Period Doubling Motions of a Nonlinear Rotating Beam at 1:1 Resonance

2014 ◽  
Vol 24 (12) ◽  
pp. 1450159 ◽  
Author(s):  
Fengxia Wang ◽  
Yuhui Qu
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Balance Method ◽  
Rotating Beam ◽  
Period 2 ◽  
Geometric Stiffening ◽  
Torsional Excitation ◽  
The Galerkin Method

A rotating beam subjected to a torsional excitation is studied in this paper. Both quadratic and cubic geometric stiffening nonlinearities are retained in the equation of motion, and the reduced model is obtained via the Galerkin method. Saddle-node bifurcations and Hopf bifurcations of the period-1 motions of the model were obtained via the higher order harmonic balance method. The period-2 and period-4 solutions, which are emanated from the period-1 and period-2 motions, respectively, are obtained by the combined implementation of the harmonic balance method, Floquet theory, and Discrete Fourier transform (DFT). The analytical periodic solutions and their stabilities are verified through numerical simulation.

Analysis of the Periodic Damping Coefficient Equation Based on Floquet Theory

2017 ◽  
Author(s):  
Fatemeh Afzali ◽  
Gizem D. Acar ◽  
Brian F. Feeny
Keyword(s):  
Approximate Solution ◽  
Harmonic Balance ◽  
Floquet Theory ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Damping Coefficient ◽  
Harmonic Series ◽  
Periodic Part ◽  
Damped System ◽  
Linear Differential

In this paper, we study the response of a linear differential equation, for which the damping coefficient varies periodically in time. We use Floquet theory combined with the harmonic balance method to find the approximate solution and capture the stability criteria. Based on Floquet theory the approximate solution includes the exponential part having an unknown exponent, and a periodic part, which is expressed using a truncated series of harmonics. After substituting the assumed response in the equation, the harmonic balance method is applied. We use the characteristic equation of the truncated harmonic series to obtain the Floquet exponents. The free response and stability characteristics of the damped system for a set of parameters are shown.

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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