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.

DYNAMIC ANALYSIS FOR NONLINEAR VIBRATION OF PRESTRESSED ORTHOTROPIC MEMBRANES WITH VISCOUS DAMPING

2013 ◽  
Vol 13 (02) ◽  
pp. 1350018 ◽  
Author(s):  
CHANG-JIANG LIU ◽  
ZHOU-LIAN ZHENG ◽  
LONG JUN ◽  
JIAN-JUN GUO ◽  
KUI WU
Keyword(s):  
Nonlinear Vibration ◽  
Large Amplitude ◽  
Lateral Displacement ◽  
Membrane Surface ◽  
Displacement Function ◽  
Viscous Damping ◽  
Membrane Structures ◽  
Time Curve ◽  
Convenient Approach ◽  
Computational Basis

This paper is concerned with the nonlinear damped vibration of prestressed orthotropic membrane structures. The Krylov–Bogolubov–Mitropolsky (KBM) perturbation method is employed for solving the governing equations of large amplitude nonlinear vibration of rectangular orthotropic membranes with viscous damping. Presented herein are asymptotic analytical solutions for the frequency and displacement function of large amplitude nonlinear damped vibration of rectangular orthotropic membranes with four edges simply supported or fixed. Through the computational example, we compared and analyzed the frequency results. Meanwhile, the vibration mode of the membrane and the displacement and time curve of each feature point on the membrane surface were analyzed. The results obtained herein provide a simple and convenient approach to calculate the frequency and lateral displacement of large amplitude nonlinear vibration of rectangular orthotropic membranes with low viscous damping. In addition, the results provide some computational basis for the vibration control and dynamic design of membrane structures.

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 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.

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.

Analytical Solutions for Nonlinear Free Vibration of Micro-Scale Timoshenko Beams

2016 ◽  
Author(s):  
Zia Saadatnia ◽  
Ebrahim Esmailzadeh ◽  
Davood Younesian
Keyword(s):  
Free Vibration ◽  
Analytical Solutions ◽  
Equations Of Motion ◽  
Variational Iteration Method ◽  
Strain Gradient Theory ◽  
Physical Parameters ◽  
Gradient Theory ◽  
Beam Model ◽  
Governing Equations ◽  
Nonlinear Free Vibration

Nonlinear free vibration of micro-beams based on the Timoshenko beam model is studied. The governing equations of motion using the strain gradient theory, Von Kármán strain tensors and Hamiltonian principle, are developed. The Galerkin method is applied to the governing equations and the coupled nonlinear ordinary differential equations of system are obtained. The variational iteration method is utilized to determine the time responses of the micro-beam and also a close form expression for the frequency-amplitude is found. The analytical solutions obtained for different values of parameters are compared with those found from different numerical methods. The effects of geometrical and physical parameters on the dynamics of micro-beam are also examined. Moreover, the analytical formulation for frequency ratio, i.e., the ratio of nonlinear natural frequency to the linear one is obtained and the sensitivity of this ratio to the variations of various parameters is evaluated. It is proved that the proposed solution methods and the results obtained are accurate and reliable when dynamics of such micro structures are studied.

Nonlinear Free Vibration of Hyperelastic Beams Based on Neo-Hookean Model

2019 ◽  
Vol 20 (01) ◽  
pp. 2050015 ◽  
Author(s):  
Wei Chen ◽  
Lin Wang ◽  
Huliang Dai
Keyword(s):  
Free Vibration ◽  
Large Strain ◽  
Slenderness Ratio ◽  
Taylor Series Expansion ◽  
Soft Materials ◽  
Free Vibrations ◽  
Phase Portraits ◽  
Governing Equations ◽  
Nonlinear Free Vibration ◽  
Initial Displacement

The investigation of hyperelastic responses of soft materials and structures is essential for understanding of the mechanical behaviors and for the design of soft systems. In this paper, by considering both the material and geometrical nonlinearities, a new neo-Hookean model for the hyperelastic beam is developed with focus on its nonlinear free vibration with large strain deformations. The neo-Hookean model is employed to capture the large strain deformation of the hyperelastic beam. The governing equations of the hyperelastic beam are derived by using Hamilton’s principle. To avoid expensive calculations for solving the nonlinear governing equations, a simplified Taylor-series expansion model is proposed. The effects of two key system parameters, i.e. the initial displacement amplitude and the slenderness ratio, on the nonlinear free vibrations of the hyperelastic beam are numerically analyzed. The bifurcation diagrams, displacement time traces, phase portraits and power spectral diagrams are presented for the nonlinear free vibrations of the hyperelastic beam. For small initial displacement amplitudes, it is found that the hyperelastic beam will undergo limit cycle oscillations, depending on the initial amplitude employed. For initial displacement amplitudes large enough, interestingly, the free vibration of the hyperelastic beam will become quasi-periodic or chaotic, which were rarely reported for the free vibration of linearly elastic beams. Also observed is the traveling wave feature of oscillating shapes of the hyperelastic beam, indicating that higher-order modes of the beam are excited even for free vibrations. All these new features in the nonlinear free vibrations of hyperelastic beams indicate that the material and geometric nonlinearities play a great role in the dynamic analysis of hyperelastic beams.

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