Journal articles: 'Non-linear vibrations of plate'

1

Boukhzer, A. Abdenbi, Khalid El Bikri, and Benamar Rhali. "Homogenization Technique for Non-Linear Free Vibrations Analysis of FGM Rectangular Plates." Advanced Materials Research 971-973 (June 2014): 516–33. http://dx.doi.org/10.4028/www.scientific.net/amr.971-973.516.

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In the present study, the problem of geometrically nonlinear free vibrations of functionally graded rectangular plates (FGRP) is studied. A homogenization technique has been developed to reduce the FGRP problem under consideration to that of isotropic homogeneous rectangular plate. The material properties of the functionally graded composites examined herein are assumed to be graded in the thickness direction of the plate and estimated through the rule of mixture. The proposed theoretical model is based on the classical plate theory and the Von Karman relationships, and the amplitude equation is derived in the form of a set of non-linear algebraic equation using Hamilton’s principle and a multimode approach. The fundamental nonlinear frequency parameters and the bending stress are then calculated using the iterative and explicit methods of solution to show the effect of the vibration amplitudes and the material distributions. The results obtained in this study are found to be in a good agreement with the published ones dealing with the problem of large vibration of functionally graded plates.

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Rachid, El Kaak, El Bikri Khalid, and Benamar Rhali. "Geometrically Nonlinear Free Axisymmetric Vibrations Analysis of Thin Circular Functionally Graded Plates." Advanced Materials Research 971-973 (June 2014): 489–506. http://dx.doi.org/10.4028/www.scientific.net/amr.971-973.489.

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This paper deals with nonlinear free axisymmetric vibrations of functionally graded thin circular plates whose properties vary through its thickness. The inhomogeneity of the plate is characterized by a power law variation of the Young’s modulus and mass density of the material along the thickness direction, whereas Poisson’s ratio is assumed to be constant. The theoretical model is based on Hamilton’s principle and spectral analysis using a basis of admissible Bessel’s functions to yield the frequencies of the circular plates under clamped boundary conditions on the basis of the classical plate theory. The large vibration amplitudes problem, reduced to a set of non-linear algebraic equations, is solved numerically. The non-linear to linear frequency ratios are presented for various values of the volume fraction index n showing hardening type non-linearity. The distribution of the radial bending stress associated to the non-linear mode shape is also given for various vibration amplitudes, and is compared with those predicted by the linear theory.

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Chuaqui, Tomás R. C., and Pedro Ribeiro. "On displacement based non-local models for non-linear vibrations of thin nano plates." MATEC Web of Conferences 148 (2018): 07004. http://dx.doi.org/10.1051/matecconf/201814807004.

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This paper addresses the formulation of displacement based, non-linear, plate models adopting Eringen's non-local elasticity, to study the modes of vibration of thin, nano plates. Plate models governed by ordinary differential equations of motion with generalized displacements as unknowns have some advantages over mixed type formulations, but difficulties arise in the development of such non-linear models when non-local effects are taken into account. To circumvent those difficulties, approximations of debatable justification can be imposed. Different approximations are discussed here and the accuracy of the best non-local, non-linear displacement based model achieved is put to test, by carrying out comparisons with a model based on Airy’s stress function.

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Lee, W. K., and M. H. Yeo. "Non-linear interactions in asymmetric vibrations of a circular plate." Journal of Sound and Vibration 263, no. 5 (June 2003): 1017–30. http://dx.doi.org/10.1016/s0022-460x(03)00271-2.

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5

Asadi, Hamed, Mohammad M. Aghdam, and Mahmoud Shakeri. "Vibration analysis of axially moving line supported functionally graded plates with temperature-dependent properties." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 6 (July 26, 2013): 953–69. http://dx.doi.org/10.1177/0954406213498033.

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Vibration analysis of axially moving functionally graded plates with internal line supports and temperature-dependent properties is investigated using harmonic differential quadrature method. The plate is subjected to static in-plane forces while out-of-plane loading is dynamic. Stability of an axially moving plate, traveling at a constant velocity between different supports and experiencing small transverse vibrations are considered. The series of internal rigid line supports parallel to the plate edges are considered together with various arbitrary combinations of boundary conditions. Material properties of the plate are assumed temperature-dependent which is a non-linear function of temperature and differ continuously through thickness according to a power-law distribution of the volume fractions of the plate constituents. Two types of micromechanical models, namely, the Voigt and Mori–Tanaka models are considered. Based on the classical plate theory, the governing equations are obtained for functionally graded plate using the Hamilton’s principle. In the frame of a general dynamic analysis, it is shown that the onset of instability takes place in the form of divergence. The plate may experience divergence or flutter instability at a super critical velocity. Results for dynamic analysis of isotropic and laminated plates are validated with available data in the existing literature, which show excellent agreement. Furthermore, some new results are presented for vibration analysis of functionally graded material plates to study effects of the location of line supports, material properties, volume fraction, temperature, loading, aspect ratio and speed.

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Rachid, El Kaak, Khalid El Bikri, and Benamar Rhali. "A Semi-Analytical Study of Geometrically Nonlinear Free Axisymmetric Vibrations of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution." Applied Mechanics and Materials 704 (December 2014): 131–36. http://dx.doi.org/10.4028/www.scientific.net/amm.704.131.

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. This paper deals with nonlinear free axisymmetric vibrations of functionally graded thin circular plates (FGCP) whose properties vary through its thickness. The inhomogeneity of the plate is characterized by a power law variation of the Young’s modulus and mass density of the material along the thickness direction, whereas Poisson’s ratio is assumed to be constant. The theoretical model is based on Hamilton’s principle and spectral analysis using a basis of admissible Bessel’s functions to yield the frequencies of the circular plates under clamped boundary conditions on the basis of the classical plate theory. The large vibration amplitudes problem, reduced to a set of non-linear algebraic equations, is solved numerically. The non-linear to linear frequency ratios are presented. Then, explicit analytical solutions are presented, based on the semi-analytical model previously developed by EL Kadiri et al. [1-2] for beams and rectangular plates, which allow direct and easy calculation for the first non-linear axisymmetric mode shape, with their associated non-linear frequencies of FG circular plates and which are expected to be very useful in engineering applications and in further analytical developments. An excellent agreement is found with the results obtained by the iterative method.

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Chandrakar, Bhupesh K., NK Jain, and Ankur Gupta. "Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 236, no. 3 (December 20, 2021): 1406–29. http://dx.doi.org/10.1177/09544062211019776.

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The present work aims to study the non-linear vibrations in a cracked orthotropic tapered micro-plate. Linear and parabolic variation in the plate thickness is assumed in one as well as two directions. The partial crack is located in the centre, and it is continuous; this crack’s location is arbitrary and can be varied within the centre-line. Based on classical plate theory, the equilibrium principle is applied, and the governing equation of tapered orthotropic plate is derived. Additionally, the microstructure’s effect has been included in the governing equation using the non-classical modified couple stress theory. The simplified line spring model is used to consider the impact of partial crack on the plate dynamics and is incorporated using in-plane forces and bending moments. The introduction of Berger’s formulation brings the non-linearity in the model in terms of in-plane forces. Here, Galerkin’s method has been chosen for converting the derived governing equation into time-dependent modal coordinates, which uses an approximate solution technique to solve the non-linear Duffing equation. The crack is considered along the fibres and across the fibres to show the effect of orthotropy. Results are presented for an orthotropic cracked plate with non-uniform thickness. The effects of the variation of taper constants, crack location, crack length, internal material length scale parameter on the fundamental frequency are obtained for two different boundary conditions. The non-linear frequency response curves are plotted to show the effect of non-linearity on the system dynamics using the method of multiple scales, and the contribution of taper constants and crack parameters on non-linearity is shown with bending-hardening and bending-softening phenomenon. It has been found that vibration characteristics are affected by the taper parameters and fibre direction for a cracked orthotropic plate.

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8

Abdeddine, E., A. Majid, Z. Beidouri, and Kh Zarbane. "Experimental investigation for non-linear vibrations of free supported and cantilever FFF rectangular plates." Archives of Materials Science and Engineering 116, no. 2 (August 1, 2022): 49–56. http://dx.doi.org/10.5604/01.3001.0016.1189.

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The aim of this paper is to investigate experimentally the effect of large vibration of a cantilever and a fully free rectangular plate made by a Fused Filament Fabrication process. Furthermore, this investigation attempts to compare our measurements and those obtained in the literature experimentally. For this purpose, a test rig was designed and manufactured for all experimental trials. The plate was excited randomly and harmonically at large displacement respectively, to obtain the linear and non-linear frequencies parameter. The non-linear dynamic behaviour of our structure at forced vibration is figured by exciting the plate at large displacement. The dependence of frequency and amplitude vibration are examined for the first, second, and third mode shapes. The non-linear dynamic behaviour of cantilever plates is compared with literature to illustrate the convergence of our results by using our specific mechanical properties, printing parameters, and process. Furthermore, the non-dimensional comparison is shown by 33.38%, 5.83%, and 20.58% for the first, second, and third mode shapes, respectively. Experimental tests will be performed on a 3D-printed metal plate to improve the present work. This work is intended to determine the dynamic proprieties of our parts in order to manufacture a safe and comfort machine. Actually, the dynamic behaviour of our 3D printing plates is compared with the obtained in the case of the isotropic plate for the aim to predict the convergence of both structures.

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9

Ioannides, E., and P. Grootenhuis. "A Finite Element Analysis of the Harmonic Response of Damped Five-Layer Plates." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 199, no. 4 (October 1985): 311–17. http://dx.doi.org/10.1243/pime_proc_1985_199_128_02.

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Solutions have been obtained for the harmonic vibrations of five-layer plates by means of a finite element method. This method is an extension of a previously developed analysis for three-layer plates. The five-layer plates contain two constrained viscoelastic layers which provide the damping. The degenerate case when the thickness of the middle elastic layer becomes zero and the plate is reduced to a four-layer one has also been included in the solution procedure. Moreover, the method allows for the study of both torsional and transverse vibrations of five (or four)-layer beams treated as vibrating plates with a large aspect ratio. As in the case of three-layer plates, triangular finite elements are used to allow for a greater variety of shapes. In the analysis damping is introduced by replacing the real moduli of the viscoelastic material by complex equivalent moduli which account for the phase difference between strain and stress. The present method allows for the non-linear stress-strain behaviour of the viscoelastic layers, the effects of the rotatory inertia and the extension within the viscoelastic layers. The finite element computations have been verified by comparison with experimental results for four-layer and five-layer beams in transverse and torsional vibrations and a five-layer square plate in transverse vibration.

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10

Rajagopal, S. V., G. Singh, Y. V. K. Sadasiva Rao, and S. Narayanan. "Non-linear vibrations of sandwich plates." Journal of Sound and Vibration 110, no. 2 (October 1986): 261–69. http://dx.doi.org/10.1016/s0022-460x(86)80209-7.

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11

Al-Rajihy, Ahmed A. "THE AXISYMMETRIC DYNAMICS OF ISOTROPIC CIRCULAR PLATES WITH VARIABLE THICKNESS UNDER THE EFFECT OF LARGE AMPLITUDES." Journal of Engineering 14, no. 01 (March 1, 2008): 2302–13. http://dx.doi.org/10.31026/j.eng.2008.01.18.

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This paper presents a study of the geometrically non-linear vibrations of clamped circular plates with variable thickness by taking the effect of large amplitude motion. The maximum thickness is considered to be at the plate center and it is taken to be twice the value of thickness at the edge. The problem is solved by the numerical iteration procedure to obtain the results of vibration amplitudes up to twice the maximum plate thickness. The results are presented for the first two modes of vibration. The obtained results indicate that increasing the ratio of thickness has the effect of increasing the nonlinear frequency and modify the corresponding mode shape.

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12

Kou, Haijiang, and Huiqun Yuan. "Rub-induced non-linear vibrations of a rotating large deflection plate." International Journal of Non-Linear Mechanics 58 (January 2014): 283–94. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.10.005.

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13

Ji, Ming, Yi-Chuang Wu, and Chien-Ching Ma. "In-plane-dominated vibration characteristics of piezoelectric thick circular plates based on higher-order plate theories." Journal of Mechanics 38 (2022): 410–32. http://dx.doi.org/10.1093/jom/ufac034.

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ABSTRACT Numerous engineering applications exist for the piezoelectric effect, which results from the electromechanical coupling between electrical and mechanical fields. In-plane vibrations of piezoelectric plates’ resonance frequencies and associated mode shapes have been thoroughly investigated. However, analytical solutions for in-plane-dominated vibrations of thick piezoelectric circular plates are limited. In this paper, higher-order plate theories for the in-plane-dominated vibration characteristics of piezoelectric circular thick plates under fully clamped and completely free boundary conditions are presented. The resonant frequencies and associated mode shapes were investigated based on two higher-order plate theories: second-order shear deformation plate theory and third-order shear deformation plate theory, as well as simplified third-order linear piezoelectric theory. Hamilton's principle was applied to derive equations of motion and boundary conditions. In the theoretical analysis, the resonant frequencies, associated mode shapes and distribution of electric displacements for various radius-to-thickness ratios were calculated. The numerical results obtained by the finite element method were compared with those obtained from theoretical analysis. Excellent agreement was found between the theoretical and numerical results for the thick piezoelectric circular plates.

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14

Bambill, D. V., and C. A. Rossit. "Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading." Journal of Engineering 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/937596.

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As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.

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15

Maslov, Alexander, and Janar Batsaikhan. "The research of the operating mode of the concrete mixture plane depth compactor with a circular vibration exciter." ACADEMIC JOURNAL Series: Industrial Machine Building, Civil Engineering 1, no. 52 (July 5, 2019): 5–12. http://dx.doi.org/10.26906/znp.2019.52.1662.

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The paper proposes an essentially new design of a concrete mixture depth compactor made in the form of a vertical plate witha circular vibration exciter mounted on its upper part. A design diagram of the «plane depth compactor – concrete mixture»dynamic system is presented and the vertical plate movement equations describing its linear vibrations in the horizontal planeand torsional vibrations in relation to the center of gravity are provided. It has been determined the regularity of the verticalplate motion during the compaction of concrete mixtures. The provided results of the research enable the substantiation of therational parameters of the plane depth compactor performing spatial vibrations and the efficient modes of the vibratory actionon the concrete mixtures of different consistence.

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16

Pyatetska, O. V. "Bending vibrations of viscoelastic plates within the Kirchhoff-Love model." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 4 (2019): 64–67. http://dx.doi.org/10.17721/1812-5409.2019/4.8.

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Within the framework of the hypotheses of the classical Kirchhoff-Love theory, complete systems of resolving equations are constructed to determine the stress-strain state and the temperature of dissipative heating under steady transverse vibrations of plates made of a linear viscoelastic material, the properties of which depend on the frequency of external excitation and temperature. The equations were obtained without any preliminary suggestions about the law of temperature variation over the plate thickness. This law is determined in the process of solving the problem. The unrelated problem of vibrational bending of viscoelastic plates for complicated way of fixing a contour and different types of thermal boundary conditions is considered. Mathematical models of problems on the steady-state transverse vibrations of plates made of a linear viscoelastic material, the properties of which depend on temperature for an arbitrary law of its change over the thickness of the object. If the material characteristics depend on temperature, investigation of the influence of temperature of dissipative heating is reduced to solution of complicated non-linear systems of differential equations.

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17

Awrejcewicz, J., I. V. Papkova, E. U. Krylova, and V. A. Krysko. "Wavelet-Based Analysis of the Regular and Chaotic Dynamics of Rectangular Flexible Plates Subjected to Shear-Harmonic Loading." Shock and Vibration 19, no. 5 (2012): 979–94. http://dx.doi.org/10.1155/2012/658298.

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We investigate non-linear dynamics of flexible rectangular plates subjected to external shear harmonic load action. We show that an application of the classical and widely used Fourier analysis does not allow to obtain real picture of the frequency vibration characteristics in each time instant. On the other hand, we show that application of the wavelets approach allows to follow frequency time evolutions. Our numerical results indicate that vibrations in different plate points occur with the same frequencies set although their power is different. Hence, the vibration characteristics can be represented by one arbitrary taken plate point. Furthermore, using wavelets scenarios of transitions from regular to chaotic dynamics are illustrated and discussed including two novel scenarios not reported so far in the existing literature.

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Singh, G., and Y. V. K. Sadasiva Rao. "Non-linear vibrations of thick composite plates." Journal of Sound and Vibration 115, no. 2 (June 1987): 367–71. http://dx.doi.org/10.1016/0022-460x(87)90481-0.

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19

Sherif, H. A. "Non-linear forced flexural vibrations of a clamped circular unsymmetrical sandwich plate." Journal of Sound and Vibration 182, no. 3 (May 1995): 495–503. http://dx.doi.org/10.1006/jsvi.1995.0212.

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YEO, M. H., and W. K. LEE. "CORRECTED SOLVABILITY CONDITIONS FOR NON-LINEAR ASYMMETRIC VIBRATIONS OF A CIRCULAR PLATE." Journal of Sound and Vibration 257, no. 4 (October 2002): 653–65. http://dx.doi.org/10.1006/jsvi.2002.5037.

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21

Gupta, Arun Kumar, and Shanu Sharma. "Free Transverse Vibration of Orthotropic Thin Trapezoidal Plate of Parabolically Varying Thickness Subjected to Linear Temperature Distribution." Shock and Vibration 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/392325.

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The present paper deals with the free transverse vibration of orthotropic thin trapezoidal plate of parabolically varying thickness inx-direction subjected to linear temperature distribution inx-direction through a numerical method. The deflection function is defined by the product of the equations of the prescribed continuous piecewise boundary shape. Rayleigh-Ritz method is used to evaluate the fundamental frequencies. The equations of motion, governing the free transverse vibrations of orthotropic thin trapezoidal plates, are derived with boundary condition CSCS. Frequency corresponding to the first two modes of vibration is calculated for the orthotropic thin trapezoidal plate having CSCS edges for different values of thermal gradient, taper constant, and aspect ratio. The proposed method is applied to solve orthotropic thin trapezoidal plate of variable thickness with C-S-C-S boundary conditions. Results are shown by figures for different values of thermal gradient, taper constant, and aspect ratio for the first two modes of vibrations.

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22

Lal, R., and Yajuvindra Kumar. "Boundary Characteristic Orthogonal Polynomials in the Study of Transverse Vibrations of Nonhomogeneous Rectangular Plates with Bilinear Thickness Variation." Shock and Vibration 19, no. 3 (2012): 349–64. http://dx.doi.org/10.1155/2012/694746.

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The free transverse vibrations of thin nonhomogeneous rectangular plates of variable thickness have been studied using boundary characteristic orthogonal polynomials in the Rayleigh-Ritz method. Gram-Schmidt process has been used to generate these orthogonal polynomials in two variables. The thickness variation is bidirectional and is the cartesian product of linear variations along two concurrent edges of the plate. The nonhomogeneity of the plate is assumed to arise due to linear variations in Young's modulus and density of the plate material with the in-plane coordinates. Numerical results have been computed for four different combinations of clamped, simply supported and free edges. Effect of the nonhomogeneity and thickness variation with varying values of aspect ratio on the natural frequencies of vibration is illustrated for the first three modes of vibration. Three dimensional mode shapes for all the four boundary conditions have been presented. A comparison of results with those available in the literature has been made.

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23

Rasangika, Ambagaha Hewage Dona Kalpani, Mohammad Shakir Nasif, and Rafat Al-Waked. "Comparison of Forced Convective Heat-Transfer Enhancement of Conventional and Thin Plate-Fin Heat Sinks under Sinusoidal Vibration." Applied Sciences 13, no. 21 (October 31, 2023): 11909. http://dx.doi.org/10.3390/app132111909.

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Applying sinusoidal vibration to heat sinks has proven to be a promising technique for improving heat transfer by disrupting the thermal boundary layer. However, applying sinusoidal vibration to the base of thin plate-fin heat sinks can cause a flapping motion within the fins, further enhancing heat transfer. Therefore, the current study numerically investigates and compares the effects of sinusoidal vibrations on the thermal performance of conventional and thin plate-fin heat sinks. The study concludes that increased vibrational amplitude and frequency (f ˃ 30 Hz) increases the vibration-assisted thermal performance. It was found that the thin plate-fin heat sink provides higher thermal performance compared to the conventional heat sink at every level of vibrational characteristics. The study found that the application of vibration enhances the Nusselt number up to a maximum of 20% and 15% in thin plate-fin and conventional heat sinks, respectively. Furthermore, the Reynolds number is reduced by 33.3% and 28% with thin plate-fin and conventional heat sinks compared with non-vibrating heat sinks, indicating a potential reduction of the size of the cooling system or fin size.

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VeisiAra, Abdollah, Hamid Mohammad-Sedighi, and Arash Reza. "Computational analysis of the nonlinear vibrational behavior of perforated plates with initial imperfection using NURBS-based isogeometric approach." Journal of Computational Design and Engineering 8, no. 5 (September 15, 2021): 1307–31. http://dx.doi.org/10.1093/jcde/qwab043.

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Abstract In this article, an isogeometric analysis through NURBS basis functions is presented to study the nonlinear vibrational behavior of perforated plates with initial imperfection. In this regard, the governing equations of plate dynamics, as well as the displacement–strain relations, are derived using the Mindlin–Reissner plate theory by considering von Karman nonlinearity. The geometry of the structure is formed by selecting the order of NURBS basis functions and the number of control points according to the physics of the problem. Since similar basis functions are utilized to estimate the accurate geometry and displacement field of the domain, the order of the basic functions and the number of control points are optimized for the proper approximation of the unknown field variables. By utilizing the energy approach and Hamilton principle and discretizing the equations of motion, the vibrational response of the perforated imperfect plate is extracted through an eigenvalue problem. The results of linear vibrations, geometrically nonlinear vibrations, and nonlinear vibrations of imperfect plates are separately validated by considering the previously reported findings, which shows a satisfactory agreement. Thereafter, a coefficient of the first mode shape is considered as the initial imperfection and the vibrational analysis is reexamined. Furthermore, the nonlinear vibrations of the perforated plate with initial imperfection are analysed using an iterative approach. The effects of the perforated hole, initial imperfection, and geometric nonlinearity are also addressed and discussed.

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Hamdani, M., M. El Kadiri, and R. Benamar. "Non-linear vibrations of a simply supported rectangular plate carrying a centric mass." IOP Conference Series: Materials Science and Engineering 1070, no. 1 (February 1, 2021): 012112. http://dx.doi.org/10.1088/1757-899x/1070/1/012112.

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Yang, X. L., and P. R. Sethna. "Non-linear phenomena in forced vibrations of a nearly square plate: Antisymmetric case." Journal of Sound and Vibration 155, no. 3 (June 1992): 413–41. http://dx.doi.org/10.1016/0022-460x(92)90710-f.

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Hamdani, Mustapha ,., Mounia El Kadiri, and Rhali Benamar. "Geometrically Non-Linear Dynamic Behavior of Simply Supported Rectangular Plates Carrying a Concentrated Mass." WSEAS TRANSACTIONS ON APPLIED AND THEORETICAL MECHANICS 17 (December 31, 2022): 175–83. http://dx.doi.org/10.37394/232011.2022.17.22.

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Simply supported plates carrying an added point mass are encountered in many engineering fields, like circuit boards or slabs carrying machines at different locations. Determination of the plate modified dynamic characteristics is a quite laborious task, especially in the non-linear regime, which is rarely treated in the literature. The added mass effect on the plate linear parameters was first examined using Hamilton’s principle and spectral analysis. The modified plate's non-linear fundamental mode was then calculated and its non-linear response to high levels of harmonic excitation was determined. The non-linear formulation, involving a fourth order tensor due to the membrane forces induced in the plate mid-plane by large vibration amplitudes, led to a non-linear algebraic amplitude equation. The iterative solution gave the free vibration case a better qualitative understanding and a quantitative evaluation of the effect of the added mass. The non-linear forced response of the modified plate, examined for a wide frequency range, shows that the added eccentric mass induces changes in the area between the mass location and the simple supports and decreases the non-linear hardening effect. The numerical results, covering new situations, are expected to be useful in engineering applications necessitating for some reason the addition to the plate of a point mass or an adaptation of the plate frequencies in order to avoid the occurrence of undesirable resonances.

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Saood, Ahmad, Sanan H. Khan, and Abdel Hamid I. Mourad. "Influence of Fiber Angle on the Steady State Periodic Response of Annular Sectorial Plates." Journal of Physics: Conference Series 2478, no. 2 (June 1, 2023): 022019. http://dx.doi.org/10.1088/1742-6596/2478/2/022019.

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Abstract Structural elements in the form of annular sectorial plates are widely used in aeronautical, biomedical, and marine engineering. When these components are exposed to dynamic load, large-amplitude vibrations occur. The vibration response analysis based on the linear strain-displacement relation typically yields a conservative estimate and can be used as a first approximation to the actual solution for thin structural elements. As a result, geometric nonlinearity must be incorporated for the efficient and fail-proof design of such elements. This paper shows non-linear and linear steady-state forced vibration responses of the annular sectorial plates. The governing equations of motion have been solved in the time domain by using a modified shooting method and an arc-length/pseudo-arc length continuation technique at the bifurcation point to obtain the complete response curve comprised of stable and unstable branches. This work investigates the effect of fibre angle on the non-linear steady-state forced vibration response of annular sectorial plates. The strain/stress fluctuation throughout the thickness of the laminated annular sectorial plate is determined to explain why hardening nonlinearity has increased. The cyclic fluctuation of the non-linear steady-state normal stress during a time period at the centre of the top and bottom surface is also provided in relation to the forcing frequency ratio of peak amplitude in the non-linear response. Due to the change in restoring forces, the frequency spectra reveal much increased harmonic involvement along with the fundamental harmonic for all fibre angles.

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Xia, Z. Q., and S. Łukasiewicz. "Non-linear, Free, Damped Vibrations Of Sandwich Plates." Journal of Sound and Vibration 175, no. 2 (August 1994): 219–32. http://dx.doi.org/10.1006/jsvi.1994.1324.

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Xu, L., and L. Sun. "Electromechanical coupled non-linear vibration of the microplate." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 224, no. 6 (June 1, 2010): 1383–96. http://dx.doi.org/10.1243/09544062jmes1620.

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In this article, the non-linear electromechanical coupled dynamic equation of the micro-plate subjected to an electrostatic force is presented. The non-linear free vibration of the electromechanical coupled micro-plate system and the non-linear forced response of the system to voltage excitation are investigated. Non-linear vibration frequencies of the micro-plate for every mode are calculated and compared with the linear natural frequencies. Changes in the non-linear frequencies with the system parameters are presented. The non-linear vibration displacements of the micro-plate at some typical cross-section are calculated for the first three modes. Effects of the system parameters on the displacements are analysed. The changes in the vibration amplitudes with respect to the exciting frequency under different modes and parameters are investigated. Moreover, the effects of the boundary conditions on the electromechanical coupled micro-plate system are also investigated.

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Lal, Roshan, and Rahul Saini. "On radially symmetric vibrations of functionally graded non-uniform circular plate including non-linear temperature rise." European Journal of Mechanics - A/Solids 77 (September 2019): 103796. http://dx.doi.org/10.1016/j.euromechsol.2019.103796.

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FEDOTENKOV, Grigory V., Andrey V. GRITSKOV GRITSKOV, Dmitry Y. LEVITSKIY, Yana A. VAHTEROVA, and Ying SUN. "Timoshenko beam and plate non-stationary vibrations." INCAS BULLETIN 13, S (August 3, 2021): 41–56. http://dx.doi.org/10.13111/2066-8201.2021.13.s.5.

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The problems of Timoshenko beams and plates lateral vibrations under the influence of unsteady loads are considered. Both beam and plate is supposed to be unlimited. In case of the plate the problem has been simply studied. The approach to the solution was based on dominant function method and principle of superposition. Integral models of solutions with cores as dominant functions were built which could be analytically found with the help of the Fourier and Laplace integral transforms. Two original analytical methods for Fourier and Laplace transforms were offered and realized. The examples of calculations were given.

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33

Popov, V. S., A. A. Popova, and A. V. Christoforova. "Non-Linear Oscillations of the Channel Wall Filled with a Viscous Liquid Induced by the Vibrating Foundation." Journal of Physics: Conference Series 2182, no. 1 (March 1, 2022): 012058. http://dx.doi.org/10.1088/1742-6596/2182/1/012058.

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Abstract The paper considers the issues of mathematical modelling nonlinear vibrations for the wall of a narrow parallel walled channel filled with a viscous incompressible liquid. The upper channel wall is a rigid rectangular plate supported by nonlinear spring with a cubic restoring force, while the bottom one is a fixed rigid rectangular plate. The case of nonlinear oscillations of the upper channel wall due to channel’s foundation vibration has been investigated in the frame of hydroelasticity problem. The main attention is paid to the consideration of steady-state nonlinear oscillations for the upper channel wall and the creeping motion of the liquid in the channel. The liquid layer reaction acting on the upper channel wall is found and the channel’s wall nonlinear oscillation equation is obtained taking into account the energy dissipation due to the liquid viscosity. It was shown that this equation coincided with the Duffing one. The solution of the nonlinear oscillation equation was carried out by the harmonic balance method. Based on the found solution, the hydroelastic response of the channel wall on a nonlinear elastic suspension to the channel foundation vibration was determined.

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Warmiński, Jerzy, and Andrzej Teter. "Non-linear parametric vibrations of a composite column under uniform compression." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 226, no. 8 (February 10, 2012): 1921–38. http://dx.doi.org/10.1177/0954406212438494.

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Parametric oscillations of a prismatic, thin-walled composite column with a channel section are considered in the article. The simply supported column is made of a seven-layer composite with a symmetric ply alignment. The non-linear problem of buckling is solved with Koiter’s asymptotic theory within the first-order approximation by adoption of a plate model. The asymptotic approximation leads to non-linear equations allowing evaluation of the two mode buckling effect. Parametric vibrations, produced by a periodically changing load component, are investigated near the principal parametric resonances. The approximate analytical solutions are determined by the multiple time scales of method. The influence of amplitude and frequency of parametric excitation on the structure response is investigated. An example bifurcation scenario and a possible transition to chaotic oscillations are also presented.

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35

Challamel, Noël, Huu Phu Nguyen, Chien Ming Wang, and Giuseppe Ruta. "In-Plane Vibrations of Elastic Lattice Plates and Their Continuous Approximations." Mathematics 12, no. 15 (July 24, 2024): 2312. http://dx.doi.org/10.3390/math12152312.

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This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding continuous plate. The considered lattice converges to a 2D linear isotropic elastic continuum at the asymptotic limit for a sufficiently small lattice spacing. This continuum has a free Poisson’s ratio, which must be lower than that foreseen by the rare-constant theory, to preserve the definite positiveness of the associated discrete energy. Exact solutions for the in-plane eigenfrequencies and modes are analytically derived for the discrete plate. The stiffness characterising the lattice interactions at the boundary is corrected to preserve the symmetry properties of the discrete displacement field. Two classes of constraints are considered, i.e., sliding supports at the nodes, one normal and the other parallel to the boundary. For both boundary conditions, a single equation for the eigenfrequency spectrum is derived, with two families of eigenmodes. Such behaviour of the lattice plate is like that of the continuous plate, the eigenfrequency spectrum of which has been given by Rayleigh. The convergence of the spectrum of the lattice plate towards the spectrum of the continuous plate from below is confirmed. Two continuous size-dependent plate models, considering the strain gradient elasticity and non-local elasticity, respectively, are built from the lattice difference equations and are shown to approximate the plane lattice accurately.

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Khakimov, A. G. "Determining the velocity of a moving three-layered plate and the thickness of its filler using natural frequencies of flexural vibrations." Proceedings of the Mavlyutov Institute of Mechanics 12, no. 1 (2017): 109–14. http://dx.doi.org/10.21662/uim2017.1.016.

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Research has been performed on natural transverse vibrations of a portion of a constant length in a straight threelayered plate with a filler moving along the neutral line of a non-deformed state. The movement takes place between two rigidly fixed coaxial guides (clamps), the distance between them equalling the length of the vibrating portion. The longitudinal force is assumed to constantly act along the neutral line. It has been found that an increase in the natural frequencies of flexural vibrations of the plate occurs with an increase in the thickness of the filler. Using two frequencies of flexural vibrations, we can determine the velocity of the moving three-layered plate and the thickness of its filler. The results of the research work can find technological use in the problems on dynamics and strength of machines and mechanisms in manufacturing three-layered plate with fillers and can be used to determine the velocity of a plate and the thickness of its filler using two frequencies of flexural vibrations.

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Tubaldi, E., F. Alijani, and M. Amabili. "Non-linear vibrations and stability of a periodically supported rectangular plate in axial flow." International Journal of Non-Linear Mechanics 66 (November 2014): 54–65. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.12.004.

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Żur, Krzysztof, and Piotr Jankowski. "Multiparametric Analytical Solution for the Eigenvalue Problem of FGM Porous Circular Plates." Symmetry 11, no. 3 (March 22, 2019): 429. http://dx.doi.org/10.3390/sym11030429.

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Free vibration analysis of the porous functionally graded circular plates has been presented on the basis of classical plate theory. The three defined coupled equations of motion of the porous functionally graded circular/annular plate were decoupled to one differential equation of free transverse vibrations of plate. The one universal general solution was obtained as a linear combination of the multiparametric special functions for the functionally graded circular and annular plates with even and uneven porosity distributions. The multiparametric frequency equations of functionally graded porous circular plate with diverse boundary conditions were obtained in the exact closed-form. The influences of the even and uneven distributions of porosity, power-law index, diverse boundary conditions and the neglected effect of the coupling in-plane and transverse displacements on the dimensionless frequencies of the circular plate were comprehensively studied for the first time. The formulated boundary value problem, the exact method of solution and the numerical results for the perfect and imperfect functionally graded circular plates have not yet been reported.

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Mazur, Krzysztof, and Marek Pawełczyk. "Active Noise Control with a Single Nonlinear Control Filter for a Vibrating Plate with Multiple Actuators." Archives of Acoustics 38, no. 4 (December 1, 2013): 537–45. http://dx.doi.org/10.2478/aoa-2013-0063.

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Abstract Vibrating plates can be used in Active Noise Control (ANC) applications as active barriers or as secondary sources replacing classical loudspeakers. The system with vibrating plates, especially when nonlinear MFC actuators are used, is nonlinear. The nonlinearity in the system reduces performance of classical feedforward ANC with linear control filters systems, because they cannot cope with harmonics generated by the nonlinearity. The performance of the ANC system can be improved by using nonlinear control filters, such as Artificial Neural Networks or Volterra filters. However, when multiple actuators are mounted on a single plate, which is a common practice to provide effective control of more vibration modes, each actuator should be driven by a dedicated nonlinear control filter. This significantly increases computational complexity of the control algorithm, because adaptation of nonlinear control filters is much more computationally demanding than adaptation of linear FIR filters. This paper presents an ANC system with multiple actuators, which are driven with a single nonlinear filter. To avoid destructive interference of vibrations generated by different actuators the control signal is filtered by appropriate separate linear filters. The control system is experimentally verified and obtained results are reported.

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Ribeiro, Pedro, and Hamed Akhavan. "Non-linear vibrations of variable stiffness composite laminated plates." Composite Structures 94, no. 8 (July 2012): 2424–32. http://dx.doi.org/10.1016/j.compstruct.2012.03.025.

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41

Pyatetska, O. "Numerical-analytic solution of the problem about the damping isothermal vibrations of viscoelastic rectangular plates." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 3 (2018): 77–80. http://dx.doi.org/10.17721/1812-5409.2018/3.11.

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The damping of vibrations of rectangular plates by means of both viscoelastic layers and using piezoelectric inclusions is considered. For modeling viscoelastic properties of passive and piezoelectric materials, linear models of integral type viscoelasticity are used, which are most effective for simulating the dissipative properties of materials in the linear region. In the case of taking into account the influence of the piezoelectric inclusions on the rigid characteristics of the passive plate and in other types of boundary conditions (for example, with rigid fixing of the ends), the finite element method was used to solve the problem of damping. The solutions of concrete problems of damping of stationary and non-stationary vibrations of plates using analytical and finite element methods are given. On the basis of the aforementioned approach, algorithms for solving dynamical problems with both fully and partially electrodes are implemented.

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Lal, Roshan, and Renu Saini. "Mode shapes and frequencies of thin rectangular plates with arbitrarily varying non-homogeneity along two concurrent edges." Journal of Vibration and Control 23, no. 17 (January 22, 2016): 2841–65. http://dx.doi.org/10.1177/1077546315623710.

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Analysis and numerical results are presented for free transverse vibrations of isotropic rectangular plates having arbitrarily varying non-homogeneity with the in-plane coordinates along the two concurrent edges on the basis of Kirchhoff plate theory. For the non-homogeneity, a general type of variation for Young’s modulus and density of the plate material has been assumed. Generalized differential quadrature method has been used to obtain the eigenvalue problem for such model of plates for four different combinations of boundary conditions at the edges namely, (i) fully clamped, (ii) two opposite edges are clamped and other two are simply supported, (iii) two opposite edges are clamped and other two are free, and (iv) two opposite edges are simply supported and other two are free. By solving these eigenvalue problems using software MATLAB, the lowest three eigenvalues have been reported as the first three natural frequencies for the first three modes of vibration. The effect of various plate parameters on the vibration characteristics has been analysed. Three dimensional mode shapes have been plotted. A comparison of results with those available in literature has been presented.

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Papkov, S. O. "Vibrations of a Cantilevered Thick Plate." PNRPU Mechanics Bulletin, no. 2 (December 15, 2021): 106–17. http://dx.doi.org/10.15593/perm.mech/2021.2.10.

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It has been for the first time that an analytical solution to the problem of free vibrations of a cantilevered thick orthotropic plate is presented. This problem is quite cumbersome for using the exact methods of the theory of elasticity; therefore, methods based on the variational approach were developed to solve it. The paper suggests using the superposition method to construct a general solution of the vibration equations of a plate in the series form of particular solutions obtained with the help of a variables separation. The particular solutions of one of the coordinates are built in the form of trigonometric functions of a special type (modified trigonometric system). The constructed solution, in contrast to the solutions known in the literature on the basis of the variational approach, accurately satisfies the equations of vibrations. The use of a modified trigonometric system of functions makes it possible to obtain uniform formulas for even and odd vibration shapes and to reduce the quantity of boundary conditions on the plate sides from twelve to nine ones, while five of the nine boundary conditions are also accurately satisfied. The structure of the presented solution on the plate boundary is such that, each of the kinematic or force characteristics of the plate is represented as a sum of two series, i.e. a trigonometric series and a series in hyperbolic functions. Remaining boundary conditions make it possible to obtain an infinite system of linear algebraic equations with respect to the unknown coefficients of the series representing the solution. The convergence of the solution by the reduction method of the infinite system is investigated numerically. Examples of the numerical implementation are given; numerical studies of the spectrum of natural frequencies of the cantilevered thick plate were carried out based on the obtained solution, both with varying elastic characteristics of the material and with varying geometric parameters.

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44

Zamanian, M., S. E. Khadem, and S. N. Mahmoodi. "Analysis of non-linear vibrations of a microresonator under piezoelectric and electrostatic actuations." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 223, no. 2 (December 1, 2008): 329–44. http://dx.doi.org/10.1243/09544062jmes1147.

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In this article, the non-linear vibrations of a piezoelectrically actuated microresonator is studied. The microresonator is assumed as a clamped—clamped Bernouli—Euler microbeam. In contrast to previous researches in which the piezoelectric layer has been deposited on the integral length of the microbeam, here it is assumed that the piezoelectric layer is deposited on a part of microbeam length with equal distance from two ends. The microbeam is actuated by an AC voltage between upper and lower sides of the piezoelectric layer. Also, an electrostatic actuation is applied between the microbeam and an electrode plate, for the first time. The non-linear equation of motion has been derived by using the Hamilton principle by stretching the neutral axis assumption. The obtained equations are solved using the Galerkin, Rayleigh—Ritz, and multiple scale perturbation methods. It is shown that the sensitivity and natural frequency of the piezoelectrically actuated microresonator may be altered and controlled conveniently by applying an electrostatic actuation to the microresonator. Also, it has been shown that the system shows softening or hardening behaviour depending on the value of piezoelectric actuation, electrostatic actuation, axial load, thickness, length, and elasticity module of piezoelectric layer; thickness of microbeam; and its distance from the electrode plate. It is shown that non-linear behaviour of the piezoelectrically actuated microbeam may be changed to a linear behaviour by applying a suitable electrostatic actuation to the microbeam.

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Ng, C. F. "Testing Techniques for Chaotic Vibration of Buckled Aircraft Structures." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 210, no. 3 (July 1996): 281–90. http://dx.doi.org/10.1243/pime_proc_1996_210_371_02.

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A simple model of non-linear and chaotic dynamic behaviour of buckled plates is presented. Special procedures are described for setting up the tests of flat and buckled plates with vibrations of large amplitude. The difficulties of measuring chaotic motion include the problems in designing the clamping fixtures, controlling the excitation and analysing the results. Case studies of tests with shaker direct attachment, base excitation, and acoustic excitation are used to illustrate the procedures and ways to solve the various problems. The important non-linear dynamic characteristics of curved plates are found to be the transition from softening spring effects to hardening spring effects through the intermediate region of chaotic motion. The chaotic motion is a large amplitude dynamic snap-through motion between the two static buckled positions of the plate.

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KOUCHORO, Biaou Jean-Baptiste, Anissa MEZIANE, Philippe MICHEAU, Mathieu RéNIER, and Nicolas QUAEGEBEUR. "Experimental analysis of Local Defect Resonance in an aluminum plate." INTER-NOISE and NOISE-CON Congress and Conference Proceedings 270, no. 7 (October 4, 2024): 4295–302. http://dx.doi.org/10.3397/in_2024_3440.

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In the context of non-destructive evaluation and testing of structures, the use of Local Defect Resonance is promising for the detection of delamination-type defects in composites. In this context, an experimental analysis Local Defect Resonance is proposed. It is here carried out on a model system: a thin aluminum plate with a Flat Bottom Hole (FBH) defect consisting in a self-adhesive aluminum sheet (a shim) adhesively bonded over a circular through hole. First, the local defect resonance frequencies and mode shapes are analyzed assuming linear vibrations of the structure. Then, a nonlinear approach is used to compare two types of FBH defects, a first flawless one and a second one including a damaged area. For this purpose a high amplitude vibration around the resonance frequencies are used in order to generate high order harmonics. This enhancement is identified as associated with mainly quadratic nonlinearity.

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Noor, Ahmed K., and W. Scott Burton. "Three-Dimensional Solutions for the Free Vibrations and Buckling of Thermally Stressed Multilayered Angle-Ply Composite Plates." Journal of Applied Mechanics 59, no. 4 (December 1, 1992): 868–77. http://dx.doi.org/10.1115/1.2894055.

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Analytic three-dimensional elasticity solutions are presented for the free vibration and buckling of thermally stressed, multilayered, angle-ply composite plates. Sensitivity derivatives are also evaluated and used to study the sensitivity of the vibration and buckling responses to variations in the different lamination and material parameters of the plate. The plates are assumed to have rectangular geometry and an antisymmetric lamination with respect to the middle plane. The temperature is assumed to be independent of the surface coordinates, but has an arbitrary symmetric variation through the thickness of the plate. A linear, Duhamel-Neumann type constitutive model is used, and the material properties are assumed to be independent of temperature. The thermal plate response is subjected to time-varying perturbation displacements, strains, and stresses. A mixed formulation is used with the fundamental unknowns consisting of the six perturbation stress components and the three perturbation displacement components of the plate. The initial thermal deformations are accounted for. Each of the plate variables is decomposed into symmetric and antisymmetric components in the thickness direction, and is expressed in terms of a double Fourier series in the Cartesian surface coordinates. Numerical results are presented showing the effects of variations in material characteristics and fiber orientation of different layers, as well as the effects of initial thermal deformations on the vibrational and buckling responses of the plate, as well as their sensitivity derivatives.

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Nageswara Rao, B., and S. R. R. Pillai. "Non-linear vibrations of a simply supported rectangular antisymmetric cross-ply plate with immovable edges." Journal of Sound and Vibration 152, no. 3 (February 1992): 568–72. http://dx.doi.org/10.1016/0022-460x(92)90490-o.

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Hanane, Moulay Abdelali, Khalid El Bikri, and Benamar Rhali. "Geometrically Non-Linear Free Vibration of Fully Clamped FGM Skew Plates Using Homogenization Technique." Advanced Materials Research 1105 (May 2015): 370–80. http://dx.doi.org/10.4028/www.scientific.net/amr.1105.370.

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The present work concerns the geometrically non-linear free vibration of fully clamped functionally graded skew plates (FGSP). The theoretical model based on Hamilton’s principle and spectral analysis is used. A homogenization technique has been developed to reduce the FGSP problem under consideration to that of an isotropic homogeneous skew plate. The material properties of the skew plate examined herein are assumed to be graded in the thickness direction of the plate according to the power-law distribution in terms of volume fractions of the constituents. Results are given for the linear and non-linear fundamental frequency considering different parameters. The non-linear mode shapes exhibit a maximum value in the bending stress at the centre of the plate. It is found also that the non-linear frequencies increase with increasing the amplitude of vibration and increasing the skew angle, which corresponds to the hardening type effect. A good agreement is found with published results.

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EL KADIRI, M., R. BENAMAR, and R. G. WHITE. "THE NON-LINEAR FREE VIBRATION OF FULLY CLAMPED RECTANGULAR PLATES: SECOND NON-LINEAR MODE FOR VARIOUS PLATE ASPECT RATIOS." Journal of Sound and Vibration 228, no. 2 (November 1999): 333–58. http://dx.doi.org/10.1006/jsvi.1999.2410.

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Journal articles on the topic 'Non-linear vibrations of plate'

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