Stability Analysis of Parametrically Excited Systems Using the Energy-Growth Exponent/Coefficient

2017 ◽  
Vol 17 (02) ◽  
pp. 1750018 ◽  
Author(s):  
Yuchun Li ◽  
Lishi Wang ◽  
Yanqing Yu
Keyword(s):  
Parametric Instability ◽  
Mathieu Equation ◽  
Energy Criterion ◽  
Growth Exponent ◽  
Parametrically Excited ◽  
Floquet Exponent ◽  
Energy Growth ◽  
Triple Pendulum ◽  
Analytical Formulae ◽  
The Stability

In this paper, the energy exponent is used to study the instability of parametrically excited systems governed by the damped Mathieu equation. Through the numerical tests, an energy-growth exponent (EGE) is adopted to evaluate the instability intensity and instability boundary of the system. The EGE can be expressed as a product of the modal frequency and a dimensionless coefficient, defined as the energy-growth coefficient (EGC). Based on the Floquet theory, the relationship between the EGE, Floquet exponent and Lyapunov exponent are derived. An energy criterion of parametric instability is proposed by using the EGE. Using a simple pendulum as an example, the geometric characteristics of the EGC are investigated, and approximate analytical formulae of the EGC/EGE for three different unstable patterns are developed. The EGC/EGE formulae are applicable to the parametrically excited systems governed by the damped Mathieu equation. The unstable behaviors and properties of parametric vibrations are analyzed and discussed in details with EGE/EGC for three systems including a triple pendulum, two-dimensional sloshing fluid, and a two-span continuous beam. The stability boundaries established by using EGE/EGC agree well with those by the conventional theory and experiment. As a practical tool, the EGE/EGC formulae can be easily applied to analyzing the unstable intensities and boundaries of the parametrically excited systems.

Stability and Limit Cycles of Parametrically Excited, Axially Moving Strings

1996 ◽  
Vol 118 (3) ◽  
pp. 346-351 ◽  
Author(s):  
E. M. Mockensturm ◽  
N. C. Perkins ◽  
A. Galip Ulsoy
Keyword(s):  
Limit Cycles ◽  
Parametric Instability ◽  
Parametrically Excited ◽  
Weakly Nonlinear ◽  
Closed Form Solutions ◽  
Belt Drives ◽  
Existence And Stability ◽  
Instability Regions ◽  
Axially Moving ◽  
The Stability

Tension fluctuations are the dominant source of excitation in automotive belts. In particular designs, these fluctuations may parametrically excite large amplitude transverse belt vibrations and adversely impact belt life. This paper evaluates an efficient discrete model of a parametrically excited translating belt. The efficiency derives from the use of translating string eigenfunctions as a basis for a Galerkin discretization of the equations of transverse belt response. Accurate and low-order models lead to simple closed-form solutions for the existence and stability of limit cycles near parametric instability regions. In particular, simple expressions are found for the stability boundaries of the general nth-mode principal parametric instability regions and the first summation and difference parametric instability regions. Subsequent evaluation of the weakly nonlinear equation of motion leads to an analytical expression for the amplitudes (and stability) of nontrivial limit cycles that exist around the nth-mode principal parametric instability regions. Example results highlight important conclusions concerning the response of automotive belt drives.

scholarly journals Parametric Instability of a Traveling Plate Partially Supported by a Laterally Moving Elastic Foundation

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
V. Kartik ◽  
J. A. Wickert
Keyword(s):  
Data Storage ◽  
Parametric Instability ◽  
Primary Resonance ◽  
Free Edge ◽  
Moving Plate ◽  
Torsional Mode ◽  
Plane Vibration ◽  
Out Of Plane ◽  
Axially Moving ◽  
The Stability

The parametric excitation of an axially moving plate is examined in an application where a partial foundation moves in the plane of the plate and in a direction orthogonal to the plate’s transport. The stability of the plate’s out-of-plane vibration is of interest in a magnetic tape data storage application where the read/write head is substantially narrower than the tape’s width and is repositioned during track-following maneuvers. In this case, the model’s equation of motion has time-dependent coefficients, and vibration is excited both parametrically and by direct forcing. The parametric instability of out-of-plane vibration is analyzed by using the Floquet theory for finite values of the foundation’s range of motion. For a relatively soft foundation, vibration is excited preferentially at the primary resonance of the plate’s fundamental torsional mode. As the foundation’s stiffness increases, multiple primary and combination resonances occur, and they dominate the plate’s stability; small islands, however, do exist within unstable zones of the frequency-amplitude parameter space for which vibration is marginally stable. The plate’s and foundation’s geometry, the foundation’s stiffness, and the excitation’s amplitude and frequency can be selected in order to reduce undesirable vibration that occurs along the plate’s free edge.

Dynamic Instability in Quartering Seas—Part III: Nonlinear Effects on Periodic Motions

1997 ◽  
Vol 41 (03) ◽  
pp. 210-223 ◽  
Author(s):  
K. J. Spyrou
Keyword(s):  
Periodic Motion ◽  
Horizontal Plane ◽  
Dynamic Instability ◽  
Parametric Instability ◽  
Nonlinear Effects ◽  
Ship Motions ◽  
Nonlinear Phenomena ◽  
Specific Sequence ◽  
The Stability ◽  
Two Stages

The loss of stability of the horizontal-plane periodic motion of a steered ship in waves is investigated. In earlier reports we referred to the possibility of a broaching mechanism that will be intrinsic to the periodic mode, whereby there will exist no need for the ship to go through the surf-riding stage. However, about this point the discussion was essentially conjectural. In order to provide substance we present here a theoretical approach that is organized in two stages: Initially, we demonstrate the existence of a mechanism of parametric instability of yaw on the basis of a rudimentary, single-degree model of maneuvering motion in waves. Then, with a more elaborate model, we identify the underlying nonlinear phenomena that govern the large-amplitude horizontal ship motions, considering the ship as a multi-degree, nonlinear oscillator. Our analysis brings to light a very specific sequence of phenomena leading to cumulative broaching that involves a change in the stability of the ordinary periodic motion on the horizontal plane, a transition towards subharmonic response and, ultimately, a sudden jump to resonance. Possible means for controlling the onset of such undesirable behavior are also investigated.

Influence of Axial Excitations and of Boundary Conditions on the Parametric Instability of a Beam

1995 ◽  
Author(s):  
Régis Dufour ◽  
Alain Berlioz ◽  
Thomas Streule
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Parametric Instability ◽  
Experimental Tests ◽  
Time Varying ◽  
Periodic Force ◽  
Modal Reduction ◽  
Time Varying Parameter ◽  
The Stability

Abstract In this paper the stability of the lateral dynamic behavior of a pinned-pinned, clamped-pinned and clamped-clamped beam under axial periodic force or torque is studied. The time-varying parameter equations are derived using the Rayleigh-Ritz method. The stability analysis of the solution is based on Floquet’s theory and investigated in detail. The Rayleigh-Ritz results are compared to those of a finite element modal reduction. It shows that the lateral instabilities of the beam depend on the forcing frequency, the type of excitation and the boundary conditions. Several experimental tests enable the validation of the numerical results.

A Quasiperiodic Mathieu Equation

1995 ◽  
Author(s):  
Richard Rand ◽  
Rachel Hastings
Keyword(s):  
Numerical Integration ◽  
Mathieu Equation ◽  
Stability Regions ◽  
Regions Of Stability ◽  
Extensive Computation ◽  
The Stability ◽  
Direct Numerical Integration

Abstract In this work we investigate the following quasiperiodic Mathieu equation: x ¨ + ( δ + ϵ cos ⁡ t + ϵ cos ⁡ ω t ) x = 0 We use numerical integration to determine regions of stability in the δ–ω plane for fixed ϵ. Graphs of these stability regions are presented, based on extensive computation. In addition, we use perturbations to obtain approximations for the stability regions near δ=14 for small ω, and we compare the results with those of direct numerical integration.

Use of diagrams in computing the stability boundaries for the Mathieu equation

1977 ◽  
Vol 17 (3) ◽  
pp. 360-360
Author(s):  
R. Banene ◽  
V. Lesauskis
Keyword(s):  
Mathieu Equation ◽  
Stability Boundaries ◽  
The Stability

Parametric instability of internal gravity waves

1975 ◽  
Vol 67 (4) ◽  
pp. 667-687 ◽  
Author(s):  
A. D. McEwan ◽  
R. M. Robinson
Keyword(s):  
Internal Wave ◽  
Large Scale ◽  
Wave Spectrum ◽  
Parametric Instability ◽  
Mathieu Equation ◽  
Internal Gravity Waves ◽  
Small Scale ◽  
Preliminary Calculation ◽  
Theoretical Predictions ◽  
Oceanic Internal Wave

A continuously stratified fluid, when subjected to a weak periodic horizontal acceleration, is shown to be susceptible to a form of parametric instability whose time dependence is described, in its simplest form, by the Mathieu equation. Such an acceleration could be imposed by a large-scale internal wave field. The growth rates of small-scale unstable modes may readily be determined as functions of the forcing-acceleration amplitude and frequency. If any such mode has a natural frequency near to half the forcing frequency, the forcing amplitude required for instability may be limited in smallness only by internal viscous dissipation. Greater amplitudes are required when boundaries constrain the form of the modes, but for a given bounding geometry the most unstable mode and its critical forcing amplitude can be defined.An experiment designed to isolate the instability precisely confirms theoretical predictions, and evidence is given from previous experiments which suggest that its appearance can be the penultimate stage before the traumatic distortion of continuous stratifications under internal wave action.A preliminary calculation, using the Garrett & Munk (197%) oceanic internal wave spectrum, indicates that parametric instability could occur in the ocean at scales down to that of the finest observed microstructure, and may therefore have a significant role to play in its formation.

Lyapunov and marginal instability in Hamiltonian systems

1999 ◽  
Vol 77 (8) ◽  
pp. 603-633 ◽  
Author(s):  
J Grindlay
Keyword(s):  
Lyapunov Exponents ◽  
Nonlinear Oscillator ◽  
Linear Chain ◽  
Mathieu Equation ◽  
Singular Values ◽  
Space Velocity ◽  
Singular Value ◽  
The Stability ◽  
Value Decomposition ◽  
Simple Systems

The variational equations and the evolution matrix are introduced and used to discuss the stability of a bound Hamiltonian trajectory. Singular-value decomposition is applied to the evolution matrix. Singular values and Lyapunov exponents are defined and their properties described. The singular-value expansion of the phase-space velocity is derived. Singular values and Lyapunov exponents are used to characterize the stability behaviour of five simple systems, namely, the nonlinear oscillator with cubic anharmonicity, the quasi-periodic Mathieu equation, the Hénon-Heilesmodel, the 4+2 linear chain with cubic anharmonicity, and an integrable system of arbitrary order.PACS Nos.: 03.20, 05.20

Investigation of Power Harvesting via Parametric Excitations

2008 ◽  
Vol 20 (5) ◽  
pp. 545-557 ◽  
Author(s):  
Mohammed F. Daqaq ◽  
Christopher Stabler ◽  
Yousef Qaroush ◽  
Thiago Seuaciuc-Osório
Keyword(s):  
Output Power ◽  
Coupling Coefficient ◽  
Electromechanical Coupling ◽  
Multiple Scales ◽  
Broad Band ◽  
Parametric Instability ◽  
Load Resistance ◽  
Resistive Load ◽  
Parametrically Excited ◽  
Critical Excitation

This article presents an analytical and experimental investigation of energy harvesting via parametrically excited cantilever beams. To that end, we consider a lumped-parameter non-linear model that describes the first-mode dynamics of a parametrically excited cantilever-type harvester. The model accounts for the beam's geometric and inertia non-linearities as well as non-linearities representing air drag. Using the method of multiple scales, we obtain approximate analytical expressions describing the beam response, voltage drop across a purely resistive load, and output power in the vicinity of the first principle parametric resonance. Using these expressions, we study the effect of the electromechanical coupling and load resistance on the output power. We show that these parameters play an imperative role in determining the magnitude of the output power and characterizing the broad-band properties of the harvester. Specifically, we show that the region of parametric instability wherein energy can be harvested shrinks as the coupling coefficient increases. Furthermore, we show that there exists a coupling coefficient beyond which the peak power decreases. We also demonstrate that there is a critical excitation level below which no energy can be harvested. The amplitude of this critical excitation increases with the coupling coefficient and is maximized for a given load resistance. Theoretical findings that were compared to experimental results show good agreement and reflect the general trends.

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