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.

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.

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.

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.

scholarly journals On Nonlinear Dynamics of Planar Shear Indeformable Beams

1986 ◽  
Vol 53 (3) ◽  
pp. 619-624 ◽  
Author(s):  
A. Luongo ◽  
G. Rega ◽  
F. Vestroni
Keyword(s):  
Displacement Component ◽  
Forced Oscillations ◽  
Beam Model ◽  
Constraint Condition ◽  
Unified Approach ◽  
Response Curves ◽  
Nonlinear Beam ◽  
Planar Shear ◽  
The Stability ◽  
Steady State Solutions

The planar forced oscillations of shear indeformabie beams with either movable or immovable supports are studied through a unified approach. An exact nonlinear beam model is referred to and a consistent procedure up to order three nonlinearities is followed. By eliminating the longitudinal displacement component through a constraint condition and assuming one mode, the problem is reduced to one nonlinear differential equation. A perturbational solution in the neighborhood of the resonant frequency is determined and the stability of the steady-state solutions is studied. The dependence of the phenomenon on the geometrical and mechanical characteristics of the system is put into light and the frequency-response curves for different boundary conditions are furnished.

Sub-fixed-time control for a class of second order system

2020 ◽  
pp. 014233122092100
Author(s):  
Boyan Jiang ◽  
Hua Chen ◽  
Bo Li ◽  
Xuewu Zhang
Keyword(s):  
Initial Conditions ◽  
Equilibrium Points ◽  
Fixed Time ◽  
Second Order ◽  
Order System ◽  
Sufficient Condition ◽  
Time Stability ◽  
Time Control ◽  
Initial States ◽  
The Stability

In this paper, a new concept “sub-fixed-time stability” (SFTS) is proposed and studied, which means the states can converge to a region of equilibrium points in a fixed time for any initial states’ values. Then, a sufficient condition for it is given and proven. Though SFTS is similar to “practical fixed-time stability” (PFTS), they are not the same, and the sufficient condition for SFTS is much clearer and simpler than PFTS. Next, a sub-fixed-time controller is proposed for a class of second order system. The stability analyses are given in the case without disturbance and with disturbance, respectively. Finally, to illustrate the robustness of the proposed sub-fixed-time controller to different initial conditions, 100 numerical simulations are conducted for 100 initial states’ values.

scholarly journals Existence and Stability Results for Second-Order Neutral Stochastic Differential Equations With Random Impulses and Poisson Jumps

2021 ◽  
Vol 1 (1) ◽  
pp. 1-18
Author(s):  
K. Ravikumar ◽  
K. Ramkumar ◽  
Dimplekumar Chalishajar
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Banach Contraction ◽  
Existence And Stability ◽  
Random Impulses ◽  
Functional Differential ◽  
The Stability ◽  
Stability Results

The objective of this paper is to investigate the existence and stability results of secondorder neutral stochastic functional differential equations (NSFDEs) in Hilbert space. Initially, we establish the existence results of mild solutions of the aforementioned system using the Banach contraction principle. The results are formulated using stochastic analysis techniques. In the later part, we investigate the stability results through the continuous dependence of solutions on initial conditions.

scholarly journals Passive Dynamic Bounding Control using Symmetry Condition Control Laws

2019 ◽  
pp. 451-457
Author(s):  
P. Murali Krishna ◽  
R. Prasanth Kumar
Keyword(s):  
Fixed Points ◽  
Initial Conditions ◽  
Legged Locomotion ◽  
Zero Energy ◽  
Additional Energy ◽  
Control Laws ◽  
Quadruped Robots ◽  
Existence And Stability ◽  
The Stability ◽  
Bounding Gait

Legged locomotion is preferred over the wheeled locomotion as it can be used both for flat and rough terrains. Quadruped robots are preferred since they can offer better stability with considerable reliability. In recent years, passive dynamics has been used to obtain near zero-energy bounding gaits. Although theoretically such gaits consume no energy, in practice some additional energy is required to overcome losses. Existence and stability of such gaits have been thoroughly studied in literature for quadruped models with the assumption that the mass distribution and stiffness in the front and back legs are symmetric. Fixed points found using Poincare map indicate touchdown angle-liftoff angle symmetry between front and back legs. This property can be used to search for fixed points with ease. However, the range of initial conditions where the bounding gait is stable is highly limited. Control laws based on symmetry conditions observed are proposed in this paper to improve the stability region. One such control law based on body-fixed touchdown angles theoretically allows redesign of quadruped robot with physical cross coupling between legs to achieve inherent stability without leg actuation.

Periodic and Nonperiodic Thermomechanical Responses of Shape-Memory Oscillators

2001 ◽  
Author(s):  
Walter Lacarbonara ◽  
Davide Bernardini ◽  
Fabrizio Vestroni
Keyword(s):  
Shape Memory ◽  
Numerical Procedure ◽  
Period Doubling ◽  
Hysteresis Loops ◽  
Response Curves ◽  
Restoring Force ◽  
Codimension One ◽  
Nonlinear Responses ◽  
Gradient Based ◽  
The Stability

Abstract Nonlinear responses of shape-memory oscillators are investigated systematically using a numerical procedure and a modified Ivshin-Pence model for the restoring force. Due to the discontinuities in the tangent stiffness, classical gradient-based shooting techniques for determining periodic responses are not applicable. Herein the implemented algorithm searches the periodic solutions as fixed points of the Poincaré map. The Jacobian of the map is calculated via a central finite-difference scheme and its eigenvalues are computed to ascertain the stability of the solutions and the associated codimension-one bifurcations. A number of frequency-response curves are constructed for some meaningful shape-memory oscillators (characterized by different hysteresis loops) and for various excitation levels in the primary and superharmonic frequency ranges. The investigations are conducted both in isothermal and non-isothermal conditions. A rich class of solutions and bifurcations — including jump phenomena, pitchfork, period doubling, complete or incomplete bubble structures with a variety of nonperiodic responses — is found and discussed.

Institutional Reforms and Interaction of Local Governments

2015 ◽  
Vol 3 (2) ◽  
pp. 25-33
Author(s):  
Georges Sarafopoulos ◽  
Panagiotis G. Ioannidis
Keyword(s):  
Nash Equilibrium ◽  
Local Governments ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Replicator Dynamics ◽  
Discrete Dynamical System ◽  
Period Doubling ◽  
Box Dimension ◽  
Existence And Stability ◽  
The Stability

The paper considers the interaction between regions during the implementation of a reform, on regional development through a discrete dynamical system based on replicator dynamics. The existence and stability of equilibria of this system are studied. The authors show that the parameter of the local prosperity may change the stability of equilibrium and cause a structure to behave chaotically. For the low values of this parameter the game has a stable Nash equilibrium. Increasing these values, the Nash equilibrium becomes unstable, through period-doubling bifurcation. The complex dynamics, bifurcations and chaos are displayed by computing numerically Lyapunov numbers, sensitive dependence on initial conditions and the box dimension.

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