On Instability Pockets and Influence of Damping in Parametrically Excited Systems

2018 ◽  
Vol 140 (5) ◽  
Author(s):  
Ashu Sharma ◽  
S. C. Sinha
Keyword(s):  
Degrees Of Freedom ◽  
State Transition ◽  
Degree Of Freedom ◽  
Transition Matrices ◽  
Stability Boundaries ◽  
Parametrically Excited ◽  
Parametric Space ◽  
Stability Diagrams ◽  
Wide Range ◽  
The Stability

In most parametrically excited systems, stability boundaries cross each other at several points to form closed unstable subregions commonly known as “instability pockets.” The first aspect of this study explores some general characteristics of these instability pockets and their structural modifications in the parametric space as damping is induced in the system. Second, the possible destabilization of undamped systems due to addition of damping in parametrically excited systems has been investigated. The study is restricted to single degree-of-freedom systems that can be modeled by Hill and quasi-periodic (QP) Hill equations. Three typical cases of Hill equation, e.g., Mathieu, Meissner, and three-frequency Hill equations, are analyzed. State transition matrices of these equations are computed symbolically/analytically over a wide range of system parameters and instability pockets are observed in the stability diagrams of Meissner, three-frequency Hill, and QP Hill equations. Locations of the intersections of stability boundaries (commonly known as coexistence points) are determined using the property that two linearly independent solutions coexist at these intersections. For Meissner equation, with a square wave coefficient, analytical expressions are constructed to compute the number and locations of the instability pockets. In the second part of the study, the symbolic/analytic forms of state transition matrices are used to compute the minimum values of damping coefficients required for instability pockets to vanish from the parametric space. The phenomenon of destabilization due to damping, previously observed in systems with two degrees-of-freedom or higher, is also demonstrated in systems with one degree-of-freedom.

scholarly journals A method for identification of non-linear multi-degree-of-freedom systems

1998 ◽  
Vol 212 (4) ◽  
pp. 287-298 ◽  
Author(s):  
G Dimitriadis ◽  
J E Cooper
Keyword(s):  
System Identification ◽  
Degree Of Freedom ◽  
Accurate Identification ◽  
Wide Range ◽  
Non Linear ◽  
Linear System Identification ◽  
The Stability ◽  
Non Linear System ◽  
Identification Techniques ◽  
Flutter Testing

System identification methods for non-linear aeroelastic systems could find uses in many aeroelastic applications such as validating finite element models and tracking the stability of aircraft during flight flutter testing. The effectiveness of existing non-linear system identification techniques is limited by various factors such as the complexity of the system under investigation and the type of non-linearities present. In this work, a new approach is introduced which can identify multi-degree-of-freedom systems featuring any type of non-linear function, including discontinuous functions. The method is shown to yield accurate identification of three mathematical models of aeroelastic systems containing a wide range of structural non-linearities.

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.

An Analytical and Experimental Study of Automobile Dynamics With Random Roadway Inputs

1977 ◽  
Vol 99 (4) ◽  
pp. 284-292 ◽  
Author(s):  
A. J. Healey ◽  
E. Nathman ◽  
C. C. Smith
Keyword(s):  
Experimental Study ◽  
Degrees Of Freedom ◽  
Low Frequency ◽  
Vertical Direction ◽  
Degree Of Freedom ◽  
Vehicle Body ◽  
Acceleration Response ◽  
Wide Range ◽  
Frequency Components ◽  
The Common

This paper presents the results of an analytical and experimental study of ride vibrations in an automobile over roads of various degrees of roughness. Roadway roughness inputs were measured. Three different linear mathematical models were employed to predict the acceleration response of the vehicle body. The models used included two, four, and seven degrees of freedom, primarily for vertical direction motion. The results show that the prime source of errors in predicting responses of this type lies in the common assumptions made for roadway roughness spectra. With adequate description of the roadway inputs, the results showed that the seven degree of freedom model accurately predicted the low frequency response (up to 10 Hz). Using the seven degree of freedom model, predicted accelerations compare well with measured data for a wide range of roadways in the low frequency range. Higher frequency components in the measured acceleration response are significant and are illustrated here.

scholarly journals On the Stability of Reconstruction of Irregularly Sampled Diffraction Fields

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Vladislav Uzunov ◽  
Atanas Gotchev ◽  
Karen Egiazarian
Keyword(s):  
Degrees Of Freedom ◽  
Light Field ◽  
Monochromatic Light ◽  
Three Dimensional ◽  
Regularized Inversion ◽  
Volume Of Interest ◽  
Wide Range ◽  
Data Points ◽  
Data Point ◽  
The Stability

This paper addresses the problem of reconstruction of a monochromatic light field from data points, irregularly distributed within a volume of interest. Such setting is relevant for a wide range of three-dimensional display and beam shaping applications, which deal with physically inconsistent data. Two finite-dimensional models of monochromatic light fields are used to state the reconstruction problem as regularized matrix inversion. The Tikhonov method, implemented by the iterative algorithm of conjugate gradients, is used for regularization. Estimates of the model dimensionality are related to the number of degrees of freedom of the light field as to show how to control the data redundancy. Experiments demonstrate that various data point distributions lead to ill-poseness and that regularized inversion is able to compensate for the data point inconsistencies with good numerical performance.

scholarly journals Analyzing the Stability for the Motion of an Unstretched Double Pendulum Near Resonance

2021 ◽  
Vol 11 (20) ◽  
pp. 9520
Author(s):  
Tarek S. Amer ◽  
Roman Starosta ◽  
Adelkarim S. Elameer ◽  
Mohamed A. Bek
Keyword(s):  
Degrees Of Freedom ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Analytic Solutions ◽  
Double Pendulum ◽  
Nonlinear Dynamical ◽  
Near Resonance ◽  
Wide Range ◽  
The Stability ◽  
The Impact

This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics.

A Study of the Parametrically Excited Behavior of Passenger and Freight Railway Vehicles Using Linear Models

1991 ◽  
Vol 113 (2) ◽  
pp. 336-338 ◽  
Author(s):  
J. Lieh ◽  
I. Haque
Keyword(s):  
Parametric Resonance ◽  
Parametric Excitation ◽  
Floquet Theory ◽  
Linear Models ◽  
Stability Boundaries ◽  
Parametrically Excited ◽  
Critical Speeds ◽  
Railway Vehicles ◽  
Principal Parametric Resonance ◽  
The Stability

This paper presents a study of the parametrically excited behavior of passenger and freight vehicles on tangent track due to harmonic variations in conicity using linear models. The effect of primary and secondary stiffnesses on parametric excitation is also studied. Floquet theory is used to find the stability boundaries. The results show that wavelengths associated with conicity variation that are in the vicinity of half the kinematic wavelengths of the vehicles can lead to significant reductions in critical speeds. Results also show that the primary and warp stiffnesses can affect the severity of principal parametric resonance depending on the vehicle models and magnitude of stiffnesses chosen.

Vibrations of an Elastically Mounted Spinning Rotor Partially Filled With Liquid

1982 ◽  
Vol 104 (2) ◽  
pp. 389-396 ◽  
Author(s):  
G. Lichtenberg
Keyword(s):  
Equations Of Motion ◽  
Constant Angular Velocity ◽  
Two Dimensional ◽  
Field Equations ◽  
Stability Boundaries ◽  
Inviscid Incompressible Fluid ◽  
Wide Range ◽  
The Stability ◽  
Elastically Supported ◽  
Dimensional Surface

The stability of a rotor with a cylindrical cavity, spinning with constant angular velocity and partially filled with an inviscid, incompressible fluid is studied. The rotor is elastically supported on a vertically mounted massless shaft in overhung position. A set of coupled linearized spatial equations of motion of the rotor and field equations, as well as boundary conditions of the liquid, is established and solved, leading to a characteristic equation. First numerical results predict a wide range of rotor speeds, where the system performs unstable motions caused by a two-dimensional surface wave of the liquid. The stability boundaries are calculated for a flat rotor in dependence on the mass of the contained liquid and agree extremely well with experimental data.

scholarly journals Stability boundaries of roll and square convection in binary fluid mixtures with positive separation ratio

2000 ◽  
Vol 408 ◽  
pp. 121-147 ◽  
Author(s):  
B. HUKE ◽  
M. LÜCKE ◽  
P. BÜCHEL ◽  
CH. JUNG
Keyword(s):  
Three Dimensional ◽  
Stationary Solutions ◽  
Stability Boundaries ◽  
Binary Fluid ◽  
Fluid Mixtures ◽  
Separation Ratio ◽  
Wide Range ◽  
The Stability ◽  
Binary Fluid Mixtures ◽  
Rayleigh Benard

Rayleigh–Bénard convection in horizontal layers of binary fluid mixtures heated from below with realistic horizontal boundary conditions is studied theoretically using multi-mode Galerkin expansions. For positive separation ratios the main difference between the mixtures and pure fluids lies in the existence of stable three-dimensional patterns near onset in a wide range of the parameter space. We evaluated the stationary solutions of roll, crossroll, and square convection and we determined the location of the stability boundaries for many parameter combinations thereby obtaining the Busse balloon for roll and square patterns.

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