HIGHER-DIMENSIONAL PERIODIC AND CHAOTIC OSCILLATIONS FOR VISCOELASTIC MOVING BELT WITH MULTIPLE INTERNAL RESONANCES

2007 ◽  
Vol 17 (05) ◽  
pp. 1637-1660 ◽  
Author(s):  
W. ZHANG ◽  
C. Z. SONG
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Oscillations ◽  
Chaotic Oscillations ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
First Order ◽  
Period 2 ◽  
Higher Dimensional ◽  
First Time

In this paper, higher-dimensional periodic and chaotic oscillations for a parametrically excited viscoelastic moving belt with multiple internal resonances are investigated for the first time. The external damping and internal damping of the material for the viscoelastic moving belt are considered simultaneously. First, the nonlinear governing equation of planar motion for the viscoelastic moving belt with the external damping is given. Then, the transverse nonlinear oscillations of the viscoelastic moving belt are considered. The method of multiple scales and the Galerkin approach are applied directly to the governing partial differential equation of motion for the viscoelastic moving belt to obtain an eight-dimensional averaged equation for the case of 1:2:3:4 internal resonances for the first-, the second-, the third- and the fourth-order modes and primary parametric resonance of the first-order mode. Finally, numerical method is used to investigate higher-dimensional periodic and chaotic motions of the viscoelastic moving belt. The results of numerical simulation demonstrate that there exist the period, period 2, period 4, multiple period and chaotic motions of the viscoelastic moving belt. The multipulse chaotic motions of the viscoelastic moving belt are observed from numerical simulations.

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.

Reduced Order Model for MEMS Cantilever Resonators Under Soft AC Voltage Near Natural Frequency

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Israel Martinez
Keyword(s):  
Natural Frequency ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
First Order ◽  
Electrostatically Actuated

The nonlinear response of an electrostatically actuated cantilever beam microresonator is investigated. The AC voltage is of frequency near resonator’s natural frequency. A first order fringe correction of the electrostatic force and viscous damping are included in the model. The dynamics of the resonator is investigated using the Reduced Order Model (ROM) method, based on Galerkin procedure. Steady-state motions are found. Numerical results for the uniform microresonator are compared with those obtained via the Method of Multiple Scales (MMS).

Second Order Perturbation Analysis of a Forced Nonlinear Mathieu Equation

2012 ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mathieu Equation ◽  
Second Order ◽  
Order Perturbation ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Order Expansion ◽  
Direct Forcing

The present study deals with the response of a forced nonlinear Mathieu equation. The equation considered has parametric excitation at the same frequency as direct forcing and also has cubic nonlinearity and damping. A second-order perturbation analysis using the method of multiple scales unfolds numerous resonance cases and system behavior that were not uncovered using first-order expansions. All resonance cases are analyzed. We numerically plot the frequency response of the system. The existence of a superharmonic resonance at one third the natural frequency was uncovered analytically for linear system. (This had been seen previously in numerical simulations but was not captured in the first-order expansion.) The effect of different parameters on the response of the system previously investigated are revisited.

scholarly journals An Evolutionary Analytic Method of Multi-DOF Nonlinear Coupling Dynamic Model for Controllable Close-Chain Linkage Mechanism System

2011 ◽  
Vol 2011 ◽  
pp. 1-17
Author(s):  
Ru-Gui Wang ◽  
Gan-Wei Cai ◽  
Xiao-Rong Zhou
Keyword(s):  
Approximate Solution ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Dynamic Responses ◽  
Analytic Method ◽  
Nonlinear Coupling ◽  
Linkage Mechanism ◽  
First Order ◽  
Coupling System ◽  
Coupling Dynamic

The 2-DOF controllable close-chain linkage mechanism is investigated in this paper. Based on the characteristics of the multi-DOF nonlinear coupling dynamic equation of the system established by the finite element method, an analytic method of multiple-scales Newmark is presented after thinking about the method of perturbation and the method of numerical analysis. Firstly, the first-order approximate solution of the dynamic responses of the system at the time of t is calculated by the multiple scales method. Then, taken the first-order approximate solution as the initialization of the generalized coordinate of the system, the stable dynamic response of the system is obtained by the implicit Newmark method. The simulation and experimental results are given in the end. The studies indicate that the method of multiple-scales Newmark is correct and practicable to study the dynamic characteristics of such kind of multi-DOF nonlinear coupling system.

scholarly journals Multipulse Heteroclinic Orbits and Chaotic Dynamics of the Laminated Composite Piezoelectric Rectangular Plate

2013 ◽  
Vol 2013 ◽  
pp. 1-27 ◽  
Author(s):  
Minghui Yao ◽  
Wei Zhang
Keyword(s):  
Rectangular Plate ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Plate Theory ◽  
Equations Of Motion ◽  
Laminated Composite ◽  
Heteroclinic Orbits ◽  
Phase Method ◽  
Chaotic Motions

This paper investigates the multipulse global bifurcations and chaotic dynamics for the nonlinear oscillations of the laminated composite piezoelectric rectangular plate by using an energy phase method in the resonant case. Using the von Karman type equations, Reddy’s third-order shear deformation plate theory, and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1 : 2 internal resonance and primary parametric resonance. The energy phase method is used for the first time to investigate the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The paper demonstrates how to employ the energy phase method to analyze the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Numerical simulations show that for the nonlinear oscillations of the laminated composite piezoelectric rectangular plate, the Shilnikov type multipulse chaotic motions can occur. Overall, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists.

Nonlinear Dynamics of a Travelling Beam Subjected to Multi-Frequency Parametric Excitation

2014 ◽  
Vol 592-594 ◽  
pp. 2076-2080 ◽  
Author(s):  
Bamadev Sahoo ◽  
L.N. Panda ◽  
Goutam Pohit
Keyword(s):  
Parametric Excitation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Reduced Equations ◽  
Stable Periodic ◽  
Bifurcation And Stability ◽  
First Order Differential Equations

This paper deals with two frequency parametric excitation in presence of internal resonance. The cubic nonlinearity is inserted into the equation of motion by considering the mid-line stretching of the beam. The perturbation method of multiple scales is applied directly to the governing nonlinear fourth order integro-partial differential equation of motion. This leads to a set of first order differential equations known as the reduced equations or normalized reduced equations, which are utilized to determine the additional instability zones, appeared in the trivial state stability plot, the bifurcation and stability of fixed-points, periodic, quasi-periodic, mixed mode and chaotic responses. The transition of system behaviour from stable periodic to unstable chaotic occurs through intermittency route

On Nonlinear Motions of Two-Degree-of-Freedom Nonlinear Systems with Repeated Linearized Natural Frequencies

2019 ◽  
Vol 29 (10) ◽  
pp. 1950132
Author(s):  
Hua-Zhen An ◽  
Xiao-Dong Yang ◽  
Feng Liang ◽  
Wei Zhang ◽  
Tian-Zhi Yang ◽  
...  
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Phase Portraits ◽  
Chaotic Motions ◽  
Nonlinear Parameters ◽  
Energy Ratios ◽  
Two Degree Of Freedom ◽  
Nonlinear Motions ◽  
Phase Differences

In this paper, we investigate systematically the vibration of a typical 2DOF nonlinear system with repeated linearized natural frequencies. By application of Descartes’ rule of signs, we demonstrate that there are 14 types of roots describing different modal motions for varying nonlinear parameters. The method of multiple scales is used to obtain the amplitude-phase portraits by introducing the energy ratios and phase differences. The typical nonlinear in-unison and elliptic out-of-unison modal motions are located for the 14 types of roots and then validated by numerical simulations. It is found that the normal in-unison modal motions, elliptic out-of-unison modal motions are analogous to the polarization of classical optic theory. Further, some kinds of periodic and chaotic motions under out-of-unison and in-unison excitations are investigated numerically. The result of this study offers a detailed discussion of nonlinear modal motions and responses of 2DOF systems with cubic nonlinear terms.

scholarly journals Nonlinear Response of Cantilever Beams to Combination and Subcombination Resonances

1998 ◽  
Vol 5 (5-6) ◽  
pp. 277-288 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Nonlinear Ordinary Differential Equations ◽  
Response Curves ◽  
First Order ◽  
Aspect Ratios

The nonlinear planar response of cantilever metallic beams to combination parametric and external subcombination resonances is investigated, taking into account the effects of cubic geometric and inertia nonlinearities. The beams considered here are assumed to have large length-to-width aspect ratios and thin rectangular cross sections. Hence, the effects of shear deformations and rotatory inertia are neglected. For the case of combination parametric resonance, a two-mode Galerkin discretization along with Hamilton’s extended principle is used to obtain two second-order nonlinear ordinary-differential equations of motion and associated boundary conditions. Then, the method of multiple scales is applied to obtain a set of four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two excited modes. For the case of subcombination resonance, the method of multiple scales is applied directly to the Lagrangian and virtual-work term. Then using Hamilton’s extended principle, we obtain a set of four first-order nonlinear ordinary-differential equations governing the amplitudes and phases of the two excited modes. In both cases, the modulation equations are used to generate frequency- and force-response curves. We found that the trivial solution exhibits a jump as it undergoes a subcritical pitchfork bifurcation. Similarly, the nontrivial solutions also exhibit jumps as they undergo saddle-node bifurcations.

Multi-Pulse Heteroclinic Orbits and Chaotic Dynamics of Laminated Composite Piezoelectric Rectangular Plate

2009 ◽  
Author(s):  
Ming-Hui Yao ◽  
Wei Zhang ◽  
Dong-Xing Cao
Keyword(s):  
Normal Form ◽  
Chaotic Dynamics ◽  
Multiple Scales ◽  
Laminated Composite ◽  
Heteroclinic Orbits ◽  
Global Dynamics ◽  
Rectangular Plates ◽  
Internal Resonances ◽  
Chaotic Motions ◽  
Simply Supported

The multi-pulse orbits and chaotic dynamics of the simply supported laminated composite piezoelectric rectangular plates under combined parametric excitation and transverse loads are studied in detail. It is assumed that different layers are perfectly bonded to each other with piezoelectric actuator patches embedded. The nonlinear equations of motions for the laminated composite piezoelectric rectangular plates are derived from von Karman-type equation and third-order shear deformation laminate theory of Reddy. The four-dimensional averaged equation under the case of primary parametric resonance and 1:2 internal resonances is obtained by directly using the method of multiple scales and Galerkin approach to the partial differential governing equation of motion for the laminated composite piezoelectric rectangular plates. The system is transformed to the averaged equation. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Based on normal form obtained, the extended Melnikov method is utilized to analyze the multi-pulse global bifurcations and chaotic dynamics for the laminated composite piezoelectric rectangular plates. The analysis of global dynamics indicates that there exist the multi-pulse jumping orbits in the perturbed phase space of the averaged equation. From the averaged equations obtained, the chaotic motions and the Shilnikov type multi-pulse orbits of the laminated composite piezoelectric rectangular plates are found by using numerical simulation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense for the simply supported laminated composite piezoelectric rectangular plates.

scholarly journals Free Vibration and Hardening Behavior of Truss Core Sandwich Beam

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
J. E. Chen ◽  
W. Zhang ◽  
M. Sun ◽  
M. H. Yao
Keyword(s):  
Natural Frequencies ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Sandwich Beam ◽  
Third Order ◽  
First Order ◽  
Hardening Behavior ◽  
Truss Core ◽  
Averaged Equations ◽  
Core Height

The dynamic characteristics of simply supported pyramidal truss core sandwich beam are investigated. The nonlinear governing equation of motion for the beam is obtained by using a Zig-Zag theory. The averaged equations of the beam with primary, subharmonic, and superharmonic resonances are derived by using the method of multiple scales and then the corresponding frequency response equations are obtained. The influences of strut radius and core height on the linear natural frequencies and hardening behaviors of the beam are studied. It is illustrated that the first-order natural frequency decreases continuously and the second-order and third-order natural frequencies initially increase and then decrease with the increase of strut radius, and the first three natural frequencies all increase with the rise of the core height. Furthermore, the results indicate that the hardening behaviors of the beam become weaker with the increase of the rise of strut radius and core height. The mechanisms of variations in hardening behavior of the sandwich beam with the three types of resonances are detailed and discussed.

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