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.

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.

scholarly journals Oscillation theorems for second order nonlinear differential equations with deviating arguments

1987 ◽  
Vol 10 (1) ◽  
pp. 35-45 ◽  
Author(s):  
S. R. Grace ◽  
B. S. Lalli
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Oscillatory Behaviour ◽  
Deviating Arguments ◽  
Oscillation Criteria ◽  
Oscillation Theorems

New oscillation criteria for the oscillatory behaviour of the differential(a(t)x·(t)) ·+p(t)x·(t)+q(t)f(x[g(t)])=0                ,( · =ddt)and(a(t)ψ(x(t))x·(t)) ·+p(t)x·(t)+q(t)f(x[g(t)])=0,are established

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