Visualization of Scenarios for the Transition of Oscillations from Harmonic to Chaotic for a Micropolar Kirchhoff-Love Cylindrical Meshed Panel E.Yu. Krylova1, I.V. Papkova2, O. A. Saltykova2,V.A. Krysko2 Kat.krylova@bk.ru|ikravzova@mail.ru|olga_a_saltykova@mail.ru| tak@sun.ru 1 Saratov State University, Saratov, Russia; 2 Yuri Gagarin State Technical University of Saratov, Saratov, Russia; On the basis of the kinematic hypotheses of the Kirchhoff-Love built a mathematical model of micropolar cylindrical meshed panels vibrations under the action of a normal distributed load. In order to take into account the size-dependent behavior, the panel material is considered as a Cosser’s pseudocontinuum with constrained particle rotation. The mesh structure is taken into account by the phenomenological continuum model of G. I. Pshenichnov. For a cylindrical panel consisting of two systems of mutually perpendicular edges, a scenario of transition of oscillations from harmonic to chaotic is constructed. It is shown that in the study of the behavior of cylindrical micropolar meshed panels it is necessary to study the nature of the oscillations of longitudinal waves. Keywords:visualization of scenarios for the transition of oscillations into chaos, a mesh structure, a cylindrical panel, micropolar theory, the Kirchhoff-Love model. To account for the size-dependent behavior, a non-classical 1. Introduction continuum model based on the Cosser medium is considered, To solve the problems of static and dynamic mesh plates, where, along with the usual stress field, torque stresses are also panels and shells, mainly two computational models are used. It taken into account. This assumes that the displacement and is a phenomenological continuum model and a discrete model. In rotation fields are not independent. In this case, the components the continuum model, a mesh object consisting of a regular of the symmetric bending-torsion tensor are written as follows: system of frequently located edges of one material is replaced by 2w 2w 1  2w 2w  an equivalent solid object having some averaged stiffness  xx  ;  yy   ;  xy   2  2  ; depending on the arrangement of the edges and their stiffness xy yx 2  y x  [1,3]. In the discrete model, the edges are represented by beam, 1   2 v  2u  1   2 v  2u  shell, or three-dimensional finite elements [2,5,7,11].  xz   2   ;  yz    . Eachoftheseapproacheshasitsadvantages [4]. 4  x xy  4  yx y 2  Progress in micro-and nano-technologies leads to the interest We take the defining relations for the panel material in the of scientists not only to the behavior of full-size mechanical 2 systems in the form of plates and shells [13,14,16], but also the form:  mxx , mxу , mzx   1El    xx ,  xу ,  zx  , need to create mathematical models that take into account the scale effects at the micro and nano level [10,12,19]. In most E E  xx  exx   eyy  , x y,  xy  e , где  ij - the works on this subject linear models are used for numerical 1  2  1    xy analysis [15,17,18,21,22]. However, there are experimental data confirming the need to take into account the nonlinearity in components of the stress tensor, mij components of the moment modeling the behavior of the objects under consideration [20]. tensor of higher order, E - Young’s modulus,  - Poisson’s Despite the large number of works devoted to the size- dependent behavior of mechanical objects in the form of plates, ratio. panels and shells, studies of the behavior of mesh plates and The equations of motion of a smooth plate element shells based on theories that take into account the effects of scale equivalent to a mesh one, boundary and initial conditions are is very small [6,8,9]. obtained from the Hamilton – Ostrogradsky energy principle: N xx T 1  Yyz 1  2Yxz  2u 2 2. Problemstatement     h 2 , x y 2 y 2 2 xy t A mathematical model of oscillations of a micro-polar N yy T 1  2Yxz 1  Yyz 2  2v flexible rectangular cylindrical panel under the action of a     h 2 , 3 y x 2 x 2 2 xy t transverse distributed pressure occupying a region in space R (1)  h h N xx w  2 w N yy w  2w 2H T w area   0  x  c;0  y  b;   z   is constructed.  N xx 2   N yy 2  2 2   2 2 x x x y y y xy x y The panel consists n of sets of densely arranged edges of the T w  2 w  2 M xx  M yy 2H 2 same material, which allows the use of a phenomenological 2  4T  2  k y N yy  continuum model. Taking into account the Kirchhoff-love y x xy x 2 y 2 xy  2Yxx  Yyy  Yxy  Yxy 2w w 2 2 2 hypotheses, the strain tensor components are written as:      2q   h 2   h . u 1  w   w 2 v 1  w  xy yx x y t t 2  w 2 2 2 2 exx      z 2 ; eyy      kyw  z 2 ; x 2  x  x y 2  x  y Boundary conditions: 1  u v  w w 2w exy     z . 2  y x  x y xy Where u , v, w - axial displacements of the middle surface of the plate in the directions x, y , z respectively, k y - geometric curvature parameter. Copyright © 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).  w M x w Yxy H Yxx Yyy Yxy  N xxs  A40 N xx  A22 N yy  A31T ;  N xx   2T  2       x x y x y y y x n N yys  A22 N xx  A04 N yy  A13T ; x  w M y w H Yxx Yyy Yxy  T s  A31 N xx  A13 N yy  A22T ;   N yy   2T 2     0  y y x x x x y n M xxs  A40 M xx  A22 M yy  A31H ; y M  Y   2H  Y  Y   0; xx xy n x xx yy n y M yy  A22 M xx  A04 M yy  A13 H ; 2H  Y  Y   M  Y   0; xx yy n yy xy n H s  A31M xx  A13 M yy  A22 H ; (2) Y  A40Yxx  A22Yyy  A31Yxy ; x y s 1 Yyz 1 Yxz  xx  1 Yxz    N xx    T     0; Yyys  A22Yxx  A04Yyy  A13Yxy ;  2 y nx  2 y 2 x n y Yxys  A31Yxx  A13Yyy  A22Yxy ;  1 Yxz 1 Yyz   1 Yyz  1   N xy      N yy    0;  Yxz   0; Yyzs  A11Yxz  A02Yyz  A01Yzz ;  2 x 2 y n  2 x n  2 n y x y Yxzs  A20Yxz  A11Yyz  A10Yzz .   Yxz nx  Yyz ny  0;  1 Yxz    1 Yyz   0;  1 Yyz   0.  2  nx Substituting expressions (2) into equations (1), we obtain a  2 nx  2 ny resolving system of equations of motion for a smooth micropolar Here the expression for the classical force and torque: cylindrical Kirchhoff-Love panel equivalent to the original mesh h 2 panel.  N , N ,T     , ,  dz, xx yy xx yy xy In this model, the rigidity of the rods to bend in a plane  h tangent to the middle surface of the panel is not taken into 2 account, so the orders of systems of differential equations h 2 describing the behavior of grid and solid panels coincide. At the  M , M , H     , ,  zdz, xx yy xx yy xy same time, the formulations of the boundary conditions of the h  corresponding boundary value problems coincide. 2 The scenario of transition of oscillations from harmonic to as well as expressions for the forces caused by instantaneous chaotic cylindrical panel with two families of edges is stresses: h h h investigated 1  45o ,2  135o , 1  2   , a1  a2  a (Fig 2 2 2 Yxx   mxx dz , Yxy   mxy dz , Yxz   mxz dz , h2 x y. .1). Taking into account dimensionless parameters: u  u,  h  h  h c 2 2 2 h2 h Eh 4  1 E The stresses arising in the equivalent smooth panel v v , k y  2 k y , q  2 2 q ;. t  b t,  , associated with the stresses in the edges that make up the angles b b cb E b   j with the abscissa axis will have the form: x  cx , y  b y , w  hw ,   h , a  ha , l  hl , where  - n  xj j Cos  j Sin  j n  xj j Cos 2  j dissipation factor,  - the density of the panel material,  xy   ,  xx   q  q0 Sin  pt  - external normal load, q0 and  p - its intensity , j 1 aj j 1 aj n  xj j Sin 2  j n mxj j Cos  j Sin  j and frequency, t - time. The equations of motion of the element  yy   , mxy   , of the considered micropolar mesh cylindrical panel will take the j 1 aj j 1 aj form (the line above the dimensionless variables is omitted): n mxj j Cos 2  j n mxj j Sin 2  j mxx   , myy   , j 1 aj j 1 aj n mzxj  j Cos  j n mzxj  j Sin  j mxz   , myz   , where a j - j 1 aj j 1 aj distance between edges of j-th family,  j – the thickness of the ribs, voltage index j are rods. The physical relations for the mesh plate are determined based on the Lagrange multiplier method:  xj   xx Cos 2  j   yy Sin 2  j   xy Cos  j Sin  j ;  j   xz Cos  j   yz Sin  j ; mxj  mxx Cos 2  j  m yy Sin 2  j  mxy Cos  j Sin  j ; mzj  mxz Cos  j  m yz Sin  j . Fig.1.Panel mesh geometry. n  j Cos s  j Sin k  j Subject to designation: Ask   ; s, k  0,4 j 1 aj expressions for classical forces and moments, as well as the forces caused by the moment stresses of the cylindrical mesh panel will take the form: (the upper index shows the account of the mesh structure):  h 2  4u h 2  4 v h 2  4u h 2  4v  3. Scenarios of transition of oscillations of a l 2   1   2 4  2  2 2 2  2 3  cylindrical panel to chaos  b y b xy c x y c x y  3 u2 v 2 b u 2 2 To visualize the scenarios of transition of oscillations of a 2   1 2  2  3     4   1 2 2  micropolar mesh cylindrical panel from harmonic to chaotic for y xy c x deflection and displacement, the following characteristics were b w  2 w w w  2 w 4k y   1  4   1 2 8  constructed and analyzed: signal, Fourier spectrum, wavelet 2D h x y x y xy and 3D spectra based on the mother wavelet Morle, phase and b 2 w  2 w 8a   1 h 2  2u 2 modal portraits, signs of largestLyapunov exponents. 4 1    2  The following is a scenario of transition of oscillations of a c x x 2  c 2 t 2 grid cylindrical micropolar panel from harmonic to chaotic  h 2  4u h 2  4v h 2  4u h 2  4v  (Table 1-3). l 2   1  2  2 2 2  2 3  2 4  The parameters of the experiment: l  0.5 , c  b  1 ,  b xy b x y c x y c x  3 h  0.2 ,   1 ,   a  0.2 ,   0.3 ,  p  5 , t  [0;512] , c w 2 c v 2 2  2u 4k y 1     4 1    2 2  2  3     q0 [0;200] .From the data collected in the tables it can be seen bh y b y xy that in addition to the characteristics of the deflection, the nature  2v c 2 w  2 w w  2 w 2 1     4 1    2 8  of the oscillations of the longitudinal waves should be studied, x 2 b y y 2 x yx which will allow a more accurate picture of the nature of the w  2 w 8a   1 h 2  2v oscillations of the system. At load amplitude 𝑞0 = 0.1, the 2 4   1  Fourier power spectrum for the deflection shows harmonic y x 2  b 2 t 2 oscillations, but the Lyapunov exponent for the deflection is c  w 2 4 c  w 2 4 1     6l 1    bh  4 1  l   1 positive. This discrepancy is explained by the fact that the signal 2 2  of the displacement function u has a chaotic component at low y 4 b x y 2 2 2 frequencies. Harmonics at the same frequencies are present in the b2  4w c2  1     6l 2 1    2 4  12k y2 1    2 w  deflection signal, but the Fourier spectrum does not display them. c x b These frequencies demonstrate the wavelet spectrum, so it is 2 c 2 v b 2 u c 2  w  necessary to consider the Fourier spectrum and the wavelet 12k y 1     12k y 1    2  6k y 1       spectrum together. As the load increased, a harmonic appeared bh y h x bh  y  in the signal at an independent frequency 𝜔1 . When the c 2 w  2v c2  2w c 2 v  2 w amplitude of the load 𝑞0 = 190 phase portrait of the deflection 12 1     12k y 1    w 2  12 1    2  b y y 2 2 bh y b y y 2 shows chaotic oscillations and the power spectrum of the Fourier 2 transform of the oscillations at two frequencies. Thus, to  w   2 w  2 w u  2u  w 18 1       12   1 2  12   1 2  determine the type of deflection oscillations, it is also necessary  y  y y x y x 2 to consider the function of moving by 0x or 0y. b  w   2 w  w  w  2u 2 2 6k y   1    6  5  3  2    24  Table1. The characteristics of the deflection function w and h  x  y  x  y xy the displacement function u w  2v u  2 w v  2 w 𝑞0 = 0.1, 𝜔𝑝 = 5 24  24   1  24   1  Fourierspectrum Phaseportrait x xy y xy x xy w(0.5;0.5,t) w w  2 w b 2 w  2u 24  3  5   12 1    2 -3   4 x 10 y x xy c x x 2 p 0 2 b w  2v b 2w c 2 v  2 w 12   1  12 k   1 w  12   1  w' t S c y x 2 h x 2 b 2 y x 2 y -5 0 2 -2  w   2 w b 2  w   2 w 2 6  3  5    18 1    2    -10 1 2 3 4 5 -4  y  x c  x  x 2 2  -5 0 -4 5 w u  2 w 24a   1   2 w x 10  2 w u(0.5;0.5,t) 12 1      2   2q  x x 2    -11  t t  x 10 -8 1 Boundary conditions – rigid sealing at all ends of the panel: -10 0.5 u u v v t u  v  w  0,  0,  0,  0,  0, . S w' -12 0 x y x y -0.5 -14 w w  0,  0 при x  1, y  1. -16 -1 x y 1 2 3 4 5 -10 -5 0 5 -11  w x 10 Initial conditions – zero. Table2. The characteristics of the deflection function w and The nonlinear partial differential problem in spatial the displacement function u coordinates is reduced to an ordinary differential problem by the 𝑞0 = 130, 𝜔𝑝 = 5 finite difference method with the second-order approximation of accuracy. To do this, the derivatives of spatial variables are Fourierspectrum Phaseportrait replaced by finite central differences. The Cauchy time problem w(0.5;0.5,t) is solved by the Runge-Kutta method of the fourth order of accuracy. 4 architecture and civil engineering. Series: Construction and 8  p architecture. 2014. № 37 (56). P. 121-128. (in Russian) 2 6 [4] Burnysheva T. V., Kaledin V. O. Comparison of discrete  w' t 1 and continuous approach to the calculation of the stress state of S 0 4 mesh shell structures under static loading //Journal Scientific and 2 -2 technical of the Volga region. 2011. № 4. P. 113-116. (in 0 -4 Russian) 1 2 3 4 5 -1 0 1 [5] Burnysheva T. V., Steinbrecher, O. A., Ulyanov A. D.  w Aspects of specifying boundary conditions in the simulation u(0.5;0.5,t) mesh anisogamy designs // Bulletin of the South Ural state 0.02 University. Series: Mathematical modeling and programming. 5 2018. V. 11. № 1. P. 137-144. (in Russian) 0 [6] Eremeev V. A On a nonlinear model of the mesh shell // t S 0 w' Izvestiya of the Russian Academy of Sciences. Solid mechanics. -0.02 2018. № 4. P. 127-133 (in Russian) [7] Zinin A.V., Azik N. With. The model of destruction process -5 1 2 3 4 5 -0.04 of composite structures anisakidae problems //Mechanical -0.2 -0.1 0 0.1  w engineering and reliability of machines.2018. № 5. С. 49-56. (in Russian) Table2. The characteristics of the deflection function w and [8] E. Yu. Krylova, V. I. Papkova, O. A. Saltykova, O. A. the displacement function u Sinichkina, Krys'ko V. A. Mathematical model of the 𝑞0 = 190, 𝜔𝑝 = 5 fluctuations of the size-dependent cylindrical shells mesh Fourierspectrum Phaseportrait structure given the hypotheses of the Kirchhoff-Love //non-linear w(0.5;0.5,t) world. 2018. Vol. 16. № 4. P. 17-28(in Russian) 10 [9] Krylov E. Y., Papkova I. V., Saltykova, O. A., Yakovleva  8 p T. V., Krysko V. A.-jr Analysis of the natural frequencies of the 6  micropolar, mesh for cylindrical panel of the Kirchhoff-love In w' t 1 the collection: Problems of mechanical engineering proceedings S 4 0 of the III International scientific-technical conference. In 2 parts. 2 Scientific editor P. D. Balakin. 2019. P. 278-282.(in Russian) 0 1 2 3 4 5 -10 [10] NikabadzeM.U. Some variants of the equations of the -1 0 1 micropolar theory of shells Applied mathematics and  w mathematical physics.2015. Vol. 1. № 1. P. 101-118. (in u(0.5;0.5,t) Russian) 0.04 [11] Trushin S.I., Zhuravleva T.A., Sysoeva E.V. Dynamic loss 6 of stability of nonlinear deformable grid plates made of 4 0.02 composite material with different lattice configurations // 2 w' t Scientific review. 2016. No. 4. P. 44-51. S 0 0 [12] Awrejcewicz J., Krysko V.A., Sopenko A.A., Zhigalov -2 -0.02 M.V., Kirichenko A.V., Krysko A.V. Mathematical modelling of -4 1 2 3 4 5 -0.04 physically/geometrically non-linear micro-shells with account of -0.4 -0.2 0 0.2  coupling of temperature and deformation fields // Chaos, Solitons w & Fractals. 2017. V. 104. P. 635-654. [13] Awrejcewicz J., Mrozowski J., Krysko A.V., Papkova I.V., Zakharov V.M., Erofeev N.P., Krylova E.Y., Krysko V.A. 4. Conclution Chaotic dynamics of flexible beams driven by external white A mathematical model of nonlinear oscillations of a cylindrical noise //Mechanical Systems and Signal Processing. 2016. Т. 79. panel of a grid structure is constructed. For a deep analysis of the С. 225-253. behavior of a micropolar mesh cylindrical panel, it is necessary [14] Krylova E Y, Papkova I V, Erofeev N P, Zakharov V M, to visualize the characteristics of not only the deflection function, Krysko V A Сomplex fluctuations of flexible plates under but also the displacement function, as well as to consider the longitudinal loads with account for white noise // Journal of entire apparatus of nonlinear dynamics in the aggregate. Applied Mechanics and Technical Physics. 2016. V. 57. № 4. P. 714-719. 5. Acknowledgements [15] Krylova E Yu, Papkova I V, Sinichkina A O, Yakovleva T B, Krysko-yang V A Mathematical model of flexible dimension- The study was supported by grants: RFBR №18-01-00351 а dependent mesh plates // IOP Conf. Series: Journal of Physics: and №18-41-700001 r_а. Conf. Series 1210 (2019) 012073 doi:10.1088/1742- 6596/1210/1/012073 6. 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