2;2 ðÞ ¼ ηðÞ;rr þ ðÞ;r þ 2 ðÞ;θθ þðÞ;zz 2 ðÞ; r r r L2;3 ðÞ ¼
(15a)
(15d) (15e)
ðÞ;θz ;
(15f)
1 þ 2ν ðÞ;z ; r
(15g)
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
L3;2 ðÞ ¼
1 þ 2ν ðÞ;θz ; r
η
289
(15h)
η
L3;3 ðÞ ¼ ηðÞ;rr þ ðÞ;r þ 2 ðÞ;θθ þ2ð1 þ νÞðÞ;zz : r r
(15i)
Eqs. (15a)–(15c), Eqs. (15d)–(15f) and Eqs. (15g)–(15i) are associated with the radial, circumferential and length directional governing field equations, respectively, and Eqs. (13a)–(13c) represent the boundary/continuity conditions expressed with the linear operators in matrix forms. In Eqs. (13), Br and Bz are associated with the radial and length directional boundary conditions, respectively, and Bθ is with circumferential directional continuity conditions. Note that the nonsymmetric nature of matrix L manifests the fact that the present system is non-self-adjoint. The adjoint system and biorthogonality conditions between the original and the adjoint systems are derived. To derive biorthogonality conditions, the inner product of two vector functions has to be defined first. We here define it as Z Z
a; b ¼ aU b dΩ ¼ ða1 b1 þ a2 b2 þ a3 b3 ÞdΩ; (16) Ω
Ω
where T a ¼ a1 r; θ; z ; a2 r; θ; z ; a3 r; θ; z ; T b ¼ b1 r; θ; z ; b2 r; θ; z ; b3 r; θ; z : Taking inner product on both sides of Eq. (10a) with ΨD : D E D E m 2 n n Ψm D ; LΦD ¼ λn ΨD ; MΦD m
(17)
and using the divergence theorem along D E with the given boundary conditions, we can obtain the following relationships: m n Left side of Eq. (17), i.e., ΨD ;LΦD : R m R m m n n n Ω ΨD U LΦD dΩ Γ 1 ΨB1 U Br ΦB1 r ¼ Ro ΨB2 U Br ΦB2 r ¼ Ri dΓ 1 Z
Z
m m n n n n Ψm Ψm dΓ 2 dΓ 3 B3 U Bθ ΦB3 ΨB4 U Bθ ΦB4 B5 U Bz ΦB5 ΨB6 U Bz ΦB6 Γ2 Γ3 Z
Z
m m m n n n ¼ Ln ΨD U ΦD dΩ Bnr ΨB1 U ΦB1 Bnr ΨB2 U ΦB2 dΓ 1 r ¼ Ro r ¼ Ri Ω Γ1 Z
m m n n Bnθ ΨB3 U ΦB3 Bnθ ΨB4 U ΦB4 dΓ 2 Γ Z 2
m m n n Bnz ΨB5 U ΦB5 Bnz ΨB6 U ΦB6 dΓ 3 : (18) Right side of Eq. (17), i.e., λ
2 n
D
Γ3
Ψ
Φ
m D ;M
λ2n
Z
n D
E
:
Ω
n2 n Ψm D U ΜΦD dΩ ¼ λm
Z
Ω
m n Mn ΨD U ΦD dΩ;
(19)
where ΦD , ΦB1 , ΦB2 , ΦB3 , ΦB4 , ΦB5 , ΦB6 are defined in Eqs. (14), and the vectors of the adjoint system h iT θ ðr Þ Ψm ψ ðm Þ r; θ; z ; ψ ðmzÞ r; θ; z ; D ¼ ψ m r; θ ; z ; n
n
n
n
n
n
n
h
ψ ðm Þ Ro ; θ; z ;
h
ψ ðm Þ Ri ; θ; z ;
ψ ðmzÞ Ri ; θ; z
;
(20c)
ψ ðm Þ ðr; 2π ; zÞ;
ψ ðmzÞ ðr; 2π ; zÞ ;
(20d)
ψ ðm Þ ðr; 0; zÞ;
ψ ðmzÞ ðr; 0; zÞ ;
ðrÞ Ψm B1 ¼ ψ m Ro ; θ ; z ;
ðr Þ Ψm B2 ¼ ψ m Ri ; θ ; z ;
h
ðr Þ Ψm B3 ¼ ψ m ðr; 2π ; zÞ;
h
ðrÞ Ψm B4 ¼ ψ m ðr; 0; zÞ;
h
θ
θ
θ
ψ ðm Þ r; θ; H ;
ψ ðm Þ r; θ; 0 ;
ðr Þ Ψm B6 ¼ ψ m r; θ ; 0 ;
are defined similarly to those of the original system.
ðrÞ Ψm B5 ¼ ψ m r; θ ; H ;
h
θ
θ
θ
iT
iT
(20a)
ψ ðmzÞ Ro ; θ; z
iT
iT
iT
iT
ψ ðmzÞ r; θ; H ψ ðmzÞ r; θ; 0
;
(20b)
(20e) ;
(20f) (20g)
290
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
In Eqs. (18) and (19), Ln , Bnj and Mn mean the adjoint of L, Bj and M, respectively, and Γ 1 , Γ 2 , and Γ 3 mean the boundary n domains of the radial, circumferential and length directions, respectively, and λm and ψ ðmjÞ are the eigenvalue and eigenfunction of the mth mode of the adjoint system. Ln and Bn obtained from Eq. (18) can be expressed as 2 n 3 L1;1 ðÞ Ln1;2 ðÞ Ln1;3 ðÞ 6 7 Ln ðÞ Ln2;2 ðÞ Ln2;3 ðÞ 7; (21) Ln ¼ 6 4 2;1 5 Ln3;1 ðÞ Ln3;2 ðÞ Ln3;3 ðÞ 2
ρðÞ
0
6 Mn ¼ 4 0 0 2 6 Bnr ¼ 6 4
3
0 7 5; ρðÞ
ρðÞ 0
2ηð1 þ νÞðÞ;r 2rηðÞ
(22)
2ην r ðÞ;θ ðÞ;r 2rηðÞ
η
2νðÞ;z
η
r ðÞ;θ
ηðÞ;z
0
ηðÞ;r ηr ðÞ
0
2 η ðÞ þ 2rηðÞ r 2 ;θ 6 2ην 2η Bnθ ¼ 6 4 r ðÞ;r þ r2 ðÞ 0 2
0
η
2η r ðÞ;r r 2 ðÞ
2η ð1 þ r2
η
0
νÞðÞ;θ
2 r ðÞ;z
η
r ðÞ;z
ðÞ;z
r2
6 Bnz ¼ 6 4 0
0
ðÞ;r 1r ðÞ
ðÞ;z
1 r ðÞ;θ
0
0
ðÞ
ðÞ;θ
3 7 7; 5
(23)
3 7 7; 5
(24)
3 7 7; 5
(25)
where Ln1;1 ðÞ ¼ 2ηð1 þ νÞðÞ;rr Ln1;2 ðÞ ¼
2ηð1 þ νÞ η ðÞ;r þ 2 ðÞ;θθ þ ðÞ;zz ; r r
ηð1 þ 2νÞ r
ðÞ;rθ
Ln1;3 ðÞ ¼ ð1 þ 2νÞðÞ;rz Ln2;1 ðÞ ¼
ηð1 þ2νÞ r
η
4ηð1 þ νÞ ðÞ;θ ; r2
(26b)
ð1 þ 2νÞ ðÞ;z ; r
(26c)
ðÞ;rθ þ
Ln2;2 ðÞ ¼ ηðÞ;rr ðÞ;r þ r
(26a)
2η ðÞ ; r 2 ;θ
2ηð1 þ νÞ ðÞ;θ ; r2
(26d)
(26e)
ð1 þ2νÞ ðÞ;θz ; r
(26f)
Ln3;1 ðÞ ¼ ηð1 þ 2νÞðÞ;rz ;
(26g)
Ln2;3 ðÞ ¼
Ln3;2 ðÞ ¼ Ln3;3 ðÞ ¼ ηðÞ;rr þ
ηð1 þ2νÞ r
ðÞ;θz ;
η η 1 ðÞ þ 2ð1 þ νÞðÞ;zz ðÞ;r þ 2 ðÞ: r r r 2 ;θθ
(26h)
(26i)
In order to simplify Eq. (18), we define a new form of the inner product operator hh;ii for the extended system shown in Eq. (18) that can be obtained from the inner product operator after applying the divergence theorem, as follows:
R R A;B ¼ Ω AD U BD dΩ Γ 1 ðAB1 UBB1 AB2 UBB2 ÞdΓ 1 Z Z ðAB3 U BB3 AB4 U BB4 ÞdΓ 2 ðAB5 UBB5 AB6 UBB6 ÞdΓ 3 ; (27) Γ2
Γ3
where h iT A ¼ ATD ; ATB1 ; ATB2 ; ATB3 ; ATB4 ; ATB5 ; ATB6 ;
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
291
h iT B ¼ BTD ; BTB1 ; BTB2 ; BTB3 ; BTB4 ; BTB5 ; BTB6 and AD ; AB1 ; AB2 ; AB3 ; AB4 ; AB5 ; AB6 , BD ; BB1 ; BB2 ; BB3 ; BB4 ; BB5 ; BB6 are 3 1 vectors. Note that the operator hh;ii basically produces the same results as the original inner product operator h;i, but it has a remarkable merit that enables us to explicitly treat the boundary/continuity conditions. Using Eq. (27), Eqs. (18) and (19) can be reformulated as follows:
Ψm ; TΦn ¼ Tn Ψm ; Φn ; (28)
λ2n Ψm ; SΦn
n2
n
¼ λm
S Ψm ; Φn
(29)
where h
iT
Φn ¼ ΦnT ΦnT ΦnT ΦnT ΦnT ΦnT ΦnT ; D B1 B2 B3 B4 B5 B6 h
iT
Ψm ¼ ΨmT ΨmT ΨmT ΨmT ΨmT ΨmT ΨmT ; D B1 B2 B3 B4 B5 B6 2
0
⋯
⋯
⋯
⋯
Br jr ¼ Ro
0
⋯
⋯
⋯
L
60 6 6 60 6 6 T¼60 6 60 6 60 4 0
0
Br jr ¼ Ri
0
⋯
⋯
⋯
0
Bθ
0
⋯
⋯
⋯
0
Bθ
0
⋯ ⋯
⋯ ⋯
⋯ ⋯
0 ⋯
Bz 0
" S¼ 2 6 6 6 6 6 6 n T ¼6 6 6 6 6 6 4
Ln 0
M 0183
0
0
⋯
⋯
0 0
⋯ ⋯
0 ⋯
Bnθ 0
⋯ 0
0
⋯
⋯
⋯
0 Bnθ
0
⋯
⋯
⋯
Br
r ¼ Ro
r r ¼ Ri
" n
S ¼
0 7 7 7 0 7 7 0 7 7; 7 0 7 7 0 7 5 Bz
⋯ ⋯
⋯ ⋯
⋯ ⋯
Mn
0318
0183
01818
(32)
(33)
0
0
(31)
3
# 0318 ; 01818
⋯ 0 n B
n
0
(30)
0
Bnz
⋯
0
3 0 0 7 7 7 0 7 7 7 0 7 7; 7 0 7 7 0 7 5 Bnz
(34)
# :
(35)
Using Eqs. (32)–(35) the equations for the eigenvalue problems of the adjoint system and original system can be obtained, respectively, as
Ψm ; TΦn ¼ λ2n Ψm ; SΦn ; (36)
n n2 T Ψm ; Φn ¼ λm Sn Ψm ; Φn : Using Eqs. (28)–(29) and (36)–(37), the following biorthogonality conditions can be derived:
λ2n λnm2 Ψm ; SΦn ¼ Ψm ; TΦn Tn Ψm ; Φn ¼ 0:
(37)
(38)
For distinct eigenvalues of the original system (and also its adjoint system) two sets of biorthogonality conditions of the eigenvector functions can be finally obtained, after a proper normalization process, in the forms
Ψm ; TΦn ¼ Tn Ψm ; Φn ¼ λ2n δmn ; (39a)
Ψm ; SΦn
¼
n S Ψm ; Φn ¼ δmn :
Note that both the original and its adjoint systems satisfy the positive definiteness. Using Eqs. (26a)–(26i), the field equations of the adjoint system can be readily obtained:
(39b)
292
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
Radial direction: 2ηð1 þ νÞunr;rr þ unr;zz
ηu ηð1 þ 2νÞuθ;rθ 4ð1 þ νÞηuθ;θ 2ηð1 þ νÞunr;r þ rr;2θθ þ þ ð1 þ 2 r r r2 n
n
n
νÞunz;rz
ð1 þ 2νÞunz;z n ¼ ρu€ r : r
(40a)
Circumferential direction:
ηð1 þ 2νÞunr;rθ 2ηunr;θ 2ηð1 þ νÞunθ;θθ þ
r
r2
þ
r2
þ ηunθ;rr þ unθ;zz
ηunθ;r ð1 þ2νÞunz;θz þ
r
r
¼ ρu€ θ : n
(40b)
Length direction:
ηð1 þ 2νÞunr;rz þ
ð1 þ 2νÞηunθ;θz r
þ2ð1 þ νÞunz;zz þ ηunz;rr þ
ηunz;θθ ηunz;r ηunz
r2
r
þ
r2
¼ ρu€ z : n
(40c)
Considering the boundary/continuity conditions given in Eqs. (8), the displacement functions for the original system (refer to Eq. (9)) can be assumed as ϕðrÞ ¼ vðrÞ ðrÞ cos ξz eikθ ;
ϕðθÞ ¼ vðθÞ ðrÞ cos ξz eikθ ;
ϕðzÞ ¼ vðzÞ ðr Þ sin ξz eikθ ;
(41) pffiffiffiffiffiffiffiffi ðjÞ where i is 1, v ðr Þ are functions of r to be determined, k and ξ are the wavenumbers in θ and z directions, respectively. The substitution of Eqs. (9) and (17) into Eq. (2) yields the field equations only functions of r as Radial direction: ðθÞ η k2 2ð1 þ νÞ 2ηð1 þ νÞvð;rrÞ kηð1 þ 2νÞv;r 2 2 rÞ 2ηð1 þ νÞvð;rr þ þ ρλ ξ vðrÞ þ i r r r2 kηð3 þ 2νÞvðθÞ i þ ξηð1 þ 2νÞvð;rzÞ ¼ 0: r2 Circumferential direction:
ηvð;rrÞ þ ηr vð;r Þ þ ρλ2 θ
θ
η 2k2 ð1 þ νÞ þ 1 r2
ξ
2
(42a)
kηð1 þ 2νÞvð;rrÞ kηð3 þ 2νÞvðr Þ þi vðθÞ þ i 2 r
r
kηξð1 þ 2νÞvðzÞ ¼ 0: þi r
(42b)
Length direction: ξð1 þ 2νÞvð;rrÞ ( þ
ξð1 þ 2νÞvðrÞ r
i
kξð1 þ 2νÞvðθÞ þ r
ρλ 2ξ ð1 þ νÞ 2
2
ηk2 r2
)
ηvð;rrzÞ þ ηr vð;rzÞ
vðzÞ ¼ 0:
(42c)
Eqs. (42a)–(42c) are coupled linear ordinary differential equation with respect to r. Since the existence of the singular point at r ¼ 0 in Eqs. (42) causes numerous convergence problems in the series solutions of Eqs. (42), here, the series solutions are obtained about a point other than r ¼ 0, for which expansion about a regular point of Eqs. (42) with variable coefficients can yield six independent exact power series solutions about that point, which are well behaved in the region of interest [41]. In particular, the point r~ located on the middle line of the hollow cylinder is particularly convenient for use in this work. r ¼ r~ þ r 0 ;
(43)
r n0 Ro þ Ri ¼ : R 2ðRo Ri Þ
(44)
where r0 ¼
Introducing this translated coordinate, the differential Eqs. (42a)–(42c) can be transformed into the following forms: Radial direction:
h n oi 2 2 2ηð1 þ νÞðr~ þr 0 Þ2 vð;rr~ Þr~ þ ðr~ þr 0 Þ2 ρω2 ξ η k 2ð1 þ νÞ vðrÞ þ 2ηð1 þ νÞðr~ þr 0 Þvð;rr~ Þ ðθÞ þ ikηðr~ þ r 0 Þð1 þ 2νÞv;r~ ikηð3 þ2νÞvðθÞ þ ξηðr~ þ r 0 Þ2 ð1 þ 2νÞvð;rz~ Þ ¼ 0:
(45a)
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
293
Circumferential direction:
h n oi ðθ Þ 2 2 ikηðr~ þ r 0 Þð1 þ 2νÞvð;rr~ Þ þ ðr~ þr 0 Þ2 ρω2 ξ η 2k ð1 þ νÞ þ 1 vðθÞ þ ηðr~ þ r 0 Þ2 v;r~ r~ ðθ Þ þ ηðr~ þ r 0 Þv;r~ þ ikηξðr~ þ r 0 Þð1 þ 2νÞvðzÞ þ ikηð3 þ 2νÞvðrÞ ¼ 0:
(45b)
Length direction: ξðr~ þ r 0 Þ2 ð1 þ2νÞvð;rr~ Þ ξðr~ þ r 0 Þð1 þ 2νÞvðrÞ ikξðr~ þ r 0 Þð1 þ 2νÞvðθÞ þ ηðr~ þ r 0 Þ2 vð;rz~ rÞ~ hn i o 2 2 þ ρω2 2ξ ð1 þ νÞ ðr~ þ r 0 Þ ηk vðzÞ þ ηðr~ þ r 0 Þvð;rz~ Þ ¼ 0;
(45c)
where ω ¼ λi, which means the frequency of the cylinder. Note that Eqs. (45a)–(45c) have been transformed into the functions of r~ . It is well known that the displacements in the differential equations of the original system (Eq. (5)) can be easily decomposed into irrotational and solenoidal parts by introducing two types of potential functions. The resulting differential equations can then be solved in terms of Bessel functions [15]. However, the decomposition of the adjoint system (Eq. (40)) cannot be performed easily because of the additional terms originating from the boundary conditions. On the other hand, once the variables are separated, the resulting set of ordinary differential equations in the radial direction (Eq. (42)) can be readily solved using classical methods. Because the present problem requires the simultaneous solution of the original and adjoint problem and given that the use of the Fröbenius method, expanded at a regular point, guarantees a yield of six independent solutions for each system with fast convergence, we opted to derive and use the Fröbenius-type power series solutions in this study. Note that this method does not require decomposition and can be used to directly solve the displacement equations of motion, which is obviously very convenient and efficient for the present problem. Furthermore, as can be seen in Table A1 in Appendix A, while the Bessel-type solutions are dependent on the argument interval, the series solutions can take their own forms regardless of their argument. In order to solve vðjÞ in coupled Eqs. (45a)–(45c), the method of Fröbenius [40,41] are applied by assuming the solution functions to be M
vðrÞ ðr~ Þ ¼ ∑ αm r~ χ þ m ; m¼0
M
vðθÞ ðr~ Þ ¼ ∑ β m r~ χ þ m ; m¼0 M
vðzÞ ðr~ Þ ¼ ∑ γ m r~ χ þ m ;
(46)
m¼0
where αm , βm and γ m are coefficients of r~ χ þ m for the function vðjÞ , respectively, and M is the number of terms sufficiently large to assure the satisfactory convergence of vðjÞ . Substituting Eqs. (46) into Eqs. (45), after a series of calculations and collecting of the coefficients of the like powers of r~ in each equation, we can obtain three indicial equations using the coefficients of the lowest power terms, r~ χ 2 , as Radial direction: χ r20 ð1 þ νÞ χ 1 α0 ¼ 0: (47a) Circumferential direction:
ηχ χ 1 β0 ¼ 0:
(47b)
Length direction:
ηχ χ 1 r20 γ 0 ¼ 0:
(47c)
Eqs. (47a)–(47c) are the indicial equations of the directions indicated. Equations (47) clearly show that with χ ¼ 0 and χ ¼ 1, there are six independent solutions in total. Note that Eqs. (47) are satisfied regardless of α0 , β0 and γ 0 with either χ ¼ 0 or χ ¼ 1. To the best convenience of calculations, α0 , β0 and γ 0 may be selected in the particularly convenient form, in which the first terms are uncoupled, i.e., one with α0 ¼ 1, β0 ¼ 0, γ 0 ¼ 0, another with α0 ¼ 0, β0 ¼ 1, γ 0 ¼ 0 and the other with α0 ¼ 0, β0 ¼ 0, γ 0 ¼ 1, so that we can obtain six independent series solutions; with the following six cases χ ; α0 ; β0 ; γ 0 ¼ ð0; 1; 0; 0Þ, ð0; 0; 1; 0Þ, ð0; 0; 0; 1Þ, ð1; 1; 0; 0Þ, ð1; 0; 1; 0Þ, ð1; 0; 0; 1Þ. Note that these six cases are confirmed to be linearly independent by investigating the Wronskian determinant. ðpÞ Then, we can calculate αðmpÞ , βm and γ ðmpÞ (p ¼ 1; :::; 6) for the six solution functions, where the superscript p means the case number. With the inclusion of arbitrary constants Ap that are to be determined later, vðjÞ can be expressed as 6
vðrÞ ¼ ∑ Ap vðprÞ ; p¼1
6 ðθ Þ vðθÞ ¼ ∑ Ap vp ; p¼1
6
vðzÞ ¼ ∑ Ap vðpzÞ ; p¼1
(48)
294
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
where M
M ðθ Þ ðpÞ vp ¼ ∑ βm r~ χ þ m ;
vðprÞ ¼ ∑ αðmpÞ r~ χ þ m ; m¼0
m¼0
M
vðpzÞ ¼ ∑ γ ðmpÞ r~ χ þ m :
(49)
m¼0
The length directional boundary conditions and the circumferential directional continuity conditions are automatically satisfied with the displacement functions given in Eqs. (41), so all the actual boundary conditions needed in the eigenanalysis are the six radial directional boundary conditions. The boundary conditions expressed in functions of r~ can be obtained through the use of Eqs. (9), (41) and (48) as 6 1
1 ðθ Þ (50a, b) Ap ¼ 0; at r~ ¼ 7 ; ik vp þ vðprÞ ∑ ð1 þ νÞvðp;rÞr~ þ ν ξ vðpzÞ þ r~ þr 0 2 p¼1
1 ðθ Þ ðθ Þ ik vðprÞ vp þ vp;r~ Ap ¼ 0; p¼1 r 6
∑
6
∑
p¼1
h
i
1 at r~ ¼ 7 ; 2
1 at r~ ¼ 7 : 2
ηvðp;zÞr~ ξvðprÞ Ap ¼ 0;
(50c, d)
(50e, f)
Eqs. (50a)–(50f) can be expressed in terms of Ap multiplied by their coefficients, i.e., as the following six homogeneous algebraic equations: 6
∑ Bpq Ap ¼ 0
p¼1
ðq ¼ 1; …; 6Þ;
(51)
where index p corresponds to the boundary conditions given in Eqs. (50a)–(50f), respectively. Expressing Eq. (51) in matrix form, we have BA ¼ 0:
(52)
In order to have nontrivial solutions for Ap , the determinant of matrix B must be vanished, which makes up of dispersion relations, the relations between the frequency ω and z-directional wavenumber ξ, and yields the amplitude ratios. The natural frequencies (the frequencies that correspond to ξ ¼0; π ; 2π ; …) and mode shapes can be calculated using the dispersion curves, which is the graphical representation of the dispersion relations. Then, the nth mode displacement functions can be represented by using the natural frequency and mode shapes as ðrÞ
ðrÞ
uðrnÞ ¼ ϕn eiωn t þ ϕn e iωn t ; ðθ Þ ðθ Þ uθðnÞ ¼ ϕn eiωn t þ ϕn e iωn t ; ðzÞ
uðznÞ ¼ ϕn eiωn t þ ϕn e iωn t ; ðzÞ
(53)
where ωn ¼ λn i, which denotes the nth natural frequency of the cylinder. In a similar manner, the displacement functions of the adjoint system can be expressed as unr ðnÞ ¼ ψ ðnrÞ eiωn t þ ψ ðnrÞ e iωn t ; ðθ Þ ðθÞ unθðnÞ ¼ ψ n eiωn t þ ψ n e iωn t ; unz ðnÞ ¼ ψ ðnzÞ eiωn t þ ψ ðnzÞ e iωn t ;
(54)
where
ψ ðnrÞ ¼ vnnðrÞ ðr Þ cos ξn z eikn θ ;
(55a)
θ n θ ψ ðn Þ ¼ vnð Þ ðrÞ cos ξn z eikn θ ;
(55b)
ψ ðnzÞ ¼ vnnðzÞ ðr Þ sin ξn z eikn θ ; nðjÞ
and vn
(55c)
can be obtained using Eqs. (40) following the same steps as for the original system.
2.3. Discretization of the governing field equations Using the separation of variables method and the modal expansion theorem, the appropriate displacement functions can be written in the forms 1 ðrÞ ur ¼ ∑ ϕn r; θ; z qn ðt Þ; n¼1
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
295
1 ðθ Þ uθ ¼ ∑ ϕn r; θ; z qn ðt Þ; n¼1 1
ðzÞ uz ¼ ∑ ϕn r; θ; z qn ðt Þ;
(56)
n¼1
where n denotes the mode number of the original linear system and qn is the nth modal coordinate of the original linear system. Using the linear operators defined in Eqs. (32) and (33), the original linear field equations and boundary/continuity conditions can be represented by € þ FðlÞ ; Tu ¼ Su
(57)
where 1
u ¼ ∑ Φn qn ;
(58)
FðlÞ ¼ ½016 ; σ~ rr ; 0114 ;
(59)
n¼1
andσ~ rr is expressed in Eq. (7). Taking the inner product hh;ii with Ψm , the mth mode vector eigenfunction for the extended adjoint system defined in Eq. (31), we obtain the linear discretized differential equation as q€ a þ ω2a qa ¼ F Ri ; where
Z Z F Ri ¼
θ z
ψ ðarÞ Ri ; θ; z σ~ rr dz dθ:
(60)
(61)
Note that the inner product hh;ii is defined in Eq. (27) for the extended system that includes both the volumetric and boundary parts separately in Eq. (57). Substituting Eqs. (55) and (7) into Eq. (61), we can calculate F Ri as F Ri ¼ fR1i vnaðrÞ ðRi Þ cos ξa Vt eikθ1 þ fR1i vnaðrÞ ðRi Þ cos ξa Vt e ikθ1 þ
f 2 nðrÞ f v ðRi Þ cos ξa Vt eikθ2 þ 2 vnaðrÞ ðRi Þ cos ξa Vt e ikθ2 ; Ri a Ri 1 0 r θ1 ; θ2 r 2π and 0 r t r ; V
where θ1 and θ2 represent the circumferential locations of the dual traveling force components. The analytic solutions for Eq. (62) combined with Eq. (60) can be readily obtained as follows: cos ξa Vt f 1 vna ðRi Þeika θ1 þvna e ika θ1 þ f 2 vna ðRi Þeika θ2 þ vna e ika θ2
; qðt Þ ¼ 2 Ri ξa V 2 ω2a
(62)
(63)
where the initial conditions qð0Þ and q_ ð0Þ have been set to be zero. The vanishing denominator of Eq. (63) yields the critical speed as Vc ¼ 7
ωa ; ξa
(64)
at which the response increases indefinitely and the situation resembles the occurrence of a resonance. Note that in Eq. (63), when the value of ξa is zero, then the solution becomes independent of the traveling velocity, so the associated critical speed does not exist. 3. Illustrative examples – Eigenanalysis on the hollow cylinder and its critical speeds The dispersion curves and mode shapes are obtained for several illustrative examples in this section. The structural steel was selected as the material of the hollow cylinder, and the dimensionless material properties and geometric parameters selected for these illustrations are given in Table 1. In this study, we investigated the effects of the cylinder geometry by considering three different r 0 values (refer to Eq. (44)): (a) r 0 ¼ 1, (b) r 0 ¼ 2, (c) r 0 ¼ 3. When the thickness of hollow cylinder is fixed, the inner and outer radii tend to increase with r 0 . In this section, an eigenanalysis is performed on the hollow cylinder and the dispersion curves and several major mode shapes of concern are obtained with the three r 0 values. The eigenanalysis results for the thickness, flexural and torsional modes are discussed in this section, and the natural frequencies obtained from this eigenanalysis are compared with the results obtained using an FEM package, ANSYS [42] and the Bessel-type solutions [15]. Using the dispersion relations given in Eq. (52), we can obtain the dispersion curves as depicted in Fig. 2. Fig. 2(a), (b) and (c) represent the dispersion curves for k ¼ 0, k ¼ 1 and k ¼ 2, respectively, and the two vertical dashed lines represent ξ ¼ π and ξ ¼ 2π . Note that the frequency
296
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
Table 1 Geometrical and material properties of the hollow cylinder. Nondimensional parameter
Value
v ρ
0.75
r0 η
3:6738 10 2 (a) 1, (b) 2, (c) 3 9
Fig. 2. Dispersion curves for the circumferential wavenumber of (a) k ¼ 0, (b) k ¼ 1, and (c) k ¼ 2.
values associated with the crossed points of ξ ¼ nπ ðn ¼ 0; 1; 2; …Þ line with the dispersion curves represent the natural frequencies for the modes with wavelength H=n in the z-direction. To obtain the dispersion curves, we first attempted to find frequencies for the thickness modes. The thickness modes can be found by Eqs. (B.6) in Appendix B. Note that in Figs. 2(a), and (b) for k ¼ 0 and k ¼ 1, ξ ¼ 0 corresponds to the rigid body motions which is not interested in this study. The nondimensional natural frequencies of the thickness modes are marked with the symbol x in Fig. 2. As can be expected, these frequencies tend to decrease as r 0 increases. The same trend can also be found for k ¼ 1 and k ¼ 2, as shown in Table 2. In Fig. 2(b) and (c), the start points of the real branches connected with the associated complex branches (not shown in this figure) are marked with closed circles. Note that since the boundary conditions treated in this study is a Lévy type, only real branches are of concern and other complex (including imaginary) branches are not necessary to solve the present eigenproblem. In Fig. 2(a), the dotted curves are obtained from the dispersion relations described in Eqs. (B.2)–(B.3), which are for the torsional modes. The natural frequencies obtained from the present eigenanalysis and those obtained from an FEM package, ANSYS [42], and the Bessel-type solutions [15] are shown and compared in Table 2. As is manifested in this table, these three results are almost identical, which strongly supports the validity of the present eigenanalysis. Note that the natural frequencies of the torsional modes are identical regardless of the value of r 0 . Figs. 3–5 show three representative mode shapes of three different r 0 values corresponding to Table 2. The subscripts of ω in parenthesis, (nr ; nθ ; nz ) denote the number of nodal lines,
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
297
Table 2 Dimensionless natural frequencies computed with the present model, ANSYS and Ref. [15]. Branch #
(a) r 0 ¼ 1 1
2
3
(b) r 0 ¼ 2, 1
2
3
(c) r 0 ¼ 3. 1
2
3
ξ (nπ)
k¼0
k¼1
k ¼2
Present study
ANSYS
Ref. [15]
Present study
ANSYS
Ref. [15]
Present study
ANSYS
Ref. [15]
0 1 2 0 1 2 0 1 2
0 16.3912 32.7810 0 22.9852 31.5848 29.5315 32.4303 55.0199
0 16.3905 32.7812 0 22.9857 31.5862 29.5321 32.4305 55.0201
0 16.3905 32.7810 0 22.9857 31.5861 29.5321 32.4305 55.0199
0 11.5829 27.0283 – 27.2190 39.1052 30.8459 37.7313 56.1715
0 11.5877 27.0351 – 27.2208 39.1084 30.8462 37.7316 56.1723
0 11.5877 27.0347 – 27.2207 39.1083 30.8458 37.7314 56.1719
15.2723 18.2222 30.0045 – 36.8365 47.3622 43.8694 49.7432 62.1242
15.2756 18.3307 30.0009 – 36.8498 47.3865 43.8695 49.7451 62.1309
15.2749 18.3303 30.0003 – 36.8488 47.3856 43.8678 49.7443 62.1309
0 1 2 0 1 2 0 1 2
0 16.3907 32.7811 0 13.9063 24.4334 13.6072 27.9326 53.0862
0 16.3905 32.7827 0 13.9064 24.4364 13.6072 27.9328 53.0889
0 16.3905 32.7810 0 13.9063 24.4334 13.6072 27.9326 53.0862
0 11.4750 24.5447 – 19.5032 33.9863 17.9541 30.7556 53.9753
0 11.4753 24.5480 – 19.5032 33.9878 17.9542 30.7557 53.9780
0 11.4750 24.5447 – 19.5032 33.9863 17.9541 30.7556 53.9753
4.6139 11.6101 25.6188 – 24.0774 37.0219 27.2722 37.3052 56.3584
4.6141 11.6107 25.6224 – 24.0776 37.0229 27.2727 37.3057 56.3612
4.6139 11.6101 25.6188 – 24.0774 37.0219 27.2722 37.3052 56.3584
0 1 2 0 1 2 0 1 2
0 16.3912 32.7811 0 10.6805 22.9787 8.9303 27.6339 52.8154
0 16.3905 32.7828 0 10.6807 22.9821 8.9303 27.6340 52.8181
0 16.3905 32.7810 0 10.6805 22.9787 8.9303 27.6339 52.8154
0 10.2225 23.2134 – 17.5483 33.2509 12.2642 28.9569 53.2993
0 10.2228 23.2168 – 17.5484 33.2525 12.2642 28.9570 53.3020
0 10.2225 23.2134 – 17.5483 33.2509 12.2642 28.9569 53.2993
2.1636 10.2570 23.9837 – 20.0891 34.5887 19.0737 32.4999 54.6665
2.1637 10.2575 23.9872 – 20.0892 34.5901 19.0738 32.5001 54.6692
2.1636 10.2570 23.9837 – 20.0891 34.5887 19.0737 32.4999 54.6665
Fig. 3. Three representative mode shapes and their corresponding natural frequencies of the hollow cylinder with r0 ¼ 1.
i.e., nr : in radial direction, nθ : in circumferential direction and nz : in length direction. For the modes coupled with torsional modes, the subscript t was used as in Fig. 3(c). Table 3 shows the critical speeds (refer to Eq. (64)) for all the cases treated preciously. As shown in Fig. 2(a), the branch for torsional modes is a straight line in the wavenumber (ξ) range of concern, so the torsional waves are nondispersive and the critical speeds of all the torsional mode are identical. Note that the critical speeds are the highest for the modes with ξ ¼ π on the first branch because the critical speed tends to be proportional to the natural frequency, as can be seen in Eq. (64) and the natural frequencies for these modes are the highest.
298
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
Fig. 4. Three representative mode shapes and their corresponding natural frequencies of the hollow cylinder with r 0 ¼2.
Fig. 5. Three representative mode shapes and their corresponding natural frequencies of the hollow cylinder with r 0 ¼3.
Table 3 Critical speeds of each mode. Branch #
1 2 3
ξ ðnπ Þ
1 2 1 2 1 2
r0 ¼ 1
r0 ¼ 2
r0 ¼ 3
k¼0
k ¼1
k¼2
k ¼0
k¼1
k ¼2
k ¼0
k ¼1
k ¼2
5.217 5.217 7.317 5.027 10.323 8.757
3.688 4.303 8.665 6.210 12.010 8.940
5.835 4.775 11.729 7.542 15.834 9.888
5.217 5.217 4.426 3.889 8.891 8.449
3.653 3.906 6.208 5.409 9.790 9.068
3.696 4.077 7.664 5.892 11.875 8.970
5.217 5.217 3.400 3.657 8.796 8.406
3.254 3.694 5.586 5.292 9.217 8.483
3.265 3.817 6.395 5.505 10.345 8.701
4. Conclusions In this study, we first mathematically derived governing field equations of a hollow cylinder subjected to dual traveling forces imposed on its inner surface in the cylindrical coordinates. A structural steel was used as the material of the hollow cylinder. In order to investigate the effect of the cylinder geometry, three different r 0 values were used: r 0 ¼1, 2, and 3. In order to solve these governing field differential equations, the biorthogonality conditions, along with the adjoint system of equations, were derived and then applied to obtain the discretized equations. Using the field equations and homogeneous boundary conditions, we could find dispersion curves and eigensolutions. Through the application of these eigenfunctions and biorthogonality conditions to the inhomogeneous boundary conditions that include the boundary forcing term, the discretized differential equations could be obtained. Using the solutions of the discretized equations, the critical speed for each mode could be found. By inspecting the results of this analysis, the following conclusions could be made: 1. The natural frequencies of the first thickness modes increase as r 0 decreases. In the dispersion curves, comparing with the nth branch of each r 0 case, the natural frequencies increase as r 0 decreases in the range of concern of this study except for the branch associated with the torsional modes.
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
299
2. The critical speed is proportional to the natural frequency and inversely proportional to the length directional wavenumber of each mode. When the length directional wavenumber is zero (thickness mode), the associated critical speed does not exist. 3. In the dispersion curves, each branch for torsional modes appear as a straight line and has the same values as r 0 , which indicates that the torsional waves are nondispersive. Due to the nondispersive nature of the torsional modes, the critical speeds of all the torsional modes become identical.
Acknowledgments This work was supported by the Industrial Strategic Technology Development Program (10039982, Development of next generation multi-functional machining systems for eco/bio components) funded by the Ministry of Trade, Industry and Energy (MI, Korea). This research was also supported by the Chung-Ang University Excellent Student Scholarship in 2014. Appendix A. Bessel-type solutions for original system (Ref. [15]) Gazis [15] solved the governing field equations of linear elasticity in invariant form using the Helmholtz displacement potentials. The three-dimensional differential equation in invariant form is
μ∇2 u þ λ þ μ ∇ ∇ U μ ¼ ρ
∂2 u ; ∂t 2
(A.1)
where u is the displacement vector, ρ is the density, and λ and μ are Lamé constants. Because u is the Helmholtz displacement vector, it can be decomposed into a dilatational scalar potential ϕ and an equivoluminal vector potential H as
with the condition of
u ¼ ∇ϕ þ ∇ H
(A.2)
∇ U H ¼ F r; θ; z; t
(A.3)
as the gauge invariance. In Eq. (A.3), F r; θ; z; t is an arbitrary function that may be assumed to be zero. By substituting Eq. (A.2) into Eq. (A.1), we can obtain two wave equations for potentials ϕ and H:
λ þ 2μ 2 ∂2 ∇ ϕ ¼ 2 ϕ; ρ ∂t
(A.4a)
μ 2 ∂2 ∇ H ¼ 2 H: ρ ∂t
(A.4b)
For the present problem, we can define these potentials by using the method of separation of variables as ϕ ¼ f ðrÞ cos nθ cos ωt þ ξz ; H r ¼ g r ðr Þ sin nθ sin ωt þ ξz ; H θ ¼ g θ ðr Þ cos nθ sin ωt þ ξz ; H z ¼ g 3 ðr Þ sin nθ cos ωt þ ξz :
(A.5)
The substitution of the solution functions given in Eq. (A.5) into (A.4) gives
2 ∇2 þ ωv2 ϕ ¼ 0; 1 ! 1 ω2 2 ∂ ∇2 2 þ 2 H r 2 H θ ¼ 0; r r ∂θ v2 ! 1 ω2 2 ∂ 2 ∇ 2 þ 2 H θ þ 2 H r ¼ 0; r r ∂θ v2 ! ∇2 þ
ω2 v22
H z ¼ 0;
(A.6)
μ ρ
(A.7)
where, v21 ¼
λ þ 2μ ; ρ
v22 ¼ :
300
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
For the convenience of the subsequent analyses, the following differential operator is introduced: 2 2 ∂ 1 ∂ n 2 1 Λn;x ¼ 2 þ x ∂x ∂x x Using Eq. (A.8), we can express Eq. (A.6) as
(A.8)
Λn;αr f ¼ 0;
(A.9a)
Λn þ 1;βr gr gθ ¼ 0;
(A.9b)
Λn 1;βr gr þ gθ ¼ 0;
(A.9c)
Λn;βr g3 ¼ 0;
(A.9d)
where
α2 ¼
ω2 v21
ξ ; 2
β2 ¼
ω2 v22
ξ : 2
(A.10)
Eq. (A.9) can be solved in terms of Bessel functions J and Y, or the modified Bessel functions I and K, which are determined according to whether α and β in Eq. (A.10) are imaginary or real. Note that, when the value of both α and β are real, the arguments of the Bessel functions and the modified Bessel functions become αr and β r. The detailed Bessel functions used at different intervals are expressed in Table A1, where α1 r and β 1 r denote jαr j and β r , respectively. Using the solution functions given in Table A1, the general solutions given in Eq. (A.9) can be expressed as f ¼ AZ n ðα1 r Þ þBW n ðα1 r Þ;
(A.11a)
2g 1 ¼ g r g θ ¼ 2A1 Z n þ 1 β1 r þ 2B1 W n þ 1 β1 r ;
(A.11b)
2g 2 ¼ g r þg θ ¼ 2A2 Z n 1 β1 r þ 2B2 W n 1 β1 r ;
(A.11c)
g 3 ¼ A3 Z n β1 r þ B3 W n β1 r ;
(A.11d)
where Z denotes the J or I function and W denotes the Y or K function. The property of the gauge invariance (Eq. (A.3)) can now be utilized to eliminate two of the integration constants in Eq. (A.11) [43]. For instance, by setting g 2 ¼ 0, we obtain g1 ¼ gr ¼ gθ ; and the displacement fields shown in Eqs. (A.2) as ur ¼ f ;r þ nrg 3 þ ξg 1 cos nθ cos ωt þ ξz ; h n i uθ ¼ f þ ξg 1 g 3;r sin nθ cos ωt þ ξz ; r h g i uz ¼ ξf g 1;r ðn þ 1Þ 1 cos nθ sin ωt þ ξz : r
(A.12)
(A.13)
By using the stress–strain and strain–displacement relationships given in Eq. (1) and (3), respectively, and applying the boundary conditions in the radial direction given in Eq. (6a), we can solve the eigenproblem consisting of the coefficients of the Bessel functions in Eq. (A.11),A; B; A1 ; B1 ; A3 ; B3 . Appendix B. Special cases Using Eq. (45) with k ¼ 0 or ξ ¼ 0, we can find the equations for the associated special cases. B.1 For k ¼ 0 First, we derive the equation for the case of k ¼ 0. Substituting k ¼ 0 into Eq. (45), the resulting equations can be expressed as Table A1 Bessel functions used at different intervals. Case α2 40, β2 40 α2 o 0, β2 4 0 α2 o 0, β2 o 0
Functions used J ðα1 rÞ; Y ðα1 r Þ; J β1 r ; Y β1 r Iðα1 r Þ; K ðα1 r Þ; J β1 r ; Y β1 r Iðα1 r Þ; K ðα1 r Þ; I β1 r ; K β1 r
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
Radial direction
301
h i 2 2ηð1 þ νÞðr~ þr 0 Þ2 vð;rr~ Þr~ þ 2ηð1 þ νÞðr~ þr 0 Þvð;rr~ Þ þ ðr~ þr 0 Þ2 ρω2 ξ þ2ηð1 þ νÞ vðrÞ þ ξηðr~ þ r 0 Þ2 ð1 þ2νÞvð;rz~ Þ ¼ 0:
Circumferential direction:
n
(B.1a)
o
ηðr~ þ r0 Þ2 vð;r~ r~Þ þ ηðr~ þ r0 Þvð;r~ Þ þ ðr~ þ r 0 Þ2 ρω2 ξ2 η vðθÞ ¼ 0:
(B.1b)
ξðr~ þ r 0 Þ2 ð1 þ2νÞvð;rr~ Þ ξðr~ þ r 0 Þð1 þ 2νÞvðrÞ þ ηðr~ þ r 0 Þ2 vð;rz~ rÞ~ þ ηðr~ þr 0 Þvð;rz~ Þ n o 2 þ ρω2 2ξ ð1 þ νÞ ðr~ þ r 0 ÞvðzÞ ¼ 0:
(B.1c)
θ
θ
Length direction:
Note that in Eq. (B.1), vðθÞ is decoupled out of the other two equations and thus, Eq. (B.1b) happens to only include vðθÞ . Using Eq. (B.1b), we can obtain the vðθÞ as the Bessel functions: 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 ξ2 2 ρω k0 k0 θ ð Þ A þ A BesselY @1; ðr 0 þ r~ Þ ρω ξ A: (B.2) v ¼ A1 BesselJ @1; ðr 0 þ r~ Þ 2
η
η
The decoupled boundary conditions for vðθÞ are reduced to vðθÞ ðθÞ þ v;r~ ¼ 0; r~ þ r 0
1 (B.3) atr~ ¼ 7 : 2 Using Eqs. (B.2) and (B.3) we can obtain the displacement function, vðθÞ for k ¼ 0. The displacements vðrÞ and vðzÞ can be obtained using Eqs. (B.1a) and (B.1c) combined with their reduced boundary conditions, Eqs. (B.4) and (B.5), with the € application of Frobenius method explained in Section 2.2. 1 ðrÞ 1 (B.4) ð1 þ νÞvð;rr~ Þ þ ν ξvðzÞ þ v ¼ 0; at r~ ¼ 7 ; r~ þ r 0 2
ηvð;rz~ Þ ξvðrÞ ¼ 0;
1 at r~ ¼ 7 : 2
(B.5)
B.2 For ξ ¼ 0 For the case of ξ ¼ 0, Eqs. (45) can be expressed as the following two equations: Radial direction: h n oi 2 2ηð1 þ νÞðr~ þr 0 Þ2 vð;rr~ Þr~ þ 2ηð1 þ νÞðr~ þ r 0 Þvð;rr~ Þ þ ðr~ þr 0 Þ2 ρω2 η k 2ð1 þ νÞ vðrÞ ðθ Þ þikηðr~ þ r 0 Þð1 þ2νÞv;r~ ikηð3 þ 2νÞvðθÞ ¼ 0:
(B.6a)
Circumferential direction: ðθ Þ ðθÞ ikηðr~ þr 0 Þð1 þ2νÞvð;rr~ Þ þ ikηð3 þ 2νÞvðrÞ þ ηðr~ þ r 0 Þ2 v;r~ r~ þ ηðr~ þ r 0 Þv;r~ h n oi 2 þ ðr~ þ r 0 Þ2 ρω2 η 2k ð1 þ νÞ þ1 vðθÞ ¼ 0:
(B.6b)
In this case, the length directional displacement becomes zero, so that the length directional equation simply vanishes. Similarly, the boundary conditions associated with σ rz and σ θz vanish, so the coupled equations for vðrÞ and vðθÞ only remain. The radial and circumferential directional displacements can be found using same steps followed in Section 2.2. References [1] C.W. Cai, Y.K. Cheung, H.C. Chan, Dynamic response of infinite continuous beams subjected to a moving force – an exact method, Journal of Sound and Vibration 123 (3) (1988) 461–472. [2] M. Olsson, On the fundamental moving load problem, Journal of Sound and Vibration 145 (2) (1991) 299–307. [3] T.N. Shiau, K.H. Huang, F.C. Wang, K.H. Chen, C.P. Kuo, Dynamic response of a rotating ball screw subject to a moving regenerative force, Applied Mathematical Modelling 34 (2010) 1721–1731. [4] M.R. Shadnam, M. Mofid, J.E. Akin, On the dynamic response of rectangular plate, with moving mass, Thin-Walled Structures 39 (2001) 797–896. [5] R. Katz, C.W. Lee, A.G. Ulsoy, R.A. Scott, The dynamic response of a rotating shaft subject to a moving load, Journal of Sound and Vibration 122 (1) (1988) 131–148. [6] H.S. Zibdeh, H.S. Juma, Dynamic response of a rotating beam subjected to a random moving load, Journal of Sound and Vibration 223 (5) (1999) 741–758. [7] M.A. Hilal, H.S. Zibdeh, Vibration analysis of beams with general boundary conditions traversed by a moving force, Journal of Sound and Vibration 229 (2) (2000) 377–388. [8] L.Y. Tommy, H.T. Chan, Moving force identification based on the frequency–time domain method, Journal of Sound and Vibration 261 (2003) 329–349. [9] G. Sinclair, The patterns of slotted-cylinder antennas, Proc. Inst. Radio Eng. 36 (12) (1948) 1487–1492.
302
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
[10] C.T. Swift, P.B. Gooderum, S.L. Castellow, Experimental investigation of a plasma covered, axially slotted cylinder antenna, IEEE Trans. Antennas Propag. 17 (5) (1969) 598–605. [11] A.K. Hamid, Multi-dielectric loaded axially slotted antenna on circular or elliptic cylinder, Journal of Electromagnetic Waves and Applications 20 (9) (2006) 1259–1271. [12] J.E. Greenspon, Flexural vibrations of a thick walled circular cylinder, in: Proceedings of the Third U.S. National congress of Applied Mechanics, vol. 1, 1958, pp. 163–173. [13] J.E. Greenspon, Flexural vibrations of a thick-walled circular cylinder according to the exact theory of elasticity, Journal of the Aero/Space Sciences 27 (1) (1960) 37–40. [14] L. Pochhammer, On the propagation of small free vibrations in an infinite circular solid cylinder, Crelle's Journal 81 (1876) 324–326. [15] D.C. Gazis, Three-dimensional investigation of the propagation of waves in hollow circular cylinders. I. Analytical foundation, Journal of the Acoustical Society of America 31 (5) (1959) 568–573. [16] D.C. Gazis, Three-dimensional investigation of the propagation of waves in hollow circular cylinders. II. Numerical results, Journal of the Acoustical Society of America 31 (5) (1959) 573–578. [17] I. Mirsky, Wave propagation in transversely isotropic circular cylinders. Part I: Theory, Journal of the Acoustical Society of America 37 (6) (1965) 1016–1021. [18] J.Q. Ye, K.P. Soldatos, Three-dimensional vibrations of cross-ply laminated hollow cylinders with clamped edge boundaries, Journal of Vibration and Acoustics 119 (3) (1997) 317–323. [19] J. So, A.W. Leissa, Free vibrations of thick hollow circular cylinders from three-dimensional analysis, Journal of Vibration and Acoustics 119 (1) (1997) 89–95. [20] S. Brischetto, Three-dimensional exact free vibration analysis of spherical, cylindrical, and flat one-layered panels, Shock and Vibration (2014) (2014) 479738. [21] K.P. Soldatos, V.P. Hadjigeorgiou, Three-dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels, Journal of Sound and Vibration 137 (3) (1990) 369–384. [22] C.T. Loy, K.Y. Lam, Vibration of thick cylindrical shells on the basis of three-dimensional theory of elasticity, Journal of Sound and Vibration 226 (4) (1999) 719–737. [23] T.Y. Ng, K.Y. Lam, Vibration and critical speed of a rotating cylindrical shell subjected to axial loading, Applied Acoustics 56 (1999) 273–282. [24] J. Padovan, Traveling waves vibrations and buckling of rotating anisotropic shells of revolution by finite elements, International Journal of Solids and Structures 11 (1975) 1367–1380. [25] K.M. Lew, T.Y. Ng, X. Zhao, J.N. Reddy, Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells, Computer Methods in Applied Mechanics and Engineerings 191 (37–38) (2002) 4141–4157. [26] P.G. Bhuta, Transient response of a thin elastic cylindrical shell to a moving shock wave, Journal of the Acoustical Society of America 35 (1) (1963) 25–30. [27] M.J. Forrestal, G. Herrmann, Response of a submerged cylindrical shell to an axially propagating step wave, Journal of Applied Mechanics 32 (4) (1965) 788–792. [28] H. Reismann, J. Padlog, Forced, axi-symmetric motions of cylindrical shells, Journal of The Franklin Institute 284 (5) (1967) 308–319. [29] R. Parnes, Response of an infinite elastic medium to traveling loads in a cylindrical bore, Journal of Applied Mechanics 36 (1) (1969) 51–58. [30] C.C. Huang, Moving loads on elastic cylindrical shells, Journal of Sound and Vibration 49 (2) (1976) 215–220. [31] E. Mangrum, J.J. Burns, Orthotropic cylindrical shells under dynamic loadings, Journal of Mechanical Design 101 (1979) 322–329. [32] S. Chonan, Dynamic response of a cylindrical shell imperfectly bonded to a surrounding continuum of infinite extent, Journal of Sound and Vibration 78 (2) (1981) 257–267. [33] S.K. Datta, T. Chakraborty, A.H. Shah, Dynamic response of pipelines to moving loads, Earthquake Engineering & Structural Dynamics 12 (1) (1984) 59–72. [34] C.W. Bert, V. Birman, Parametric instability of thick, orthotropic, circular cylindrical shells, Acta Mechanica 71 (1988) 61–76. [35] R. Panneton, A. Berry, F. Laville, Vibrations and sound radiation of a cylindrical shell under a circumferentially moving force, Journal of the Canadian Acoustical Association 20 (3) (1992) 25–26. [36] V.P. Singh, J.P. Dwivedi, P.C. Upadhyay, Effect of fluid presence on the nonaxisymmetric dynamic response of buried orthotropic cylindrical shells under a moving load, Journal of Vibration and Control 6 (2000) 823–847. [37] K.M. Liew, Y.G. Hu, T.Y. Ng, X. Zhao, Dynamic stability of rotating cylindrical shells subjected to periodic axial loads, International Journal of Solids and Structures 43 (2006) 7553–7570. [38] Y.Q. Wang, X.H. Guo, Y.G. Li, J. Li, Nonlinear traveling wave vibration of a circular cylindrical shell subjected to a moving concentrated harmonic force, Journal of Sound and Vibration 329 (2010) 338–352. [39] S.C. Huang, B.S. Hsu, Resonant phenomena of a rotating cylindrical shell subjected to a harmonic moving load, Journal of Sound and Vibration 136 (2) (1990) 215–228. [40] J. Seok, H.F. Tiersten, Free vibrations of annular sector cantilever plates. Part 1: Out-of-plane motion, Journal of Sound and Vibration 271 (2004) 757–772. [41] J. Seok, H.F. Tiersten, Free vibrations of annular sector cantilever plates. Part 2: In-plane motion, Journal of sound and Vibration 271 (2004) 773–787. [42] ANSYS v 12, ANSYS, Inc. [43] A.E. Armenakas, propagation of harmonic waves in composite circular cylindrical shells-I: Theoretical investigations, The American Institute of Aeronautics and Astronautics 5 (1967) 740–744. [44] H.H. Sherief, M.N. Anwar, A problem in generalized thermoelasticity for an infinitely long annular cylinder composed of two different materials, Acta Mechanica 80 (1989) 137–149. [45] Z.Y. Lee, C.K. Chen, C.I. Hung, Transient thermal stress analysis of multilayered hollow cylinder, Acta Mechanica 151 (2001) 75–88. [46] H. Cho, G.A. Kardomateas, Thermal shock stresses due to heat convection at a bounding surface in a thick orthotropic cylindrical shell, International Journal of Solids and Structures 38 (2001) 2769–2788. [47] H.H. Sherief, A.E.M. Elmisiery, M.A. Elhagary, Generalized thermoelastic problem for an infinitely long hollow cylinder for short times, Journal of Thermal Stresses 27 (2004) 885–902. [48] Z.Y. Lee, Magnetothermoelastic transient response of a multilayered hollow cylinder subjected to magnetic and vapor fields, Acta Mechanica 193 (2007) 1–16. [49] B. Bai, Consolidation solutions of a saturated porothermoelastic hollow cylinder with infinite length, Scientific Research 2 (1) (2010) 37–45. [50] M.A. Elhagary, Generalized thermoelastic diffusion problem for an infinitely long hollow cylinder for short times, Acta Mechanica 218 (2011) 205–215.
Contents lists available at ScienceDirect
Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi
Dynamic analysis of a hollow cylinder subject to a dual traveling force imposed on its inner surface Sooyoung Lee, Jongwon Seok n School of Mechanical Engineering, College of Engineering, Chung-Ang University, 84 HeukSeok-Ro, DongJak-Gu, Seoul 156-756, Republic of Korea
a r t i c l e i n f o
abstract
Article history: Received 31 May 2014 Received in revised form 24 November 2014 Accepted 25 November 2014 Handling Editor: S. Ilanko Available online 23 December 2014
The dynamic behavior of a hollow cylinder under a dual traveling force applied to the inner surface is investigated in this study. The cylinder is constrained at both the top and bottom surfaces not to move in the length direction but free in other directions. And a dual force travels at a constant velocity along the length direction on the inner surface of the hollow cylinder. The resulting governing field equations and the associated boundary conditions are ruled by the general Hooke's law. Due to the nature of the field equations, proper adjoint system of equations and biorthogonality conditions were derived in a precise and detailed manner. To solve these field equations in this study, the method of € separation of variable is used and the method of Frobenius is employed for the differential equations in the radial direction. Using the field equations, the eigenanalyses on both the original and its adjoint system were performed with great care, which results in the eigenfunction sets of both systems. The biorthogonality conditions were applied to the field equations to obtain the discretized equation for each mode. Using the solutions of the discretized equations that account for the boundary forcing terms, the critical speed for a dual traveling force for each mode could be computed. & 2014 Elsevier Ltd. All rights reserved.
1. Introduction The dynamic behaviors of the structures subjected to a variety types of traveling forces have been studied for a long time in various engineering fields [1–8]. In particular, hollow cylindrical structure is used in many machines, which motivates the researches on the investigation of its dynamics under a traveling force. Single or dual slotting of a hollow cylinder is an illustrative example [9–11]. Although few in number, several studies dealing with the three-dimensional eigenanalysis of thick hollow cylinders have been conducted. Greenspon [12,13] addressed the flexural vibrations of a thick-walled circular cylinder of finite length using the work of Pochhammer [14]. In his study, he obtained the curves of the frequency parameter as a function of the length and thickness ratios. Similarly, Gazis [15,16] studied the analytical foundation for an investigation of the harmonic waves in a hollow circular cylinder of infinite extent using the work described in [14]. Mirsky [17] obtained the most general type of harmonic waves in both hollow and solid circular cylinders of transversely isotropic material. Ye and Soldatos [18] analyzed the free vibration of cross-ply laminated hollow cylinders based on a recursive solution. So and Leissa [19] performed an analysis of the free vibrations of a thick cylinder using a Ritz analysis. Brischetto [20] obtained an
n
Corresponding author. Tel.: þ82 2 820 5729; fax: þ 82 2 3280 9982. E-mail address: [email protected] (J. Seok).
http://dx.doi.org/10.1016/j.jsv.2014.11.032 0022-460X/& 2014 Elsevier Ltd. All rights reserved.
284
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
exact solution to a free vibration problem of single-layer spherical, cylindrical, and flat panels. Further, Soldatos and Hadjigeorgiou [21] solved a free vibration problem of a homogeneous isotropic thick cylindrical shell using a new iterative approach. This new approach allows the accurate prediction of the frequencies of vibration. Loy and Lam [22] performed an approximate analysis of the vibration of thick circular cylindrical shells using a concept that is similar to the finite-strip method. Owing to the difficulties faced in solving a three-dimensional problem, the hollow cylinder is often simplified to a twodimensional cylindrical shell equivalent. And the dynamic problem of the cylindrical shell under a traveling force has long been treated by many researchers [23–25] due to the relative simplicity compared to that of its three-dimensional counterpart. The dynamic response of shells based on the classical shell theory was studied by Bhuta [26], using the theory of a thin cylindrical shell. And the response of an infinitely long, circular, cylindrical shell axially propagating step wave was investigated by Forrestal and Hermann [27]. Later, Reismann and Padlog [28] presented a formal solution of the dynamic response of a cylindrical shell of finite length by the use of the method of eigenfunction expansion. Raymond Parnes [29] obtained the response of an infinite elastic cylindrical bore to a traveling load. Huang [30] investigated the dynamic response of the shell of infinite length acted upon by a ring load traveling at a constant velocity using the Fourier transform method in conjunction with the contour integral method. Also, Mangrum and Burns [31] studied the dynamic response of orthotropic cylindrical shells under an axial moving load using small deflection shell theory. Chonan [32] studied the dynamic response of a cylindrical shell imperfectly bonded to a surrounding elastic continuum of infinite extent. In his study [32], an axisymmetric ring pressure moving in the axial direction along the interior of the shell was treated as the applied load. Datta et al. [33] obtained the dynamic response of a cylindrical shell to moving disturbances. Bert and Birman [34] performed the dynamic stability analyses of a simply supported, finite-length, circular cylindrical thick shell subjected to a parametrically excited periodic axial loading. Panneton et al. [35] investigated the vibrations and sound radiation by the finite cylindrical shells subjected to a circumferentially moving force. Singh et al. [36] studied the nonaxisymmetric dynamic behavior of the orthotropic cylindrical shells subjected to a load moving along the axis of the shell. They used a thick shell model that includes the effects of shear deformation and rotary inertia. Liew et al. [37] studied the dynamic stability of the rotating cylindrical shells under static and periodic axial forces using the energy formulation. In their study [37], they used a combination of the Ritz method and Bolotin's first approximation. Wang et al. [38] obtained the nonlinear traveling wave response of a cantilever circular cylindrical shell subjected to a concentrated harmonic force moving in a concentric circular path. Donnell's nonlinear shallow-shell theory, together with the consideration of geometric nonlinearities, were used in their study [38]. Huang et al. [39] investigated the resonance behavior of a rotating cylindrical shell caused by the action of harmonic moving loads. They used the shell equations derived from the Love–Timoshenko theory for a nonrotating thin cylindrical shell. On the other hand, the Laplace transform technique was used to solve axially symmetrical thermoelastic problems. Sherief and Anwar [44] studied the heat conduction of an infinitely long thermoelastic cylindrical medium, for which a onedimensional solution was sought in the Laplace transform domain. Lee et al. [45] obtained a transient response for a onedimensional axisymmetric quasistatic thermoelastic problem. Cho and Kardomateas [46] investigated thermal shock stress caused by heat transfer into a medium at a constant temperature in a thick orthotropic cylindrical shell. Sherief et al. [47] studied a one-dimensional problem for an infinitely long hollow cylinder with one relaxation time. Lee [48] investigated one-dimensional axisymmetric quasi-state coupled magnetothermoelastic problems. Bai [49] obtained an analytical solution for a hollow cylinder of porothermoelastic media of infinite length. Elhagary [50] considered a homogeneous isotropic thermoelastic hollow cylinder with one relaxation time. Most of the abovementioned analytic approaches on the cylindrical shell problem subjected to a traveling load can provide valuable insight into the dynamic response of thin shell-like cylindrical structures caused by a traveling force. However, in the cases that the hollow cylinder is thick enough not to be considered as a thin shell, the radial mode can appear in a relatively low frequency range, so that the thin shell theory cannot provide a useful way to accurately compute its dynamic behaviors. In the present study, we treated a general hollow cylinder and solved its dynamic problems in an analytic way. As an illustrative example in this work, the hollow cylinder subjected to a dual traveling force on its inner surface is treated. For the analysis of this study, proper three-dimensional nonlinear governing field equations were derived. Using these equations, an eigenanalysis was performed first with great care. Due to the nonsymmetric nature of the field equations, the present eigensystem is non-self-adjoint, so that the biorthogonality conditions, along with the adjoint system of equations, were obtained. During this process, series form of eigensolutions, with respect to the radial coordinate, was attempted to € obtain using the Frobenius method [40,41], which results in the dispersion relations. It is well known that a threedimensional linear elasticity problem can be solved in terms of Bessel functions [15,16], after decomposing the problem into irrotational and solenoidal parts by introducing two types of potential functions. However, the decomposition of the adjoint system cannot be performed easily because of the additional terms originating from the boundary conditions, and therefore, we opted to derive and use the Fröbenius-type power series solutions in this study. Using the eigensolutions, we obtained the mode shape for each mode. Then, the same approach is applied to the adjoint system to solve its eigenproblem. Using the biorthogonality conditions between the eigenfunctions of the original and its adjoint systems, the field equations that account for the inhomogeneous boundary forcing terms could be successfully discretized. The critical speed was obtained for a set of several representative modes.
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
285
Fig. 1. Schematics of the hollow cylinder.
2. Mathematical model of a hollow cylinder In this study, the hollow cylinder subjected to a dual traveling force imposed on its inner surface is investigated. Fig. 1 shows a schematic of the hollow cylinder, where the inner radius is denoted as Ri , the outer radius as Ro and the height as H. Here, the cylindrical coordinate system (r, θ, z) with unit vectors (e^ r , e^ θ , e^ z ) respectively, are used as shown in Fig. 1. 2.1. Derivation of the governing field equations and boundary conditions in cylindrical coordinates The cylinder is fixed only in z-direction (uz ¼ 0) at z ¼ 0 and H and the forces travels at a constant velocity v, along the z-direction on the inner surface of the hollow cylinder (r ¼ Ri ) at θ ¼ θ1 and θ2 . Note that subscripts or superscripts i, j and k are used to denotes the coordinates, and this symbolism will be understood hereafter. The equations for the stress components in terms of the strain components in the cylindrical coordinate system can be expressed with Einstein's index notation as below:
σ^ ij ¼ Eε^ ij þ Eνδij ε^ kk ;
(1)
where E ¼ E=ð1 þ νÞ, ν ¼ ν=ð1 2νÞ, δij is the Kronecker delta function, E and ν are Young's modulus and Poisson's ratio, respectively. The stress equations of motion of the problem at hand in cylindrical coordinates can be obtained as the following usual forms: σ^ rr σ^ θθ σ^ ¼ ρu€^ r : Radial direction: σ^ rr;r þ rθ;θ þ σ^ rz;z þ (2a) r^ r^
σ^ 2σ^ Circumferential direction: σ^ rθ;r þ θθ;θ þ σ^ θz;z þ rθ ¼ ρu€^ θ : r^ r^
(2b)
σ^ σ^ rz ¼ ρu€^ z : Length direction: σ^ rz;r þ θz;θ þ σ^ zz;z þ r^ r^
(2c)
where ρ is the mass density of the cylinder material, ui means the i directional displacement component, dot is used over a symbol to represent the time derivative and a comma followed by subscript(s) denotes partial differentiation with respect to the spatial coordinate indicated by the subscript(s). The following strain–displacement relations are employed in this study:
ε^ rr ¼ u^ r;r ;
(3a)
u^ þ u^ θ;θ ^εθθ ¼ r ; r^
(3b)
ε^ zz ¼ u^ z;z ;
(3c)
1 u^ r;θ u^ θ þ u^ θ;r ; ε^ rθ ¼ ^ 2 r
(3d)
ε^ θz ¼
u^ 1 u^ θ;z þ z;θ ; ^ 2 r
(3e)
286
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
ε^ rz ¼
1 u^ r;z þ u^ z;r : 2
(3f)
For the convenience of the subsequent analyses, the following dimensionless parameters are introduced: ur ¼
z^ t^ 2ρR2 H2 u^ r u^ u^ z r^ ; uθ ¼ θ ; uz ¼ ; r ¼ ; z ¼ ; t ¼ ; ρ ¼ ; η ¼ 2; 2 H R R R H t R Et
where R ¼ Ro Ri . Using Eqs. (1)–(4), the displacement equations of motion can be derived in dimensionless forms u ηð3 þ 2νÞuθ;θ Radial direction: 2ηð1 þ νÞur;rr þ ur;zz þ ηr r;rθθ þ 2ð1 þ νÞ ur;r urr r2
uθ;rθ ¼ ρu€ r ; þ ηð1 þ 2νÞ uz;rz þ r Circumferential direction:
2ηð1 þ νÞuθ;θθ þ r2
þ
(4)
(5a)
ηuθ;rr þ uθ;zz þ ηr uθ;r ur2θ þ ηð3 þr22νÞur;θ
ηð1 þ 2νÞ uz;θz þ ur;rθ r
¼ ρu€ θ :
(5b)
ðu þ u Þ Length direction: 2ð1 þ νÞuz;zz þ η uz;rr þ urz;r þ urz:2θθ þur;rz þ r;z r θ;θz ur;z þuθ;θz ¼ ρu€ z : þ 2ν ur;rz þ r
(5c)
The number of the boundary/continuity conditions of the present model are 18 in total and they can be expressed as follows: Radial directional boundary conditions: σ rr Ro ; θ; z ¼ 0; σ rθ Ro ; θ; z ¼ 0; σ rz Ro ; θ; z ¼ 0; (6a)
σ rr Ri ; θ; z ¼ 0;
σ rθ Ri ; θ; z ¼ 0;
σ rz Ri ; θ; z ¼ 0:
(6b)
σ rθ ðr; 2π ; zÞ ¼ σ rθ ðr; 0; zÞ; σ θθ ðr; 2π ; zÞ ¼ σ θθ ðr; 0; zÞ; σ θz ðr; 2π ; zÞ ¼ σ θz ðr; 0; zÞ;
(6c)
Circumferential directional continuity conditions:
ur ðr; 2π ; zÞ ¼ ur ðr; 0; zÞ;
uθ ðr; 2π ; zÞ ¼ uθ ðr; 0; zÞ;
uz ðr; 2π ; zÞ ¼ uz ðr; 0; zÞ: Length directional boundary conditions: σ rz r; θ; H ¼ 0;
uz r; θ; H ¼ 0;
σ θz r; θ; H ¼ 0;
σ rz r; θ; 0 ¼ 0; σ θz r; θ; 0 ¼ 0; uz r; θ; 0 ¼ 0:
(6d)
(6e) (6f)
Eq. (6a)–(f) are the boundary/continuity conditions on the hollow cylinder in its natural state. In case that the traveling forces exist, the following boundary forcing term σ~ rr should be superposed to the σ rr Ri ; θ; z condition. σ~ rr Ri ; θ; z ¼ f 1 δ Ri θ Ri θ1 þ f 2 δ Ri θ Ri θ2 δðz Vt Þ; (7) where, δðÞ denotes the Dirac delta function and V is the traveling velocity of the forces in the z-direction. Substituting Eqs. (1) and (3) into Eqs. (6a)–(6f), the boundary/continuity conditions can be expressed in terms of displacements and their gradients as follows: Radial directional boundary conditions: uθ;θ Ro ; θ; z þur Ro ; θ; z 2ηð1 þ νÞur;r Ro ; θ; z þ 2ην uz;z Ro ; θ; z þ ¼ 0; (8a) Ro ur;θ Ro ; θ; z uθ Ro ; θ; z þ 2ηuθ;r Ro ; θ; z ¼ 0; 2η Ro
(8b)
ur;z Ro ; θ; z þ ηuz;r Ro ; θ; z ¼ 0;
(8c)
uθ;θ Ri ; θ; z þ ur Ri ; θ; z 2ηð1 þ νÞur;r Ri ; θ; z þ2ην uz;z Ri ; θ; z þ ¼ 0; Ri
(8d)
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
287
ur;θ Ri ; θ; z uθ Ri ; θ; z 2η þ 2ηuθ;r Ri ; θ; z ¼ 0; Ri
(8e)
ur;z Ri ; θ; z þ ηuz;r Ri ; θ; z ¼ 0:
(8f)
Circumferential directional continuity conditions: ur;θ ðr;2π ;zÞ uθ ðr;2π ;zÞ 2ηuθ;r ðr;2π ;zÞ þ r r2
2η
u ðr; 0; zÞ uθ ðr; 0; zÞ 2ηuθ;r ðr; 0; zÞ ; ¼ 2η r;θ þ r r2
(8g)
2ηð1 þ νÞðuθ;θ ðr;2π ;zÞ þ ur ðr;2π ;zÞÞ 2ηνfur;r ðr;2π ;zÞ þ uz;z ðr;2π ;zÞg þ r r2
¼
2ηð1 þ νÞ uθ;θ ðr; 0; zÞ þ ur ðr; 0; zÞ 2ην ur;r ðr; 0; zÞ þuz;z ðr; 0; zÞ ; þ r r2
(8h)
uθ;z ðr; 2π ; zÞ 2ηuz;θ ðr; 2π ; zÞ uθ;z ðr; 0; zÞ 2ηuz;θ ðr; 0; zÞ þ þ ¼ ; r r r2 r2
(8i)
ur ðr; 2π ; zÞ ¼ ur ðr; 0; zÞ;
(8j)
uθ ðr; 2π ; zÞ ¼ uθ ðr; 0; zÞ;
(8k)
uz ðr; 2π ; zÞ ¼ uz ðr; 0; zÞ:
(8l)
Length directional boundary conditions:
ur;z r; θ; H þ ηuz;r r; θ; H ¼ 0;
(8m)
2ηuz;θ r; θ; H ¼ 0; uθ;z r; θ; H þ r uz r; θ; H ¼ 0; ur;z r; θ; 0 þ ηuz;r r; θ; 0 ¼ 0;
(8n) (8o) (8p)
2ηuz;θ r; θ; 0 ¼ 0; uθ;z r; θ; 0 þ r uz r; θ; 0 ¼ 0:
(8q) (8r)
2.2. Eigenanalysis and derivation of the biorthogonality conditions In this section, an eigenanalysis is performed using the governing field equations together with the homogeneous boundary conditions. Assuming the displacement functions in nth mode eigenstate as ðr Þ
ðr Þ
ur ¼ ϕ eλt þ ϕ
eλt ;
ðθ Þ θ uθ ¼ ϕð Þ eλt þ ϕ eλt ; ðzÞ
uz ¼ ϕ eλt þ ϕ eλt ; ðzÞ
ðjÞ
(9)
ðjÞ
where ϕ is the eigenfunction of the jth mode and ϕ is its complex conjugate. Substituting Eqs. (9) into Eqs. (5) and (6), ðjÞ ðjÞ followed by replacing ϕ and λ with ϕn and λn , respectively, the following equations for the eigenvalue problem of the original system can be obtained as LΦD ¼ λn MΦD ; n
2
n
(10a)
n n Br ΦB1 r ¼ Ro ¼ Br ΦB2 r ¼ R ¼ 0;
(10b)
Bθ ΦB3 ¼ Bθ ΦB4 ;
(10c)
Bz ΦB5 ¼ Bz ΦB6 ¼ 0;
(10d)
i
n
n
n
n
288
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
where 2
L1;1 ðÞ 6 L ¼ 4 L2;1 ðÞ L3;1 ðÞ
L1;2 ðÞ L3;2 ðÞ
ρðÞ 6 M¼4 0 0 2 6 Br ¼ 6 4
L2;3 ðÞ 7 5;
L2;2 ðÞ
2
3
0
0
0 7 5; ρðÞ
2ηð1 þ νÞðÞ;r þ 2ην r ðÞ
2ηνðÞ;θ
η
ðÞ;z
ðÞ;r 2η r ðÞ;θ
0
ðÞ
0
ϕðθÞ r; θ; z
h
ϕðθÞ Ro ; θ; z
h
ϕðθÞ Ri ; θ; z
ΦB1 ¼ ϕðrÞ Ro ; θ; z
ΦB2 ¼ ϕðrÞ Ri ; θ; z
ηðÞ;r
(13a)
3 7
(13b)
3 7 5;
(13c)
7 7; 5
7 2ην 7; r ðÞ;z 5 2η 2η ðÞ þ r ðÞ r 2 ;θ
ϕðzÞ r; θ; z
3
0
2ηð1 þ νÞ ðÞ;θ þ 2rηðÞ r2 1 r ðÞ;z
ðÞ;z 6 Bz ¼ 4 0 0
0
2η 2η r ðÞ;r r2 ðÞ
2
h
2ηνðÞ;z
0
2η ðÞ þ 2rηðÞ r 2 ;θ 6 6 2ηð1 þ νÞ Bθ ¼ 6 r2 ðÞ þ 2ην r ðÞ;r 4 0
ΦD ¼ ϕðrÞ r; θ; z
(12)
2rηðÞ
ðÞ;z
2
(11)
L3;3 ðÞ
0 ρðÞ
2η r ðÞ;θ
3
L1;3 ðÞ
iT
;
(14a)
iT
iT
ϕðzÞ Ro ; θ; z ϕðzÞ Ri ; θ; z
;
(14b)
;
(14c)
ΦB3 ¼ ϕðrÞ ðr; 2π ; zÞ ϕðθÞ ðr; 2π ; zÞ ϕðzÞ ðr; 2π ; zÞ ;
(14d)
h
iT
h
iT
ΦB4 ¼ ϕðrÞ ðr; 0; zÞ ϕðθÞ ðr; 0; zÞ ϕðzÞ ðr; 0; zÞ ; h
ϕðθÞ r; θ; H
h
ϕðθÞ r; θ; 0
ΦB5 ¼ ϕðrÞ r; θ; H
ΦB6 ¼ ϕðrÞ r; θ; 0
(14e)
iT
iT
ϕðzÞ r; θ; H ϕðzÞ r; θ; 0
;
(14f)
;
(14g)
andthe linear operators associated with the governing field equations are defined as L1;1 ðÞ ¼ 2ηð1 þ νÞðÞ;rr L1;2 ðÞ ¼
2ηð1 þ νÞ η 2ηð1 þ νÞ ðÞ;r þ 2 ðÞ;θθ þðÞ;zz ðÞ; r r2 r2
ηð1 þ 2νÞ r
ðÞ;rθ
ηð3 þ2νÞ r2
ðÞ;θ ;
L1;3 ðÞ ¼ ηð1 þ 2νÞðÞ;rz ; L2;1 ðÞ ¼
ηð1 þ 2νÞ r
ðÞ;rθ þ
ηð3 þ2νÞ r2
ηð1 þ2νÞ r
L3;1 ðÞ ¼ ð1 þ2νÞðÞ;rz þ
(15b) (15c)
ðÞ;θ ;
η 2η η L2;2 ðÞ ¼ ηðÞ;rr þ ðÞ;r þ 2 ðÞ;θθ þðÞ;zz 2 ðÞ; r r r L2;3 ðÞ ¼
(15a)
(15d) (15e)
ðÞ;θz ;
(15f)
1 þ 2ν ðÞ;z ; r
(15g)
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
L3;2 ðÞ ¼
1 þ 2ν ðÞ;θz ; r
η
289
(15h)
η
L3;3 ðÞ ¼ ηðÞ;rr þ ðÞ;r þ 2 ðÞ;θθ þ2ð1 þ νÞðÞ;zz : r r
(15i)
Eqs. (15a)–(15c), Eqs. (15d)–(15f) and Eqs. (15g)–(15i) are associated with the radial, circumferential and length directional governing field equations, respectively, and Eqs. (13a)–(13c) represent the boundary/continuity conditions expressed with the linear operators in matrix forms. In Eqs. (13), Br and Bz are associated with the radial and length directional boundary conditions, respectively, and Bθ is with circumferential directional continuity conditions. Note that the nonsymmetric nature of matrix L manifests the fact that the present system is non-self-adjoint. The adjoint system and biorthogonality conditions between the original and the adjoint systems are derived. To derive biorthogonality conditions, the inner product of two vector functions has to be defined first. We here define it as Z Z
a; b ¼ aU b dΩ ¼ ða1 b1 þ a2 b2 þ a3 b3 ÞdΩ; (16) Ω
Ω
where T a ¼ a1 r; θ; z ; a2 r; θ; z ; a3 r; θ; z ; T b ¼ b1 r; θ; z ; b2 r; θ; z ; b3 r; θ; z : Taking inner product on both sides of Eq. (10a) with ΨD : D E D E m 2 n n Ψm D ; LΦD ¼ λn ΨD ; MΦD m
(17)
and using the divergence theorem along D E with the given boundary conditions, we can obtain the following relationships: m n Left side of Eq. (17), i.e., ΨD ;LΦD : R m R m m n n n Ω ΨD U LΦD dΩ Γ 1 ΨB1 U Br ΦB1 r ¼ Ro ΨB2 U Br ΦB2 r ¼ Ri dΓ 1 Z
Z
m m n n n n Ψm Ψm dΓ 2 dΓ 3 B3 U Bθ ΦB3 ΨB4 U Bθ ΦB4 B5 U Bz ΦB5 ΨB6 U Bz ΦB6 Γ2 Γ3 Z
Z
m m m n n n ¼ Ln ΨD U ΦD dΩ Bnr ΨB1 U ΦB1 Bnr ΨB2 U ΦB2 dΓ 1 r ¼ Ro r ¼ Ri Ω Γ1 Z
m m n n Bnθ ΨB3 U ΦB3 Bnθ ΨB4 U ΦB4 dΓ 2 Γ Z 2
m m n n Bnz ΨB5 U ΦB5 Bnz ΨB6 U ΦB6 dΓ 3 : (18) Right side of Eq. (17), i.e., λ
2 n
D
Γ3
Ψ
Φ
m D ;M
λ2n
Z
n D
E
:
Ω
n2 n Ψm D U ΜΦD dΩ ¼ λm
Z
Ω
m n Mn ΨD U ΦD dΩ;
(19)
where ΦD , ΦB1 , ΦB2 , ΦB3 , ΦB4 , ΦB5 , ΦB6 are defined in Eqs. (14), and the vectors of the adjoint system h iT θ ðr Þ Ψm ψ ðm Þ r; θ; z ; ψ ðmzÞ r; θ; z ; D ¼ ψ m r; θ ; z ; n
n
n
n
n
n
n
h
ψ ðm Þ Ro ; θ; z ;
h
ψ ðm Þ Ri ; θ; z ;
ψ ðmzÞ Ri ; θ; z
;
(20c)
ψ ðm Þ ðr; 2π ; zÞ;
ψ ðmzÞ ðr; 2π ; zÞ ;
(20d)
ψ ðm Þ ðr; 0; zÞ;
ψ ðmzÞ ðr; 0; zÞ ;
ðrÞ Ψm B1 ¼ ψ m Ro ; θ ; z ;
ðr Þ Ψm B2 ¼ ψ m Ri ; θ ; z ;
h
ðr Þ Ψm B3 ¼ ψ m ðr; 2π ; zÞ;
h
ðrÞ Ψm B4 ¼ ψ m ðr; 0; zÞ;
h
θ
θ
θ
ψ ðm Þ r; θ; H ;
ψ ðm Þ r; θ; 0 ;
ðr Þ Ψm B6 ¼ ψ m r; θ ; 0 ;
are defined similarly to those of the original system.
ðrÞ Ψm B5 ¼ ψ m r; θ ; H ;
h
θ
θ
θ
iT
iT
(20a)
ψ ðmzÞ Ro ; θ; z
iT
iT
iT
iT
ψ ðmzÞ r; θ; H ψ ðmzÞ r; θ; 0
;
(20b)
(20e) ;
(20f) (20g)
290
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
In Eqs. (18) and (19), Ln , Bnj and Mn mean the adjoint of L, Bj and M, respectively, and Γ 1 , Γ 2 , and Γ 3 mean the boundary n domains of the radial, circumferential and length directions, respectively, and λm and ψ ðmjÞ are the eigenvalue and eigenfunction of the mth mode of the adjoint system. Ln and Bn obtained from Eq. (18) can be expressed as 2 n 3 L1;1 ðÞ Ln1;2 ðÞ Ln1;3 ðÞ 6 7 Ln ðÞ Ln2;2 ðÞ Ln2;3 ðÞ 7; (21) Ln ¼ 6 4 2;1 5 Ln3;1 ðÞ Ln3;2 ðÞ Ln3;3 ðÞ 2
ρðÞ
0
6 Mn ¼ 4 0 0 2 6 Bnr ¼ 6 4
3
0 7 5; ρðÞ
ρðÞ 0
2ηð1 þ νÞðÞ;r 2rηðÞ
(22)
2ην r ðÞ;θ ðÞ;r 2rηðÞ
η
2νðÞ;z
η
r ðÞ;θ
ηðÞ;z
0
ηðÞ;r ηr ðÞ
0
2 η ðÞ þ 2rηðÞ r 2 ;θ 6 2ην 2η Bnθ ¼ 6 4 r ðÞ;r þ r2 ðÞ 0 2
0
η
2η r ðÞ;r r 2 ðÞ
2η ð1 þ r2
η
0
νÞðÞ;θ
2 r ðÞ;z
η
r ðÞ;z
ðÞ;z
r2
6 Bnz ¼ 6 4 0
0
ðÞ;r 1r ðÞ
ðÞ;z
1 r ðÞ;θ
0
0
ðÞ
ðÞ;θ
3 7 7; 5
(23)
3 7 7; 5
(24)
3 7 7; 5
(25)
where Ln1;1 ðÞ ¼ 2ηð1 þ νÞðÞ;rr Ln1;2 ðÞ ¼
2ηð1 þ νÞ η ðÞ;r þ 2 ðÞ;θθ þ ðÞ;zz ; r r
ηð1 þ 2νÞ r
ðÞ;rθ
Ln1;3 ðÞ ¼ ð1 þ 2νÞðÞ;rz Ln2;1 ðÞ ¼
ηð1 þ2νÞ r
η
4ηð1 þ νÞ ðÞ;θ ; r2
(26b)
ð1 þ 2νÞ ðÞ;z ; r
(26c)
ðÞ;rθ þ
Ln2;2 ðÞ ¼ ηðÞ;rr ðÞ;r þ r
(26a)
2η ðÞ ; r 2 ;θ
2ηð1 þ νÞ ðÞ;θ ; r2
(26d)
(26e)
ð1 þ2νÞ ðÞ;θz ; r
(26f)
Ln3;1 ðÞ ¼ ηð1 þ 2νÞðÞ;rz ;
(26g)
Ln2;3 ðÞ ¼
Ln3;2 ðÞ ¼ Ln3;3 ðÞ ¼ ηðÞ;rr þ
ηð1 þ2νÞ r
ðÞ;θz ;
η η 1 ðÞ þ 2ð1 þ νÞðÞ;zz ðÞ;r þ 2 ðÞ: r r r 2 ;θθ
(26h)
(26i)
In order to simplify Eq. (18), we define a new form of the inner product operator hh;ii for the extended system shown in Eq. (18) that can be obtained from the inner product operator after applying the divergence theorem, as follows:
R R A;B ¼ Ω AD U BD dΩ Γ 1 ðAB1 UBB1 AB2 UBB2 ÞdΓ 1 Z Z ðAB3 U BB3 AB4 U BB4 ÞdΓ 2 ðAB5 UBB5 AB6 UBB6 ÞdΓ 3 ; (27) Γ2
Γ3
where h iT A ¼ ATD ; ATB1 ; ATB2 ; ATB3 ; ATB4 ; ATB5 ; ATB6 ;
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
291
h iT B ¼ BTD ; BTB1 ; BTB2 ; BTB3 ; BTB4 ; BTB5 ; BTB6 and AD ; AB1 ; AB2 ; AB3 ; AB4 ; AB5 ; AB6 , BD ; BB1 ; BB2 ; BB3 ; BB4 ; BB5 ; BB6 are 3 1 vectors. Note that the operator hh;ii basically produces the same results as the original inner product operator h;i, but it has a remarkable merit that enables us to explicitly treat the boundary/continuity conditions. Using Eq. (27), Eqs. (18) and (19) can be reformulated as follows:
Ψm ; TΦn ¼ Tn Ψm ; Φn ; (28)
λ2n Ψm ; SΦn
n2
n
¼ λm
S Ψm ; Φn
(29)
where h
iT
Φn ¼ ΦnT ΦnT ΦnT ΦnT ΦnT ΦnT ΦnT ; D B1 B2 B3 B4 B5 B6 h
iT
Ψm ¼ ΨmT ΨmT ΨmT ΨmT ΨmT ΨmT ΨmT ; D B1 B2 B3 B4 B5 B6 2
0
⋯
⋯
⋯
⋯
Br jr ¼ Ro
0
⋯
⋯
⋯
L
60 6 6 60 6 6 T¼60 6 60 6 60 4 0
0
Br jr ¼ Ri
0
⋯
⋯
⋯
0
Bθ
0
⋯
⋯
⋯
0
Bθ
0
⋯ ⋯
⋯ ⋯
⋯ ⋯
0 ⋯
Bz 0
" S¼ 2 6 6 6 6 6 6 n T ¼6 6 6 6 6 6 4
Ln 0
M 0183
0
0
⋯
⋯
0 0
⋯ ⋯
0 ⋯
Bnθ 0
⋯ 0
0
⋯
⋯
⋯
0 Bnθ
0
⋯
⋯
⋯
Br
r ¼ Ro
r r ¼ Ri
" n
S ¼
0 7 7 7 0 7 7 0 7 7; 7 0 7 7 0 7 5 Bz
⋯ ⋯
⋯ ⋯
⋯ ⋯
Mn
0318
0183
01818
(32)
(33)
0
0
(31)
3
# 0318 ; 01818
⋯ 0 n B
n
0
(30)
0
Bnz
⋯
0
3 0 0 7 7 7 0 7 7 7 0 7 7; 7 0 7 7 0 7 5 Bnz
(34)
# :
(35)
Using Eqs. (32)–(35) the equations for the eigenvalue problems of the adjoint system and original system can be obtained, respectively, as
Ψm ; TΦn ¼ λ2n Ψm ; SΦn ; (36)
n n2 T Ψm ; Φn ¼ λm Sn Ψm ; Φn : Using Eqs. (28)–(29) and (36)–(37), the following biorthogonality conditions can be derived:
λ2n λnm2 Ψm ; SΦn ¼ Ψm ; TΦn Tn Ψm ; Φn ¼ 0:
(37)
(38)
For distinct eigenvalues of the original system (and also its adjoint system) two sets of biorthogonality conditions of the eigenvector functions can be finally obtained, after a proper normalization process, in the forms
Ψm ; TΦn ¼ Tn Ψm ; Φn ¼ λ2n δmn ; (39a)
Ψm ; SΦn
¼
n S Ψm ; Φn ¼ δmn :
Note that both the original and its adjoint systems satisfy the positive definiteness. Using Eqs. (26a)–(26i), the field equations of the adjoint system can be readily obtained:
(39b)
292
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
Radial direction: 2ηð1 þ νÞunr;rr þ unr;zz
ηu ηð1 þ 2νÞuθ;rθ 4ð1 þ νÞηuθ;θ 2ηð1 þ νÞunr;r þ rr;2θθ þ þ ð1 þ 2 r r r2 n
n
n
νÞunz;rz
ð1 þ 2νÞunz;z n ¼ ρu€ r : r
(40a)
Circumferential direction:
ηð1 þ 2νÞunr;rθ 2ηunr;θ 2ηð1 þ νÞunθ;θθ þ
r
r2
þ
r2
þ ηunθ;rr þ unθ;zz
ηunθ;r ð1 þ2νÞunz;θz þ
r
r
¼ ρu€ θ : n
(40b)
Length direction:
ηð1 þ 2νÞunr;rz þ
ð1 þ 2νÞηunθ;θz r
þ2ð1 þ νÞunz;zz þ ηunz;rr þ
ηunz;θθ ηunz;r ηunz
r2
r
þ
r2
¼ ρu€ z : n
(40c)
Considering the boundary/continuity conditions given in Eqs. (8), the displacement functions for the original system (refer to Eq. (9)) can be assumed as ϕðrÞ ¼ vðrÞ ðrÞ cos ξz eikθ ;
ϕðθÞ ¼ vðθÞ ðrÞ cos ξz eikθ ;
ϕðzÞ ¼ vðzÞ ðr Þ sin ξz eikθ ;
(41) pffiffiffiffiffiffiffiffi ðjÞ where i is 1, v ðr Þ are functions of r to be determined, k and ξ are the wavenumbers in θ and z directions, respectively. The substitution of Eqs. (9) and (17) into Eq. (2) yields the field equations only functions of r as Radial direction: ðθÞ η k2 2ð1 þ νÞ 2ηð1 þ νÞvð;rrÞ kηð1 þ 2νÞv;r 2 2 rÞ 2ηð1 þ νÞvð;rr þ þ ρλ ξ vðrÞ þ i r r r2 kηð3 þ 2νÞvðθÞ i þ ξηð1 þ 2νÞvð;rzÞ ¼ 0: r2 Circumferential direction:
ηvð;rrÞ þ ηr vð;r Þ þ ρλ2 θ
θ
η 2k2 ð1 þ νÞ þ 1 r2
ξ
2
(42a)
kηð1 þ 2νÞvð;rrÞ kηð3 þ 2νÞvðr Þ þi vðθÞ þ i 2 r
r
kηξð1 þ 2νÞvðzÞ ¼ 0: þi r
(42b)
Length direction: ξð1 þ 2νÞvð;rrÞ ( þ
ξð1 þ 2νÞvðrÞ r
i
kξð1 þ 2νÞvðθÞ þ r
ρλ 2ξ ð1 þ νÞ 2
2
ηk2 r2
)
ηvð;rrzÞ þ ηr vð;rzÞ
vðzÞ ¼ 0:
(42c)
Eqs. (42a)–(42c) are coupled linear ordinary differential equation with respect to r. Since the existence of the singular point at r ¼ 0 in Eqs. (42) causes numerous convergence problems in the series solutions of Eqs. (42), here, the series solutions are obtained about a point other than r ¼ 0, for which expansion about a regular point of Eqs. (42) with variable coefficients can yield six independent exact power series solutions about that point, which are well behaved in the region of interest [41]. In particular, the point r~ located on the middle line of the hollow cylinder is particularly convenient for use in this work. r ¼ r~ þ r 0 ;
(43)
r n0 Ro þ Ri ¼ : R 2ðRo Ri Þ
(44)
where r0 ¼
Introducing this translated coordinate, the differential Eqs. (42a)–(42c) can be transformed into the following forms: Radial direction:
h n oi 2 2 2ηð1 þ νÞðr~ þr 0 Þ2 vð;rr~ Þr~ þ ðr~ þr 0 Þ2 ρω2 ξ η k 2ð1 þ νÞ vðrÞ þ 2ηð1 þ νÞðr~ þr 0 Þvð;rr~ Þ ðθÞ þ ikηðr~ þ r 0 Þð1 þ 2νÞv;r~ ikηð3 þ2νÞvðθÞ þ ξηðr~ þ r 0 Þ2 ð1 þ 2νÞvð;rz~ Þ ¼ 0:
(45a)
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
293
Circumferential direction:
h n oi ðθ Þ 2 2 ikηðr~ þ r 0 Þð1 þ 2νÞvð;rr~ Þ þ ðr~ þr 0 Þ2 ρω2 ξ η 2k ð1 þ νÞ þ 1 vðθÞ þ ηðr~ þ r 0 Þ2 v;r~ r~ ðθ Þ þ ηðr~ þ r 0 Þv;r~ þ ikηξðr~ þ r 0 Þð1 þ 2νÞvðzÞ þ ikηð3 þ 2νÞvðrÞ ¼ 0:
(45b)
Length direction: ξðr~ þ r 0 Þ2 ð1 þ2νÞvð;rr~ Þ ξðr~ þ r 0 Þð1 þ 2νÞvðrÞ ikξðr~ þ r 0 Þð1 þ 2νÞvðθÞ þ ηðr~ þ r 0 Þ2 vð;rz~ rÞ~ hn i o 2 2 þ ρω2 2ξ ð1 þ νÞ ðr~ þ r 0 Þ ηk vðzÞ þ ηðr~ þ r 0 Þvð;rz~ Þ ¼ 0;
(45c)
where ω ¼ λi, which means the frequency of the cylinder. Note that Eqs. (45a)–(45c) have been transformed into the functions of r~ . It is well known that the displacements in the differential equations of the original system (Eq. (5)) can be easily decomposed into irrotational and solenoidal parts by introducing two types of potential functions. The resulting differential equations can then be solved in terms of Bessel functions [15]. However, the decomposition of the adjoint system (Eq. (40)) cannot be performed easily because of the additional terms originating from the boundary conditions. On the other hand, once the variables are separated, the resulting set of ordinary differential equations in the radial direction (Eq. (42)) can be readily solved using classical methods. Because the present problem requires the simultaneous solution of the original and adjoint problem and given that the use of the Fröbenius method, expanded at a regular point, guarantees a yield of six independent solutions for each system with fast convergence, we opted to derive and use the Fröbenius-type power series solutions in this study. Note that this method does not require decomposition and can be used to directly solve the displacement equations of motion, which is obviously very convenient and efficient for the present problem. Furthermore, as can be seen in Table A1 in Appendix A, while the Bessel-type solutions are dependent on the argument interval, the series solutions can take their own forms regardless of their argument. In order to solve vðjÞ in coupled Eqs. (45a)–(45c), the method of Fröbenius [40,41] are applied by assuming the solution functions to be M
vðrÞ ðr~ Þ ¼ ∑ αm r~ χ þ m ; m¼0
M
vðθÞ ðr~ Þ ¼ ∑ β m r~ χ þ m ; m¼0 M
vðzÞ ðr~ Þ ¼ ∑ γ m r~ χ þ m ;
(46)
m¼0
where αm , βm and γ m are coefficients of r~ χ þ m for the function vðjÞ , respectively, and M is the number of terms sufficiently large to assure the satisfactory convergence of vðjÞ . Substituting Eqs. (46) into Eqs. (45), after a series of calculations and collecting of the coefficients of the like powers of r~ in each equation, we can obtain three indicial equations using the coefficients of the lowest power terms, r~ χ 2 , as Radial direction: χ r20 ð1 þ νÞ χ 1 α0 ¼ 0: (47a) Circumferential direction:
ηχ χ 1 β0 ¼ 0:
(47b)
Length direction:
ηχ χ 1 r20 γ 0 ¼ 0:
(47c)
Eqs. (47a)–(47c) are the indicial equations of the directions indicated. Equations (47) clearly show that with χ ¼ 0 and χ ¼ 1, there are six independent solutions in total. Note that Eqs. (47) are satisfied regardless of α0 , β0 and γ 0 with either χ ¼ 0 or χ ¼ 1. To the best convenience of calculations, α0 , β0 and γ 0 may be selected in the particularly convenient form, in which the first terms are uncoupled, i.e., one with α0 ¼ 1, β0 ¼ 0, γ 0 ¼ 0, another with α0 ¼ 0, β0 ¼ 1, γ 0 ¼ 0 and the other with α0 ¼ 0, β0 ¼ 0, γ 0 ¼ 1, so that we can obtain six independent series solutions; with the following six cases χ ; α0 ; β0 ; γ 0 ¼ ð0; 1; 0; 0Þ, ð0; 0; 1; 0Þ, ð0; 0; 0; 1Þ, ð1; 1; 0; 0Þ, ð1; 0; 1; 0Þ, ð1; 0; 0; 1Þ. Note that these six cases are confirmed to be linearly independent by investigating the Wronskian determinant. ðpÞ Then, we can calculate αðmpÞ , βm and γ ðmpÞ (p ¼ 1; :::; 6) for the six solution functions, where the superscript p means the case number. With the inclusion of arbitrary constants Ap that are to be determined later, vðjÞ can be expressed as 6
vðrÞ ¼ ∑ Ap vðprÞ ; p¼1
6 ðθ Þ vðθÞ ¼ ∑ Ap vp ; p¼1
6
vðzÞ ¼ ∑ Ap vðpzÞ ; p¼1
(48)
294
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
where M
M ðθ Þ ðpÞ vp ¼ ∑ βm r~ χ þ m ;
vðprÞ ¼ ∑ αðmpÞ r~ χ þ m ; m¼0
m¼0
M
vðpzÞ ¼ ∑ γ ðmpÞ r~ χ þ m :
(49)
m¼0
The length directional boundary conditions and the circumferential directional continuity conditions are automatically satisfied with the displacement functions given in Eqs. (41), so all the actual boundary conditions needed in the eigenanalysis are the six radial directional boundary conditions. The boundary conditions expressed in functions of r~ can be obtained through the use of Eqs. (9), (41) and (48) as 6 1
1 ðθ Þ (50a, b) Ap ¼ 0; at r~ ¼ 7 ; ik vp þ vðprÞ ∑ ð1 þ νÞvðp;rÞr~ þ ν ξ vðpzÞ þ r~ þr 0 2 p¼1
1 ðθ Þ ðθ Þ ik vðprÞ vp þ vp;r~ Ap ¼ 0; p¼1 r 6
∑
6
∑
p¼1
h
i
1 at r~ ¼ 7 ; 2
1 at r~ ¼ 7 : 2
ηvðp;zÞr~ ξvðprÞ Ap ¼ 0;
(50c, d)
(50e, f)
Eqs. (50a)–(50f) can be expressed in terms of Ap multiplied by their coefficients, i.e., as the following six homogeneous algebraic equations: 6
∑ Bpq Ap ¼ 0
p¼1
ðq ¼ 1; …; 6Þ;
(51)
where index p corresponds to the boundary conditions given in Eqs. (50a)–(50f), respectively. Expressing Eq. (51) in matrix form, we have BA ¼ 0:
(52)
In order to have nontrivial solutions for Ap , the determinant of matrix B must be vanished, which makes up of dispersion relations, the relations between the frequency ω and z-directional wavenumber ξ, and yields the amplitude ratios. The natural frequencies (the frequencies that correspond to ξ ¼0; π ; 2π ; …) and mode shapes can be calculated using the dispersion curves, which is the graphical representation of the dispersion relations. Then, the nth mode displacement functions can be represented by using the natural frequency and mode shapes as ðrÞ
ðrÞ
uðrnÞ ¼ ϕn eiωn t þ ϕn e iωn t ; ðθ Þ ðθ Þ uθðnÞ ¼ ϕn eiωn t þ ϕn e iωn t ; ðzÞ
uðznÞ ¼ ϕn eiωn t þ ϕn e iωn t ; ðzÞ
(53)
where ωn ¼ λn i, which denotes the nth natural frequency of the cylinder. In a similar manner, the displacement functions of the adjoint system can be expressed as unr ðnÞ ¼ ψ ðnrÞ eiωn t þ ψ ðnrÞ e iωn t ; ðθ Þ ðθÞ unθðnÞ ¼ ψ n eiωn t þ ψ n e iωn t ; unz ðnÞ ¼ ψ ðnzÞ eiωn t þ ψ ðnzÞ e iωn t ;
(54)
where
ψ ðnrÞ ¼ vnnðrÞ ðr Þ cos ξn z eikn θ ;
(55a)
θ n θ ψ ðn Þ ¼ vnð Þ ðrÞ cos ξn z eikn θ ;
(55b)
ψ ðnzÞ ¼ vnnðzÞ ðr Þ sin ξn z eikn θ ; nðjÞ
and vn
(55c)
can be obtained using Eqs. (40) following the same steps as for the original system.
2.3. Discretization of the governing field equations Using the separation of variables method and the modal expansion theorem, the appropriate displacement functions can be written in the forms 1 ðrÞ ur ¼ ∑ ϕn r; θ; z qn ðt Þ; n¼1
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
295
1 ðθ Þ uθ ¼ ∑ ϕn r; θ; z qn ðt Þ; n¼1 1
ðzÞ uz ¼ ∑ ϕn r; θ; z qn ðt Þ;
(56)
n¼1
where n denotes the mode number of the original linear system and qn is the nth modal coordinate of the original linear system. Using the linear operators defined in Eqs. (32) and (33), the original linear field equations and boundary/continuity conditions can be represented by € þ FðlÞ ; Tu ¼ Su
(57)
where 1
u ¼ ∑ Φn qn ;
(58)
FðlÞ ¼ ½016 ; σ~ rr ; 0114 ;
(59)
n¼1
andσ~ rr is expressed in Eq. (7). Taking the inner product hh;ii with Ψm , the mth mode vector eigenfunction for the extended adjoint system defined in Eq. (31), we obtain the linear discretized differential equation as q€ a þ ω2a qa ¼ F Ri ; where
Z Z F Ri ¼
θ z
ψ ðarÞ Ri ; θ; z σ~ rr dz dθ:
(60)
(61)
Note that the inner product hh;ii is defined in Eq. (27) for the extended system that includes both the volumetric and boundary parts separately in Eq. (57). Substituting Eqs. (55) and (7) into Eq. (61), we can calculate F Ri as F Ri ¼ fR1i vnaðrÞ ðRi Þ cos ξa Vt eikθ1 þ fR1i vnaðrÞ ðRi Þ cos ξa Vt e ikθ1 þ
f 2 nðrÞ f v ðRi Þ cos ξa Vt eikθ2 þ 2 vnaðrÞ ðRi Þ cos ξa Vt e ikθ2 ; Ri a Ri 1 0 r θ1 ; θ2 r 2π and 0 r t r ; V
where θ1 and θ2 represent the circumferential locations of the dual traveling force components. The analytic solutions for Eq. (62) combined with Eq. (60) can be readily obtained as follows: cos ξa Vt f 1 vna ðRi Þeika θ1 þvna e ika θ1 þ f 2 vna ðRi Þeika θ2 þ vna e ika θ2
; qðt Þ ¼ 2 Ri ξa V 2 ω2a
(62)
(63)
where the initial conditions qð0Þ and q_ ð0Þ have been set to be zero. The vanishing denominator of Eq. (63) yields the critical speed as Vc ¼ 7
ωa ; ξa
(64)
at which the response increases indefinitely and the situation resembles the occurrence of a resonance. Note that in Eq. (63), when the value of ξa is zero, then the solution becomes independent of the traveling velocity, so the associated critical speed does not exist. 3. Illustrative examples – Eigenanalysis on the hollow cylinder and its critical speeds The dispersion curves and mode shapes are obtained for several illustrative examples in this section. The structural steel was selected as the material of the hollow cylinder, and the dimensionless material properties and geometric parameters selected for these illustrations are given in Table 1. In this study, we investigated the effects of the cylinder geometry by considering three different r 0 values (refer to Eq. (44)): (a) r 0 ¼ 1, (b) r 0 ¼ 2, (c) r 0 ¼ 3. When the thickness of hollow cylinder is fixed, the inner and outer radii tend to increase with r 0 . In this section, an eigenanalysis is performed on the hollow cylinder and the dispersion curves and several major mode shapes of concern are obtained with the three r 0 values. The eigenanalysis results for the thickness, flexural and torsional modes are discussed in this section, and the natural frequencies obtained from this eigenanalysis are compared with the results obtained using an FEM package, ANSYS [42] and the Bessel-type solutions [15]. Using the dispersion relations given in Eq. (52), we can obtain the dispersion curves as depicted in Fig. 2. Fig. 2(a), (b) and (c) represent the dispersion curves for k ¼ 0, k ¼ 1 and k ¼ 2, respectively, and the two vertical dashed lines represent ξ ¼ π and ξ ¼ 2π . Note that the frequency
296
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
Table 1 Geometrical and material properties of the hollow cylinder. Nondimensional parameter
Value
v ρ
0.75
r0 η
3:6738 10 2 (a) 1, (b) 2, (c) 3 9
Fig. 2. Dispersion curves for the circumferential wavenumber of (a) k ¼ 0, (b) k ¼ 1, and (c) k ¼ 2.
values associated with the crossed points of ξ ¼ nπ ðn ¼ 0; 1; 2; …Þ line with the dispersion curves represent the natural frequencies for the modes with wavelength H=n in the z-direction. To obtain the dispersion curves, we first attempted to find frequencies for the thickness modes. The thickness modes can be found by Eqs. (B.6) in Appendix B. Note that in Figs. 2(a), and (b) for k ¼ 0 and k ¼ 1, ξ ¼ 0 corresponds to the rigid body motions which is not interested in this study. The nondimensional natural frequencies of the thickness modes are marked with the symbol x in Fig. 2. As can be expected, these frequencies tend to decrease as r 0 increases. The same trend can also be found for k ¼ 1 and k ¼ 2, as shown in Table 2. In Fig. 2(b) and (c), the start points of the real branches connected with the associated complex branches (not shown in this figure) are marked with closed circles. Note that since the boundary conditions treated in this study is a Lévy type, only real branches are of concern and other complex (including imaginary) branches are not necessary to solve the present eigenproblem. In Fig. 2(a), the dotted curves are obtained from the dispersion relations described in Eqs. (B.2)–(B.3), which are for the torsional modes. The natural frequencies obtained from the present eigenanalysis and those obtained from an FEM package, ANSYS [42], and the Bessel-type solutions [15] are shown and compared in Table 2. As is manifested in this table, these three results are almost identical, which strongly supports the validity of the present eigenanalysis. Note that the natural frequencies of the torsional modes are identical regardless of the value of r 0 . Figs. 3–5 show three representative mode shapes of three different r 0 values corresponding to Table 2. The subscripts of ω in parenthesis, (nr ; nθ ; nz ) denote the number of nodal lines,
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
297
Table 2 Dimensionless natural frequencies computed with the present model, ANSYS and Ref. [15]. Branch #
(a) r 0 ¼ 1 1
2
3
(b) r 0 ¼ 2, 1
2
3
(c) r 0 ¼ 3. 1
2
3
ξ (nπ)
k¼0
k¼1
k ¼2
Present study
ANSYS
Ref. [15]
Present study
ANSYS
Ref. [15]
Present study
ANSYS
Ref. [15]
0 1 2 0 1 2 0 1 2
0 16.3912 32.7810 0 22.9852 31.5848 29.5315 32.4303 55.0199
0 16.3905 32.7812 0 22.9857 31.5862 29.5321 32.4305 55.0201
0 16.3905 32.7810 0 22.9857 31.5861 29.5321 32.4305 55.0199
0 11.5829 27.0283 – 27.2190 39.1052 30.8459 37.7313 56.1715
0 11.5877 27.0351 – 27.2208 39.1084 30.8462 37.7316 56.1723
0 11.5877 27.0347 – 27.2207 39.1083 30.8458 37.7314 56.1719
15.2723 18.2222 30.0045 – 36.8365 47.3622 43.8694 49.7432 62.1242
15.2756 18.3307 30.0009 – 36.8498 47.3865 43.8695 49.7451 62.1309
15.2749 18.3303 30.0003 – 36.8488 47.3856 43.8678 49.7443 62.1309
0 1 2 0 1 2 0 1 2
0 16.3907 32.7811 0 13.9063 24.4334 13.6072 27.9326 53.0862
0 16.3905 32.7827 0 13.9064 24.4364 13.6072 27.9328 53.0889
0 16.3905 32.7810 0 13.9063 24.4334 13.6072 27.9326 53.0862
0 11.4750 24.5447 – 19.5032 33.9863 17.9541 30.7556 53.9753
0 11.4753 24.5480 – 19.5032 33.9878 17.9542 30.7557 53.9780
0 11.4750 24.5447 – 19.5032 33.9863 17.9541 30.7556 53.9753
4.6139 11.6101 25.6188 – 24.0774 37.0219 27.2722 37.3052 56.3584
4.6141 11.6107 25.6224 – 24.0776 37.0229 27.2727 37.3057 56.3612
4.6139 11.6101 25.6188 – 24.0774 37.0219 27.2722 37.3052 56.3584
0 1 2 0 1 2 0 1 2
0 16.3912 32.7811 0 10.6805 22.9787 8.9303 27.6339 52.8154
0 16.3905 32.7828 0 10.6807 22.9821 8.9303 27.6340 52.8181
0 16.3905 32.7810 0 10.6805 22.9787 8.9303 27.6339 52.8154
0 10.2225 23.2134 – 17.5483 33.2509 12.2642 28.9569 53.2993
0 10.2228 23.2168 – 17.5484 33.2525 12.2642 28.9570 53.3020
0 10.2225 23.2134 – 17.5483 33.2509 12.2642 28.9569 53.2993
2.1636 10.2570 23.9837 – 20.0891 34.5887 19.0737 32.4999 54.6665
2.1637 10.2575 23.9872 – 20.0892 34.5901 19.0738 32.5001 54.6692
2.1636 10.2570 23.9837 – 20.0891 34.5887 19.0737 32.4999 54.6665
Fig. 3. Three representative mode shapes and their corresponding natural frequencies of the hollow cylinder with r0 ¼ 1.
i.e., nr : in radial direction, nθ : in circumferential direction and nz : in length direction. For the modes coupled with torsional modes, the subscript t was used as in Fig. 3(c). Table 3 shows the critical speeds (refer to Eq. (64)) for all the cases treated preciously. As shown in Fig. 2(a), the branch for torsional modes is a straight line in the wavenumber (ξ) range of concern, so the torsional waves are nondispersive and the critical speeds of all the torsional mode are identical. Note that the critical speeds are the highest for the modes with ξ ¼ π on the first branch because the critical speed tends to be proportional to the natural frequency, as can be seen in Eq. (64) and the natural frequencies for these modes are the highest.
298
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
Fig. 4. Three representative mode shapes and their corresponding natural frequencies of the hollow cylinder with r 0 ¼2.
Fig. 5. Three representative mode shapes and their corresponding natural frequencies of the hollow cylinder with r 0 ¼3.
Table 3 Critical speeds of each mode. Branch #
1 2 3
ξ ðnπ Þ
1 2 1 2 1 2
r0 ¼ 1
r0 ¼ 2
r0 ¼ 3
k¼0
k ¼1
k¼2
k ¼0
k¼1
k ¼2
k ¼0
k ¼1
k ¼2
5.217 5.217 7.317 5.027 10.323 8.757
3.688 4.303 8.665 6.210 12.010 8.940
5.835 4.775 11.729 7.542 15.834 9.888
5.217 5.217 4.426 3.889 8.891 8.449
3.653 3.906 6.208 5.409 9.790 9.068
3.696 4.077 7.664 5.892 11.875 8.970
5.217 5.217 3.400 3.657 8.796 8.406
3.254 3.694 5.586 5.292 9.217 8.483
3.265 3.817 6.395 5.505 10.345 8.701
4. Conclusions In this study, we first mathematically derived governing field equations of a hollow cylinder subjected to dual traveling forces imposed on its inner surface in the cylindrical coordinates. A structural steel was used as the material of the hollow cylinder. In order to investigate the effect of the cylinder geometry, three different r 0 values were used: r 0 ¼1, 2, and 3. In order to solve these governing field differential equations, the biorthogonality conditions, along with the adjoint system of equations, were derived and then applied to obtain the discretized equations. Using the field equations and homogeneous boundary conditions, we could find dispersion curves and eigensolutions. Through the application of these eigenfunctions and biorthogonality conditions to the inhomogeneous boundary conditions that include the boundary forcing term, the discretized differential equations could be obtained. Using the solutions of the discretized equations, the critical speed for each mode could be found. By inspecting the results of this analysis, the following conclusions could be made: 1. The natural frequencies of the first thickness modes increase as r 0 decreases. In the dispersion curves, comparing with the nth branch of each r 0 case, the natural frequencies increase as r 0 decreases in the range of concern of this study except for the branch associated with the torsional modes.
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
299
2. The critical speed is proportional to the natural frequency and inversely proportional to the length directional wavenumber of each mode. When the length directional wavenumber is zero (thickness mode), the associated critical speed does not exist. 3. In the dispersion curves, each branch for torsional modes appear as a straight line and has the same values as r 0 , which indicates that the torsional waves are nondispersive. Due to the nondispersive nature of the torsional modes, the critical speeds of all the torsional modes become identical.
Acknowledgments This work was supported by the Industrial Strategic Technology Development Program (10039982, Development of next generation multi-functional machining systems for eco/bio components) funded by the Ministry of Trade, Industry and Energy (MI, Korea). This research was also supported by the Chung-Ang University Excellent Student Scholarship in 2014. Appendix A. Bessel-type solutions for original system (Ref. [15]) Gazis [15] solved the governing field equations of linear elasticity in invariant form using the Helmholtz displacement potentials. The three-dimensional differential equation in invariant form is
μ∇2 u þ λ þ μ ∇ ∇ U μ ¼ ρ
∂2 u ; ∂t 2
(A.1)
where u is the displacement vector, ρ is the density, and λ and μ are Lamé constants. Because u is the Helmholtz displacement vector, it can be decomposed into a dilatational scalar potential ϕ and an equivoluminal vector potential H as
with the condition of
u ¼ ∇ϕ þ ∇ H
(A.2)
∇ U H ¼ F r; θ; z; t
(A.3)
as the gauge invariance. In Eq. (A.3), F r; θ; z; t is an arbitrary function that may be assumed to be zero. By substituting Eq. (A.2) into Eq. (A.1), we can obtain two wave equations for potentials ϕ and H:
λ þ 2μ 2 ∂2 ∇ ϕ ¼ 2 ϕ; ρ ∂t
(A.4a)
μ 2 ∂2 ∇ H ¼ 2 H: ρ ∂t
(A.4b)
For the present problem, we can define these potentials by using the method of separation of variables as ϕ ¼ f ðrÞ cos nθ cos ωt þ ξz ; H r ¼ g r ðr Þ sin nθ sin ωt þ ξz ; H θ ¼ g θ ðr Þ cos nθ sin ωt þ ξz ; H z ¼ g 3 ðr Þ sin nθ cos ωt þ ξz :
(A.5)
The substitution of the solution functions given in Eq. (A.5) into (A.4) gives
2 ∇2 þ ωv2 ϕ ¼ 0; 1 ! 1 ω2 2 ∂ ∇2 2 þ 2 H r 2 H θ ¼ 0; r r ∂θ v2 ! 1 ω2 2 ∂ 2 ∇ 2 þ 2 H θ þ 2 H r ¼ 0; r r ∂θ v2 ! ∇2 þ
ω2 v22
H z ¼ 0;
(A.6)
μ ρ
(A.7)
where, v21 ¼
λ þ 2μ ; ρ
v22 ¼ :
300
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
For the convenience of the subsequent analyses, the following differential operator is introduced: 2 2 ∂ 1 ∂ n 2 1 Λn;x ¼ 2 þ x ∂x ∂x x Using Eq. (A.8), we can express Eq. (A.6) as
(A.8)
Λn;αr f ¼ 0;
(A.9a)
Λn þ 1;βr gr gθ ¼ 0;
(A.9b)
Λn 1;βr gr þ gθ ¼ 0;
(A.9c)
Λn;βr g3 ¼ 0;
(A.9d)
where
α2 ¼
ω2 v21
ξ ; 2
β2 ¼
ω2 v22
ξ : 2
(A.10)
Eq. (A.9) can be solved in terms of Bessel functions J and Y, or the modified Bessel functions I and K, which are determined according to whether α and β in Eq. (A.10) are imaginary or real. Note that, when the value of both α and β are real, the arguments of the Bessel functions and the modified Bessel functions become αr and β r. The detailed Bessel functions used at different intervals are expressed in Table A1, where α1 r and β 1 r denote jαr j and β r , respectively. Using the solution functions given in Table A1, the general solutions given in Eq. (A.9) can be expressed as f ¼ AZ n ðα1 r Þ þBW n ðα1 r Þ;
(A.11a)
2g 1 ¼ g r g θ ¼ 2A1 Z n þ 1 β1 r þ 2B1 W n þ 1 β1 r ;
(A.11b)
2g 2 ¼ g r þg θ ¼ 2A2 Z n 1 β1 r þ 2B2 W n 1 β1 r ;
(A.11c)
g 3 ¼ A3 Z n β1 r þ B3 W n β1 r ;
(A.11d)
where Z denotes the J or I function and W denotes the Y or K function. The property of the gauge invariance (Eq. (A.3)) can now be utilized to eliminate two of the integration constants in Eq. (A.11) [43]. For instance, by setting g 2 ¼ 0, we obtain g1 ¼ gr ¼ gθ ; and the displacement fields shown in Eqs. (A.2) as ur ¼ f ;r þ nrg 3 þ ξg 1 cos nθ cos ωt þ ξz ; h n i uθ ¼ f þ ξg 1 g 3;r sin nθ cos ωt þ ξz ; r h g i uz ¼ ξf g 1;r ðn þ 1Þ 1 cos nθ sin ωt þ ξz : r
(A.12)
(A.13)
By using the stress–strain and strain–displacement relationships given in Eq. (1) and (3), respectively, and applying the boundary conditions in the radial direction given in Eq. (6a), we can solve the eigenproblem consisting of the coefficients of the Bessel functions in Eq. (A.11),A; B; A1 ; B1 ; A3 ; B3 . Appendix B. Special cases Using Eq. (45) with k ¼ 0 or ξ ¼ 0, we can find the equations for the associated special cases. B.1 For k ¼ 0 First, we derive the equation for the case of k ¼ 0. Substituting k ¼ 0 into Eq. (45), the resulting equations can be expressed as Table A1 Bessel functions used at different intervals. Case α2 40, β2 40 α2 o 0, β2 4 0 α2 o 0, β2 o 0
Functions used J ðα1 rÞ; Y ðα1 r Þ; J β1 r ; Y β1 r Iðα1 r Þ; K ðα1 r Þ; J β1 r ; Y β1 r Iðα1 r Þ; K ðα1 r Þ; I β1 r ; K β1 r
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
Radial direction
301
h i 2 2ηð1 þ νÞðr~ þr 0 Þ2 vð;rr~ Þr~ þ 2ηð1 þ νÞðr~ þr 0 Þvð;rr~ Þ þ ðr~ þr 0 Þ2 ρω2 ξ þ2ηð1 þ νÞ vðrÞ þ ξηðr~ þ r 0 Þ2 ð1 þ2νÞvð;rz~ Þ ¼ 0:
Circumferential direction:
n
(B.1a)
o
ηðr~ þ r0 Þ2 vð;r~ r~Þ þ ηðr~ þ r0 Þvð;r~ Þ þ ðr~ þ r 0 Þ2 ρω2 ξ2 η vðθÞ ¼ 0:
(B.1b)
ξðr~ þ r 0 Þ2 ð1 þ2νÞvð;rr~ Þ ξðr~ þ r 0 Þð1 þ 2νÞvðrÞ þ ηðr~ þ r 0 Þ2 vð;rz~ rÞ~ þ ηðr~ þr 0 Þvð;rz~ Þ n o 2 þ ρω2 2ξ ð1 þ νÞ ðr~ þ r 0 ÞvðzÞ ¼ 0:
(B.1c)
θ
θ
Length direction:
Note that in Eq. (B.1), vðθÞ is decoupled out of the other two equations and thus, Eq. (B.1b) happens to only include vðθÞ . Using Eq. (B.1b), we can obtain the vðθÞ as the Bessel functions: 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 ξ2 2 ρω k0 k0 θ ð Þ A þ A BesselY @1; ðr 0 þ r~ Þ ρω ξ A: (B.2) v ¼ A1 BesselJ @1; ðr 0 þ r~ Þ 2
η
η
The decoupled boundary conditions for vðθÞ are reduced to vðθÞ ðθÞ þ v;r~ ¼ 0; r~ þ r 0
1 (B.3) atr~ ¼ 7 : 2 Using Eqs. (B.2) and (B.3) we can obtain the displacement function, vðθÞ for k ¼ 0. The displacements vðrÞ and vðzÞ can be obtained using Eqs. (B.1a) and (B.1c) combined with their reduced boundary conditions, Eqs. (B.4) and (B.5), with the € application of Frobenius method explained in Section 2.2. 1 ðrÞ 1 (B.4) ð1 þ νÞvð;rr~ Þ þ ν ξvðzÞ þ v ¼ 0; at r~ ¼ 7 ; r~ þ r 0 2
ηvð;rz~ Þ ξvðrÞ ¼ 0;
1 at r~ ¼ 7 : 2
(B.5)
B.2 For ξ ¼ 0 For the case of ξ ¼ 0, Eqs. (45) can be expressed as the following two equations: Radial direction: h n oi 2 2ηð1 þ νÞðr~ þr 0 Þ2 vð;rr~ Þr~ þ 2ηð1 þ νÞðr~ þ r 0 Þvð;rr~ Þ þ ðr~ þr 0 Þ2 ρω2 η k 2ð1 þ νÞ vðrÞ ðθ Þ þikηðr~ þ r 0 Þð1 þ2νÞv;r~ ikηð3 þ 2νÞvðθÞ ¼ 0:
(B.6a)
Circumferential direction: ðθ Þ ðθÞ ikηðr~ þr 0 Þð1 þ2νÞvð;rr~ Þ þ ikηð3 þ 2νÞvðrÞ þ ηðr~ þ r 0 Þ2 v;r~ r~ þ ηðr~ þ r 0 Þv;r~ h n oi 2 þ ðr~ þ r 0 Þ2 ρω2 η 2k ð1 þ νÞ þ1 vðθÞ ¼ 0:
(B.6b)
In this case, the length directional displacement becomes zero, so that the length directional equation simply vanishes. Similarly, the boundary conditions associated with σ rz and σ θz vanish, so the coupled equations for vðrÞ and vðθÞ only remain. The radial and circumferential directional displacements can be found using same steps followed in Section 2.2. References [1] C.W. Cai, Y.K. Cheung, H.C. Chan, Dynamic response of infinite continuous beams subjected to a moving force – an exact method, Journal of Sound and Vibration 123 (3) (1988) 461–472. [2] M. Olsson, On the fundamental moving load problem, Journal of Sound and Vibration 145 (2) (1991) 299–307. [3] T.N. Shiau, K.H. Huang, F.C. Wang, K.H. Chen, C.P. Kuo, Dynamic response of a rotating ball screw subject to a moving regenerative force, Applied Mathematical Modelling 34 (2010) 1721–1731. [4] M.R. Shadnam, M. Mofid, J.E. Akin, On the dynamic response of rectangular plate, with moving mass, Thin-Walled Structures 39 (2001) 797–896. [5] R. Katz, C.W. Lee, A.G. Ulsoy, R.A. Scott, The dynamic response of a rotating shaft subject to a moving load, Journal of Sound and Vibration 122 (1) (1988) 131–148. [6] H.S. Zibdeh, H.S. Juma, Dynamic response of a rotating beam subjected to a random moving load, Journal of Sound and Vibration 223 (5) (1999) 741–758. [7] M.A. Hilal, H.S. Zibdeh, Vibration analysis of beams with general boundary conditions traversed by a moving force, Journal of Sound and Vibration 229 (2) (2000) 377–388. [8] L.Y. Tommy, H.T. Chan, Moving force identification based on the frequency–time domain method, Journal of Sound and Vibration 261 (2003) 329–349. [9] G. Sinclair, The patterns of slotted-cylinder antennas, Proc. Inst. Radio Eng. 36 (12) (1948) 1487–1492.
302
S. Lee, J. Seok / Journal of Sound and Vibration 340 (2015) 283–302
[10] C.T. Swift, P.B. Gooderum, S.L. Castellow, Experimental investigation of a plasma covered, axially slotted cylinder antenna, IEEE Trans. Antennas Propag. 17 (5) (1969) 598–605. [11] A.K. Hamid, Multi-dielectric loaded axially slotted antenna on circular or elliptic cylinder, Journal of Electromagnetic Waves and Applications 20 (9) (2006) 1259–1271. [12] J.E. Greenspon, Flexural vibrations of a thick walled circular cylinder, in: Proceedings of the Third U.S. National congress of Applied Mechanics, vol. 1, 1958, pp. 163–173. [13] J.E. Greenspon, Flexural vibrations of a thick-walled circular cylinder according to the exact theory of elasticity, Journal of the Aero/Space Sciences 27 (1) (1960) 37–40. [14] L. Pochhammer, On the propagation of small free vibrations in an infinite circular solid cylinder, Crelle's Journal 81 (1876) 324–326. [15] D.C. Gazis, Three-dimensional investigation of the propagation of waves in hollow circular cylinders. I. Analytical foundation, Journal of the Acoustical Society of America 31 (5) (1959) 568–573. [16] D.C. Gazis, Three-dimensional investigation of the propagation of waves in hollow circular cylinders. II. Numerical results, Journal of the Acoustical Society of America 31 (5) (1959) 573–578. [17] I. Mirsky, Wave propagation in transversely isotropic circular cylinders. Part I: Theory, Journal of the Acoustical Society of America 37 (6) (1965) 1016–1021. [18] J.Q. Ye, K.P. Soldatos, Three-dimensional vibrations of cross-ply laminated hollow cylinders with clamped edge boundaries, Journal of Vibration and Acoustics 119 (3) (1997) 317–323. [19] J. So, A.W. Leissa, Free vibrations of thick hollow circular cylinders from three-dimensional analysis, Journal of Vibration and Acoustics 119 (1) (1997) 89–95. [20] S. Brischetto, Three-dimensional exact free vibration analysis of spherical, cylindrical, and flat one-layered panels, Shock and Vibration (2014) (2014) 479738. [21] K.P. Soldatos, V.P. Hadjigeorgiou, Three-dimensional solution of the free vibration problem of homogeneous isotropic cylindrical shells and panels, Journal of Sound and Vibration 137 (3) (1990) 369–384. [22] C.T. Loy, K.Y. Lam, Vibration of thick cylindrical shells on the basis of three-dimensional theory of elasticity, Journal of Sound and Vibration 226 (4) (1999) 719–737. [23] T.Y. Ng, K.Y. Lam, Vibration and critical speed of a rotating cylindrical shell subjected to axial loading, Applied Acoustics 56 (1999) 273–282. [24] J. Padovan, Traveling waves vibrations and buckling of rotating anisotropic shells of revolution by finite elements, International Journal of Solids and Structures 11 (1975) 1367–1380. [25] K.M. Lew, T.Y. Ng, X. Zhao, J.N. Reddy, Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells, Computer Methods in Applied Mechanics and Engineerings 191 (37–38) (2002) 4141–4157. [26] P.G. Bhuta, Transient response of a thin elastic cylindrical shell to a moving shock wave, Journal of the Acoustical Society of America 35 (1) (1963) 25–30. [27] M.J. Forrestal, G. Herrmann, Response of a submerged cylindrical shell to an axially propagating step wave, Journal of Applied Mechanics 32 (4) (1965) 788–792. [28] H. Reismann, J. Padlog, Forced, axi-symmetric motions of cylindrical shells, Journal of The Franklin Institute 284 (5) (1967) 308–319. [29] R. Parnes, Response of an infinite elastic medium to traveling loads in a cylindrical bore, Journal of Applied Mechanics 36 (1) (1969) 51–58. [30] C.C. Huang, Moving loads on elastic cylindrical shells, Journal of Sound and Vibration 49 (2) (1976) 215–220. [31] E. Mangrum, J.J. Burns, Orthotropic cylindrical shells under dynamic loadings, Journal of Mechanical Design 101 (1979) 322–329. [32] S. Chonan, Dynamic response of a cylindrical shell imperfectly bonded to a surrounding continuum of infinite extent, Journal of Sound and Vibration 78 (2) (1981) 257–267. [33] S.K. Datta, T. Chakraborty, A.H. Shah, Dynamic response of pipelines to moving loads, Earthquake Engineering & Structural Dynamics 12 (1) (1984) 59–72. [34] C.W. Bert, V. Birman, Parametric instability of thick, orthotropic, circular cylindrical shells, Acta Mechanica 71 (1988) 61–76. [35] R. Panneton, A. Berry, F. Laville, Vibrations and sound radiation of a cylindrical shell under a circumferentially moving force, Journal of the Canadian Acoustical Association 20 (3) (1992) 25–26. [36] V.P. Singh, J.P. Dwivedi, P.C. Upadhyay, Effect of fluid presence on the nonaxisymmetric dynamic response of buried orthotropic cylindrical shells under a moving load, Journal of Vibration and Control 6 (2000) 823–847. [37] K.M. Liew, Y.G. Hu, T.Y. Ng, X. Zhao, Dynamic stability of rotating cylindrical shells subjected to periodic axial loads, International Journal of Solids and Structures 43 (2006) 7553–7570. [38] Y.Q. Wang, X.H. Guo, Y.G. Li, J. Li, Nonlinear traveling wave vibration of a circular cylindrical shell subjected to a moving concentrated harmonic force, Journal of Sound and Vibration 329 (2010) 338–352. [39] S.C. Huang, B.S. Hsu, Resonant phenomena of a rotating cylindrical shell subjected to a harmonic moving load, Journal of Sound and Vibration 136 (2) (1990) 215–228. [40] J. Seok, H.F. Tiersten, Free vibrations of annular sector cantilever plates. Part 1: Out-of-plane motion, Journal of Sound and Vibration 271 (2004) 757–772. [41] J. Seok, H.F. Tiersten, Free vibrations of annular sector cantilever plates. Part 2: In-plane motion, Journal of sound and Vibration 271 (2004) 773–787. [42] ANSYS v 12, ANSYS, Inc. [43] A.E. Armenakas, propagation of harmonic waves in composite circular cylindrical shells-I: Theoretical investigations, The American Institute of Aeronautics and Astronautics 5 (1967) 740–744. [44] H.H. Sherief, M.N. Anwar, A problem in generalized thermoelasticity for an infinitely long annular cylinder composed of two different materials, Acta Mechanica 80 (1989) 137–149. [45] Z.Y. Lee, C.K. Chen, C.I. Hung, Transient thermal stress analysis of multilayered hollow cylinder, Acta Mechanica 151 (2001) 75–88. [46] H. Cho, G.A. Kardomateas, Thermal shock stresses due to heat convection at a bounding surface in a thick orthotropic cylindrical shell, International Journal of Solids and Structures 38 (2001) 2769–2788. [47] H.H. Sherief, A.E.M. Elmisiery, M.A. Elhagary, Generalized thermoelastic problem for an infinitely long hollow cylinder for short times, Journal of Thermal Stresses 27 (2004) 885–902. [48] Z.Y. Lee, Magnetothermoelastic transient response of a multilayered hollow cylinder subjected to magnetic and vapor fields, Acta Mechanica 193 (2007) 1–16. [49] B. Bai, Consolidation solutions of a saturated porothermoelastic hollow cylinder with infinite length, Scientific Research 2 (1) (2010) 37–45. [50] M.A. Elhagary, Generalized thermoelastic diffusion problem for an infinitely long hollow cylinder for short times, Acta Mechanica 218 (2011) 205–215.