ON A "CROSSROAD AREA–SPRING AREA" TRANSITION OCCURRING IN A DUFFING–RAYLEIGH EQUATION WITH A PERIODICAL EXCITATION

1993 ◽  
Vol 03 (04) ◽  
pp. 1029-1037 ◽  
Author(s):  
CHRISTIAN MIRA ◽  
MOHAMMED QRIOUET
Keyword(s):  
Excitation Frequency ◽  
Period Doubling ◽  
Qualitative Change ◽  
Parameter Plane ◽  
Rayleigh Equation ◽  
Flip Bifurcation ◽  
External Excitation ◽  
Spring Area ◽  
Bifurcation Structures ◽  
Parameter Coefficient

The bifurcation structures considered in this paper are given by a Duffing–Rayleigh equation in the presence of a periodic external excitation. The first one is related to a cascade of fold lips generated by period doubling at subharmonic oscillations, which is obtained in a parameter plane defined by the excitation frequency and its amplitude. When a third parameter (coefficient of the linear approximation of the damping) varies, a qualitative change of the parameter plane occurs. It is related to a new mechanism of "crossroad area–spring area" transition, the areas corresponding to typical arrangements of fold and flip bifurcation curves around a fold cusp.

"CROSSROAD AREA–SPRING AREA" TRANSITION (II) FOLIATED PARAMETRIC REPRESENTATION

1991 ◽  
Vol 01 (02) ◽  
pp. 339-348 ◽  
Author(s):  
C. MIRA ◽  
J. P. CARCASSÈS ◽  
M. BOSCH ◽  
C. SIMÓ ◽  
J. C. TATJER
Keyword(s):  
Complete Information ◽  
Parametric Representation ◽  
Periodic Points ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Cusp Point ◽  
Dimensional Representation ◽  
Bifurcation Structure ◽  
Spring Area

The areas considered are related to two different configurations of fold and flip bifurcation curves of maps, centred at a cusp point of a fold curve. This paper is a continuation of an earlier one devoted to parameter plane representation. Now the transition is studied in a thee-dimensional representation by introducing a norm associated with fixed or periodic points. This gives rise to complete information on the map bifurcation structure.

NONSMOOTH AND SMOOTH BIFURCATIONS IN A DISCONTINUOUS PIECEWISE MAP

2012 ◽  
Vol 22 (05) ◽  
pp. 1250112 ◽  
Author(s):  
ZHIYING QIN ◽  
YUEJING ZHAO ◽  
JICHEN YANG
Keyword(s):  
Fixed Point ◽  
Bifurcation Diagram ◽  
Bifurcation Point ◽  
Peculiar Feature ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Border Collision Bifurcation ◽  
Bifurcation Structures ◽  
Piecewise Map ◽  
Two Parameter

In this paper, a piecewise map with singularity of the power (-1/2) is introduced. For this piecewise map, there is an infinite discontinuous gap on the origin. The conditions of nonsmooth border-collision bifurcation and smooth fold or flip bifurcation are analytically derived. For period-1 fixed point, two-parameter-plane can be divided into seven ranges according to different bifurcation structures. For period-n orbits, codimension-2 bifurcation point may lead to different period-increment sequence, and a peculiar feature is found that there are two different period-increment sequences in the same bifurcation diagram.

"CROSSROAD AREA–SPRING AREA" TRANSITION (I) PARAMETER PLANE REPRESENTATION

1991 ◽  
Vol 01 (01) ◽  
pp. 183-196 ◽  
Author(s):  
J. P. CARCASSES ◽  
C. MIRA ◽  
M. BOSCH ◽  
C. SIMÓ ◽  
J. C. TATJER
Keyword(s):  
Parameter Plane ◽  
Flip Bifurcation ◽  
Two Dimensional ◽  
Bifurcation Structure ◽  
One Dimensional ◽  
Codimension Two ◽  
Transition Mechanism ◽  
Spring Area ◽  
Cusp Points ◽  
Codimension Two Bifurcation

This paper is devoted to the bifurcation structure of a parameter plane related to one- and two-dimensional maps. Crossroad area and spring area correspond to a characteristic organization of fold and flip bifurcation curves of the parameter plane, involving the existence of cusp points (fold codimension-two bifurcation) and flip codimension-two bifurcation points. A transition "mechanism" (among others) from one area type to another one is given from a typical one-dimensional map.

ON THE "CROSSROAD AREA–SADDLE AREA" AND "CROSSROAD AREA–SPRING AREA" TRANSITIONS

1991 ◽  
Vol 01 (03) ◽  
pp. 641-655 ◽  
Author(s):  
C. MIRA ◽  
J. P. CARCASSÈS
Keyword(s):  
Three Dimensional ◽  
Parameter Plane ◽  
Flip Bifurcation ◽  
Two Dimensional ◽  
Cusp Point ◽  
Dimensional Representation ◽  
One Dimensional ◽  
Transition Mechanism ◽  
Three Dimensional Representation ◽  
Spring Area

Let T be a one-dimensional or two-dimensional map. The three considered areas are related to three different configurations of fold and flip bifurcation curves, centred at a cusp point of a fold curve in the T parameter plane (b, c). The two transitions studied here occur via a codimension-three bifurcation defined in each case, when varying a third parameter a. The transition "mechanism," from an area type to another one, is given with a three-dimensional representation describing the sheet configuration of the parameter plane.

“CROSSROAD AREA- DISSYMMETRICAL SPRING AREA — SYMMETRICAL SPRING AREA,” AND “DOUBLE CROSSROAD AREA — DOUBLE SPRING AREA” TRANSITIONS

1993 ◽  
Vol 03 (02) ◽  
pp. 429-435 ◽  
Author(s):  
REZK ALLAM ◽  
CHRISTIAN MIRA
Keyword(s):  
Parameter Plane ◽  
Flip Bifurcation ◽  
Two Dimensional ◽  
Cusp Point ◽  
Symmetrical Configuration ◽  
Spring Area

In a parameter plane, crossroad areas and spring areas are two typical organizations of fold and flip bifurcation curves centred at a fold cusp point. Till now only spring areas in a “symmetrical” configuration have been described. This letter introduces another type of spring area for which such a “symmetry” does not exist. It is called a dissymmetrical spring area. When a third parameter is varied, qualitative modifications of the parameter plane are considered, and an example of a two-dimensional diffeomorphism is given.

scholarly journals Complex response analysis of a non-smooth oscillator under harmonic and random excitations

2021 ◽  
Vol 42 (5) ◽  
pp. 641-648
Author(s):  
Shichao Ma ◽  
Xin Ning ◽  
Liang Wang ◽  
Wantao Jia ◽  
Wei Xu
Keyword(s):  
Excitation Frequency ◽  
System Stability ◽  
Response Analysis ◽  
External Excitation ◽  
Stochastic Response ◽  
Impact Oscillator ◽  
Nonlinear Parameters ◽  
Bifurcation Phenomena ◽  
Mc Simulation ◽  
Smooth System

AbstractIt is well-known that practical vibro-impact systems are often influenced by random perturbations and external excitation forces, making it challenging to carry out the research of this category of complex systems with non-smooth characteristics. To address this problem, by adequately utilizing the stochastic response analysis approach and performing the stochastic response for the considered non-smooth system with the external excitation force and white noise excitation, a modified conducting process has proposed. Taking the multiple nonlinear parameters, the non-smooth parameters, and the external excitation frequency into consideration, the steady-state stochastic P-bifurcation phenomena of an elastic impact oscillator are discussed. It can be found that the system parameters can make the system stability topology change. The effectiveness of the proposed method is verified and demonstrated by the Monte Carlo (MC) simulation. Consequently, the conclusions show that the process can be applied to stochastic non-autonomous and non-smooth systems.

Two Groups of Chaotic Vibrations in a Single-Degree-of-Freedom Maglev System

1997 ◽  
Author(s):  
Zhixiang Xu ◽  
Hideyuki Tamura
Keyword(s):  
Magnetic Levitation ◽  
Excitation Frequency ◽  
Power Spectra ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Moving Base ◽  
Period Doubling Bifurcations

Abstract In this paper, a single-degree-of-freedom magnetic levitation dynamic system, whose spring is composed of a magnetic repulsive force, is numerically analyzed. The numerical results indicate that a body levitated by magnetic force shows many kinds of vibrations upon adjusting the system parameters (viz., damping, excitation amplitude and excitation frequency) when the system is excited by the harmonically moving base. For a suitable combination of parameters, an aperiodic vibration occurs after a sequence of period-doubling bifurcations. Typical aperiodic vibrations that occurred after period-doubling bifurcations from several initial states are identified as chaotic vibration and classified into two groups by examining their power spectra, Poincare maps, fractal dimension analyses, etc.

Analysis of Stochastic Nonlinear Dynamics in the Gear Transmission System with Backlash

2015 ◽  
Vol 16 (2) ◽  
pp. 111-121 ◽  
Author(s):  
Jingyue Wang ◽  
Haotian Wang ◽  
Lixin Guo
Keyword(s):  
Time Course ◽  
Damping Ratio ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Spur Gear ◽  
Transmission System ◽  
Gear Transmission ◽  
Stochastic Dynamic ◽  
Random Disturbance ◽  
Gear Transmission System

AbstractIn order to study the different backlash, gear damping ratio and random disturbance on dynamic behavior of gear transmission system, stochastic dynamic equations of the three-degree-of-freedom spur gear transmission system are established considering random disturbances of a low-frequency external excitation induced by torque fluctuation, gear damping ratio, gear backlash, excitation frequency and meshing stiffness. Using bifurcation diagram, phase diagram, time course diagram, Poincaré map and power spectrum of the system, the dynamic characteristics of the gear transmission system with different backlash under gear damping ratio changing, and the influence of the random disturbance of gear damping ratio on the bifurcation characteristic of system are analyzed. Numerical simulation shows that the gear transmission system will be from periodic motion with a noisy disturbance to chaotic-like motion by period-doubling bifurcation with decreasing gear damping ratio. In the small damping ratio range, the backlash has great effect on the motion characteristics. Random disturbance has an important effect on the bifurcation characteristics.

Utilizing Off-Resonance and Dual-Frequency Excitation to Distinguish Attractive and Repulsive Surface Forces in Atomic Force Microscopy

2010 ◽  
Vol 6 (3) ◽  
Author(s):  
Andrew J. Dick ◽  
Santiago D. Solares
Keyword(s):  
Fundamental Frequency ◽  
Period Doubling ◽  
Qualitative Change ◽  
Surface Forces ◽  
Resonance Excitation ◽  
Atomic Interaction ◽  
Beam Model ◽  
Dual Frequency ◽  
Atomic Force ◽  
Frequency Excitation

A beam model is developed and discretized to study the dynamic behavior of the cantilever probe of an atomic force microscope. Atomic interaction force models are used with a multimode approximation in order to simulate the probe’s response. The system is excited at two-and-a-half times the fundamental frequency and with a dual-frequency signal consisting of the AFM probe’s fundamental frequency and two-and-a-half times the fundamental frequency. A qualitative change in the response in the form of period doubling is observed for the harmonic off-resonance excitation when significantly influenced by repulsive surface forces. Through the use of dual-frequency excitation, standard response characteristics are maintained, while the inclusion of the off-resonance frequency component results in an identifiable qualitative change in the response. By monitoring specific frequency components, the influence of attractive and repulsive surface forces may be distinguished. This information could then be used to distinguish between imaging regimes when bistability occurs or to operate at the separation distance between surface force regimes to minimize force levels.

Vibration Analysis of Nonlinear Isolator’s Characteristics by ADAMS

2012 ◽  
Vol 152-154 ◽  
pp. 1077-1081 ◽  
Author(s):  
Zhao Qi He ◽  
Yu Chao Song ◽  
Hong Liang Yu
Keyword(s):  
Nonlinear Vibration ◽  
Vibration Isolation ◽  
Excitation Frequency ◽  
Vibration Isolator ◽  
Vibration System ◽  
Linear Vibration ◽  
External Excitation ◽  
Mass Model ◽  
Adams Software ◽  
Isolation Systems

A nonlinear spring-mass model is established to study the dynamic characteristics of nonlinear vibration isolator. By use of ADAMS software, the influence of stiffness, foundation displacement excitation and frequency of external excitation on the nonlinear vibration isolation systems are analyzed. Results indicate that the linear vibration system needs 4s to achieve stability, but the nonlinear vibration system only needs 0.1s. The response value increases with the increase of excitation frequency, the response pick value increases by 61.58% and 102.35% and each corresponding stable value increases by 159.35% and 309.87%.

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