Study of the Oscillations of a Nonlinearly Supported String Using Nonsmooth Transformations

1998 ◽  
Vol 120 (2) ◽  
pp. 434-440 ◽  
Author(s):  
V. N. Pilipchuk ◽  
A. F. Vakakis
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Leading Order ◽  
Perturbation Expansions ◽  
Regular Perturbation ◽  
Nonlinear Dynamical ◽  
Singular Terms ◽  
Time Periodic ◽  
Infinite String

An analytical method for analyzing the oscillations of a linear infinite string supported by a periodic array of nonlinear stiffnesses is developed. The analysis is based on nonsmooth transformations of a spatial variable, which leads to the elimination of singular terms (generalized functions) from the governing partial differential equation of motion. The transformed set of equations of motion are solved by regular perturbation expansions, and the resulting set of modulation equations governing the amplitude of the motion is studied using techniques from the theory of smooth nonlinear dynamical systems. As an application of the general methodology, localized time-periodic oscillations of a string with supporting stiffnesses with cubic nonlinearities are computed, and leading-order discreteness effects in the spatial distribution of the slope of the motion are detected.

Dynamic Characteristics of a Hydrostatic Gas Bearing Driven by Oscillating Exhaust Pressure

1984 ◽  
Vol 106 (4) ◽  
pp. 477-483 ◽  
Author(s):  
C. B. Watkins ◽  
H. D. Branch ◽  
I. E. Eronini
Keyword(s):  
Reynolds Equation ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Source Model ◽  
Regular Perturbation ◽  
Response Characteristics ◽  
Finite Difference Approximations ◽  
Bode Plots ◽  
Gas Bearing

Vibration of a statically loaded, inherently compensated hydrostatic journal bearing due to oscillating exhaust pressure is investigated. Both angular and radial vibration modes are analyzed. The time-dependent Reynolds equation governing the pressure distribution between the oscillating journal and sleeve is solved together with the journal equation of motion to obtain the response characteristics of the bearing. The Reynolds equation and the equation of motion are simplified by applying regular perturbation theory for small displacements. The numerical solutions of the perturbation equations are obtained by discretizing the pressure field using finite-difference approximations with a discrete, nonuniform line-source model which excludes effects due to feeding hole volume. An iterative scheme is used to simultaneously satisfy the equations of motion for the journal. The results presented include Bode plots of bearing-oscillation gain and phase for a particular bearing configuration for various combinations of parameters over a range of frequencies, including the resonant frequency.

Analytic Study of Discreteness Effects in a String Resting on a Periodic Array of Vibro-Impact Supports

1997 ◽  
Author(s):  
Gary D. Salenger ◽  
Alexander F. Vakakis
Keyword(s):  
Nonlinear Boundary ◽  
Equations Of Motion ◽  
Forced Oscillations ◽  
System Of Equations ◽  
Solitary Wave Solutions ◽  
Governing Equations ◽  
Nonlinear Boundary Value Problems ◽  
Regular Perturbation ◽  
Discrete Nature ◽  
Infinite String

Abstract We analyze the forced oscillations of an infinite string supported by an array of vibro-impact supports. The envelope of the excitation possesses ‘slow’ and ‘fast’ scales and is periodic with respect to the ‘fast’ scale. The ‘fast’ spatial scale is defined by the distance between adjacent nonlinear supports. To eliminate the singularities from the governing equations of motion that arise due to the discrete nature of the supports, we employ the nonsmooth transformations of the spatial variable first introduced in (Pilipchuk, 1985) and (Pilipchuk, 1988). Thus, we convert the problem to a set of two nonhomogeneous nonlinear boundary value problems which we solve by means of perturbation theory. The boundary conditions of these problems arise from ‘smoothness conditions’ that are imposed to guarantee sufficient differentiability of the results. The transformed system of equations is simplified using regular perturbation and harmonic balancing. Standing solitary wave solutions reflecting the discreteness effects inherent in the discrete foundation are calculated numerically for the unforced system.

The strange instability of the equatorial Kelvin wave

2020 ◽  
Author(s):  
Stephen Griffiths
Keyword(s):  
Growth Rate ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Numerical Verification ◽  
Kelvin Wave ◽  
Perturbation Expansions ◽  
Wave Instability ◽  
Regular Perturbation ◽  
Confluent Hypergeometric Functions ◽  
Linear Equations Of Motion

<p>The Kelvin wave is perhaps the most important of the equatorially trapped waves in the terrestrial atmosphere and ocean, and plays a role in various phenomena such as tropical convection and El Nino. Theoretically, it can be understood from the linear dynamics of a stratified fluid on an equatorial β-plane, which, with simple assumptions about the disturbance structure, leads to wavelike solutions propagating along the equator, with exponential decay in latitude. However, when the simplest possible background flow is added (with uniform latitudinal shear), the Kelvin wave (but not the other equatorial waves) becomes unstable. This happens in an extremely unusual way: there is instability for arbitrarily small nondimensional shear <em>λ</em>, and the growth rate is proportional to exp(-1/λ^2) as λ → 0. This in contrast to most hydrodynamic instabilities, in which the growth rate typically scales as a positive power of λ-λ<sub>c</sub> as the control parameter λ passes through a critical value λ<sub>c</sub>.</p><p>This Kelvin wave instability has been established numerically by Natarov and Boyd, who also speculated as to the underlying mathematical cause by analysing a quantum harmonic oscillator perturbed by a potential with a remote pole. Here we show how the growth rate and full spatial structure of the Kelvin wave instability may be derived using matched asymptotic expansions applied to the (linear) equations of motion. This involves an adventure with confluent hypergeometric functions in the exponentially-decaying tails of the Kelvin waves, and a trick to reveal the exponentially small growth rate from a formulation that only uses regular perturbation expansions. Numerical verification of the analysis is also interesting and challenging, since special high-precision solutions of the governing ordinary differential equations are required even when the nondimensional shear is not that small (circa 0.5). </p>

Amplitude-Dependent Stiffness Method for Studying Frequency and Damping Variations in Nonlinear Dynamical Systems

2017 ◽  
Author(s):  
Mohammad A. AL-Shudeifat
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical Systems ◽  
Energy Content ◽  
Equations Of Motion ◽  
A Priori ◽  
Nonlinear Dynamical System ◽  
Nonlinear Coupling ◽  
Nonlinear Dynamical ◽  
Coupling Stiffness ◽  
Nonlinear Energy

A method is introduced here for extracting the fundamental backbone branches of the frequency energy plot in which the obtained nonlinear frequencies of the nonlinear dynamical system are plotted with respect to the nonlinear energy content. The proposed method is directly applied to the equations of motion where the solution is not required to be known a priori. The method is based on linearizing the nonlinear coupling force where a scaled amplitude-dependent coupling stiffness force is obtained to replace the original nonlinear coupling stiffness force. Accordingly, the backbone branches in the frequency-nonlinear-energy plot are extracted from the eigensolution of the mass-normalized amplitude-dependent global stiffness matrix of the nonlinear dynamical system. Moreover, the variations in the damping content under the effect of the nonlinear coupling stiffness are also studied. Interesting behavior of damping content under the effect of the amplitude-dependent stiffness has been observed and discussed.

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