On Multiple Hopf Bifurcations of Airflow Excited Vibration of a Translating String

2007 ◽  
Author(s):  
Yuefang Wang ◽  
Lefeng Lu¨ ◽  
Yingxi Liu
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Equilibrium Configuration ◽  
Transverse Motion ◽  
Hopf Bifurcations ◽  
Flip Bifurcation ◽  
Excited Vibration ◽  
Incremental Harmonic Balance ◽  
The Stability ◽  
Bifurcation Behavior

This paper presents the stability and bifurcation of transverse motion of translating strings excited by a steady wind flowfield. The stability of the equilibrium configuration is presented for loss of stability and generation of limit cycles via the Hopf bifurcation. It is demonstrated that there are single, double and quadruple Hopf bifurcations in the parametric space that lead to the limit cycle motion. The method of Incremental Harmonic Balance is used to solve the limit cycle response of which the stability is determined by computation of the Floquet multipliers. For the forced vibration, it is pointed out that the periodic and quasi-periodic motions exist as parameters are changed. The quench frequency and the Neimark-Sacker (NS) bifurcation and flip bifurcation are obtained. The continuity software MATCONT is adopted and the Resonance 1:1, 1:3 and 1:4 as well as NS to NS bifurcations are presented. The bifurcation behavior reveals the complexity of the string’s motion response induce by aerodynamic excitations.

NONLINEAR VIBRATION ANALYSIS FOR AN AIRFLOW-EXCITED TRANSLATING STRING

2012 ◽  
Vol 09 (04) ◽  
pp. 1250051 ◽  
Author(s):  
YUEFANG WANG ◽  
LEFENG LÜ ◽  
LIHUA HUANG
Keyword(s):  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Equilibrium Configuration ◽  
Excitation Frequency ◽  
Transverse Motion ◽  
Hopf Bifurcations ◽  
Flip Bifurcation ◽  
Wind Force ◽  
Incremental Harmonic Balance ◽  
The Stability

The nonlinear vibration of transverse motion of a translating string excited by steady wind force is investigated in this paper. The stability of the equilibrium configuration is analyzed and the generation of limit cycles via multiple Hopf bifurcations is presented. Single-, double-, and quadruple-Hopf bifurcations are determined in the parametric space. The limit-cycle response is solved through the method of Incremental Harmonic Balance, with its stability determined by Floquet multipliers. For the forced vibration, the coexistence of periodic and quasiperiodic motions is found with varying excitation frequency and amplitude. The Neimark–Sacker (NS) bifurcation and the flip bifurcation are demonstrated in an example. The continuation software MATCONT is adopted to identify the fold and NS bifurcations of periodic motions, as well as other codim-2 bifurcations of NS–NS, the Chenciner and the 1:1, 1:3, and 1:4 resonances. These bifurcations present the complexity of the string dynamics induced by steady wind excitations.

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.

Period Motions and Limit Cycle in a Periodically Forced, Plunged Galloping Oscillator

2017 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Limit Cycle ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Hopf Bifurcations ◽  
Periodic Motions ◽  
Stability And Bifurcations ◽  
Generalized Harmonic Balance Method

In this paper, periodic motions of a periodically forced, plunged galloping oscillator are investigated. The analytical solutions of stable and unstable periodic motions are obtained by the generalized harmonic balance method. Stability and bifurcations of the periodic motions are discussed through the eigenvalue analysis. The saddle-node and Hopf bifurcations of periodic motions are presented through frequency-amplitude curves. The Hopf bifurcation generates the quasiperiodic motions. Numerical simulations of stable and unstable periodic motions are illustrated.

Nonlinear Dynamics of High-Speed Milling—Analyses, Numerics, and Experiments

2004 ◽  
Vol 127 (2) ◽  
pp. 197-203 ◽  
Author(s):  
Gabor Stepan ◽  
Robert Szalai ◽  
Brian P. Mann ◽  
Philip V. Bayly ◽  
Tamas Insperger ◽  
...  
Keyword(s):  
Small Parameter ◽  
High Speed ◽  
Period Doubling ◽  
Hopf Bifurcations ◽  
Cutting Condition ◽  
Flip Bifurcation ◽  
Machining Parameters ◽  
High Speed Milling ◽  
The Stability ◽  
Discrete Mathematical Model

High-speed milling is often modeled as a kind of highly interrupted machining, when the ratio of time spent cutting to not cutting can be considered as a small parameter. In these cases, the classical regenerative vibration model, playing an essential role in machine tool vibrations, breaks down to a simplified discrete mathematical model. The linear analysis of this discrete model leads to the recognition of the doubling of the so-called instability lobes in the stability charts of the machining parameters. This kind of lobe-doubling is related to the appearance of period doubling vibrations originated in a flip bifurcation. This is a new phenomenon occurring primarily in low-immersion high-speed milling along with the Neimark-Sacker bifurcations related to the classical self-excited vibrations or Hopf bifurcations. The present work investigates the nonlinear vibrations in the case of period doubling and compares this to the well-known subcritical nature of the Hopf bifurcations in turning processes. The identification of the global attractor in the case of unstable cutting leads to contradiction between experiments and theory. This contradiction draws the attention to the limitations of the small parameter approach related to the highly interrupted cutting condition.

On Bifurcation Control in Time Delay Feedback Systems

1998 ◽  
Vol 08 (04) ◽  
pp. 713-721 ◽  
Author(s):  
M. Basso ◽  
A. Evangelisti ◽  
R. Genesio ◽  
A. Tesi
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Bifurcation Control ◽  
Flip Bifurcation ◽  
Delay Model ◽  
Feedback Systems ◽  
First Order ◽  
Balance Analysis ◽  
Feedback Control Systems ◽  
Limit Cycle Bifurcation

The paper addresses bifurcations of limit cycles for a class of feedback control systems depending on parameters. A set of simple approximate analytical conditions characterizing all the generic limit cycle bifurcations is determined via a first-order harmonic balance analysis in a suitable frequency band. Based on the results of this analysis, an approach to limit cycle bifurcation control is proposed. In particular, an example concerning a biological delay model is developed, where a flip bifurcation control is designed via a modified Pyragas technique.

Stability Analysis of TLP Tethers Under Vortex-Induced Oscillations

2008 ◽  
Vol 131 (1) ◽  
Author(s):  
A. K. Banik ◽  
T. K. Datta
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Equation Of Motion ◽  
Incremental Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
The Stability ◽  
Lock In ◽  
Continuation Technique ◽  
Mode Response ◽  
Stability Of The Solution

The vortex-induced oscillation of TLP tether is investigated in the vicinity of lock-in condition. The vortex shedding is caused purely due to current, which may vary across the depth of the sea. The vibration of TLP is modeled as a SDOF problem by assuming that the first mode response of the tether dominates the motion. Nonlinearity in the equation of motion is produced due to the relative velocity squared drag force. In order to trace different branches of the response curve and investigate different instability phenomena that may exist, an arc-length continuation technique along with the incremental harmonic balance method (IHBC) is employed. A procedure for treating the nonlinear term using distribution theory is presented so that the equation of motion is transformed to a form amenable to the application of IHBC. The stability of the solution is investigated by the Floquet theory using Hsu’s scheme.

Nonlinear Stability of a Fluid-Conveying Cantilevered Pipe With End Mass in Case of Horizontal Excitation at the Upper End

2010 ◽  
Author(s):  
Hiroaki Furuya ◽  
Kiyotaka Yamashita ◽  
Hiroshi Yabuno
Keyword(s):  
Limit Cycle ◽  
Excitation Frequency ◽  
Internal Flow ◽  
Frequency Component ◽  
Single Frequency ◽  
Dynamic Features ◽  
Cycle Frequency ◽  
Excited Vibration ◽  
Cantilevered Pipe ◽  
The Stability

Nonplanar vibrations of a cantilevered pipe with an end mass is studied. We have already clarified the nonplanar vibrations with a single frequency component when the pipe conveys fluid whose velocity is slightly over the critical value, above which the lateral vibration of the pipe is self-excited due to the internal flow. Moreover, for the case that the upper end of the pipe is excited periodically in the horizontal direction, we have shown in the previous paper that the nonplanar limit cycle motions start complex spatial transients and settle down to stationary planar forced-excited vibration when the excitation frequency is near the nonplanar limit cycle frequency. The purpose of this paper is to examine the stability of the nonplanar pipe vibrations when the nonplanar self-excited pipe vibrations are subjected to the excitation at the upper end. A set of ordinary differential equations, which govern the amplitudes and phases of unstable mode vibration and contain the effect of excitation at the upper end are derived. Stability analysis of these equations clarifies the nonlinear interactions between nonplanar self-excited pipe vibrations and the forced excitation. Second, the experiments are conducted with a silicon rubber pipe conveying water, confirming the dynamic features of pipe vibrations for the horizontal excitation.

scholarly journals State-Space Analysis on The Stability of Limit Cycle Predicted by Harmonic Balance

2011 ◽  
Vol 6 (5) ◽  
pp. 697-705
Author(s):  
Byung-Jin Lee ◽  
Suk-Chang Yun ◽  
Chang-Joo Kim ◽  
Jung-Keun Park ◽  
Sang-Kyung Sung
Keyword(s):  
State Space ◽  
Limit Cycle ◽  
Harmonic Balance ◽  
Space Analysis ◽  
State Space Analysis ◽  
The Stability
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