scholarly journals A Harmonic-Based Method for Computing the Stability of Periodic Oscillations of Non-Linear Structural Systems

2010 ◽  
Author(s):  
O. Thomas ◽  
A. Lazarus ◽  
C. Touze´
Keyword(s):  
Numerical Method ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Computation Time ◽  
Fourier Series Expansion ◽  
Simple Strategy ◽  
The Stability ◽  
Linear Periodic System ◽  
Continuation Technique

In this paper, we present a validation on a practical example of a harmonic-based numerical method to determine the local stability of periodic solutions of dynamical systems. Based on Floquet theory and Fourier series expansion (Hill method), we propose a simple strategy to sort the relevant physical eigenvalues among the expanded numerical spectrum of the linear periodic system governing the perturbed solution. By mixing the Harmonic Balance Method and Asymptotic Numerical Method continuation technique with the developed Hill method, we obtain a purely-frequency based continuation tool able to compute the stability of the continued periodic solutions in a reduced computation time. This procedure is validated by considering an externally forced string and computing the complete bifurcation diagram with the stability of the periodic solutions. The particular coupled regimes are exhibited and found in excellent agreement with results of the literature, allowing a method validation.

Bifurcation Analysis of a Belousov–Zhabotinsky Reaction Model

2015 ◽  
Vol 25 (06) ◽  
pp. 1550093 ◽  
Author(s):  
Xiaoli Wang ◽  
Yu Chang ◽  
Dashun Xu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Reaction Model ◽  
Bifurcation Phenomena ◽  
Route To Chaos ◽  
The Stability ◽  
Belousov Zhabotinsky Reaction

We investigate the bifurcation phenomena in a Belousov–Zhabotinsky reaction model by applying Hopf bifurcation theory in frequency domain and harmonic balance method. The high accurate predictions, i.e. fourth-order harmonic balance approximation, on frequencies, amplitudes, and approximation expressions for periodic solutions emerging from Hopf bifurcation are provided. We also detect the stability and location of these periodic solutions. Numerical simulations not only confirm the theoretical analysis results but also illustrate some complex oscillations such as a cascade of period-doubling bifurcation, quasi-periodic solution, and period-doubling route to chaos. All these results improve the understanding of the dynamics of the model.

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.

Finding Periodic Solutions of Forced Systems With Local Nonlinearities: A Mixed Shooting Harmonic Balance Method

2015 ◽  
Author(s):  
Frederic Schreyer ◽  
Remco Leine
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Shooting Method ◽  
Mechanical Systems ◽  
Harmonic Balance Method ◽  
Nonlinear Effects ◽  
Balance Method ◽  
Pros And Cons ◽  
Numerical Approaches

Several numerical approaches have been developed to capture nonlinear effects of dynamical systems. In this paper we present a mixed shooting-harmonic balance method to solve large mechanical systems with local nonlinearities efficiently. The Harmonic Balance Method as well as the shooting method have both their pros and cons. The proposed mixed shooting-HBM approach combines the efficiency of HBM and the accuracy of the shooting method and has therefore advantages of both.

Multiple Harmonic Balance Method for Aperiodic Vibration of a Piecewise-Linear System

1998 ◽  
Vol 120 (1) ◽  
pp. 181-187 ◽  
Author(s):  
Y. B. Kim
Keyword(s):  
Linear System ◽  
Harmonic Balance ◽  
Piecewise Linear ◽  
Harmonic Balance Method ◽  
Steady State Solution ◽  
Balance Method ◽  
Piecewise Linear System ◽  
Higher Dimensional ◽  
Computational Procedures ◽  
The Stability

A multiple harmonic balance method is presented in this paper for obtaining the aperiodic steady-state solution of a piecewise-linear system. As the method utilizes general and systematic computational procedures, it can be applied to analyze the multi-tone or combination-tone responses for the higher dimensional nonlinear systems such as rotors. Moreover, it is capable of informing the stability of the obtained solution using Floquet theory. To demonstrate the systematic approach of the new method, the almost periodic forced vibration of an articulated loading platform (ALP) with a piecewise-linear stiffness is computed as an example.

Efficient Damping Prediction of Bolted Structures Using Harmonic Balance Method

2012 ◽  
Author(s):  
Pascal Reuss ◽  
Jens Becker ◽  
Lothar Gaul
Keyword(s):  
Harmonic Balance ◽  
Normal Force ◽  
Contact Stiffness ◽  
Harmonic Balance Method ◽  
Computation Time ◽  
Nonlinear Effects ◽  
Element Model ◽  
Balance Method ◽  
Friction Damper ◽  
Step Size

In this paper damping induced by extensive friction occurring in the interface between bolted structures is considered by simulations and experiments. A friction damper is attached to a beam-like flexible structure by screws such that the normal force in the interface can be varied by the clamping force of the screws. Contact and friction force parameters are identified by the comparison of simulated and experimentally determined FRFs for a particular normal force. Afterward a prediction of damping for different configurations is established. For simulations a finite element model is used where suitable contact and friction models are implemented. A time simulation of the system is expensive due to the large number of DoFs of the discretized substructures and the required small step size due to the high contact stiffness. Therefore model reduction methods are used. A further reduction of the computation time can be achieved by using the Harmonic Balance Method (HBM) for a direct frequency domain computation of FRFs. This enables an efficient procedure to approximate the reachable damping as well as to search the optimal damper position and the optimal normal force. The dependency of the friction to the vibration amplitude is therefore taken into account. A more detailed investigation of the nonlinear effects, e.g. higher harmonic response, is then accomplished by transient simulations for the optimal configured system in the time domain and the results are compared to experimental results.

Parametric Stability of a Nonlinear Rotating Blade

2013 ◽  
Author(s):  
Fengxia Wang ◽  
Albert C. J. Luo
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Parametric Stability ◽  
Rotating Blade ◽  
Geometric Term ◽  
Stability And Bifurcations ◽  
The Stability ◽  
The Galerkin Method ◽  
Gyroscopic Systems ◽  
Generalized Harmonic Balance Method

The stability of period-1 motions of a rotating blade with geometric nonlinearity is studied. The roles of cubic stiffening geometric term are considered in the study of nonlinear periodic motions and its stability and bifurcations of a rotating blade. The nonlinear model of a rotating blade is reduced to the ordinary differential equations through the Galerkin method, and the gyroscopic systems with parametric excitations are obtained. The generalized harmonic balance method is employed to determine the period-1 solutions and the corresponding stability and bifurcations.

Bifurcation Analysis of a Power System Model with Three Machines and Four Buses

2016 ◽  
Vol 26 (05) ◽  
pp. 1650082 ◽  
Author(s):  
Yu Chang ◽  
Xiaoli Wang ◽  
Dashun Xu
Keyword(s):  
Power Systems ◽  
Power System ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Chaotic Behavior ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Bifurcation Phenomena ◽  
Fold Bifurcation ◽  
Approximate Expressions

The bifurcation phenomena in a power system with three machines and four buses are investigated by applying bifurcation theory and harmonic balance method. The existence of saddle-node bifurcation and Hopf bifurcation is analyzed in time domain and in frequency domain, respectively. The approach of the fourth-order harmonic balance is then applied to derive the approximate expressions of periodic solutions bifurcated from Hopf bifurcations and predict their frequencies and amplitudes. Since the approach is valid only in some neighborhood of a bifurcation point, numerical simulations and the software Auto2007 are utilized to verify the predictions and further study bifurcations of these periodic solutions. It is shown that the power system may have various types of bifurcations, including period-doubling bifurcation, torus bifurcation, cyclic fold bifurcation, and complex dynamical behaviors, including quasi-periodic oscillations and chaotic behavior. These findings help to better understand the dynamics of the power system and may provide insight into the instability of power systems.

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.

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