PARAMETRICALLY EXCITED NONLINEAR TWO-DEGREE-OF-FREEDOM SYSTEMS WITH NONSEMISIMPLE ONE-TO-ONE RESONANCE

1995 ◽  
Vol 05 (03) ◽  
pp. 725-740 ◽  
Author(s):  
C. CHIN ◽  
A.H. NAYFEH
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Degree Of Freedom ◽  
Parametrically Excited ◽  
Parametric Resonances ◽  
Quadratic Nonlinearities ◽  
One To One ◽  
Two Degree Of Freedom

The response of a parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities and a nonsemisimple one-to-one internal resonance is investigated. The method of multiple scales is used to derive four first-order differential equations governing the modulation of the amplitudes and phases of the two modes for the cases of fundamental and principal parametric resonances. Bifurcation analysis of the case of fundamental parametric resonance reveals that the quadratic nonlinearities qualitatively change the response of the system. They change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed.

Period-Doubling Bifurcation and Non-Typical Route to Chaos of a Two-Degree-Of-Freedom Vibro-Impact System

2000 ◽  
Vol 68 (4) ◽  
pp. 670-674 ◽  
Author(s):  
G. L. Wen and ◽  
J. H. Xie
Keyword(s):  
Finite Number ◽  
Period Doubling ◽  
Degree Of Freedom ◽  
Period Doubling Bifurcation ◽  
Periodic Response ◽  
Impact System ◽  
Route To Chaos ◽  
Two Degree Of Freedom ◽  
Impact Motion ◽  
Period Doubling Bifurcations

A nontypical route to chaos of a two-degree-of-freedom vibro-impact system is investigated. That is, the period-doubling bifurcations, and then the system turns out to the stable quasi-periodic response while the period 4-4 impact motion fails to be stable. Finally, the system converts into chaos through phrase locking of the corresponding four Hopf circles or through a finite number of times of torus-doubling.

Nonlinear Oscillations of a Particle in the Plane Under Longitudinal End Excitation

2003 ◽  
Author(s):  
Rajesh K. Jha ◽  
Robert G. Parker
Keyword(s):  
Nonlinear Oscillations ◽  
Low Frequency ◽  
Period Doubling ◽  
Lumped Parameter Model ◽  
Lumped Parameter ◽  
Parametric Resonances ◽  
Jump Phenomena ◽  
Route To Chaos ◽  
Chaotic Nature ◽  
Two Degree Of Freedom

We study the forced vibrations of a two degree of freedom lumped parameter model of a belt span under longitudinal excitation. The belt inertia is modelled as a particle and the belt elasticity is modelled by two identical linear springs. Numerical integration is used to calculate free responses and perform frequency and amplitude sweeps. Frequency sweep results indicate parametric resonances, jump phenomena, sub- and super-harmonic responses, quasiperiodicity and chaos. Amplitude sweep at a low frequency shows bifurcations of limit cycles and the period doubling route to chaos. Poincare sections are computed to show the chaotic nature of the responses.

Bifurcation and Chaos in Externally Excited Circular Cylindrical Shells

1996 ◽  
Vol 63 (3) ◽  
pp. 565-574 ◽  
Author(s):  
Char-Ming Chin ◽  
A. H. Nayfeh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Multiple Scales ◽  
Period Doubling ◽  
Single Mode ◽  
Equilibrium Solutions ◽  
Multiple Attractors ◽  
Partial Differential ◽  
Quadratic Nonlinearities ◽  
Mode Solution

The nonlinear response of an infinitely long cylindrical shell to a primary excitation of one of its two orthogonal flexural modes is investigated. The method of multiple scales is used to derive four ordinary differential equations describing the amplitudes and phases of the two orthogonal modes by (a) attacking a two-mode discretization of the governing partial differential equations and (b) directly attacking the partial differential equations. The two-mode discretization results in erroneous solutions because it does not account for the effects of the quadratic nonlinearities. The resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations. The response could be a single-mode solution or a two-mode solution. The equilibrium solutions of the two orthogonal third flexural modes undergo a Hopf bifurcation. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, explosive bifurcations, and crises.

Analytical Investigation of Vibration Attenuation With a Nonlinear Tuned Mass Damper

2015 ◽  
Author(s):  
Andrew J. Dick ◽  
Aaron Atzil ◽  
Satish Nagarajaiah
Keyword(s):  
Primary Structure ◽  
Multiple Scales ◽  
Tuned Mass Damper ◽  
Analytical Investigation ◽  
Degree Of Freedom ◽  
Approximate Analytical Solutions ◽  
Vibration Attenuation ◽  
Mass Damper ◽  
Two Degree Of Freedom ◽  
Nonlinear Tuned Mass Damper

Vibration attenuation devices are used to reduce the vibrations of various mechanical systems and structures. In this work, an analytical method is proposed to provide the means to investigate the influence of system parameters on the dynamic response of a system. The method of multiple scales is used to calculate an approximate broadband solution for a two degree-of-freedom system consisting of a linear primary structure and a nonlinear tuned mass damper. The model is decoupled, approximate analytical solutions are calculated, and then they are combined to produce the desired frequency-response information. The approach is initially applied to a linear two degree-of-freedom system in order to verify its performance. The approach is then applied to the nonlinear system in order to study how varying the values of parameters associated with the nonlinear absorber affect its ability to attenuate the response of the primary structure.

Parametric Resonances of a Three-Blade-Rotor System With Reference to Wind Turbines

2020 ◽  
Vol 142 (2) ◽  
Author(s):  
Gizem D. Acar ◽  
Mustafa A. Acar ◽  
Brian F. Feeny
Keyword(s):  
Multiple Scales ◽  
Rotor System ◽  
Degree Of Freedom ◽  
Rotor Speed ◽  
Horizontal Axis Wind Turbine ◽  
Direct Excitation ◽  
Parametric Resonances ◽  
Parametric Effects ◽  
Rotor Angle ◽  
First Order Method

Abstract Coupled blade-hub dynamics of a coupled three-blade-rotor system with parametric stiffness, which is similar to a horizontal-axis wind turbine, is studied. Blade equations have parametric and direct excitation terms due to gravity and are coupled through the hub equation. For a single degree-of-freedom blade model with only in-plane transverse vibrations, the reduced-order model shows parametric resonances. A small parameter is established for large blades, which enables us to treat the effect of blade motion as a perturbation on the rotor motion. The rotor speed is not constant, and the cyclic variations cannot be expressed as explicit functions of time. Therefore, it is more convenient to use the rotor angle as the independent variable. By expressing the system dynamics in the rotor angle domain and assuming small variations in rotor speed, the blade equations are decoupled from the rotor equation. The interdependent blade equations constitute a three-degree-of-freedom system with periodic parametric and direct excitation. The response is analyzed by using a first-order method of multiple scales (MMS). The system has a superharmonic and a subharmonic resonances due to direct and parametric effects introduced by gravity. Amplitude-frequency relations and stabilities of these resonances are studied. The MMS solutions are compared with numerical simulations for verification.

Vibrations of Dynamical Systems With Quadratic Nonlinearities

1965 ◽  
Vol 32 (3) ◽  
pp. 576-582 ◽  
Author(s):  
P. R. Sethna
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Method ◽  
Degree Of Freedom ◽  
System Parameters ◽  
Quadratic Nonlinearities ◽  
The Stability ◽  
Physical Phenomena ◽  
Two Degree Of Freedom

General two-degree-of-freedom dynamical systems with weak quadratic nonlinearities are studied. With the aid of an asymptotic method of analysis a classification of these systems is made and the more interesting subclasses are studied in detail. The study includes an examination of the stability of the solutions. Depending on the values of the system parameters, several different physical phenomena are shown to occur. Among these is the phenomenon of amplitude-modulated motions with modulation periods that are much larger than the periods of the excitation forces.

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