Nonlinear Vibrations of Viscoelastic Heterogeneous Solids of Finite Size: Internal Resonances and Modes Interactions

2021 ◽  
pp. 141-152
Author(s):  
Igor V. Andrianov ◽  
Vladyslav Danishevskyy ◽  
Jan Awrejcewicz
Keyword(s):  
Nonlinear Vibrations ◽  
Finite Size ◽  
Internal Resonances ◽  
Heterogeneous Solids

Internal Resonance and Nonlinear Vibrations of Eccentric Rotating Composite Laminated Circular Cylindrical Shell

2019 ◽  
Author(s):  
Tao Liu ◽  
Wei Zhang ◽  
Yan Zheng ◽  
Yufei Zhang
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Shear Deformation Theory ◽  
Internal Resonances

Abstract This paper is focused on the internal resonances and nonlinear vibrations of an eccentric rotating composite laminated circular cylindrical shell subjected to the lateral excitation and the parametric excitation. Based on Love thin shear deformation theory, the nonlinear partial differential equations of motion for the eccentric rotating composite laminated circular cylindrical shell are established by Hamilton’s principle, which are derived into a set of coupled nonlinear ordinary differential equations by the Galerkin discretization. The excitation conditions of the internal resonance is found through the Campbell diagram, and the effects of eccentricity ratio and geometric papameters on the internal resonance of the eccentric rotating system are studied. Then, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equations in the case of 1:2 internal resonance and principal parametric resonance-1/2 subharmonic resonance. Finally, we study the nonlinear vibrations of the eccentric rotating composite laminated circular cylindrical shell systems.

Analysis of Nonlinear Free Vibration of Circular Plates With Cut-Outs Using R-Function Method

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
K. V. Avramov ◽  
O. Tyshkovets ◽  
K. V. Maksymenko-Sheyko
Keyword(s):  
Function Method ◽  
Degrees Of Freedom ◽  
Nonlinear Vibrations ◽  
Ritz Method ◽  
Multiple Scale ◽  
Geometrically Nonlinear ◽  
Circular Plates ◽  
Internal Resonances ◽  
Geometrically Nonlinear Vibrations ◽  
Three Degrees Of Freedom

Geometrically nonlinear vibrations of circular plate with two cut-outs are simulated by the von Karman equations with respect to displacements. The combination of the Rayleigh–Ritz method and the R-function method, which allows satisfying all boundary conditions, is applied to obtain the vibration modes of the plate. The nonlinear vibrations are expanded using these vibrations modes. The dynamical system with three degrees of freedom is derived by Galerkin method. The influence of cut-outs size on linear and nonlinear vibrations of the plate is analyzed. For different parameters of cut-outs, different internal resonances occur in the plate. The nonlinear vibrations of the system for different internal resonances are analyzed by multiple scale method.

A beam–ring circular truss antenna restrained by means of the negative speed feedback procedure

2021 ◽  
pp. 107754632110036
Author(s):  
Ashraf T EL-Sayed Taha ◽  
Hany S Bauomy
Keyword(s):  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Nonlinear Partial Differential Equations ◽  
Time Frame ◽  
Ring Structure ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Averaged Equations ◽  
Fractional Differential

The present article contemplates the nonlinear powerful exhibitions of affecting dynamic vibration controller over a beam–ring structure for demonstrating the circular truss antenna exposed to mixed excitations. The dynamic controller comprises the included negative speed input added to the framework’s idea. By using the statue, Hamilton, the nonlinear fractional differential administering conditions of movement and the limit conditions have inferred for the shaft ring structure. Through Galerkin’s method, the nonlinear partial differential equations referred to overseeing the movement of the shaft ring structure have diminished to a coupled normal differential equations extending the nonlinearities square terms. Multiple time scales have helped in acquiring (getting) the four-dimensional averaged equations for measuring the primary and 1:2 internal resonances. This article’s controlled assessment is useful for controlling the nonlinear vibrations of the considered framework. Likewise, the controller dispenses with the framework’s oscillations in a brief time frame. The demonstrations of the numerous coefficients and the framework directed at the examined resonance case have been determined. Using MATLAB 7.0 programs has aided in completing the simulation results. At last, the numerical outcomes displayed an admirable concurrence with the methodical ones. A comparison with recently available articles has also indicated good results through using the presented controller.

scholarly journals Vibrations of nonlinear elastic lattices: low- and high-frequency dynamic models, internal resonances and modes coupling

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190532 ◽  
Author(s):  
Igor V. Andrianov ◽  
Vladyslav V. Danishevskyy ◽  
Graham Rogerson
Keyword(s):  
High Frequency ◽  
Degrees Of Freedom ◽  
Dynamic Models ◽  
Finite Size ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Internal Resonances ◽  
Energy Transfers ◽  
Fourth Order Method ◽  
Nonlinear Elastic

We aim to study how the interplay between the effects of nonlinearity and heterogeneity can influence on the distribution and localization of energy in discrete lattice-type structures. As the classical example, vibrations of a cubically nonlinear elastic lattice are considered. In contrast with many other authors, who dealt with infinite and periodic lattices, we examine a finite-size model. Supposing the length of the lattice to be much larger than the distance between the particles, continuous macroscopic equations suitable to describe both low- and high-frequency motions are derived. Acoustic and optical vibrations are studied asymptotically by the method of multiple time scales. For numerical simulations, the Runge–Kutta fourth-order method is employed. Internal resonances and energy exchange between the vibrating modes are predicted and analysed. It is shown that the decrease in the number of particles restricts energy transfers to higher-order modes and prevents the equipartition of energy between all degrees of freedom. The conditions for a possible reduction in the original nonlinear system are also discussed.

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