Applications of the Theory of Impulsive Parametric Excitation and New Treatments of General Parametric Excitation Problems

1973 ◽  
Vol 40 (1) ◽  
pp. 78-86 ◽  
Author(s):  
C. S. Hsu ◽  
W.-H. Cheng
Keyword(s):  
Parametric Excitation ◽  
Stability Theory ◽  
Mechanical Systems ◽  
General Nature ◽  
Periodic Systems ◽  
Stability Conditions ◽  
Approximate Methods ◽  
Step Functions ◽  
The Stability ◽  
New Treatments

In this paper the stability theory of impulsive parametric excitation developed in [1] is first applied to three mechanical systems. Explicit and exact stability conditions are easily found and some typical stability charts are presented. Also presented in the paper is the use of this theory and a parallel theory involving step functions as approximate methods for treating periodic parametric excitations of more general nature. Exploratory studies along this line have led us to believe that these approximate methods have promise to be very powerful and practical tools for dealing with the stability of general high-order periodic systems.

Frequency Sweeping With Concurrent Parametric Amplification

2008 ◽  
Author(s):  
Nicholas J. Miller ◽  
Steven W. Shaw
Keyword(s):  
Quality Factor ◽  
Parametric Excitation ◽  
Mechanical Systems ◽  
Parametric Amplification ◽  
Stability Conditions ◽  
Mass Sensor ◽  
Modal Interactions ◽  
Numerical Example ◽  
Mems Mass Sensor ◽  
Frequency Sweeping

In this paper we explore parametric amplification of multidegree of freedom mechanical systems. We consider frequency conditions for modal interactions and determine stability conditions for three important cases. We develop conditions under which it is possible to sweep with direct and parametric excitation to produce a sweep response with amplified effective quality factor of resonances encountered during the sweep. With this technique it is possible to improve the measurement of resonance locations in swept devices, such as those that operate on resonance shifting. A numerical example motivated by a MEMS mass sensor is given in support of the analysis.

An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Ashu Sharma ◽  
S. C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Picard Iteration ◽  
Periodic Systems ◽  
Transition Matrices ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Stability Diagrams ◽  
The Stability

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

Frequency Sweeping With Concurrent Parametric Amplification

2011 ◽  
Vol 134 (2) ◽  
Author(s):  
Nicholas J. Miller ◽  
Steven W. Shaw
Keyword(s):  
Parametric Excitation ◽  
Mechanical Systems ◽  
Parametric Amplification ◽  
Stability Conditions ◽  
Mass Sensor ◽  
Frequency Range ◽  
Quality Factors ◽  
Modal Interactions ◽  
Electromechanical Systems ◽  
Frequency Sweeping

In this paper, we explore parametric amplification of multiple resonances in multidegree-of-freedom mechanical systems, and the use of frequency sweeping with a parametric pump to amplify several adjacent resonance peaks. We develop conditions under which it is possible to sweep with direct and parametric excitation to produce a sweep response with amplified effective quality factors for all resonances over a given frequency range. We determine gain and stability conditions and include analysis for potential problematic modal interactions. This technique makes it possible to improve the measurements of resonance locations in devices, for example, sensors that rely on tracking shifts in resonance peaks. The results are demonstrated on a model for a multi-analyte micro-electromechanical systems mass sensor.

An Approximate Analysis of Quasi-Periodic Systems via Floquét Theory

2017 ◽  
Author(s):  
Ashu Sharma ◽  
Subhash C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Periodic Systems ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Approximate Approach ◽  
Varying Coefficients ◽  
P Transformation ◽  
The Stability

Parametrically excited systems are generally represented by a set of linear/nonlinear ordinary differential equations with time varying coefficients. In most cases, the linear systems have been modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. Although Floquét theory is applicable only to periodic systems, it is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to two typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are extremely close to the exact boundaries of the original quasi-periodic equations. The exact boundaries are detected by computing the maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. The coefficients of the parametric excitation terms are not necessarily small in all cases. ‘Instability loops’ or ‘Instability pockets’ that appear in the stability diagram of Meissner’s equation are also observed in one case presented here. The proposed approximate approach would allow one to construct Lyapunov-Perron (L-P) transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.

scholarly journals Lyapunov Stability of Quasiperiodic Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Sangram Redkar
Keyword(s):  
Lyapunov Stability ◽  
Parametric Excitation ◽  
State Vector ◽  
Stability Margin ◽  
The State ◽  
Stability Conditions ◽  
Quasiperiodic Systems ◽  
The Stability

We present some observations on the stability and reducibility of quasiperiodic systems. In a quasiperiodic system, the periodicity of parametric excitation is incommensurate with the periodicity of certain terms multiplying the state vector. We present a Lyapunov-type approach and the Lyapunov-Floquet (L-F) transformation to derive the stability conditions. This approach can be utilized to investigate the robustness, stability margin, and design controller for the system.

Impulsive Parametric Excitation: Theory

1972 ◽  
Vol 39 (2) ◽  
pp. 551-558 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Stability Analysis ◽  
General Theory ◽  
Parametric Excitation ◽  
Periodic Systems ◽  
Stability Criteria ◽  
Dirac Delta ◽  
Special Cases ◽  
Delta Functions ◽  
Analytic Stability ◽  
The Stability

Given in this paper is the development of a theory for dynamical systems subjected to periodic impulsive parametric excitations. By periodic impulsive parametric excitation we mean those excitations representable by periodic coefficients which consist of sequences of Dirac delta functions. It turns out that for this class of periodic systems the stability analysis can be carried out in a remarkably simple and general manner without approximation. In the paper, after giving the general theory, many special cases are examined. In many instances simple and closed-form analytic stability criteria can be easily established.

scholarly journals Causality and stability conditions of a conformal charged fluid

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Farid Taghinavaz
Keyword(s):  
Chemical Potential ◽  
Second Law Of Thermodynamics ◽  
Dense Medium ◽  
Wave Number ◽  
Second Law ◽  
Stability Conditions ◽  
Charged Fluid ◽  
Large Wave ◽  
The Stability ◽  
Large Wave Number

Abstract In this paper, I study the conditions imposed on a normal charged fluid so that the causality and stability criteria hold for this fluid. I adopt the newly developed General Frame (GF) notion in the relativistic hydrodynamics framework which states that hydrodynamic frames have to be fixed after applying the stability and causality conditions. To do this, I take a charged conformal matter in the flat and 3 + 1 dimension to analyze better these conditions. The causality condition is applied by looking to the asymptotic velocity of sound hydro modes at the large wave number limit and stability conditions are imposed by looking to the imaginary parts of hydro modes as well as the Routh-Hurwitz criteria. By fixing some of the transports, the suitable spaces for other ones are derived. I observe that in a dense medium having a finite U(1) charge with chemical potential μ0, negative values for transports appear and the second law of thermodynamics has not ruled out the existence of such values. Sign of scalar transports are not limited by any constraints and just a combination of vector transports is limited by the second law of thermodynamic. Also numerically it is proved that the most favorable region for transports $$ {\tilde{\upgamma}}_{1,2}, $$ γ ˜ 1 , 2 , coefficients of the dissipative terms of the current, is of negative values.

Basic Concepts in the Stability Theory of Thin-walled Structures

2010 ◽  
Author(s):  
A. Guran ◽  
L. Lebedev ◽  
Michail D. Todorov ◽  
Christo I. Christov
Keyword(s):  
Stability Theory ◽  
Thin Walled Structures ◽  
Thin Walled ◽  
Basic Concepts ◽  
The Stability

scholarly journals Iterative stability analysis for general polynomial control systems

2021 ◽  
Author(s):  
Bo Xiao ◽  
Hak-Keung Lam ◽  
Zhixiong Zhong
Keyword(s):  
Stability Analysis ◽  
Lyapunov Function ◽  
Control Systems ◽  
Polynomial System ◽  
Stability Conditions ◽  
General Polynomial ◽  
Main Challenge ◽  
Detailed Simulation ◽  
Feasible Solutions ◽  
The Stability

AbstractThe main challenge of the stability analysis for general polynomial control systems is that non-convex terms exist in the stability conditions, which hinders solving the stability conditions numerically. Most approaches in the literature impose constraints on the Lyapunov function candidates or the non-convex related terms to circumvent this problem. Motivated by this difficulty, in this paper, we confront the non-convex problem directly and present an iterative stability analysis to address the long-standing problem in general polynomial control systems. Different from the existing methods, no constraints are imposed on the polynomial Lyapunov function candidates. Therefore, the limitations on the Lyapunov function candidate and non-convex terms are eliminated from the proposed analysis, which makes the proposed method more general than the state-of-the-art. In the proposed approach, the stability for the general polynomial model is analyzed and the original non-convex stability conditions are developed. To solve the non-convex stability conditions through the sum-of-squares programming, the iterative stability analysis is presented. The feasible solutions are verified by the original non-convex stability conditions to guarantee the asymptotic stability of the general polynomial system. The detailed simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed approach is more capable to find feasible solutions for the general polynomial control systems when compared with the existing ones.

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