scholarly journals Steady Oscillations of Systems With Nonlinear and Unsymmetrical Elasticity

1938 ◽  
Vol 5 (4) ◽  
pp. A169-A177
Author(s):  
Manfred Rauscher
Keyword(s):  
Time Function ◽  
Free Motion ◽  
Forced Oscillations ◽  
General Criterion ◽  
Restoring Force ◽  
Forced Motion ◽  
Space Function ◽  
Steady Oscillations ◽  
The Stability ◽  
Undamped Oscillations

Abstract Steady forced oscillations are reduced to free undamped oscillations of the same amplitude. The reduction is accomplished by changing the excitation from a time function F(t) into a space function F(x) through the assumption that x and t are related as in a free motion. By combining F(x) with the elastic restoring force E(x), a new effective E(x) is obtained, to which there corresponds a new “free” motion, which in turn furnishes a second approximation for the relation between x and t, and hence a new function F(x). A new effective E(x) and a new free motion are then found; and the cycle is repeated until the relation between x and t ceases to change. The frequency of the forced motion at the assumed amplitude is then known. In general, the process converges rapidly. The accuracy can be checked at any stage of the work. A general criterion of the stability of the motions is offered.

The Existence and Stability of Ultraharmonics and Subharmonics in Forced Nonlinear Oscillations

1954 ◽  
Vol 21 (4) ◽  
pp. 327-335
Author(s):  
T. K. Caughey
Keyword(s):  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Forced Oscillations ◽  
Order System ◽  
Response Curves ◽  
Restoring Force ◽  
Amplitude Frequency Response ◽  
Existence And Stability ◽  
Linearized Theory ◽  
The Stability

Abstract A study is made of the forced oscillations of a second-order system having a small cubic nonlinearity in the restoring force. It is shown that under suitable conditions ultraharmonic or subharmonic motion exists in addition to the harmonic motion which a linearized theory would predict. By studying the stability of such motions it is shown that at points on the amplitude frequency-response curves having vertical tangents, instability and consequently “jumps” occur. A study of the dependence of the motion on the initial conditions reveals that while ultra-harmonic and harmonic motions are rather insensitive to initial conditions, the existence of subharmonic motion can be achieved only for a restricted set of initial conditions.

Stability of Forced Oscillations With Nonlinear Second-Order Terms

1959 ◽  
Vol 26 (4) ◽  
pp. 499-502
Author(s):  
Chi-Neng Shen
Keyword(s):  
Continued Fraction ◽  
Variational Equation ◽  
Second Order ◽  
Iteration Procedure ◽  
Forced Oscillations ◽  
The Stability ◽  
Boundary Of Stability

Abstract A solution is obtained for forced oscillations with nonlinear second-order terms. The stability of this solution is given by its variational equation. The boundary of stability is analyzed by both the perturbation and continued fraction methods. The amplitude of osclllation with damping terms is also determined by the iteration procedure.

Harmonic Oscillations of Nonlinear Two-Degree-of-Freedom Systems

1955 ◽  
Vol 22 (1) ◽  
pp. 107-110
Author(s):  
T. C. Huang
Keyword(s):  
Nonlinear System ◽  
Degrees Of Freedom ◽  
Forced Oscillations ◽  
Viscous Damping ◽  
Free Oscillations ◽  
Response Curves ◽  
Harmonic Oscillations ◽  
Restoring Force ◽  
Nonlinear Restoring Force ◽  
Two Degree Of Freedom

Abstract In this paper an investigation is made of equations governing the oscillations of a nonlinear system in two degrees of freedom. Analyses of harmonic oscillations are illustrated for the cases of (1) the forced oscillations with nonlinear restoring force, damping neglected; (2) the free oscillations with nonlinear restoring force, damping neglected; and (3) the forced oscillations with nonlinear restoring force, small viscous damping considered. Amplitudes of oscillations and frequency equations are derived based on the mathematically justified perturbation method. Response curves are then plotted.

A Study of Noise Impact on the Stability of Electrostatic MEMS

2020 ◽  
Vol 15 (11) ◽  
Author(s):  
Yan Qiao ◽  
Wei Xu ◽  
Hongxia Zhang ◽  
Qin Guo ◽  
Eihab Abdel-Rahman
Keyword(s):  
Noise Intensity ◽  
Initial Conditions ◽  
Basins Of Attraction ◽  
Low Noise ◽  
Intensity Level ◽  
Lumped Mass ◽  
Mass Model ◽  
Restoring Force ◽  
Electrostatic Mems ◽  
The Stability

Abstract Noise-induced motions are a significant source of uncertainty in the response of micro-electromechanical systems (MEMS). This is particularly the case for electrostatic MEMS where electrical and mechanical sources contribute to noise and can result in sudden and drastic loss of stability. This paper investigates the effects of noise processes on the stability of electrostatic MEMS via a lumped-mass model that accounts for uncertainty in mass, mechanical restoring force, bias voltage, and AC voltage amplitude. We evaluated the stationary probability density function (PDF) of the resonator response and its basins of attraction in the presence noise and compared them to that those obtained under deterministic excitations only. We found that the presence of noise was most significant in the vicinity of resonance. Even low noise intensity levels caused stochastic jumps between co-existing orbits away from bifurcation points. Moderate noise intensity levels were found to destroy the basins of attraction of the larger orbits. Higher noise intensity levels were found to destroy the basins of attraction of smaller orbits, dominate the dynamic response, and occasionally lead to pull-in. The probabilities of pull-in of the resonator under different noise intensity level are calculated, which are sensitive to the initial conditions.

Stability Analysis of Two-Point Mooring System in Surge Oscillation

2010 ◽  
Vol 5 (2) ◽  
Author(s):  
A. K. Banik ◽  
T. K. Datta
Keyword(s):  
Harmonic Wave ◽  
Period Doubling ◽  
Mooring System ◽  
Restoring Force ◽  
Incremental Harmonic Balance ◽  
Cubic Polynomial ◽  
Motion Characteristics ◽  
The Stability ◽  
Continuation Technique ◽  
Amplitude Versus

Nonlinear surge response behavior of a multipoint mooring system under harmonic wave excitation is analyzed to investigate various instability phenomena such as bifurcation, period-doubling, and subharmonic and chaotic responses. The nonlinearity of the system arises due to nonlinear restoring force, which is modeled as a cubic polynomial. In order to trace different branches at the bifurcation point on the response curve (amplitude versus frequency of excitation plot), an arc-length continuation technique along with the incremental harmonic balance (IHBC) method is employed. The stability of the solution is investigated by the Floquet theory using Hsu’s scheme. The period-one and subharmonic solutions obtained by the IHBC method are compared with those obtained by the numerical integration of the equation of motion. Characteristics of solutions from stable to unstable zones, chaotic motion, nT solutions, etc., are identified with the help of phase plots and Poincaré map sections.

Oscillations of a rigid sphere embedded in an infinite elastic solid

1967 ◽  
Vol 63 (4) ◽  
pp. 1189-1205 ◽  
Author(s):  
P. Chadwick ◽  
E. A. Trowbridge
Keyword(s):  
Elastic Solid ◽  
Forced Oscillations ◽  
Graphical Form ◽  
Rigid Sphere ◽  
Free Oscillations ◽  
Modes Of Vibration ◽  
Fixed Axis ◽  
Steady Oscillations ◽  
Angular Oscillations ◽  
Elastic Shear

AbstractA detailed study is made of angular oscillations of small amplitude about a fixed axis of a rigid sphere embedded in an infinite elastic solid. Three modes of vibration of the sphere are considered: steady oscillations arising from the application of a periodic torque; forced oscillations produced by an arbitrary time-dependent torque; and free oscillations excited by an impulsive torque. Due to the transfer of energy to the surrounding material by the radiation of an elastic shear wave, free oscillations of the sphere are damped, the principal parameter affecting the damping being the density contrast between the sphere and its surroundings. Illustrative numerical results, referring to steady and free oscillations of the sphere, are presented in graphical form.

scholarly journals Prominence Oscillations and the Influence of the Distant Photosphere

1998 ◽  
Vol 167 ◽  
pp. 147-150
Author(s):  
N.A.J. Schutgens ◽  
M. Kuperus ◽  
G.H.J. van den Oord
Keyword(s):  
Magnetic Field ◽  
Boundary Condition ◽  
The Other ◽  
Forced Oscillations ◽  
Crossing Time ◽  
Self Consistent ◽  
Normal Polarity ◽  
The Stability ◽  
The Magnetic Field ◽  
Flux Conservation

AbstractWe model vertical prominence dynamics, describing the evolution of the magnetic field in a self-consistent way. Since the photosphere imposes a boundary condition on the field (flux conservation), the Alfvén crossing time τ0/2 between prominence and photosphere has to be taken into account. Using an electrodynamical description of the prominence we are able to compare two basic prominence models: Normal Polarity (NP) and Inverse Polarity (IP).The results indicate that for IP prominences, the stability properties are sensitive to ωτ0 (ω: oscillation frequency of prominence). For ωτ0 ≳ 1 instability results. Forced oscillations of five minutes are efficiently excited in IP prominences that meet certain criteria only. NP prominences on the other hand, are insensitive to the Alfvén crossing time. Forced oscillations of five minutes are difficult to excite in NP prominences.

Nonlinear Oscillations of Nonlinear Damping Gyros: Resonances, Hysteresis and Multistability

2020 ◽  
Vol 30 (14) ◽  
pp. 2050203
Author(s):  
C. H. Miwadinou ◽  
A. V. Monwanou ◽  
L. A. Hinvi ◽  
V. Kamdoum Tamba ◽  
A. A. Koukpémèdji ◽  
...  
Keyword(s):  
Fixed Points ◽  
Parametric Excitation ◽  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Approximate Equation ◽  
Nonlinear Damping ◽  
Phase Portraits ◽  
Exact Equation ◽  
Restoring Force ◽  
The Stability

This paper addresses the issues on the dynamics of nonlinear damping gyros subjected to a quintic nonlinear parametric excitation. The fixed points and their stability are analyzed for the autonomous gyros equation. The number of fixed points of the system varies from one to six. The approximate equation of gyros is considered by expanding the nonlinear restoring force and parametric excitation for the study of the dynamics of gyros. Amplitude and frequency of possible resonances are found by using the multiple scales method. Also obtained are the principal parametric resonance and orders 4 and 6 subharmonic resonances. The stability conditions for each of these resonances are also obtained. Chaotic oscillations, multistability, hysteresis, and coexisting attractors are found using the bifurcation diagrams, the Lyapunov exponents, the phase portraits, the Poincaré section and the time histories. The effects of the damping parameter, the angular spin velocity and the parametric nonlinear excitation are analyzed. Results obtained by using the approximate gyros equation are compared to the dynamics obtained with the exact equation of gyros. The analytical investigations are complemented by numerical simulations.

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