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.

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.

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.

Response Characteristics of Systems With Two-Harmonic Parametric Excitation

2018 ◽  
Author(s):  
Fatemeh Afzali ◽  
Gizem D. Acar ◽  
Brian F. Feeny
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Initial Conditions ◽  
Second Harmonic ◽  
Mathieu Equation ◽  
Frequency Content ◽  
Periodic Term ◽  
Response Characteristics ◽  
Truncated Fourier Series ◽  
Exponential Part

Floquet theory is combined with harmonic balance to study parametrically excited systems with two harmonics of excitation, where the second harmonic has twice the frequency of the first one. An approximated solution composed of an exponential part with unknown exponents and a periodic term consisting of a truncated Fourier series is considered. When applied to a two-harmonic Mathieu equation the analysis shows that the second harmonic alters stability characteristics, particularly in the primary and superharmonic instabilities. We also look at the initial conditions response and its frequency content. The second excitation harmonic in the system with parametric damping is seen to disrupt the coexistence phenomenon which is observed in the single-harmonic case.

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.

Experimental and Theoretical Investigation on the Nonlinear Mathieu Equation Applied to a Belt-Pulley System

2007 ◽  
Author(s):  
Guilhem Michon ◽  
Lionel Manin ◽  
Robert G. Parker ◽  
Regis Dufour
Keyword(s):  
Experimental Investigation ◽  
Harmonic Balance ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Harmonic Balance Method ◽  
Mathieu Equation ◽  
Pulley System ◽  
Transverse Vibrations ◽  
Instability Regions ◽  
Set Up

This paper is devoted to the theoretical and experimental investigation of a sample automotive belt-pulley system subjected to tension fluctuations. The equation of motion for transverse vibrations leads to a nonlinear Mathieu equation. The analyzes are performed via either the harmonic balance method for establishing the instability regions or the multiple scales approach for predicting the nonlinear response. An experimental set-up gives rise to non-linear parametric instabilities. The experimental investigation shows that the model is satisfactory.

Response Characteristics of Systems With Parametric Excitation Through Damping and Stiffness

2020 ◽  
Author(s):  
Fatemeh Afzali ◽  
Brian F. Feeny
Keyword(s):  
Fourier Series ◽  
Harmonic Balance ◽  
Floquet Theory ◽  
Initial Conditions ◽  
Time Varying ◽  
Frequency Content ◽  
Periodic Term ◽  
Response Characteristics ◽  
Truncated Fourier Series ◽  
Exponential Part

Abstract Floquet theory is combined with harmonic balance to study parametrically excited systems with combination of both time varying damping and stiffness. An approximated solution having an exponential part with unknown exponents and a periodic term consisting of a truncated Fourier series is considered. When applied to a system with parametric damping and stiffness the analysis shows that combination of parametric damping and stiffness alters stability characteristics, particularly in the primary and superharmonic instabilities comparing to the system with only parametric damping or stiffness. We also look at the initial conditions response and its frequency content. The second excitation harmonic in the system with parametric damping is seen to disrupt the coexistence phenomenon which is observed in the parametric damping case.

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.

A Study of Dynamic Instability of Plates by an Extended Incremental Harmonic Balance Method

1985 ◽  
Vol 52 (3) ◽  
pp. 693-697 ◽  
Author(s):  
C. Pierre ◽  
E. H. Dowell
Keyword(s):  
Harmonic Balance ◽  
Dynamic Instability ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Mathieu Equation ◽  
Equation System ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
Nonlinear Equations Of Motion

The dynamic instability of plates is investigated with geometric nonlinearities being included in the model, which allows one to determine the amplitude of the parametric vibrations. A modal analysis allowing one spatial mode is performed on the nonlinear equations of motion and the resulting nonlinear Mathieu equation is solved by the incremental harmonic balance method, which takes several temporal harmonics into account. When viscous damping is included, a new algorithm is proposed to solve the equation system obtained by the incremental method. For this purpose, a new characterization of the parametric vibration by its total amplitude—or Euclidian norm—is introduced. This algorithm is particularly simple and convenient for computer implementation. The instability regions are obtained with a high degree of accuracy.

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