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.

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.

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.

A Floquet-Based Analysis of Parametric Excitation Through the Damping Coefficient

2020 ◽  
Vol 143 (4) ◽  
Author(s):  
Fatemeh Afzali ◽  
Gizem D. Acar ◽  
Brian F. Feeny
Keyword(s):  
Floquet Theory ◽  
Initial Conditions ◽  
Harmonic Balance Method ◽  
Damping Coefficient ◽  
Periodic Part ◽  
Response Characteristics ◽  
Floquet Solution ◽  
Mathieu’S Equation ◽  
Truncated Fourier Series ◽  
The Stability

Abstract The Floquet theory has been classically used to study the stability characteristics of linear dynamic systems with periodic coefficients and is commonly applied to Mathieu’s equation, which has parametric stiffness. The focus of this article is to study the response characteristics of a linear oscillator for which the damping coefficient varies periodically in time. The Floquet theory is used to determine the effects of mean plus cyclic damping on the Floquet multipliers. An approximate Floquet solution, which includes an exponential part and a periodic part that is represented by a truncated Fourier series, is then applied to the oscillator. Based on the periodic part, the harmonic balance method is used to obtain the Fourier coefficients and Floquet exponents, which are then used to generate the response to the initial conditions, the boundaries of instability, and the characteristics of the free response solution of the system. The coexistence phenomenon, in which the instability wedges disappear and the transition curves overlap, is recovered by this approach, and its features and robustness are examined.

Floquet-Based Analysis of General Responses of the Mathieu Equation

2016 ◽  
Vol 138 (4) ◽  
Author(s):  
Gizem Acar ◽  
Brian F. Feeny
Keyword(s):  
Infinite Series ◽  
Harmonic Balance ◽  
Series Solution ◽  
Floquet Theory ◽  
Harmonic Balance Method ◽  
Mathieu Equation ◽  
Periodic Part ◽  
Floquet Solution ◽  
Exponential Part ◽  
Stability Of The Solution

Solutions to the linear unforced Mathieu equation, and their stabilities, are investigated. Floquet theory shows that the solution can be written as a product between an exponential part and a periodic part at the same frequency or half the frequency of excitation. In the current work, an approach combining Floquet theory with the harmonic balance method is investigated. A Floquet solution having an exponential part with an unknown exponential argument and a periodic part consisting of a series of harmonics is assumed. Then, performing harmonic balance, frequencies of the response are found and stability of the solution is examined over a parameter set. The truncated solution is consistent with an existing infinite series solution for the undamped case. The truncated solution is then applied to the damped Mathieu equation and parametric excitation with two harmonics.

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