One-to-One Internal Resonance of Symmetric Crossply Laminated Shallow Shells

2000 ◽  
Vol 68 (4) ◽  
pp. 640-649 ◽  
Author(s):  
A. Abe ◽  
Y. Kobayashi ◽  
G. Yamada
Keyword(s):  
Internal Resonance ◽  
Shear Deformation Theory ◽  
Primary Resonance ◽  
Vibration Modes ◽  
Driving Frequency ◽  
Shallow Shells ◽  
State Response ◽  
Quadratic Nonlinearities ◽  
Galerkin's Procedure ◽  
Asymmetric Vibration

This paper presents the response of symmetric crossply laminated shallow shells with an internal resonance ω2≈ω3, where ω2 and ω3 are the linear natural frequencies of the asymmetric vibration modes (2,1) and (1,2), respectively. Galerkin’s procedure is applied to the nonlinear governing equations for the shells based on the von Ka´rma´n-type geometric nonlinear theory and the first-order shear deformation theory, and the shooting method is used to obtain the steady-state response when a driving frequency Ω is near ω2. In order to take into account the influence of quadratic nonlinearities, the displacement functions of the shells are approximated by the eigenfunctions for the linear vibration mode (1,1) in addition to the ones for the modes (2,1) and (1,2). This approximation overcomes the shortcomings in Galerkin’s procedure. In the numerical examples, the effect of the (1,1) mode on the primary resonance of the (2,1) mode is examined in detail, which allows us to conclude that the consideration of the (1,1) mode is indispensable for analyzing nonlinear vibrations of asymmetric vibration modes of shells.

Dynamic Instability and Nonlinear Vibration of Doubly-Curved Cross-Ply Shallow Shells

2001 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Khaled Alhazza
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Vibration ◽  
Dynamic Instability ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Primary Resonance ◽  
Partial Differential ◽  
Shallow Shells ◽  
Doubly Curved

Abstract We consider the dynamic instability and nonlinear vibration of doubly-curved cross-ply laminated shallow shells with simply supported boundary conditions. We investigate their responses and stability to a primary resonance (i.e., Ω ≠ ω11). The governing nonlinear partial-differential equations of motion are based on the von Karman-type geometric nonlinear theory and the first-order shear-deformation theory. We use the Galerkin procedure to reduce the governing nonlinear partial-differential equations to an infinite system of nonlinear coupled ordinary-differential equations. We use a combination of a shooting technique and Floquet theory to calculate the periodic responses of the shell and investigate their bifurcations. We show that for some shell parameters, a single-mode approximation misses some important dynamics, such as period-doubling, bifurcations.

scholarly journals Theoretical and Experimental Investigation of Two-to-One Internal Resonance in MEMS Arch Resonators

2018 ◽  
Vol 14 (1) ◽  
Author(s):  
Feras K. Alfosail ◽  
Amal Z. Hajjaj ◽  
Mohammad I. Younis
Keyword(s):  
Internal Resonance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Chaotic Motions ◽  
Different Types ◽  
Quadratic Nonlinearities ◽  
Multiple Scales Perturbation Method ◽  
Good Agreement ◽  
Perturbation Solutions

We investigate theoretically and experimentally the two-to-one internal resonance in micromachined arch beams, which are electrothermally tuned and electrostatically driven. By applying an electrothermal voltage across the arch, the ratio between its first two symmetric modes is tuned to two. We model the nonlinear response of the arch beam during the two-to-one internal resonance using the multiple scales perturbation method. The perturbation solution is expanded up to three orders considering the influence of the quadratic nonlinearities, cubic nonlinearities, and the two simultaneous excitations at higher AC voltages. The perturbation solutions are compared to those obtained from a multimode Galerkin procedure and to experimental data based on deliberately fabricated Silicon arch beam. Good agreement is found among the results. Results indicate that the system exhibits different types of bifurcations, such as saddle node and Hopf bifurcations, which can lead to quasi-periodic and potentially chaotic motions.

A third-order shear deformation theory for nonlinear vibration analysis of stiffened functionally graded material sandwich doubly curved shallow shells with four material models

2017 ◽  
Vol 21 (4) ◽  
pp. 1316-1356 ◽  
Author(s):  
Dang T Dong ◽  
Dao Van Dung
Keyword(s):  
Functionally Graded Material ◽  
Vibration Analysis ◽  
Deformation Theory ◽  
Natural Frequencies ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Third Order ◽  
Shallow Shells ◽  
Graded Material ◽  
Doubly Curved

This study presents a nonlinear vibration analysis of function graded sandwich doubly curved shallow shells, which reinforced by functionally graded material stiffeners and rested on the Pasternak foundation. The shells are subjected to the combination of mechanical, thermal, and damping loading. Four models of the sandwich shells with general sigmoid and power laws distribution are considered. The governing equations are established based on the third-order shear deformation theory. Von Kármán-type nonlinearity and smeared stiffener technique are taken into account. The explicit expressions for determining natural frequencies, nonlinear frequency–amplitude relation, and time–deflection curves are obtained by employing the Galerkin method. Finally, the fourth-order Runge–Kutta method is applied to investigate the influences of functionally graded material stiffeners, the boundary conditions, the models of the shells, thermal environment, foundation and geometrical parameters on the natural frequencies and dynamic nonlinear responses of the sandwich shells.

Bifurcations in Dynamical Systems With Internal Resonance

1978 ◽  
Vol 45 (4) ◽  
pp. 895-902 ◽  
Author(s):  
P. R. Sethna ◽  
A. K. Bajaj
Keyword(s):  
Dynamical Systems ◽  
Internal Resonance ◽  
Excitation Frequency ◽  
The Other ◽  
Hopf Bifurcations ◽  
System Of Equations ◽  
Jump Phenomena ◽  
Quadratic Nonlinearities

Dynamical systems with quadratic nonlinearities and exhibiting internal resonance under periodic excitations are studied. Two types of transition from stable to unstable motions are shown to occur. One kind are shown to be associated with jump phenomena while the other kind are shown to be associated with Hopf bifurcations of the averaged system of equations. In the case of the latter, the motions are shown to be amplitude modulated motions at the excitation frequency with the amplitude of modulation determined by the motion of a point on a torus.

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