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.

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.

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.

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.

A New Multi-Pulse Chaotic Motion for Parametrically Excited Viscoelastic Axially Moving String

2005 ◽  
Author(s):  
Wei Zhang ◽  
Ming-Hui Yao ◽  
Li-Lai Bai
Keyword(s):  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Three Dimensional ◽  
Constitutive Law ◽  
Chaotic Motions ◽  
Parametrically Excited ◽  
Dimensional Phase Space ◽  
Axially Moving ◽  
First Time

In this paper, the Shilnikov type multi-pulse orbits and chaotic dynamics of parametrically excited viscoelastic moving string are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equation of motion for viscoelastic moving string with the external damping and parametric excitation is given. The four-dimensional averaged equation under primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of viscoelastic moving string. The Shilnikov type multi-pulse chaotic motions of viscoelastic moving string are also found by using numerical simulation. A new phenomenon on the multi-pulse jumping orbits and a new strange attractor are observed from three-dimensional phase space for the first time.

Three-Dimensional Dynamic Stress and Vibration Analyses of Thick Singular-Kernel Fractional-Order Viscoelastic Annular Rotating Discs Under Nonuniform Loads

2019 ◽  
Vol 20 (01) ◽  
pp. 2050007 ◽  
Author(s):  
M. Shariyat ◽  
R. Mohammadjani
Keyword(s):  
Fractional Order ◽  
Dynamic Stress ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Theory Of Elasticity ◽  
Constitutive Law ◽  
Sensitivity Analyses ◽  
Governing Equations

In this paper, the dynamic stress and radial/lateral vibration of circular/annular discs made of fractional-order viscoelastic materials under nonuniform mechanical loads are investigated for the first time, utilizing the exact 3D theory of elasticity, rather than the plate theories. The governing equations of motion of the disc are derived based on the Kelvin–Voigt fractional viscoelastic model. To solve these equations, the spatial partial and the time ordinary derivatives are replaced by adequate central, backward or forward finite difference expressions. Then the resulting Caputo-type time-dependent system of the coupled integro-differential governing equations of the fractional-order is solved by a novel numerical procedure. Namely, a time-marching procedure is employed to extract the time histories of the responses, in the space-time domain for various time and spatial distributions. Finally, comprehensive sensitivity analyses and various 3D plots are presented and discussed. In this regard, effects of the fractional-order of the constitutive law, viscoelastic parameters, material rigidity, distribution and time variation patterns of the nonuniform distributed transverse loads, and boundary conditions on the distributions of the displacement and stress components are investigated.

scholarly journals A Nonlinear Analysis of the Whirling Motions of Slender Beams Under Various Resonant Excitations

1991 ◽  
Author(s):  
In-Ming K. Shyu ◽  
Dean T. Mook ◽  
Raymond H. Plaut
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Approximate Solutions ◽  
Plane Motion ◽  
Static Deflection ◽  
Governing Equations ◽  
Principal Moments Of Inertia ◽  
Out Of Plane ◽  
Lateral Displacements

Abstract The response of a slender, elastic, cantilevered beam to a simple harmonic excitation is investigated. The nonlinear equations governing the motion of the beam are essentially those derived earlier by Crespo da Silva and Glynn (1978). In the current study, the governing equations are extended to include static deflection. These equations not only include nonlinear curvature, nonlinear inertia, inextensionality and static deflection, but also include torsional displacement. In all cases, the torsional displacement is written in terms of lateral displacements and eliminated from the governing equations. Previous derivations of equations of motion contain only the linear and cubic terms without consideration of the static displacement produced by the weight of the beam. As a result of this static deflection, there are quadratic terms in the governing equations, which introduce the possibility of superharmonic and subharmonic resonances of order two. The partial-differential equations of motion are converted into a system of coupled ordinary-differential equations in time by the application of Galerkin’s procedure. Approximate solutions of the temporal equations are determined by the method of multiple scales. The analysis reveals that only the in-plane modes directly excited by a primary or secondary resonance and the out-of-plane modes excited by an internal resonance are involved in the first approximation of the response. The amplitudes of the other modes decay. Under some circumstances, there is no steady-state (constant amplitude, constant phase) response. Instead, the amplitude and phase are slowly modulated. Under some circumstances, the modulations are harmonic and produce discrete side bands around the fundamental frequency. For other circumstances, the modulations are chaotic. Both stable and unstable whirling motions are found in every resonance when the principal moments of inertia of the cross-section are approximately equal. The longer the beam is, the more prominent the whirling motion becomes. The accuracy of some of the approximate solutions is verified by numerical integration. The analysis reveals some interesting possibilities: For example, in a subharmonic resonance of order two, it is possible for the out of plane motion to have a frequency that is exactly one half that of the in-plane motion, which has a frequency equal to that of the excitation. It is also possible for the frequency of the in-plane motion to be equal to that of the out-of-plane motion, which is the one half frequency of the excitation.

scholarly journals Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Bamadev Sahoo ◽  
L. N. Panda ◽  
G. Pohit
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Time History ◽  
Mean Value ◽  
Phase Portraits ◽  
Axially Moving Beam ◽  
Internal Resonances ◽  
Moving Beam ◽  
Axially Moving ◽  
The Stability

The nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.

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.

Linear Dynamics of an Elastic Beam Under Moving Loads

2000 ◽  
Vol 122 (3) ◽  
pp. 281-289 ◽  
Author(s):  
G. Visweswara Rao
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Moving Load ◽  
Mass Parameter ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Mass Effect ◽  
Moving Loads ◽  
Parameter Ratio ◽  
Load Mass

The dynamic response of an Euler-Bernoulli beam under moving loads is studied by mode superposition. The inertial effects of the moving load are included in the analysis. The time-dependent equations of motion in modal space are solved by the method of multiple scales. Instability regions of parametric resonance are identified and the moving mass effect is shown to significantly affect the transient response of the beam. Importance of modal interaction arising out of the possible internal resonance is highlighted. While the external resonance is due to the gravity effects of the moving load, the parametric and internal resonance solely depends on the load mass parameter—ratio of the moving load mass to the beam mass. Numerical results show the influence of the load inertia terms on the beam response under either a single moving load or a series of moving loads. [S0739-3717(00)01703-7]

Effects of Tuned Mass Damper on Fixed-Fixed 3D Nonlinear String Resting on Nonlinear Elastic Foundation

2017 ◽  
Vol 17 (04) ◽  
pp. 1750047 ◽  
Author(s):  
Yi-Ren Wang ◽  
Li-Ping Wu
Keyword(s):  
Elastic Foundation ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Maximum Amplitude ◽  
Tuned Mass Damper ◽  
Mode Shapes ◽  
Nonlinear Elastic Foundation ◽  
Mass Damper ◽  
Nonlinear Elastic

This paper studies the vibration of a nonlinear 3D-string fixed at both ends and supported by a nonlinear elastic foundation. Newton’s second law is adopted to derive the equations of motion for the string resting on an elastic foundation. Then, the method of multiple scales (MOMS) is employed for the analysis of the nonlinear system. It was found that 1:3 internal resonance exists in the first and fourth modes of the string when the wave speed in the transverse direction is [Formula: see text] and the elasticity coefficient of the foundation is [Formula: see text]. Fixed point plots are used to obtain the frequency responses of the various modes and to identify internal resonance through observation of the amplitudes and mode shapes. To prevent internal resonance and reduce vibration, a tuned mass damper (TMD) is applied to the string. The effects of various TMD masses, locations, damper coefficients ([Formula: see text]), and spring constants ([Formula: see text]) on overall damping were analyzed. The 3D plots of the maximum amplitude (3D POMAs) and 3D maximum amplitude contour plots (3D MACPs) are generated for the various modes to illustrate the amplitudes of the string, while identifying the optimal TMD parameters for vibration reduction. The results were verified numerically. It was concluded that better damping effects can be achieved using a TMD mass ratio [Formula: see text]–0.5 located near the middle of the string. Furthermore, for damper coefficient [Formula: see text], the use of spring constant [Formula: see text]–13 can improve the overall damping.

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