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A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: Primary resonance case

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A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: Primary resonance case
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  This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institutionand sharing with colleagues.Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third partywebsites are prohibited.In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further informationregarding Elsevier’s archiving and manuscript policies areencouraged to visit:http://www.elsevier.com/copyright  Author's personal copy A general solution procedure for the forced vibrations of a systemwith cubic nonlinearities: Three-to-one internal resonances withexternal excitation B. Burak O¨zhan, Mehmet Pakdemirli  Department of Mechanical Engineering, Celal Bayar University 45140, Muradiye, Manisa, T ¨urkiye a r t i c l e i n f o  Article history: Received 11 July 2009Received in revised form11 December 2009Accepted 8 January 2010Handling editor: M.P. CartmellAvailable online 9 February 2010 a b s t r a c t A general vibrational model of a continuous system with arbitrary linear and cubicoperators is considered. Approximate analytical solutions are found using the method of multiple scales. The primary resonances of the external excitation and three-to-oneinternal resonances between two arbitrary natural frequencies are treated. Theamplitude and phase modulation equations are derived. The steady-state solutionsand their stability are discussed. The solution algorithm is applied to two specificproblems: (1) axially moving Euler–Bernoulli beam, and (2) axially moving viscoelasticbeam. &  2010 Elsevier Ltd. All rights reserved. 1. Introduction Many nonlinear models for vibrations of continuous systems contain cubic nonlinearities. Nonlinearities add to thecomplexity of the systems, which make exact analytical solutions impossible to obtain. Approximate analytical solutionsare sought for such system as a second best alternative and perturbation methods are widely used for this task. Thenonlinearities may appear in a variety of forms including algebraic, differential or integral structures. Regardless of thespecific forms of the nonlinearities, they can be classified with respect to their common natures such as quadratic or cubicnonlinearities. To understand and analyze the effects of arbitrary quadratic and cubic nonlinearities on the solutions, anoperator notation suitable for perturbation analysis was developed [1]. The motivation behind the study was to comparethe direct-perturbation methods with discretization–perturbation methods. The discussion was limited to single modeapproximations of free vibrations. Later the analysis was generalized to infinite number of modes for forced vibrations [2].The advantages of direct-perturbation methods were discussed in detail. Comparison of both methods for a parameticallyexcited linear system expressed by arbitrary linear operators was also done [3]. It is concluded that while infinite moderesults agree with each other, finite mode truncations of both methods yield different results with direct-perturbationmethod producing more precise solutions. In order to compare results of different versions of method of multiple scales and decide which method yields better steady-state solutions, an arbitrary cubic nonlinear system was treated [4]. For coupled partial differential systems, a solution procedure for one-to-one internal resonances was developed [5]. Using thesame model of  [5], possible internal resonances were classified [6]. For a general cubic nonlinear system, three-to-one internal resonances were further considered [7]. Two-to-one internal resonances were analysed for arbitrary quadratic Contents lists available at ScienceDirectjournal homepage: www.elsevier.com/locate/jsvi  Journal of Sound and Vibration ARTICLE IN PRESS 0022-460X/$-see front matter  &  2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.jsv.2010.01.010  Corresponding author. Tel.: +902362412146; fax: +902362412143. E-mail addresses:  mpak@bayar.edu.tr, pakdemirli@gmail.com, pakdemirli@yahoo.com (M. Pakdemirli). Journal of Sound and Vibration 329 (2010) 2603–2615  Author's personal copy nonlinearities [8]. The new operator notation developed and used in the previous studies [1–8] was adopted to understand the effects of nonlinearities by others also (see [9,10] for example).In a very recent work [11], additional linear and cubic operators with time derivatives were incorporated into themodel. In contrast to the mentioned previous studies with cubic nonlinearities, which employ only spatial operators,additional linear and cubic operators containing spatial as well as time derivatives were included. This enables analyzing amore general class of continuous systems such as gyroscopic systems, which are encountered in axially moving media orpipes conveying fluids.In this work, the work of  [11] is extended to include three-to-one internal resonances. A general solution procedure isdeveloped for arbitrary operators first. Method of multiple scales, a perturbation technique is used in the analysis.Amplitude and phase modulation equations, steady-state solutions and their stability are discussed in a general sense. Thesolution algorithm is applied to two different problems: (1)nonlinear vibrations of stretched axially moving Euler–Bernoulli beam, and (2) nonlinear vibrations of axially moving viscoelastic beam. While the first application containintegro-differential type and the second differential type, both models possess the common feature of being cubicnonlinear, which enables the algorithm developed to be applied directly to these equations. The application problems arediscussed in detail. Stability analysis is presented and frequency-response curves are drawn to depict the effects of variousparameters on the vibrations of the system. Energy transfer between the modes are displayed in the figures. Apart from theexamples considered, the general solution procedure developed may be applied to a wide range of physical problems. 2. Equation of motion The dimensionless model considered is € w  þ L  1 ð w  Þþ L  2 ð  _ w  Þþ e L  3 ð  _ w  Þ ¼  e F  cos O t  þ e f C 1 ð w  ; w  ; w  Þþ C 2 ð  _ w  ; w  ; w  Þg  (1) B 1 ð w  Þ ¼ 0  x ¼ 0 ;  B 2 ð w  Þ ¼ 0  x ¼ 1 (2)with dependent variable  w (  x ,  t  ) representing deflection, and independent variables  x  and  t   being the spatial and timevariables, respectively. Note that there may be more than one spatial variable and a 3-D problem in spatial variables  x ,  y and  z   has not been excluded from the analysis.  L  1 ,  L  2  and  L  3  are linear differential and/or integral operators.  C 1  and  C 2  arecubic nonlinear operators.  F   and  O  represents external excitation amplitude and external excitation frequency,respectively.  B 1  and  B 2  are linear operators of boundary conditions. The representation of boundary conditions shouldbe expressed in a modified form for a 3-D problem. e  is a small dimensionless physical parameter. Dot denotes partialdifferentiation with respect to time. To capture the effects of gyroscopic systems, additional linear and cubic operators (i.e. L  2 ,  L  3  and  C 2 ) containing time derivatives are included in the model.Note that model (1) is a fairly general model and any vibration problem that can be cast into the formalism of Eq. (1) canbe solved approximately through the algorithm developed in the following analysis. A restriction of the boundary valueproblem comes from the boundary conditions, i.e. they are assumed to be linear. Furthermore, operator  L  1  is symmetricand  L  2  is skew symmetric with respect to boundary conditions. This introduces some simplifications when calculatingsolvability conditions [12]. If the specific problem contains nonlinear boundary conditions or the operators L  1  and L  2  do notpossess the mentioned properties with respect to boundary conditions, the general solution algorithm cannot be directlyapplied to it. This case needs further analysis since the solvability condition brings more terms for nonlinear boundaryconditions, which are hard to express in a general way. Although the boundary conditions given here represent a 1-Dproblem such as normalized length, in fact the solution algorithm is more general than that and can be successfully appliedto 2 or 3-D problems in spatial variables. Note that both equations of motion and boundary conditions should be expressedfirst in a non-dimensional form for applications.The cubic operators  C 1  and  C 2  may not be symmetric with respect to the inner variables and possesses the property of being multilinear. See Ref. [11] for details. 3. Perturbation solution The method of multiple scales [13] is applied directly to the model to find the general solution of Eq. (1). The followingexpansion for  w (  x ,  t  ) is assumed w  ð  x ; T  0 ; T  1 ; e Þ ¼ w  0 ð  x ; T  0 ; T  1 Þþ e w  1 ð  x ; T  0 ; T  1 Þþ     (3)where  T  0 = t   is the usual fast time scale and  T  1 = e t   is the slow time scale. Time derivatives are expressed in terms of fast andslow time scales as follows ddt   ¼ D 0  þ e D 1  þ     (4) d 2 dt  2  ¼ D 20  þ 2 e D 0 D 1  þ     (5) ARTICLE IN PRESS B. Burak O¨ zhan, M. Pakdemirli / Journal of Sound and Vibration 329 (2010) 2603–2615 2604  Author's personal copy where  D k  ¼  @=@ T  k . Inserting Eqs. (3)–(5) into Eqs. (1) and (2) and separating at each order of   e , one obtains O ð e 0 Þ D 20 w  0  þ L  1 ð w  0 Þþ L  2 ð D 0 w  0 Þ ¼ 0 (6) B 1 ð w  0 Þ ¼ 0 at  x ¼ 0 ;  B 2 ð w  0 Þ ¼ 0 at  x ¼ 1 (7) O ð e 1 Þ D 20 w  1  þ L  1 ð w  1 Þþ L  2 ð D 0 w  1 Þ ¼  2 D 0 D 1 w  0  L  2 ð D 1 w  0 Þ L  3 ð D 0 w  0 Þþ F  cos O T  0  þ C 1 ð w  0 ; w  0 ; w  0 Þþ C 2 ð D 0 w  0 ; w  0 ; w  0 Þ  (8) B 1 ð w  1 Þ ¼ 0 at  x ¼ 0 ;  B 2 ð w  1 Þ ¼ 0 at  x ¼ 1 (9)At  O ( e 0 ), two arbitrary modes of frequencies  o n  and  o m  ( o m 4 o n ) are assumed to interact through internal resonances w  0 ð  x ; T  0 ; T  1 Þ ¼  A n ð T  1 Þ e i o n T  0 Y  n ð  x Þþ  A n ð T  1 Þ e  i o n T  0 Y  n ð  x Þþ  A m ð T  1 Þ e i o m T  0 Y  m ð  x Þþ  A m ð T  1 Þ e  i o m T  0 Y  m ð  x Þ  (10)where  A n  and  A n  are complex amplitudes and their conjugates, respectively.  Y  n (  x ) satisfy the following equations andboundary conditions, L  1 ð Y  n Þ o 2 n Y  n  þ i o n L  2 ð Y  n Þ ¼ 0 ;  n ¼ 1 ; 2 ; . . .  (11) B 1 ð Y  n Þ ¼ 0 at  x ¼ 0 ;  B 2 ð Y  n Þ ¼ 0 at  x ¼ 1 (12)Due to the dissipative term at the zeroth order,  Y  n (  x ) may not be real and the complex conjugate of the function isincorporated in the zeroth-order solution (10). The above equation and boundary conditions constitute an eigenvalue–eigenfunction problem. o n  are the eigenvalues and  Y  n (  x ) are the eigenfunctions of the system, respectively. For continuoussystem, it is clear that there are infinite number of eigenvalues and corresponding eigenfunctions. Note that under somecircumstances more complex mode interactions may occur for the system considered. This work, however, is limited toexternal excitation of one of the modes and excitation of another mode via 3:1 internal resonance.Upon substitution of solution (10) to the right-hand side of (8) yields D 20 w  1  þ L  1 ð w  1 Þþ L  2 ð D 0 w  1 Þ ¼  2 D 1 ð i o n  A n e i o n T  0 Y  n  i o n  A n e  i o n T  0 Y  n  þ i o m  A m e i o m T  0 Y  m  i o m  A m e  i o m T  0 Y  m Þ L  2 ð D 1 ð  A n e i o n T  0 Y  n  þ  A n e  i o n T  0 Y  n  þ  A m e i o m T  0 Y  m þ  A m e  i o m T  0 Y  m ÞÞ L  3 ð i o n  A n e i o n T  0 Y  n  i o n  A n e  i o n T  0 Y  n  þ i o m  A m e i o m T  0 Y  m  i o m  A m e  i o m T  0 Y  m Þþ  12 F  ð e i O T  0 þ e  i O T  0 Þþ C 1 ð  A n e i o n T  0 Y  n þ  A n e  i o n T  0 Y  n  þ  A m e i o m T  0 Y  m  þ  A m e  i o m T  0 Y  m ;  A n e i o n T  0 Y  n  þ  A n e  i o n T  0 Y  n  þ  A m e i o m T  0 Y  m þ  A m e  i o m T  0 Y  m ;  A n e i o n T  0 Y  n  þ  A n e  i o n T  0 Y  n  þ  A m e i o m T  0 Y  m þ  A m e  i o m T  0 Y  m Þþ C 2 ð i o n  A n e i o n T  0 Y  n  i o n  A n e  i o n T  0 Y  n  þ i o m  A m e i o m T  0 Y  m  i o m  A m e  i o m T  0 Y  m ;  A n e i o n T  0 Y  n  þ  A n e  i o n T  0 Y  n  þ  A m e i o m T  0 Y  m  þ  A m e  i o m T  0 Y  m ;  A n e i o n T  0 Y  n  þ  A n e  i o n T  0 Y  n  þ  A m e i o m T  0 Y  m  þ  A m e  i o m T  0 Y  m Þ  (13) B 1 ð w  1 Þ ¼ 0 at  x ¼ 0 ;  B 2 ð w  1 Þ ¼ 0 at  x ¼ 1 (14)Under the assumed resonances O ¼ o n  þ es n  (15) o m  ¼ 3 o n  þ er n  (16)where the external excitation frequency is near to the  n th natural frequency and the  m th natural frequency is excited via3:1 internal resonance.  s n  and  r n  are detuning parameters of order 1. Using the multilinearity properties of the cubicoperators with the resonance conditions, Eq. (13) resumes the form D 20 w  1  þ L  1 ð w  1 Þþ L  2 ð D 0 w  1 Þ ¼  2 i o n D 1  A n e i o n T  0 Y  n  2 i o m D 1  A m e i o m T  0 Y  m  D 1  A n e i o n T  0 L  2 ð Y  n Þ D 1  A m e i o m T  0 L  2 ð Y  m Þ i o n  A n e i o n T  0 L  3 ð Y  n Þ i o m  A m e i o m T  0 L  3 ð Y  m Þþ  12 Fe i o n T  0 e i s n T  1 þ  A 3 n e i o m T  0 e  i r n T  1 f C 1 ð Y  n ; Y  n ; Y  n Þþ i o n C 2 ð Y  n ; Y  n ; Y  n Þgþ  A 2 n  A m e i o n T  0 e i r n T  1 f C 1 ð Y  m ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  m ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  m Þ i o n ð C 2 ð Y  n ; Y  m ; Y  n Þþ C 2 ð Y  n ; Y  n ; Y  m ÞÞþ i o m C 2 ð Y  m ; Y  n ; Y  n Þgþ  A 2 n  A n e i o n T  0 f C 1 ð Y  n ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  n Þþ i o n ð C 2 ð Y  n ; Y  n ; Y  n Þþ C 2 ð Y  n ; Y  n ; Y  n Þ C 2 ð Y  n ; Y  n ; Y  n ÞÞgþ  A 2 m  A m e i o m T  0 f C 1 ð Y  m ; Y  m ; Y  m Þþ C 1 ð Y  m ; Y  m ; Y  m Þþ C 1 ð Y  m ; Y  m ; Y  m Þþ i o m ð C 2 ð Y  m ; Y  m ; Y  m Þþ C 2 ð Y  m ; Y  m ; Y  m Þ C 2 ð Y  m ; Y  m ; Y  m ÞÞgþ  A n  A n  A m e i o m T  0 f C 1 ð Y  n ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  n Þþ i o n ð C 2 ð Y  n ; Y  n ; Y  m Þþ C 2 ð Y  n ; Y  m ; Y  n Þ C 2 ð Y  n ; Y  n ; Y  m Þ C 2 ð Y  n ; Y  m ; Y  n ÞÞþ i o m ð C 2 ð Y  m ; Y  n ; Y  n Þþ C 2 ð Y  m ; Y  n ; Y  n ÞÞgþ  A n  A m  A m e i o n T  0 f C 1 ð Y  m ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  m Þþ C 1 ð Y  m ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  m Þþ i o n ð C 2 ð Y  n ; Y  m ; Y  m Þþ C 2 ð Y  n ; Y  m ; Y  m ÞÞþ i o m ð C 2 ð Y  m ; Y  m ; Y  n Þþ C 2 ð Y  m ; Y  n ; Y  m Þ C 2 ð Y  m ; Y  m ; Y  n Þ C 2 ð Y  m ; Y  n ; Y  m ÞÞgþ cc þ NST (17) ARTICLE IN PRESS B. Burak O¨ zhan, M. Pakdemirli / Journal of Sound and Vibration 329 (2010) 2603–2615  2605  Author's personal copy where cc stands for complex conjugates of the preceding terms and NST stands for non-secular terms. A solution for  w 1 may be assumed in the form w 1 ð  x ; T  0 ; T  1 Þ ¼ j n ð  x ; T  1 Þ e i o n T  0 þ j m ð  x ; T  1 Þ e i o m T  0 þ cc þ W  ð  x ; T  0 ; T  1 Þ  (18)where  W  (  x ,  T  0 ,  T  1 ) represents solution associated with non-secular terms and  j n  and  j m  represent solutions associatedwith secular terms. Substituting solution (18) into Eq. (17) and (14) yields after separation to relevant modes  o 2 n j n  þ L  1 ð j n Þþ i o n L  2 ð j n Þ ¼  2 i o n D 1  A n Y  n  D 1  A n L  2 ð Y  n Þ i o n  A n L  3 ð Y  n Þþ  12 Fe i s n T  1 þ  A 2 n  A m e i r n T  1 C 1 ð Y  m ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  m ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  m Þ i o n ð C 2 ð Y  n ; Y  m ; Y  n Þ  þ C 2 ð Y  n ; Y  n ; Y  m ÞÞþ i o m C 2 ð Y  m ; Y  n ; Y  n Þgþ  A 2 n  A n f C 1 ð Y  n ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  n Þþ i o n ð C 2 ð Y  n ; Y  n ; Y  n Þþ C 2 ð Y  n ; Y  n ; Y  n Þ C 2 ð Y  n ; Y  n ; Y  n ÞÞgþ  A n  A m  A m f C 1 ð Y  m ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  m Þþ C 1 ð Y  m ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  m Þþ i o n ð C 2 ð Y  n ; Y  m ; Y  m Þþ C 2 ð Y  n ; Y  m ; Y  m ÞÞþ i o m ð C 2 ð Y  m ; Y  m ; Y  n Þþ C 2 ð Y  m ; Y  n ; Y  m Þ C 2 ð Y  m ; Y  m ; Y  n Þ C 2 ð Y  m ; Y  n ; Y  m ÞÞg  (19) B 1 ð j n Þ ¼ 0  x ¼ 0  B 2 ð j n Þ ¼ 0  x ¼ 1 (20)  o 2 m j m  þ L  1 ð j m Þþ i o m L  2 ð j m Þ ¼  2 i o m D 1  A m Y  m  D 1  A m L  2 ð Y  m Þ i o m  A m L  3 ð Y  m Þþ  A 3 n e  i r n T  1 f C 1 ð Y  n ; Y  n ; Y  n Þþ i o n C 2 ð Y  n ; Y  n ; Y  n Þgþ  A 2 m  A m f C 1 ð Y  m ; Y  m ; Y  m Þþ C 1 ð Y  m ; Y  m ; Y  m Þþ C 1 ð Y  m ; Y  m ; Y  m Þþ i o m ð C 2 ð Y  m ; Y  m ; Y  m Þþ C 2 ð Y  m ; Y  m ; Y  m Þ C 2 ð Y  m ; Y  m ; Y  m ÞÞgþ  A n  A n  A m f C 1 ð Y  n ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  n Þþ i o n ð C 2 ð Y  n ; Y  n ; Y  m Þþ C 2 ð Y  n ; Y  m ; Y  n Þ C 2 ð Y  n ; Y  n ; Y  m Þ C 2 ð Y  n ; Y  m ; Y  n ÞÞþ i o m ð C 2 ð Y  m ; Y  n ; Y  n Þþ C 2 ð Y  m ; Y  n ; Y  n ÞÞg  (21) B 1 ð j m Þ ¼ 0  x ¼ 0  B 2 ð j m Þ ¼ 0  x ¼ 1 (22)Since the homogenous parts of Eqs. (19) and (21) have non-trivial solutions, non-homogenous equations have a solutiononly if a solvability condition is satisfied [13]. For the present model, with  L  1  symmetric and  L  2  skew symmetric withrespect to linear boundary conditions (See the discussions in [12]), the solvability condition is D 1  A n þ k 1 n  A n   f  n e i s n T  1  k 2 n  A 2 n  A n  k 3 nm  A 2 n  A m e i r n T  1  k 4 nm  A n  A m  A m  ¼ 0 (23) D 1  A m  þ k 1 m  A m  k 2 m  A 2 m  A m  k 5 mn  A 3 n e  i r n T  1  k 4 mn  A n  A n  A m  ¼ 0 (24)Note that if the  m th mode is excited through external excitation, the complex amplitude modulation Eqs. (23) and (24)should be slightly modified with the forcing term appearing in (24) rather than in (23). The coefficients are k 1 n  ¼ i o n R  10 Y  n L  3 ð Y  n Þ dx 2 i o n R  10 Y  n Y  n  dx þ R  10 Y  n L  2 ð Y  n Þ dx (25)  f  n  ¼ 12   R  10 FY  n dx 2 i o n R  10 Y  n Y  n dx þ R  10 Y  n L  2 ð Y  n Þ dx (26) k 2 n  ¼ R  10 Y  n f C 1 ð Y  n ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  n Þþ i o n ð C 2 ð Y  n ; Y  n ; Y  n Þþ C 2 ð Y  n ; Y  n ; Y  n Þ C 2 ð Y  n ; Y  n ; Y  n ÞÞg dx 2 i o n R  10 Y  n Y  n  dx þ R  10 Y  n L  2 ð Y  n Þ dx (27) k 3 nm  ¼ R  10 Y  n f C 1 ð Y  m ; Y  n ; Y  n Þþ C 1 ð Y  n ; Y  m ; Y  n Þþ C 1 ð Y  n ; Y  n ; Y  m Þ i o n ð C 2 ð Y  n ; Y  m ; Y  n Þþ C 2 ð Y  n ; Y  n ; Y  m ÞÞþ i o m C 2 ð Y  m ; Y  n ; Y  n Þg dx 2 i o n R  10 Y  n Y  n  dx þ R  10 Y  n L  2 ð Y  n Þ dx (28) k 4 nm  ¼  R  10 Y  n f C 1 ð Y  m ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  m Þþ C 1 ð Y  m ; Y  m ; Y  n Þþ C 1 ð Y  m ; Y  n ; Y  m Þþ C 1 ð Y  n ; Y  m ; Y  m Þþ i o n ð C 2 ð Y  n ; Y  m ; Y  m Þþ C 2 ð Y  n ; Y  m ; Y  m ÞÞþ i o m ð C 2 ð Y  m ; Y  m ; Y  n Þþ C 2 ð Y  m ; Y  n ; Y  m Þ C 2 ð Y  m ; Y  m ; Y  n Þ C 2 ð Y  m ; Y  n ; Y  m ÞÞg dx 2 i o n R  10 Y  n Y  n  dx þ R  10 Y  n L  2 ð Y  n Þ dx (29) k 5 mn  ¼ R  10 Y  m f C 1 ð Y  n ; Y  n ; Y  n Þþ i o n C 2 ð Y  n ; Y  n ; Y  n Þg 2 i o m R  10 Y  m Y  m dx þ R  10 Y  m L  2 ð Y  m Þ dx (30) ARTICLE IN PRESS B. Burak O¨ zhan, M. Pakdemirli / Journal of Sound and Vibration 329 (2010) 2603–2615 2606
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