On the Stability of Forced Radial Oscillations of an Experimental Spring Pendulum

1993 ◽  
Author(s):  
Philip V. Bayly ◽  
Lawrence N. Virgin
Keyword(s):  
Fixed Point ◽  
Limit Cycle ◽  
Internal Resonance ◽  
Floquet Theory ◽  
Degree Of Freedom ◽  
Restoring Force ◽  
Linear Spring ◽  
Basic Ideas ◽  
The Stability ◽  
Elastic Pendulum

Abstract The elastic pendulum is a 2-degree-of-freedom, nonlinear device in which the pendulum bob may slide up and down the pendulum arm subject to the restoring force of a linear spring. In this study, radial motion (motion along the arm) is excited directly. Responses to this excitation include purely radial oscillations as well as swinging motion due to a 2:1 internal resonance. Changes in the behavior of the nonlinear spring pendulum occur when, under the control of a parameter, radial oscillations become unstable and are replaced by radial plus swinging motion. This bifurcation is explored analytically, numerically and experimentally, using the basic ideas of Floquet theory. Poincaré sampling is used to reduce the problem of describing the stability of a limit cycle to the easier task of defining the stability of the fixed point of a Poincaré map.

An empirical study of the stability of periodic motion in the forced spring-pendulum

1993 ◽  
Vol 443 (1918) ◽  
pp. 391-408 ◽  
Keyword(s):  
Radial Motion ◽  
Restoring Force ◽  
Local Modes ◽  
Transient Behaviour ◽  
Empirical Estimates ◽  
Dimensional State Space ◽  
Linear Spring ◽  
Basic Ideas ◽  
The Stability ◽  
Two Degree Of Freedom

The elastic pendulum is a two-degree-of-freedom, nonlinear device in which the primary mass slides up and down the pendulum arm subject to the restoring force of a linear spring. In this study, radial motion (motion along the arm) is excited directly. Responses to this excitation include purely radial motion as well as swinging motion due to a 2:1 internal resonance. Changes in the behaviour of the nonlinear spring-pendulum occur when, under the control of a parameter. one response becomes unstable and is replaced by another. These bifurcations are explored analytically, numerically and experimentally, using the basic ideas of Floquet theory. Poincaré sampling is used to reduce the problem of describing the stability of a limit cycle to the easier task of defining the stability of the fixed point of a Poincaré map. Empirical estimates of characteristic multipliers in four-dimensional state space are obtained by examining transient behaviour after perturbations; the Karhunen-Loeve decomposition is used to identify dominant local modes in these transients.

scholarly journals Calculating periodic vibrations of nonlinear autonomous multi-degree-of-freedom systems by the incremental harmonic balance method

2004 ◽  
Vol 26 (3) ◽  
pp. 157-166
Author(s):  
Nguyen Van Khang ◽  
Thai Manh Cau
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Van Der Pol ◽  
Incremental Harmonic Balance ◽  
The Stability

In this paper the incremental harmonic balance method is used to calculate periodic vibrations of nonlinear autonomous multip-degree-of-freedom systems. According to Floquet theory, the stability of a periodic solution is checked by evaluating the eigenvalues of the monodromy matrix. Using the programme MAPLE, the authors have studied the periodic vibrations of the system multi-degree van der Pol form.

scholarly journals On Stability of Periodic Solutions of Lienard Type Equations

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Zijian Yin ◽  
Hongbin Chen
Keyword(s):  
Asymptotic Behavior ◽  
Periodic Solutions ◽  
Exponential Decay ◽  
Floquet Theory ◽  
Linear Growth ◽  
Liénard Equation ◽  
Damping Term ◽  
Restoring Force ◽  
Stable Periodic ◽  
The Stability

We use the Floquet theory to analyze the stability of periodic solutions of Lienard type equations under the asymptotic linear growth of restoring force in this paper. We find that the existence and the stability of periodic solutions are determined primarily by asymptotic behavior of damping term. For special type of Lienard equation, the uniqueness and stability of periodic solutions are obtained. Furthermore, the sharp rate of exponential decay of the stable periodic solutions is determined under suitable conditions imposed on restoring force.

D-partition Stability Analysis of a Two-Degree-of-Freedom System with Delayed Restoring Force

1967 ◽  
Vol 9 (3) ◽  
pp. 190-197 ◽  
Author(s):  
B. Porter
Keyword(s):  
Stability Analysis ◽  
Machine Tool ◽  
Algebraic Equation ◽  
Characteristic Equation ◽  
Normal Modes ◽  
Degree Of Freedom ◽  
Central Feature ◽  
Restoring Force ◽  
The Stability ◽  
Two Degree Of Freedom

The method of D-partition is used to analyse the stability of a two-degree-of-freedom system subjected to a delayed restoring force of the kind which causes chatter in certain types of machine tool. The central feature of the analysis is the reduction of a stabliity problem involving a transcendental characteristic equation to a much simpler problem concerning the roots of a related algebraic equation. The results of the exact analysis are compared with approximate results obtained by assuming that the normal modes of the two-degree-of-freedom system can be decoupled.

A fixed point approach to the stability of sextic Lie *-derivations

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4933-4944
Author(s):  
Dongseung Kang ◽  
Heejeong Koh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Single Case ◽  
Fixed Point Method ◽  
Point Method ◽  
Fixed Point Approach ◽  
The Stability ◽  
The Given ◽  
Lie Derivations

We obtain a general solution of the sextic functional equation f (ax+by)+ f (ax-by)+ f (bx+ay)+ f (bx-ay) = (ab)2(a2 + b2)[f(x+y)+f(x-y)] + 2(a2-b2)(a4-b4)[f(x)+f(y)] and investigate the stability of sextic Lie *-derivations associated with the given functional equation via fixed point method. Also, we present a counterexample for a single case.

scholarly journals On the stability of set-valued functional equations with the fixed point alternative

2012 ◽  
Vol 2012 (1) ◽  
pp. 81 ◽  
Author(s):  
Hassan Kenary ◽  
Hamid Rezaei ◽  
Yousof Gheisari ◽  
Choonkil Park
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
The Stability

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

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.

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