NONLINEAR RESPONSES OF A STRING-BEAM COUPLED SYSTEM WITH FOUR-DEGREES-OF-FREEDOM AND MULTIPLE PARAMETRIC EXCITATIONS

2009 ◽  
Vol 19 (01) ◽  
pp. 225-243 ◽  
Author(s):  
D. X. CAO ◽  
W. ZHANG
Keyword(s):  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Coupled System ◽  
Dynamic Responses ◽  
Governing Equations ◽  
Chaotic Motions

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions.

scholarly journals Analysis of Nonlinear Vibrations and Dynamic Responses in a Trapezoidal Cantilever Plate Using the Rayleigh-Ritz Approach Combined with the Affine Transformation

2019 ◽  
Vol 2019 ◽  
pp. 1-23 ◽  
Author(s):  
Wei Tian ◽  
Zhichun Yang ◽  
Tian Zhao
Keyword(s):  
Nonlinear Dynamic ◽  
Affine Transformation ◽  
Internal Resonance ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Dynamic Responses ◽  
Geometrical Parameters ◽  
Cantilever Plate ◽  
Chaotic Motions ◽  
Trapezoidal Plate

Nonlinear vibrations of a trapezoidal cantilever plate subjected to transverse external excitation are investigated. Based on von Karman large deformation theory, the Rayleigh-Ritz approach combined with the affine transformation is developed to obtain the nonlinear ordinary differential equation of a trapezoidal plate with irregular geometries. With the variation of geometrical parameters, there exists the 1:3 internal resonance for the trapezoidal plate. The amplitude-frequency formulations of the system in three different coupled conditions are derived by using multiple scales method for 1:3 internal resonance analysis. It is found that the strong coupling of two modes can change nonlinear stiffness behaviors of modes from hardening-spring to soft-spring characteristics. The detuning parameter and excitation amplitude have significant influence on nonlinear dynamic responses of the system. The bifurcation diagrams show that there exist the periodic, quasi-periodic, and chaotic motions for the trapezoidal cantilever plate in the 1:3 internal resonance cases and the nonlinear dynamic responses are dependent on the amplitude of excitation. The possible adverse dynamic behaviors and undesired resonance can be avoided by designing appropriate excitation and system parameters.

Periodic and Chaotic Oscillations of an Axially Moving Viscoelastic Belt in Three-Dimensional Space

2007 ◽  
Author(s):  
Wei Zhang ◽  
Yan-Qi Liu ◽  
Li-Hua Chen ◽  
Ming-Hui Yao
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Governing Equations ◽  
Chaotic Motions ◽  
Out Of Plane ◽  
Axially Moving

Periodic and chaotic space oscillations of an axially moving viscoelastic belt with one-to-one internal resonance are investigated for the first time. The Kelvin viscoelastic model is introduced to describe the viscoelastic property of the belt material. The external damping and internal damping of the material for the axially moving viscoelastic belt are considered simultaneously. The nonlinear governing equations of motion of the axially moving viscoelastic belt for the in-plane and out-of-plane are derived by the extended Hamilton’s principle. The method of multiple scales and Galerkin’s approach are applied directly to the partial differential governing equations of motion to obtain four-dimensional averaged equation under the case of 1:1 internal resonance and primary parametric resonance of the first order modes for the in-plane and out-of-plane oscillations. Numerical method is used to investigate periodic and chaotic space motions of the axially moving viscoelastic belt. The results of numerical simulation demonstrate that there exist periodic, period-2, period-3, period-4, period-6, quasiperiodic and chaotic motions of the axially moving viscoelastic belt.

Multimode Interactions in Suspended Cables

2002 ◽  
Vol 8 (3) ◽  
pp. 337-387 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat ◽  
Char-Ming Chin ◽  
Walter Lacarbonara
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Internal Resonances ◽  
Out Of Plane ◽  
Space Formulation ◽  
Suspended Cables ◽  
One To One ◽  
Plane Mode

We investigate the nonlinear nonplanar responses of suspended cables to external excitations. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The sag-to-span ratio of the cable considered is such that the natural frequency of the first symmetric in-plane mode is at first crossover. Hence, the first symmetric in-plane mode is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and, simultaneously, in a two-to-one internal resonance with the first symmetric out-of-plane mode. Under these resonance conditions, we analyze the response when the first symmetric in-plane mode is harmonically excited at primary resonance. First, we express the two governing equations of motion as four first-order (i.e., state-space formulation) partial-differential equations. Then, we directly apply the methods of multiple scales and reconstitution to determine a second-order uniform asymptotic expansion of the solution, including the modulation equations governing the dynamics of the phases and amplitudes of the interacting modes. Then, we investigate the behavior of the equilibrium and dynamic solutions as the forcing amplitude and resonance detunings are slowly varied and determine the bifurcations they may undergo.

Theoretical and Experimental Investigation of an L-Shape Beam Structure

2008 ◽  
Author(s):  
Dong-Xing Cao ◽  
Wei Zhang ◽  
Ming-Hui Yao
Keyword(s):  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Space Station ◽  
Dynamic Responses ◽  
Flexible Beam ◽  
Beam Structure ◽  
Large Space ◽  
Chaotic Motions ◽  
Experimental Apparatus

Flexible multi-beam structures are significant components of large space station, architecture engineering and other structural systems. The understanding of the dynamic characteristics of these structures is essential for their design and control of vibrations. In this paper, the planar nonlinear vibrations and chaotic dynamics of an L-shape flexible beam structure will be investigated using theoretical and experimental methods. The L-shape beam structure considered here is composed of two flexible beams with right-angle. The governing equations of motion for the L-shape beam structure are established firstly. Then, the method of multiple scales is utilized to obtain a four-dimensional averaged equation. Numerical method is used to analyze the nonlinear dynamic responses and chaotic motions. Finally, The experimental apparatus and schemes for measuring the amplitude of nonlinear vibrations for the L-shape beam structure are introduced briefly. Then, the detailed analysis for experimental data and signals which represent the nonlinear responses of the beam structure are given.

Non-Ideal System With Quadratic Nonlinearities Containing a Two-to-One Internal Resonance

2016 ◽  
Author(s):  
Rodrigo T. Rocha ◽  
Jose M. Balthazar ◽  
D. Dane Quinn ◽  
Angelo M. Tusset ◽  
Jorge L. P. Felix
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Coupled System ◽  
Dynamical Behaviour ◽  
Main System ◽  
Weakly Coupled System ◽  
Ideal System ◽  
Quadratic Nonlinearities

The dynamical behaviour of a non-ideal three-degrees-of-freedom weakly coupled system associated with the quadratic nonlinearities in the equations of motion is investigated. The main system consists of two nonlinear mechanical oscillators coupling with quadratic nonlinearities and in which possess a 2:1 internal resonance between their translational movements. Under these conditions, we analyzed the response when a DC unbalanced motor with limited power supply (non-ideal system) excites the main system. When the excitation frequency is near to second natural frequency of the main system, saturation and jump phenomena are presented. Then, this work will analyze some torques of the motor, which causes the phenomena, and due to high amplitudes of motion will be possible to look for a way to harvest energy in a future work.

scholarly journals Nonlinear Dynamic Response of Ropeway Roller Batteries via an Asymptotic Approach

2021 ◽  
Vol 11 (20) ◽  
pp. 9486
Author(s):  
Andrea Arena
Keyword(s):  
Nonlinear Dynamic ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Dynamic Features ◽  
Response Curves ◽  
Internal Resonances ◽  
The Third ◽  
Fold Bifurcation

The nonlinear dynamic features of compression roller batteries were investigated together with their nonlinear response to primary resonance excitation and to internal interactions between modes. Starting from a parametric nonlinear model based on a previously developed Lagrangian formulation, asymptotic treatment of the equations of motion was first performed to characterize the nonlinearity of the lowest nonlinear normal modes of the system. They were found to be characterized by a softening nonlinearity associated with the stiffness terms. Subsequently, a direct time integration of the equations of motion was performed to compute the frequency response curves (FRCs) when the system is subjected to direct harmonic excitations causing the primary resonance of the lowest skew-symmetric mode shape. The method of multiple scales was then employed to study the bifurcation behavior and deliver closed-form expressions of the FRCs and of the loci of the fold bifurcation points, which provide the stability regions of the system. Furthermore, conditions for the onset of internal resonances between the lowest roller battery modes were found, and a 2:1 resonance between the third and first modes of the system was investigated in the case of harmonic excitation having a frequency close to the first mode and the third mode, respectively.

Bifurcations and Chaos of Composite Laminated Piezoelectric Rectangular Plate With One-to-Three Internal Resonance

2008 ◽  
Author(s):  
Zhi-Gang Yao ◽  
Wei Zhang
Keyword(s):  
Rectangular Plate ◽  
Multiple Scales ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Governing Equations ◽  
Third Order ◽  
Chaotic Motions ◽  
Piezoelectric Layers ◽  
Parametric And External Excitations ◽  
First Time

The bifurcations and chaotic motions of a simply supported symmetric cross-ply composite laminated piezoelectric rectangular plate are analyzed for the first time, which are forced by the transverse and in-plane excitations. It is assumed that different layers of symmetric cross-ply composite laminated piezoelectric rectangular plate are perfectly bonded to each other and with piezoelectric actuator layers embedded in the plate. Based on the Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are derived by using the Hamilton’s principle. The excitation loaded by piezoelectric layers is considered. The Galerkin’s approach is employed to discretize partial differential governing equations to a two-degree-of-freedom nonlinear system under combined the parametric and external excitations. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. Numerical method is used to find the periodic and chaotic motions of the composite laminated piezoelectric rectangular plate. The numerical results show the existence of the periodic and chaotic motions in the averaged equation. It is found that the chaotic responses are especially sensitive to the forcing and the parametric excitations. The influence of the transverse, in-plane and piezoelectric excitations on the bifurcations and chaotic behaviors of the composite laminated piezoelectric rectangular plate is investigated numerically.

scholarly journals Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

2021 ◽  
Author(s):  
Yunfei Liu ◽  
Zhaoye Qin ◽  
Fulei Chu
Keyword(s):  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Metal Foam ◽  
Multiple Scales ◽  
Porous Metal ◽  
Functionally Graded ◽  
Dynamic Responses ◽  
Elastic Media ◽  
The Galerkin Method

AbstractIn this article, the nonlinear dynamic responses of sandwich functionally graded (FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell’s nonlinear shell theory and Hamilton’s principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters, specifically, the radial load, core thickness, foam type, foam coefficient, structure damping, and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.

Nonlinear Vibrations of Two-Span Composite Laminated Plates with Equal and Unequal Subspan Lengths

2017 ◽  
Vol 9 (6) ◽  
pp. 1485-1505
Author(s):  
Lingchang Meng ◽  
Fengming Li
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Laminated Plates ◽  
Laminated Plate ◽  
Composite Laminated Plate ◽  
Vibration Localization ◽  
Composite Laminated Plates ◽  
Localization Phenomenon ◽  
Harmonic Resonance

AbstractThe nonlinear transverse vibrations of ordered and disordered two-dimensional (2D) two-span composite laminated plates are studied. Based on the von Karman's large deformation theory, the equations of motion of each-span composite laminated plate are formulated using Hamilton's principle, and the partial differential equations are discretized into nonlinear ordinary ones through the Galerkin's method. The primary resonance and 1/3 sub-harmonic resonance are investigated by using the method of multiple scales. The amplitude-frequency relations of the steady-state responses and their stability analyses in each kind of resonance are carried out. The effects of the disorder ratio and ply angle on the two different resonances are analyzed. From the numerical results, it can be concluded that disorder in the length of the two-span 2D composite laminated plate will cause the nonlinear vibration localization phenomenon, and with the increase of the disorder ratio, the vibration localization phenomenon will become more obvious. Moreover, the amplitude-frequency curves for both primary resonance and 1/3 sub-harmonic resonance obtained by the present analytical method are compared with those by the numerical integration, and satisfactory precision can be obtained for engineering applications and the results certify the correctness of the present approximately analytical solutions.

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.

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