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Modeling method for boundary conditions of the rectangular thin plate by using the power series polynomial

ARCHIVE OF APPLIED MECHANICS 2025年第2期95卷 1-20页

作者： Cao, Yuteng Zhang, Tianxiang Li, Wenbo Bi, Qiang Beijing Informat Sci & Technol Univ Coll Mech Engn Beijing 100192 Peoples R China Beijing Inst Control Engn Natl Key Lab Sci & Technol Space Intelligent Contr Beijing 100190 Peoples R China Beijing Inst Control Engn Beijing 100094 Peoples R China

There exist complex deformation characteristics and torque transmission mechanisms of the structure whose boundary is constrained locally. This paper proposes the power series polynomial constraining method to establish the dynamical modeling of a rectangular thin plate with a localized constraint. The boundary condition can be constructed by applying the power series polynomial with undetermined coefficients to the free boundary directly. This means that the derivation of the admissible function no longer relies on the first term associated with a specific constraint. The undetermined coefficient of the power series polynomial can be obtained while calculating the weight coefficient of the admissible function by using the Rayleigh-Ritz method. Then natural frequencies are calculated and polynomial coefficients can be further obtained. The influence of variations in boundary length, constraint length, constraint position and multiple discontinuous localized constraints on natural frequencies of the plate is studied. Convergence verification is performed for the truncated number of orthogonal polynomials and power series multipliers. Then the appropriate number of the term for the power series multiplier is determined. Natural frequencies of the cantilever plate and the opposite sides simply supported plate obtained by using the proposed method are compared with those obtained using the traditional method. Then natural frequencies of the plate with a local boundary constrained are compared with those obtained from the finite element software ***. The fairly low relative error demonstrates the validity of the proposed method. The dynamical response analysis shows the superiority of the proposed method for the boundary locally constrained boundaries, which cannot be adequately handled by the traditional method.. The power series polynomial overcomes the limitation that the traditional Lagrange multiplier method can only construct point constraints.

关键词： rectangular thin plate Complex boundary Localized constraints Power series polynomials

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Study on the Self-Synchronization and Sommerfeld Effect of rectangular thin plate Vibration System Excited by Double Induction Motors

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JOURNAL OF VIBRATION ENGINEERING & TECHNOLOGIES 2024年第2 supplement期12卷 1545-1562页

作者： Kong, Xiangxi Zeng, Fanxing Li, Wenjie Kong, Fei Wu, Tong Shenyang Univ Technol Sch Mech Engn Shenyang 110870 Peoples R China

Purpose With the development of vibration equipment towards larger scales and higher frequencies, complex structural vibrations become unavoidable. This paper delves into the Sommerfeld effect and self-synchronization in a system featuring a rectangular thin plate structure excited by two non-ideal induction motors. This research can be applied to the engineering practice of large-scale vibrating machinery such as vibrating platform. Methods The exploration encompasses theoretical analyses, simulation studies, and experimental investigations. The motion differential equations of the system are established according to the Hamilton's principle. The semi-inverse method is employed to derive the modal vibration functions and frequency equations for a rectangular thin plate. The finite element method (FEM) analysis is used for verification. The synchronization conditions and stability conditions of the synchronous operation of the system are obtained according to the linear perturbation theory, and the criterion of speed capture is given. Results The synchronization state of two induction motors changes from 0 degrees to 180 degrees before and after the natural frequency of the rectangular thin plate. Changing the installation position of the induction motor, two induction motors are on the same waveform of the second-order natural frequency mode shape, and a special phase difference change will occur, that is, the natural frequency will change from 0 degrees to 180 degrees before and after, and again to 0 degrees at higher speeds, and to 180 degrees through the second-order resonance region. Conclusion The theoretical results correspond perfectly with the simulation and experiment ones, which proves the correctness of the theoretical results and the feasibility of the theoretical method. The complex interaction between Sommerfeld effect and self-synchronization phenomenon is revealed, and the influence of motor installation position on both is discussed. It is proved t

关键词： Sommerfeld effect Synchronization analysis rectangular thin plate Non-ideal induction motors Mechanical systems

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Thermal Bend of Concrete rectangular thin plate: Two Adjacent Clamped Edges, One Simply Supported Edge, One Free Edge

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ARABIAN JOURNAL FOR SCIENCE AND ENGINEERING 2018年第10期43卷 5689-5699页

作者： Cheng, Xuansheng Gong, Lijun Fu, Xuedong Lanzhou Univ Technol Sch Civil Engn Lanzhou 730050 Gansu Peoples R China

Using the superposition and small deflection theories, thermal change perpendicular to a surface is considered. A CRTP (concrete rectangular thin plate) with two adjacent CEDs (clamped edges) and one SSED (simply supported edge) and one FED (free edge) under thermal change is seen as a RTP's superposition with three SSEDs and one FED in the case of temperature change and a RTP with three SSEDs and one FED under the bending moment on two adjacent edges. Adopting the Levy method, by supposing the deflection function with undetermined parameter on the FED, the theoretical solution of a CRTP (concrete rectangular thin plate) with three SSEDs and one FED is obtained;the theoretical solution of a CRTP with three SSEDs and one FED is obtained under thermal change;taking the theoretical solution of a CRTP with four SSEDs under the bending moment on two adjacent edges, the theoretical solution of a CRTP with three SSEDs and one FED under the bending moment on two adjacent edges is obtained;the theoretical solution for the bending moment and deflection of a CRTP with two adjacent CEDs and one SSED and one FED under temperature change is obtained.

关键词： Concrete rectangular thin plate Two adjacent clamped edges and one simply supported edge and one free edge Temperature Bending moment Deflection

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Benchmark solutions of stationary random vibration for rectangular thin plate based on discrete analytical method

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PROBABILISTIC ENGINEERING MECHANICS 2017年 50卷 17-24页

作者： Chen, Guohai Zhou, Jilei Yang, Dixiong Dalian Univ Technol State Key Lab Struct Anal Ind Equipment Dept Engn Mech Int Res Ctr Computat Mech Dalian 116024 Peoples R China Shandong Univ Technol Sch Traff & Vehicle Engn Zibo 255049 Peoples R China

This paper aims to accurately and efficiently achieve the benchmark solutions of stationary stochastic responses for rectangular thin plate. Firstly, the exact solutions of free vibration for thin plate with SSSS, SSSC, SCSC, SFSF, SSSF and SCSF boundary conditions are introduced to random vibration analysis. Based on pseudo excitation method (PEM), the analytical power spectral density (PSD) functions of the transverse deflection, velocity, acceleration and stress responses for thin plate under random base acceleration excitation are derived. Subsequently, to enhance computational efficiency, the discrete analytical method (DAM) that realizes the discretization for the modal coordinates and frequency domain is proposed. Finally, the efficiency of DAM and the accuracy of benchmark solutions are scrutinized by comparison with the analytical solutions and finite element solutions. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Stationary random vibration rectangular thin plate Discrete analytical method Benchmark solutions Pseudo excitation method

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Thermal buckling and natural vibration of a rectangular thin plate with in-plane stick-slip-stop boundaries

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JOURNAL OF VIBRATION AND CONTROL 2016年第7期22卷 1950-1966页

作者： Cui, Defu Hu, Haiyan Nanjing Univ Aeronaut & Astronaut State Key Lab Mech & Control Mech Struct Nanjing 210016 Jiangsu Peoples R China Beijing Inst Technol Sch Aerosp Engn MOE Key Lab Dynam & Control Flight Vehicle Beijing 100081 Peoples R China

This study is devoted to analyzing the thermal buckling behavior and natural vibration characteristics of a uniformly heated rectangular thin plate with two adjacent in-plane frictional sliding edges. These sliding edges are initially at a stick status under the friction forces, and may slightly slip due to the thermal expansion of the plate until the sliding edges contact the stops, i.e. the bounds of the clearances. In the study, both temperature dependencies of material properties and friction coefficients are taken into consideration. For each case, the explicit equations for the buckling temperature rise and the natural frequencies of the plate are derived. Then, some parametric studies are made to reveal the influences of both normal forces and clearances on the critical buckling temperature rise and the natural vibration characteristics of the plate. The results show that both normal forces and clearances have significant influences on the critical buckling temperature rise and the natural frequencies of the plate, especially for the square plate, which has repeated natural frequencies. These results enable one to adjust the normal forces and the clearances in design in order to improve the mechanical performance of a heated rectangular thin plate.

关键词： Clearance critical buckling temperature rise in-plane stick-slip-stop boundary natural vibration normal force rectangular thin plate

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Thermal bending of rectangular thin plate with two opposite edges clamped, one edge simply supported and one edge free

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KSCE JOURNAL OF CIVIL ENGINEERING 2016年第1期20卷 333-342页

作者： Cheng, Xuansheng Fan, Jin Lanzhou Univ Technol Sch Civil Engn Lanzhou 730050 Peoples R China

The rectangular thin plate (RTP) of two opposite edges clamped, one edge simply supported and one edge free under temperature disparity is regarded as a superposition of the RTP of three simply supported edges and one edge free under temperature disparity and the RTP under the bending moment with two opposite edges. This condition is based on small deflection theory and superposition principle, with consideration of temperature variation that is perpendicular to the surface. This study obtains the deflection and bending analytical solution for the RTP of two opposite edges clamped, one edge simply supported and one edge free under temperature disparity. Numerical results show that the bending moment and deflection can converge to the exact solutions, and the calculation coefficient table based on the concrete material can provide a theoretical base for subsequent engineering design and calculation of concrete RTP.

关键词： rectangular thin plate temperature two opposite edges clamped one edge simply supported and one edge free deflection bending moment

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Active Control of a rectangular thin plate Via Negative Acceleration Feedback

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JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS 2016年第4期11卷 041025-041025页

作者： Bauomy, H. S. El-Sayed, A. T. Zagazig Univ Fac Sci Dept Math Zagazig 44519 Egypt Prince Sattam Bin Abdulaziz Univ Coll Arts & Sci Wadi Addawasir Dept Math POB 54 Wadi Addawasir 11991 Saudi Arabia Modern Acad Engn & Technol Dept Basic Sci Mokatem 11585 Egypt

In this paper, the dynamic oscillation of a rectangular thin plate under parametric and external excitations is investigated and controlled. The motion of a rectangular thin plate is modeled by coupled second-order nonlinear ordinary differential equations. The formulas of the thin plate are derived from the von Karman equation and Galerkin's method. A control law based on negative acceleration feedback is proposed for the system. The multiple time scale perturbation technique is applied to solve the nonlinear differential equations and obtain approximate solutions up to the second-order approximations. One of the worst resonance case of the system is the simultaneous primary resonances, where Omega(1) congruent to omega(1) and Omega(2) congruent to omega(2). From the frequency response equations, the stability of the system is investigated according to the Routh-Hurwitz criterion. The effects of the different parameters are studied numerically. It is also shown that the system parameters have different effects on the nonlinear response of the thin plate. The simulation results are achieved using MATLAB 7.0 software. A comparison is made with the available published work.

关键词： rectangular thin plate active control acceleration feedback stability

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Symplectic superposition method for the free-vibrating problem of sigmoid functionally graded material rectangular thin plates clamped at four edges

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JOURNAL OF VIBRATION AND CONTROL 2025年第7-8期31卷 1416-1426页

作者： Bao, Xiaoying Bai, Eburilitu Han, Lingqing Inner Mongolia Univ Sch Math Sci 235 West Daxue Rd Hohhot 010021 Peoples R China

The symplectic superposition method (SSM) is implemented to investigate the free vibration problem of sigmoid functionally graded material (FGM) rectangular thin plates clamped at four edges. Firstly, the vibration equation of the sigmoid FGM rectangular thin plate is transformed into Hamiltonian canonical equations. Then based on the analysis of boundary conditions of the plate, the vibration problem of the sigmoid FGM rectangular thin plate clamped at four edges is decomposed as two sub-problems simply supported on two opposite edges. And then based on the symplectic elasticity method (SEM), the series expansion solutions for the two sub-vibrating problems are obtained. Afterward, by superposing the series expansion solutions for the two sub-vibrating problems, a symplectic superposition solution of the freely vibrating sigmoid FGM rectangular thin plates clamped at four edges is obtained. Finally, we apply the obtained symplectic superposition solution to research the frequencies and vibration mode functions of some specific sigmoid FGM rectangular thin plates, and the impact of the aspect ratio a/b and material gradient index k on the vibration frequencies of the sigmoid FGM rectangular thin plates is studied.

关键词： Sigmoid functionally graded material rectangular thin plate Hamiltonian canonical equations symplectic superposition method free vibration

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Bending of variable thickness rectangular thin plates resting on a double-parameter foundation: integral transform solution

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ENGINEERING COMPUTATIONS 2022年第7期39卷 2689-2704页

作者： Fu, Guangming Tuo, Yuhang Sun, Baojiang Shi, Chen Su, Jian China Univ Petr East China Shandong Key Lab Oil & Gas Storage & Transportat Qingdao Peoples R China China Univ Petr East China Sch Petr Engn Qingdao Peoples R China Harbin Inst Technol Sch Ocean Engn Weihai Campus Weihai Peoples R China Univ Fed Rio de Janeiro COPPE Nucl Engn Program Rio De Janeiro Brazil

Purpose The purpose of this study is to propose a generalized integral transform technique (GITT) to investigate the bending behavior of rectangular thin plates with linearly varying thickness resting on a double-parameter foundation. Design/methodology/approach The bending of plates with linearly varying thickness resting on a double-parameter foundation is analyzed by using the GITT for six combinations of clamped, simply-supported and free boundary conditions under linearly varying loads. The governing equation of plate bending is integral transformed in the uniform-thickness direction, resulting in a linear system of ordinary differential equations in the varying thickness direction that is solved by a fourth-order finite difference method. Parametric studies are performed to investigate the effects of boundary conditions, foundation coefficients and geometric parameters of variable thickness plates on the bending behavior. Findings The proposed hybrid analytical-numerical solution is validated against a fourth-order finite difference solution of the original partial differential equation, as well as available results in the literature for some particular cases. The results show that the foundation coefficients and the aspect ratio b/a (width in the y direction to height of plate in the x direction) have significant effects on the deflection of rectangular plates. Originality/value The present GITT method can be applied for bending problems of rectangular thin plates with arbitrary thickness variation along one direction under different combinations of loading and boundary conditions.

关键词： rectangular thin plate Bending Variable thickness plate Double-parameter foundation Generalized integral transform technique

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THE NONLINEAR RANDOM VIBRATION OF A CLAMPED rectangular thin plate IN MAGNETIC FIELD 2

THE NONLINEAR RANDOM VIBRATION OF A CLAMPED RECTANGULAR THIN...

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2nd International Conference on Machinery, Materials Science and Energy Engineering (ICMMSEE)

作者： Tu JianXin Wang ZhiRen Zhu Han Wang Ping Yanshan Univ Coll Sci Qinhuangdao Peoples R China Yanshan Univ Coll Civil Engn & Mech Qinhuangdao Peoples R China

ISBN: (纸本)9783038352280

In this paper, the magneto-elastic nonlinear random vibration of a clamped rectangular thin plate in magnetic field is studied. According to the magneto-elastic theory of plates and shells and the theory of structural random vibration, the magneto-elastic nonlinear random vibration equation of a clamped rectangular thin plate in a magnetic field is derived. Then the nonlinear random vibration equation is transferred into the Ito differential equation, and the Ito differential equation is solved using FPK equation method. Thus the numerical characteristics of displacement response and velocity response of the rectangular thin plate are obtained. Finally, through a numerical example, the influences of magnetic field parameters on the numerical characteristics are discussed, and some methods which can be used to effectively control the random vibration responses of the plate are given.

关键词： magneto-elasticity nonlinearity random vibration rectangular thin plate

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