Nonlinear Free Vibration for Cross-Ply Laminated Damaged Plates with Piezoelectric Actuators

2006 ◽  
Vol 324-325 ◽  
pp. 479-482
Author(s):  
Yu Fang Zheng ◽  
Yi Ming Fu ◽  
Kai Qi
Keyword(s):  
Free Vibration ◽  
Harmonic Balance ◽  
Piezoelectric Actuators ◽  
Laminated Plates ◽  
Governing Equations ◽  
Nonlinear Free Vibration ◽  
Response Curves ◽  
Method Of Harmonic Balance ◽  
Amplitude Frequency Response ◽  
Damage Theory

On the basis of the anisotropic damage theory and piezoelectric theory, the nonlinear free vibration governing equations for cross-ply laminated damaged plates with piezoelectric actuators are established. The Galerkin procedure furnishes an infinite system of equations for time functions which are solved by the method of harmonic balance. In the numerical results, the influences of damage parameters and piezoelectric effect on the nonlinear amplitude-frequency response curves of the laminated plates are discussed, which results reveal the inherent features about the coupled mechanics and electricity.

Effect of Piezoelectricity on Nonlinear Free Vibration of Moderately Thick Laminated Piezoelectric Plates

2011 ◽  
Vol 239-242 ◽  
pp. 1223-1226
Author(s):  
Yu Fang Zheng ◽  
Li Qiong Deng
Keyword(s):  
Free Vibration ◽  
Harmonic Balance ◽  
Infinite System ◽  
Piezoelectric Effect ◽  
Plate Theory ◽  
Laminated Plate ◽  
Governing Equations ◽  
Nonlinear Free Vibration ◽  
Method Of Harmonic Balance ◽  
Piezoelectric Plates

On the basis of laminated plate theory and piezoelectric theory, the nonlinear free vibration governing equations for symmetric cross-ply moderately thick laminated piezoelectric plates are established. The Galerkin procedure furnishes an infinite system of equations for time functions which are solved by the method of harmonic balance. In the numerical results, the influences of piezoelectric effect and various location of piezoelectric layer on the nonlinear vibrating frequency of the laminated piezoelectric plates are discussed.

Free Vibration of Buckled Laminated Plates by Finite Element Method

1997 ◽  
Vol 119 (4) ◽  
pp. 635-640 ◽  
Author(s):  
Le-Chung Shiau ◽  
Teng-Yuan Wu
Keyword(s):  
Free Vibration ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Composite Plates ◽  
Laminated Plates ◽  
Squeezing Effect ◽  
Governing Equations ◽  
Amplitude Vibration ◽  
Equilibrium Profile ◽  
Sudden Jump

Free vibration behavior of buckled composite plates are studied by using a high precision triangular plate element. This element is developed based on a simplified high order plate theory and von Ka´rma´n large deformation assumptions. The nonlinear governing equations of motion for the plates is linearized into two sets of equations by assuming small amplitude vibration of the laminates about its buckled static equilibrium profile. Results show that, in the postbuckling regime, the fundamental mode may be shifted from the first mode to the second due to squeezing effect of the in-plane force on the plate. For plate with certain boundary conditions, the natural frequency may have a sudden jump due to buckle pattern change of the plate in the postbuckling regime.

scholarly journals L-P Perturbation Solution of Nonlinear Free Vibration of Prestressed Orthotropic Membrane in Large Amplitude

2010 ◽  
Vol 2010 ◽  
pp. 1-17 ◽  
Author(s):  
Liu Chang-jiang ◽  
Zheng Zhou-lian ◽  
He Xiao-ting ◽  
Sun Jun-yi ◽  
Song Wei-ju ◽  
...  
Keyword(s):  
Free Vibration ◽  
Large Amplitude ◽  
Lateral Displacement ◽  
Membrane Surface ◽  
Displacement Function ◽  
Governing Equations ◽  
Membrane Structures ◽  
Nonlinear Free Vibration ◽  
Time Curve ◽  
Computational Basis

This paper reviewed the research on the nonlinear free vibration of pre-stressed orthotropic membrane, which is commonly applied in building membrane structures. We applied the L-P perturbation method to solve the governing equations of large amplitude nonlinear free vibration of rectangular orthotropic membranes and obtained a simple approximate analytical solution of the frequency and displacement function of large amplitude nonlinear free vibration of rectangular membrane with four edges simply supported. By giving computational examples, we compared and analyzed the frequency results. In addition, vibration mode of the membrane and displacement and time curve of each feature point on the membrane surface were analyzed in the computational example. Results obtained from this paper provide a simple and convenient method to calculate the frequency and lateral displacement of nonlinear free vibration of rectangular orthotropic membranes in large amplitude. Meanwhile, the results provide some theoretical basis for solving the response of membrane structures under dynamic loads and provide some computational basis for the vibration control and dynamic design of building membrane structures.

scholarly journals Nonlinear Free Vibration for Viscoelastic Moderately Thick Laminated Composite Plates with Damage Evolution

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Y. F. Zheng ◽  
L. Q. Deng
Keyword(s):  
Free Vibration ◽  
Transverse Shear ◽  
Superposition Principle ◽  
Laminated Composite ◽  
Composite Plates ◽  
Laminated Plates ◽  
Damage Effect ◽  
Laminated Composite Plates ◽  
Nonlinear Free Vibration ◽  
The Galerkin Method

The nonlinear free vibration for viscoelastic cross-ply moderately thick laminated composite plates under considering transverse shear deformation and damage effect is investigated. Based on the Timoshenko-Mindlin theory, strain-equivalence hypothesis, and Boltzmann superposition principle, the nonlinear free vibration governing equations for viscoelastic moderately thick laminated plates with damage are established and solved by the Galerkin method, Simpson integration, Newton-Cotes, Newmark, and iterative methods. In the numerical results, the effects of transverse shear, material viscoelasticity, span-thickness ratio, aspect ratio, and damage effect on the nonlinear free vibrating frequency of the viscoelastic cross-ply moderately thick laminated plates are discussed.

Power Series Solution of Nonlinear Free Vibration Frequency of Isotropic Rectangular Thin Plates in Large Amplitude

2011 ◽  
Vol 261-263 ◽  
pp. 883-887 ◽  
Author(s):  
Chang Jiang Liu ◽  
Zhou Lian Zheng ◽  
Wei Ju Song ◽  
Yun Ping Xu ◽  
Jun Long
Keyword(s):  
Power Series ◽  
Free Vibration ◽  
Nonlinear Vibration ◽  
Vibration Frequency ◽  
Large Amplitude ◽  
Series Solution ◽  
Thin Plates ◽  
Power Series Solution ◽  
Governing Equations ◽  
Nonlinear Free Vibration

Nonlinear vibration computational problem of isotropic thin plates in large amplitude was investigated here. We applied the Von Kármán’s theory of thin plates to derive the governing equations of nonlinear free vibration of isotropic thin plates, and solved the governing equations by direct integration method combined with power series expansion method. We obtained the power series solution of the nonlinear vibration frequency of the rectangular thin plates with four edges simply supported. Finally, the paper gave the computational example and compared the two results from the large amplitude theory and the small one, respectively. Results obtained from this paper provide a new analytical computational approach for calculating the frequency of nonlinear free vibration of isotropic thin plates in large amplitude, and provide more accurate theoretical basis for the vibration control and dynamic design of plate structures.

Nonlinear Free Vibration of a Beam With Off-Axis Attached Mass

2013 ◽  
Author(s):  
Pezhman A. Hassanpour
Keyword(s):  
Free Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
Beam Theory ◽  
Governing Equations ◽  
Attached Mass ◽  
Lumped Mass ◽  
Nonlinear Free Vibration ◽  
Euler Bernoulli Beam

A model of a clamped-clamped beam with an attached lumped mass is presented in this paper. The system is modeled using the Euler-Bernoulli beam theory. In the models presented in literature, it is assumed that the center of mass of the attached mass is located on the neutral axis of the beam. In this paper, this assumption is relaxed. The governing equations of motion are derived. It has been shown that the off-axis center of mass of the attached mass generates an amplitude-dependent transverse force in the beam, which introduces a quadratic nonlinearity. The nonlinear governing equations of motion are solved using the Multiple Scales method. The nonlinear free vibration frequencies are determined.

On Asymptotic Analysis for Large Amplitude Nonlinear Free Vibration of Simply Supported Laminated Plates

2009 ◽  
Vol 131 (5) ◽  
Author(s):  
S. K. Lai ◽  
C. W. Lim ◽  
Y. Xiang ◽  
W. Zhang
Keyword(s):  
Differential Equation ◽  
Free Vibration ◽  
Nonlinear Differential Equation ◽  
Large Amplitude ◽  
Analytical Approximation ◽  
Laminated Plates ◽  
Thin Plates ◽  
Nonlinear Free Vibration ◽  
Simply Supported ◽  
Analytical Approximations

An analytical approximation is developed for solving large amplitude nonlinear free vibration of simply supported laminated cross-ply composite thin plates. Applying Kirchhoff’s hypothesis and the nonlinear von Kármán plate theory, a one-dimensional nonlinear second-order ordinary differential equation with quadratic and cubic nonlinearities is formulated with the aid of an energy function. By imposing Newton’s method and harmonic balancing to the linearized governing equation, we establish the higher-order analytical approximations for solving the nonlinear differential equation with odd nonlinearity. Based on the nonlinear differential equation with odd and even nonlinearities, two new nonlinear differential equations with odd nonlinearity are introduced for constructing the analytical approximations to the nonlinear differential equation with general nonlinearity. The analytical approximations are mathematically formulated by combining piecewise approximate solutions from such two new nonlinear systems. The third-order analytical approximation with better accuracy is proposed here and compared with other numerical and approximate methods with respect to the exact solutions. In addition, the method presented herein is applicable to small as well as large amplitude vibrations of laminated plates. Several examples including large amplitude nonlinear free vibration of simply supported laminated cross-ply rectangular thin plates are illustrated and compared with other published results to demonstrate the applicability and effectiveness of the approach.

scholarly journals Theoretical Analysis on the Nonlinear Free Vibration of a Tri-Cross String

2017 ◽  
Vol 2017 ◽  
pp. 1-17
Author(s):  
Bo Pan ◽  
Jingda Tang ◽  
Ryuichi Tarumi ◽  
Fulin Shang ◽  
Yanbo Wang ◽  
...  
Keyword(s):  
Constitutive Equation ◽  
Theoretical Analysis ◽  
Free Vibration ◽  
Natural Frequency ◽  
Analytical Solutions ◽  
Vibration Amplitude ◽  
Natural Frequencies ◽  
Geometrical Nonlinearity ◽  
Governing Equations ◽  
Nonlinear Free Vibration

Here we present a theoretical analysis on the nonlinear free vibration of a tri-cross string system, which is an element of space net-antennas. We derived the governing equations from Hamilton’s principle and obtained a linearized solution by the standard perturbation method. The semi-analytical solutions of the governing equations have not been provided referring to the solution of plate vibrating problem. This analysis revealed that natural frequencies of the tri-cross string depend on the vibration amplitude due to the geometrical nonlinearity in the constitutive equation. The geometric parameters, such as the diameters and the lengths of the constituent strings, also affect the frequency through the nonlinearity of the tri-cross string. The nonlinear natural frequency shows coupled characteristic; that is, the natural frequency of the tri-cross string varies with that of the constituent strings, but the contribution of each constituent string to the natural frequency is in different proportions.

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