Vibrations in Physical Systems Vol.6 (0) Geometrically Nonlinear Free ransversal Vibrations of hin-walled Elongated Panels with Arbitrary Generatrix Mykhailo MARCHUK Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU 3 b Naukova Str., L viv, 79053, Ukraine and National University L viv Polytechnic Bandera Str., L viv, 7903, Ukraine mv_marchuk@ukr.net; marchuk@iapmm.lviv.ua aras GORIACHKO Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU 3 b Naukova Str., L viv, 79053, Ukraine taras.horyachko@gmail.com Vira PAKOSH Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, NASU 3 b Naukova Str., L viv, 79053, Ukraine v.pakosh@ukr.net Abstract he algorithm for finding a finite number of the first values of natural frequencies and forms of geometrically nonlinear free transverse vibrations of thin-walled elongated panels with arbitrary generatrix is proposed and verified. Under normal coordinate quadratic the approximation of displacements is used. Along the tangential coordinates used one-dimensional finite elements. he discrete variation problem is built. For its solving the perturbation method is applied. he numerical results are compared with previously obtained by other authors. Keywords: elongated panels, vibrations, nonlinearity, perturbations method. Introduction hin elongated panels with various curves as generatrix medial surfaces are widely used in the construction and hardware for various purpose. In the operating conditions they are subected to intense dynamic loading, in particular, cyclic. hese loads are causing in panels the normal displacement commensurate with their thickness. he last are causing to their geometrically nonlinear dynamic stress-strain state. o avoid resonance phenomena for the actions of cyclic loading is necessary at the design stage to determine the spectrum of frequencies of said structural elements. Issues of geometrically nonlinear vibrations of plate and shell elements of the constructions on the basis classical and shear theories thoroughly examined in [] for the definition of the fundamental frequency. Significant progress in this field together with experimental approaches is done in [, ] and some analytical results are given in [8]. However, for nonlinear oscillations in many cases it is necessary to define a number of first frequencies and forms to detect the phenomena of internal, subharmonic and combina-
5 tion resonances []. A numerical method for determining the first several frequencies and forms at geometrically nonlinear vibrations of shells is proposed in the work [5]. In this paper is developed and verified algorithm for determining a finite number of natural frequencies and forms elongated thin panels for geometrically nonlinear vibrations. For the primary relations is taken spatial equation geometrically nonlinear dynamic theory of elasticity. Used quadratic approximation of displacements by the normal coordinate and finite-element by tangential. he discrete variation problem is built. For its solving the method of perturbations is applied.. Problem statement Curved anisotropic elastic layer with thickness h we take to natural mixed system of coordinate α, α, α 3 on the median surface. his surface is formed by the motion of the line α = 0; α 3 = 0 on the segment of arbitrary generatrix. We consider that layer is significantly larger along the axis α to the length of the section arc α = 0 of the middle surface α 3 = 0. So we have an elongated panel. If the conditions of fixing the ends of the panel α = ±α 0 and the initial conditions are independent of the coordinate α, then through little influence of conditions fixing the edges α = ±α 0, the functions, that determine the characteristics of geometrically nonlinear vibration processes in the plane of the middle section, are dependent from α, α 3. o find these functions are [9]: motion equations elasticity relations ˆ U div S = ρ ; () t Σˆ = A εˆ ; () deformation relation between the strain tensor components εˆ and the components r r r of the elastic displacement vector U = u e e i i k ε i = ( iu + ui + iu uk ) ; (3) i relation between the components S of the nonsymmetrical Kirchhoff stress tensor Ŝ and the components σ of the symmetric Piola stress tensor ik Σˆ i S = σ (δ + u ). () k ik In equations () and () A tensor of elastic properties of anisotropic layer, and ρ its density. Boundary conditions on the front surface of the panel α 3 = ±h/ for the free vibrations has the form k k
S Vibrations in Physical Systems Vol.6 (0) 55 3 33 ( α, ± h /, = S ( α, ± h /, = 0, 0 α α. (5) At the elongated ends of the panel α = ±α 0 under the conditions of the fixing the hinge on the lower surface of the front α = h/ boundary conditions has the form ( 3 S i a, α, = 0, (6) u i ( a, ± h /, = 0, α 3 h /, i =, 3, a = 0, l. (7) he motion equations () together with relations () () and boundary conditions (5) (7) are describe geometrically nonlinear transverse vibrations of the middle section of the panel, if the initial conditions specify as follows: 0 u u i ( α, α3, = vi ( α, α3), i ( α, α3, = v ( α, ) t= t i α3, i =, 3, (8) 0 t 0 0 3 ( 3 α3 t= t0 0 0 3 h v α, α ) >> v ( α, ), ( α, α ) Ω = [ α, α ] [ h /, / ]. (9) 3. Discretezed problem Considered above differential formulation of the problem of geometrically nonlinear free vibrations is equivalent to the problem of minimizing the functional L [0]: L = = Ω Ω i S ui x i S i i u x i dω dω Ω Ω U ρ t U ρ t UdΩ = UdΩ min. Boundary conditions (5) and (6) for the variation formulation of the problem is a natural [0], and condition (7) must take into account during its solution. Assuming that the considering panel is thin-walled, approximate the unknown displacement at transverse coordinate [7]: where α3 p0( α3) =, h (0) ui ( α, α) = uik ( α) pk ( α3), i =, 3, () k= α3 α p( α 3) = +, 3 p ( α 3) =. h h For finding unknown coefficients u ik (α ) in () we use approximation by the tangential coordinate α on one-dimensional izoparametrical linear finite elements [0]:
56 u ik α = uikm( α) ϕm ( ξ), ξ =, () α α k, m ikm ik m where e element number; u = u ( α ) ; α e m ), m =, coordinates of the ( ) ( α ( ) element nodes; ϕ ( ξ ) = ( ξ ) ; ϕ e ( ξ) = ( + ξ). After substituting () into () and the result together with () into (0) we obtain: L = { K { + { K ( u){ + { M{ u& } min, (3) L NL (e) where { = {( vector of values of the coefficients u ikm at points in the finiteelement partition of the section [ α 0, α 0 ]; K L linear, and K NL nonlinear components of stiffness matrix; M matrix of mass [0]. Non-linear component of stiffness matrix K NL presented in the form K NL ({ ( ) = B({ ( ) { B ({ ( ). () Matrix B({() we obtain by integrating in (0) members, who are the product of partial derivatives, the displacement u i [0]. Minimum of discrete functional (3) is achieved at the point {(, where the equation is satisfied K. he method of perturbations L ({ ( ) + K ({ ( ){ ( + M ({ u& }) = 0. (5) he system of nonlinear equations (5) is written as K L NL ({ ( ) + µ K ({ ( ){ ( + M ({ u& }) = 0, (6) NL where µ (0 µ ) the parameter perturbation. At µ = 0 have a system of linear algebraic equations for the vector{, while µ = the nonlinearity is taken into account fully. he method of perturbations the desired vector of functions {( and matrix K L presented in the form u } = { + µ {..., { 0 + K L = K µ K L.... (7) he result of substituting (7) into (6) and grouping expressions under the same powers of µ are the equations M u& } + K{ 0, (8) { 0 0 = M u& } + K{ K ({ ) + K ({ ){ 0. (9) { L 0 NL 0 0 = Solution of equation (8) is written as
Vibrations in Physical Systems Vol.6 (0) 57 { t u 0 ( ) = cosωt, (0) and the solution of (9) can be written as follows. Consider equation (7), which seeks a solution in the form * { t = { + { ( ), () where { u } * і { u } solutions of homogeneous and inhomogeneous equations (9). According to [6] matrix K L can be represented as 3 K } L = B{ { u B. () After substituting (0) and () into (9) and taking into account formula [3] for finding { u } we obtain the equation solution of which we take as 3 cos ωt = 3cosωt + cos3ωt, (3) M{ u& } K{ B{ u }{ u } B { u + = }cos3ωt, () u { } = c cos3ωt. (5) After substituting (5) into equation () we arrive at a relations for determination of parameter c : }{ ( K 9ω M ) c = B{ u B { u }. (6) If the initial moment the panel is deformed on a certain law, which describes the first formula in (8) and is stationary, then the initial conditions for the functions { 0 ( and { ( can be represented as In view of (), we write: { u } 0 (0) = A, { u& } 0 (0) = 0, (7) { u } (0) = 0, { u& } (0) = 0. (8) { ( = ( A c )cosωt + c cos3ωt. (9) his allows you to build an algorithm for partial finding a finite number of the first natural frequencies and amplitudes geometrically nonlinear vibrations of the panel:. Set r = and a ( 0) = 0.
58 3. Compute K ( r ) = K + Ba( r ) a( r ) B. 3. Find the eigenvalues ω (r) and eigenvectors a (r) from the system ( ( r ) ( r) ( r) = K ω M ) a 0.. If the conditions are satisfied a ( r) a( r ) / a( r) ε, r ) ω( r ) ω( r), ω( / ε where ε and ε specified accuracy, then go to step 5, otherwise r : = r +, and go to step. 5. As the solution we accept a : = a( r), ω : = ω( r ) and find a vector c having solved the a system of algebraic equations ( K r M c ( ) 9ω ) = Baa B a. 5. Analysis of results and conclusions o verify the proposed algorithm practicing it for problem, where are known analytical and numerical solutions [6]. We consider an isotropic plate-strip elongated edges which are fixed by with stationary hinges on the lower front of the plane (see Fig. ), with characteristics: geometric l = m; h = 0. m and mechanical E = 0000 N/m ; ν = 0.3. Figure. he plate-strip with stationary hinges on the elongated edges In Figure shows graphs of free vibrations of the point that has coordinates (l / ; 0) for linear ( ), analytical ( ) and obtained using the proposed algorithm ( ).Sufficiently good correlation with the analytical solution is marked. In Figure 3 shows the first four own forms (modes) for geometrically nonlinear vibrations the considered plate-strip. In Figure shows the skeletal curves [], constructed using the proposed method (dashed line) and the results presented in the work [5] (solid line). he maximum relative error does not exceed 9%, indicating a sufficiently good approximation property of the proposed method. Subsequently, it is advisable to perform a similar study for a wider class of thin-walled elements of constructions and anisotropy of mechanical properties.
Vibrations in Physical Systems Vol.6 (0) 59 Figure. Free vibrations of point (l / ; 0) Figure 3. View panels in different modes: a) the first mode; b) second; c) third; d) the fourth Figure. Comparison of amplitude-frequency characteristics obtained from the use of perturbation method and the results of the work [5]
60 Acknowledgments his research is conducted with a partial support of the Ukrainian State Fund for Fundamental Researches (proect code: F 53./08). References. M. Amabili, Nonlinear vibrations and stability of shells and plates, Cambridge: Cambridge Univ. Press, 008.. M. Amabili, Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments, Comput. Struct., 8(00) 587 605. 3. G. Korn,. Korn, Handbook of high society mathematics, M.: Nauka, 97.. V. D. Kubenko, P. S. Kovalchuk, N. P. Podchasov, Nonlinear vibrations of cylindrical shells, K.: Vyshcha Shkola, 989. 5. L. V. Kurpa, N. A. Budnikov, Investigation of forced nonlinear vibrations of laminated shallow shells using a multimode approximation, Bulletin of Donetsk Univ. Serіes A. Natural Sciences, (03) 55 60. 6. R. Lewandowski, Analysis of strongly nonlinear free vibration of beams using perturbation method, Civil and Env. Eng. Reports, (005) 53 68. 7. M. Marchuk, I. Mukha,. Goriachko, A comparative analysis of the characteristics of geometrically nonlinear stress-strain state of composite plates and cylindrical panels on the basis refied model and elasticity theory, Modelling and Stability: Abstracts of Conference Reports, K: aras Shevchenko National Univ. of Kyiv, 0. 8. M. Marchuk, V. Pakosh, O. Lesyk, I. Hurayewska, Influence of Pliability to ransversal Deformations of Shear and Compression on Correlation Frequency from Amplitude for Nonlinear Oscillations of Composite Plates, Vibrations in Physical Systems, (006) 5 56. 9. N. F. Morozov, Selected two-dimensional problem of elasticity theory, Leningrad: Publishing House of Leningrad. University Press, 978. 0. J. N. Reddy, An introduction to nonlinear finite element analysis, Oxford Univ. Press, 00.. A. S. Volmir, Nonlinear dynamics of plates and shells, M.: Nauka, 97.