Nonlinear Vibration Response Analysis on a Composite Plate Reinforced With Carbon Nanotubes

2012 ◽  
Author(s):  
Xiang-Ying Guo ◽  
Wei Zhang ◽  
Qian Wang
Keyword(s):  
Carbon Nanotubes ◽  
Differential Equations ◽  
Nonlinear Vibration ◽  
Thin Plate ◽  
Composite Plate ◽  
Laminated Plates ◽  
Vibration Response ◽  
Response Analysis ◽  
Rectangular Thin Plate ◽  
Chaotic Motions

In order to compare nonlinear vibration response of the different enabled materials in the matrix of composites, the nonlinear vibrations of a composite plate reinforced with carbon nanotubes (CNT) are studied. In this paper, the carbon nanotubes are supposed to be long fibers. The nonlinear governing partial differential equations of motion for the composite rectangular thin plate are derived by using the Reddy’s third-order shear deformation plate theory, the von Karman type equation and the Hamilton’s principle. Then, the governing equations get reduced to ordinary differential equations in thickness direction with variable coefficients and these are solved by the Galerkin method. The case of 1:1 internal resonance is considered. The asymptotic perturbation method is employed to obtain the four-dimensional averaged equations. The numerical method is used to investigate the periodic and chaotic motions of the composite rectangular thin plate reinforced with carbon nanotubes. The results of numerical simulation demonstrate that there exist different kinds of periodic and chaotic motions of the composite plate under certain conditions. At last, the nonlinear vibration responses of the plate are compared with the same responses of angle-ply composite laminated plates.

Mulit-Pulse Orbits and Chaotic Motion of Composite Laminated Plates

2008 ◽  
Author(s):  
Xiangying Guo ◽  
Wei Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Numerical Simulation ◽  
Normal Form ◽  
Differential Equations ◽  
Thin Plate ◽  
Equations Of Motion ◽  
Laminated Plates ◽  
Phase Method ◽  
Rectangular Thin Plate ◽  
Partial Differential ◽  
Chaotic Motions

This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s three-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization approach, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the energy phase method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation. The results of numerical simulation also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.

Multi-Pulse Orbits With a Melnikov Method and Chaotic Dynamics in Motion of Composite Laminated Rectangular Thin Plate

2008 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Xiang-Ying Guo ◽  
Dong-Xing Cao
Keyword(s):  
Numerical Simulation ◽  
Normal Form ◽  
Differential Equations ◽  
Thin Plate ◽  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Melnikov Method ◽  
Rectangular Thin Plate ◽  
Partial Differential ◽  
Chaotic Motions

This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s three-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization approach, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the extended Melnikov method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation. The results of numerical simulation also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.

Periodic and Chaotic Oscillations of Composite Laminated Thin Plate With Third-Order Shear Deformation

2009 ◽  
Author(s):  
W. Zhang ◽  
X. Y. Guo
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Shear Deformation ◽  
Thin Plate ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Rectangular Thin Plate ◽  
Third Order ◽  
Partial Differential ◽  
Chaotic Motions

An analysis on the nonlinear oscillations and chaotic dynamics is presented for a simply-supported symmetric cross-ply composite laminated rectangular thin plate with parametric and forcing excitations. Based on the Reddy’s third-order shear deformation plate theory and the von Karman type equation, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. The Galerkin method is utilized to discretize the governing partial differential equations of motion to a three-degree-of-freedom nonlinear system including the cubic nonlinear terms. The case of 1:3:3 internal resonance and fundamental parametric resonance, 1/3 subharmonic resonance is considered. The method of multiple scales is employed to obtain the averaged equation. The stability analysis is given for the steady-state solutions of the averaged equation. The Numerical method is used to investigate the periodic and chaotic motions of the composite laminated rectangular thin plate. The results of numerical simulation demonstrate that there exist different kinds of periodic and chaotic motions of the composite laminated rectangular thin plate under certain conditions.

Multi-Pulse Chaotic Motion for a Non-Autonomous Buckled Plate by Using the Extended Melnikov Method

2007 ◽  
Author(s):  
Wei Zhang ◽  
Jun-Hua Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Thin Plate ◽  
Chaotic Dynamics ◽  
Melnikov Method ◽  
Type Equation ◽  
Rectangular Thin Plate ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Simply Supported ◽  
Extended Melnikov Method ◽  
Two Degree Of Freedom

The multi-pulse Shilnikov orbits and chaotic dynamics for a parametrically excited, simply supported rectangular buckled thin plate are studied by using the extended Melnikov method. Based on von Karman type equation and the Galerkin’s approach, two-degree-of-freedom nonlinear system is obtained for the rectangular thin plate. The extended Melnikov method is directly applied to the non-autonomous governing equations of the thin plate. The results obtained here show that the multipulse chaotic motions can occur in the thin plate.

Active Control of a Rectangular Thin Plate Via Negative Acceleration Feedback

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
H. S. Bauomy ◽  
A. T. EL-Sayed
Keyword(s):  
Differential Equations ◽  
Thin Plate ◽  
Nonlinear Differential Equations ◽  
Perturbation Technique ◽  
Approximate Solutions ◽  
Second Order ◽  
Multiple Time ◽  
Rectangular Thin Plate ◽  
Negative Acceleration ◽  
Acceleration Feedback

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 Kármán 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 Ω1≅ω1 and  Ω2≅ω2. 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.

Stochastic Nonlinear Vibration of Fluid-Loaded Double-Walled Carbon Nanotubes

2013 ◽  
Vol 284-287 ◽  
pp. 362-366
Author(s):  
Tai Ping Chang
Keyword(s):  
Carbon Nanotubes ◽  
Differential Equations ◽  
Nonlinear Vibration ◽  
Nonlinear Differential Equations ◽  
Mean Value ◽  
Stochastic Dynamic ◽  
Governing Equations ◽  
The Mean ◽  
Double Walled Carbon Nanotubes ◽  
Walled Carbon Nanotubes

This paper investigates the stochastic dynamic behaviors of nonlinear vibration of the fluid-loaded double-walled carbon nanotubes (DWCNTs) by considering the effects of the geometric nonlinearity and the nonlinearity of van der Waals (vdW) force. The nonlinear governing equations of the fluid-conveying DWCNTs are formulated based on the Hamilton’s principle. The Young’s modulus of elasticity of the DWCNTs is assumed as stochastic with respect to the position to actually describe the random material properties of the DWCNTs. By utilizing the perturbation technique, the nonlinear governing equations of the fluid-conveying can be decomposed into two sets of nonlinear differential equations involving the mean value of the displacement and the first variation of the displacement separately. Then we adopt the harmonic balance method in conjunction with Galerkin’s method to solve the nonlinear differential equations successively. Some statistical dynamic response of the DWCNTs such as the mean values and standard deviations of the amplitude of the displacement are computed. It is concluded that the mean value and standard deviation of the amplitude of the displacement increase nonlinearly with the increase of the frequencies.

A New Kind of Energy Transfer From the High-Frequency to Low-Frequency Mode in a Composite Laminated Plate

2010 ◽  
Author(s):  
Wei Zhang ◽  
Xiang-Ying Guo ◽  
Qian Wang ◽  
Cui-Cui Liu ◽  
Yun-cheng He
Keyword(s):  
Numerical Simulation ◽  
Natural Frequency ◽  
Thin Plate ◽  
High Frequency ◽  
Low Frequency ◽  
Frequency Mode ◽  
Rectangular Thin Plate ◽  
Chaotic Motions ◽  
Simply Supported ◽  
Low Frequency Mode

This paper focuses on the analysis on a new kind of nonlinear resonant motion with the low-frequency large-amplitude, which can be induced by the high-frequency small-amplitude mode through the mechanism of modulation of amplitude and phase. The system investigated is a simply supported symmetric cross-ply composite laminated rectangular thin plate subjected to parametric excitations. Experimental research has been carried out for the first time. The test plate was excited near the first natural frequency with parametric forces and the above mentioned high-to-low frequency mode has been observed, whose frequency is extremely lower than the first natural frequency. Theoretical job goes to analysis the above phenomenon accordingly. Based on the Reddy’s third-order shear deformation plate theory and the von Karman type equation, the nonlinear governing equations of the simply supported symmetric cross-ply composite laminated rectangular thin plate subjected to parametric excitations are formulated. The Galerkin method is utilized to discretize the governing partial differential equations into a two-degree-of-freedom nonlinear system. Numerical simulation is conducted to investigate this non-autonomous system subsequently. The results of numerical simulation demonstrate that there is a qualitative agreement between the experimental observation and the theoretical result. Besides, the multi-pulse chaotic motions are also reported in numerical simulations.

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