Analysis of Non-linear Vibrations of a Fractionally Damped Cylindrical Shell Under the Conditions of Combinational Internal Resonance

2015 ◽  
pp. 59-107 ◽  
Author(s):  
Yury Rossikhin ◽  
Marina Shitikova
Keyword(s):  
Cylindrical Shell ◽  
Internal Resonance ◽  
Non Linear ◽  
Linear Vibrations

scholarly journals NUMERICAL ANALYSIS OF NON-LINEAR VIBRATIONS OF A FRACTIONALLY DAMPED CYLINDRICAL SHELL UNDER THE ADDITIVE COMBINATIONAL INTERNAL RESONANCE

2018 ◽  
Vol 14 (4) ◽  
pp. 42-58
Author(s):  
Basem Ajarmah ◽  
Marina V. Shitikova
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Differential Equations ◽  
Surrounding Medium ◽  
Non Linear ◽  
Generalized Displacements ◽  
Linear Vibrations ◽  
Fractional Differential

Non-linear damped vibrations of a cylindrical shell subjected to the additive type combinational internal resonance are investigated numerically using two different numerical methods. The damping features of the surrounding medium are described by the fractional derivative Kelvin-Voigt model involving the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential are estimated using numerical solution of nonlinear multi-term fractional differential equations by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the additive combinational internal resonance. A good agreement in results is declared

Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

1993 ◽  
Vol 4 (5) ◽  
pp. 433-460 ◽  
Author(s):  
S. I. Chang ◽  
A. K. Bajaj ◽  
C. M. Krousgrill
Keyword(s):  
Internal Resonance ◽  
Rectangular Plates ◽  
Non Linear ◽  
Linear Vibrations ◽  
One To One

scholarly journals Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance

2006 ◽  
Vol 49 (1-2) ◽  
pp. 259-284 ◽  
Author(s):  
O. Thomas ◽  
C. Touzé ◽  
É. Luminais
Keyword(s):  
Internal Resonance ◽  
Free Edge ◽  
Spherical Shells ◽  
Non Linear ◽  
Linear Vibrations

Influence of Geometry on the Non-Linear Vibrations of Cylindrical Shells With Internal Flowing Fluid

2010 ◽  
Author(s):  
Zenon J. del Prado ◽  
Paulo B. Gonc¸alves ◽  
Michael P. Pai¨doussis
Keyword(s):  
Cylindrical Shell ◽  
Degrees Of Freedom ◽  
Lateral Displacement ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Dynamic Environment ◽  
Geometric Parameters ◽  
Flowing Fluid ◽  
Non Linear ◽  
Linear Vibrations

In this work, the influence of the characteristic geometric parameters of a cylindrical shell, such as radius-to-thickness and radius-to-length ratios, on both the linear and non-linear vibrations of a fluid-filled cylindrical shell with internal flowing fluid is studied. The Donnell non-linear shallow shell equations are used to study a simply supported cylindrical shell subjected to both lateral and axial time-dependent loads with internal flowing fluid. The fluid is assumed to be inviscid and incompressible and the flow isentropic and irrotational. An expansion with eight degrees of freedom, containing the fundamental, companion, gyroscopic and five axisymmetric modes is used to describe the lateral displacement of the shell. The Galerkin method is used to obtain the nonlinear equations of motion which are, in turn, solved by the Runge-Kutta method. First, the parametric linear equations are used to study the influence of geometry and physical properties on the natural frequencies, critical flow and critical circumferential wavenumber. Secondly, numerical methods are used to describe the influence of geometric characteristics on the non-linear frequency-amplitude relations of the shell. The results obtained show the influence of the geometric parameters on the vibration characteristics of the shell and can be used as a basic tool for design of cylindrical shells in a dynamic environment.

scholarly journals Numerical analysis of non-linear vibrations of a fractionally damped cylindrical shell under the conditions of combinational internal resonance

2018 ◽  
Vol 148 ◽  
pp. 03006 ◽  
Author(s):  
Yury A. Rossikhin ◽  
Marina V. Shitikova ◽  
Basem Ajarmah
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Differential Equations ◽  
Surrounding Medium ◽  
Non Linear ◽  
Fractional Differential ◽  
Comparison Of The Results

Non-linear damped vibrations of a cylindrical shell embedded into a fractional derivative medium are investigated for the case of the combinational internal resonance, resulting in modal interaction, using two different numerical methods with further comparison of the results obtained. The damping properties of the surrounding medium are described by the fractional derivative Kelvin-Voigt model utilizing the Riemann-Liouville fractional derivatives. Within the first method, the generalized displacements of a coupled set of nonlinear ordinary differential equations of the second order are estimated using numerical solution of nonlinear multi-term fractional differential equations by the procedure based on the reduction of the problem to a system of fractional differential equations. According to the second method, the amplitudes and phases of nonlinear vibrations are estimated from the governing nonlinear differential equations describing amplitude-and-phase modulations for the case of the combinational internal resonance. A good agreement in results is declared.

scholarly journals Influence of Time Delay on Controlling the Non-Linear Oscillations of a Rotating Blade

Symmetry ◽  
2021 ◽  
Vol 13 (1) ◽  
pp. 85
Author(s):  
Yasser Salah Hamed ◽  
Ali Kandil
Keyword(s):  
Time Delay ◽  
Natural Frequency ◽  
Multiple Scales ◽  
Control Process ◽  
Control Force ◽  
Specific Level ◽  
Loop Control ◽  
Positive Displacement ◽  
Non Linear ◽  
Linear Vibrations

Time delay is an obstacle in the way of actively controlling non-linear vibrations. In this paper, a rotating blade’s non-linear oscillations are reduced via a time-delayed non-linear saturation controller (NSC). This controller is excited by a positive displacement signal measured from the sensors on the blade, and its output is the suitable control force applied onto the actuators on the blade driving it to the desired minimum vibratory level. Based on the saturation phenomenon, the blade vibrations can be saturated at a specific level while the rest of the energy is transferred to the controller. This can be done by adjusting the controller natural frequency to be one half of the blade natural frequency. The whole behavior is governed by a system of first-order differential equations gained by the method of multiple scales. Different responses are included to show the influences of time delay on the closed-loop control process. Also, a good agreement can be noticed between the analytical curves and the numerically simulated ones.

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