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.

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 A COMPOSITE LAMINATED PLATE USING THE THIRD-ORDER SHEAR DEFORMATION PLATE THEORY

2012 ◽  
Vol 22 (05) ◽  
pp. 1250103 ◽  
Author(s):  
W. ZHANG ◽  
X. Y. GUO
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Thin Plate ◽  
Internal Resonance ◽  
Plate Theory ◽  
Laminated Plate ◽  
Rectangular Thin Plate ◽  
Third Order ◽  
Composite Laminated Plate ◽  
Averaged Equations

An analysis on nonlinear oscillations and chaotic dynamics is presented for a simply-supported symmetric cross-ply composite laminated rectangular thin plate with parametric and forcing excitations in the case of 1:3:3 internal resonance. Based on Reddy's third-order shear deformation plate theory and the von Karman-type equations, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate can be established via the Hamilton's principle. Such partial differential equations are further discretized by the Galerkin method to form a three-degree-of-freedom coupled nonlinear system including the cubic nonlinear terms. The method of multiple scales is then employed to derive a set of averaged equations. Through the stability analysis, the steady-state solutions of the averaged equations are provided. An illustrative case of 1:3:3 internal resonance and fundamental parametric resonance, 1/3 subharmonic resonance is considered. Numerical simulation is applied to investigate the intrinsically nonlinear behavior of the composite laminated rectangular thin plate. With certain external load excitations, the simulation results demonstrate that the nonlinear dynamical system of the composite laminated plate exhibits different kinds of periodic and chaotic motions.

Disk Position Nonlinearity Effects on the Chaotic Behavior of Rotating Flexible Shaft-Disk Systems

2012 ◽  
Vol 28 (3) ◽  
pp. 513-522 ◽  
Author(s):  
H. M. Khanlo ◽  
M. Ghayour ◽  
S. Ziaei-Rad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Dynamic Behavior ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Disk System ◽  
Partial Differential ◽  
Rayleigh Beam ◽  
Flexible Shaft

AbstractThis study investigates the effects of disk position nonlinearities on the nonlinear dynamic behavior of a rotating flexible shaft-disk system. Displacement of the disk on the shaft causes certain nonlinear terms which appears in the equations of motion, which can in turn affect the dynamic behavior of the system. The system is modeled as a continuous shaft with a rigid disk in different locations. Also, the disk gyroscopic moment is considered. The partial differential equations of motion are extracted under the Rayleigh beam theory. The assumed modes method is used to discretize partial differential equations and the resulting equations are solved via numerical methods. The analytical methods used in this work are inclusive of time series, phase plane portrait, power spectrum, Poincaré map, bifurcation diagrams, and Lyapunov exponents. The effect of disk nonlinearities is studied for some disk positions. The results confirm that when the disk is located at mid-span of the shaft, only the regular motion (period one) is observed. However, periodic, sub-harmonic, quasi-periodic, and chaotic states can be observed for situations in which the disk is located at places other than the middle of the shaft. The results show nonlinear effects are negligible in some cases.

Nonlinear Oscillations of a Composite Laminated Plate with Parametrically and Externally Excitations

2013 ◽  
Vol 699 ◽  
pp. 641-644
Author(s):  
Xiao Li Bian ◽  
Shuang Bao Li
Keyword(s):  
Thin Plate ◽  
Nonlinear Oscillations ◽  
Plate Theory ◽  
Laminated Plate ◽  
Rectangular Thin Plate ◽  
Third Order ◽  
Composite Laminated Plate ◽  
Strain Relationship ◽  
Relationship Of ◽  
Equations Of Motions

Nonlinear oscillations of a simply-supported symmetric cross-ply composite laminated rectangular thin plate are investigated in this paper. The rectangular thin plate is subjected to the transversal and in-plane excitations. Based on the Reddy’s third-order shear deformation plate theory and the stress-strain relationship of the composite laminated plate, a two-degree-of-freedom non-autonomous nonlinear system governing equations of motions for the composite laminated rectangular thin plate is derived by using the Galerkin’s method. Numerical simulations illustrate that there exist complex nonlinear oscillations for composite laminated rectangular thin plate.

Sign in / Sign up

Export Citation Format

Share Document