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.

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.

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.

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.

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.

Nonlinear Vibration Analysis of Viscoelastic Isolator

2012 ◽  
Vol 160 ◽  
pp. 140-144
Author(s):  
Chao Zhou ◽  
Cai Mao Zhong
Keyword(s):  
Dynamic Response ◽  
Fractional Derivative ◽  
Nonlinear Dynamic ◽  
Harmonic Balance Method ◽  
Nonlinear Damping ◽  
Vibration Isolator ◽  
Response Characteristics ◽  
Nonlinear Dynamic Equation ◽  
The Stability ◽  
Stiffness And Damping

Research on nonlinear dynamic response of passive vibration isolator, which was excited by foundation vibration and isolated by viscoelastic material was done. Nonlinear stiffness was expressed by the cubic polynomial function of deformation and nonlinear damping was characterized by viscoelastic fractional derivative operator. Then the fractional derivative nonlinear dynamic equation of passive vibration isolator was established. The dynamic response characteristics were analyzed by harmonic balance method and the frequency response equation and amplitude-frequency curve were obtained, and furthermore, the influence of nonlinearity on system was analyzed. Finally, the stability and the stable interval of the periodic solution were argued by the Floquet theory. The result s indicates that the proposed equation can precisely describe the dynamic characteristics of viscoelastic vibration isolator. The ignorance of nonlinearity of stiffness and damping will result in obvious error. The proposed method provides theoretic reference for design of viscoelastic isolator and the evaluation of its effect.

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.

scholarly journals Transition curves in a parametrically excited pendulum with a force of elliptic type

2012 ◽  
Vol 468 (2148) ◽  
pp. 3995-4007 ◽  
Author(s):  
Si Mohamed Sah ◽  
Brian Mann
Keyword(s):  
Numerical Integration ◽  
Elliptic Function ◽  
Harmonic Balance ◽  
Floquet Theory ◽  
Parameter Plane ◽  
Elliptic Type ◽  
Mathieu’S Equation ◽  
Balance Analysis ◽  
The Stability ◽  
Transition Curves

This article investigates the equilibria and stability of a pendulum when the support has a prescribed motion defined by an elliptic function. Stability charts are generated in the parameter plane for different values of the elliptic function modulus. Numerical integration and Floquet theory are used to generate stability charts that are later obtained through harmonic balance analysis. It is shown that the size and location of the instability tongues is directly linked to the elliptic function modulus. Comparisons are also made between the stability charts of Mathieu's equation and those of the pendulum when the prescribed motion is defined by an elliptic function.

Parametrically Excited Behavior of a Railway Wheelset

1988 ◽  
Vol 110 (1) ◽  
pp. 8-17 ◽  
Author(s):  
J. Lieh ◽  
I. Haque
Keyword(s):  
Floquet Theory ◽  
Initial Conditions ◽  
Excitation Frequency ◽  
Time History ◽  
Parametrically Excited ◽  
Load Variations ◽  
Rail Vehicles ◽  
Stability And Instability ◽  
Instability Regions ◽  
The Stability

The dynamic response of rail vehicles is affected by parameters such as wheel-rail geometry, track gage, and axle load. Variations in these parameters, as a rail vehicle moves down the track, can cause instabilities that are related to parametrically excited behavior. This paper reports on the use of Floquet Theory to predict the stability and instability regions for a single wheelset subjected to harmonic variations in wheel-rail geometry, track gage and axle load. Time studies showing the response of a wheelset to various initial conditions are also included. The results show that harmonic variations in the wheel-rail geometry can influence the behavior of a wheelset significantly. The system is especially susceptible to variations in conicity. Time history studies show that the response is dependent on initial conditions, the amount of variations and the magnitude of the excitation frequency.

In-Plane Parametric Instability of a Rigid Body With a Dual-Rotor System

2015 ◽  
Author(s):  
Allen Anilkumar ◽  
V. Kartik
Keyword(s):  
Rigid Body ◽  
Floquet Theory ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Harmonic Balance Method ◽  
Frequency Spectra ◽  
Stability Diagrams ◽  
The Stability ◽  
Lumped Masses

Rotating machines can be modeled at a basic level using lumped masses that are rotating about and attached using springs to an axis. Even such seemingly simple system can exhibit rich dynamics in the presence of time-varying terms in the governing differential equations. This paper investigates the dynamics of a rigid body with two attached rotors that rotate in the same plane. The system is parametrically-excited and the equations of motion are periodic in both rotor frequencies. The frequency spectra of the time responses show distinct side-band structures centered about the unforced natural frequencies. In addition to classical resonances, the stability diagrams generated using Floquet theory reveal instabilities at unexpected combinations of the forcing and natural frequencies. The harmonic balance method is employed to verify the stability boundaries obtained using Floquet theory. The study reveals safe regimes of parameter combinations that can help prevent the onset of instability in such systems.

Novel Fiber-Optic Sensing System for Detection of Methane

2013 ◽  
Vol 562-565 ◽  
pp. 1008-1015 ◽  
Author(s):  
Shu Tao Wang ◽  
Peng Wei Zhang ◽  
Quan Min Zhu
Keyword(s):  
Fiber Optic ◽  
Detection System ◽  
Step Motor ◽  
Harmonic Detection ◽  
Distributed Feedback Laser ◽  
Gas Cell ◽  
Response Characteristics ◽  
The Stability ◽  
Application Fields ◽  
Methane Detection

Based on DFBLD (Distributed Feedback Laser Diode) and harmonic detection technique, a novel fiber-optic methane detection system is constructed. The system can be applied to broad-range concentration detection of methane. Based on the approximation express of the law of Beer-Lambert, detection of methane with various concentration from 0% to 20% is completed using subtraction of background and ratio processing method, as the atmosphere surroundings are treated as background noise. The direct absorption spectra for various concentration is measured using GRIN gas cell, combined with DFBLD. The R5 line of the 2v3 band of methane is selected as the absorption peak. The system is tested online during gas mixing process and the linear relationship between system indication and concentration variation is validated. Also the stability and dynamic response characteristics are confirmed by the experiments. The sensitivity of the system can be adjusted according to the concentration level of various field environments by changing the prism distance using step motor. In the range of 0% to 20% the sensitivity of methane detection can arrive at 0.001%. So the system can be applied to various application fields and adopted as monitoring instruments for coalmine tunnel and natural pipeline.

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