Response Curves for a System With Softening Restoring Force

1955 ◽  
Vol 22 (3) ◽  
pp. 434-435
Author(s):  
H. N. Abramson
Keyword(s):  
Nonlinear System ◽  
Response Curves ◽  
Restoring Force

Abstract The purpose of this note is to present, and discuss briefly, calculated response curves for the nonlinear system titled. Several features of the response curves peculiar to this type of system, not previously discussed in the literature, are pointed out.

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.

scholarly journals Isolated Response Curves in a Base-Excited, Two-Degree-of-Freedom, Nonlinear System

2015 ◽  
Author(s):  
J. P. Noël ◽  
T. Detroux ◽  
L. Masset ◽  
G. Kerschen ◽  
L. N. Virgin
Keyword(s):  
Nonlinear System ◽  
Harmonic Balance ◽  
Global Analysis ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Basins Of Attraction ◽  
Response Curves ◽  
Restoring Force ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

In the present paper, isolated response curves in a nonlinear system consisting of two masses sliding on a horizontal guide are examined. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imposed by prescribing the displacement of their supports. Numerical simulations are carried out to study the conditions of existence of isolated solutions, their bifurcations, their merging with the main response branch and their basins of attraction. This is achieved using tools including nonlinear normal modes, energy balance, harmonic balance-based continuation and bifurcation tracking, and global analysis.

Strongly Nonlinear Vibrations of a Hyperelastic Thin-walled Cylindrical Shell Based on the Modified Lindstedt–Poincaré Method

2019 ◽  
Vol 19 (12) ◽  
pp. 1950160 ◽  
Author(s):  
Jing Zhang ◽  
Jie Xu ◽  
Xuegang Yuan ◽  
Wenzheng Zhang ◽  
Datian Niu
Keyword(s):  
Cylindrical Shell ◽  
Nonlinear System ◽  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Harmonic Excitation ◽  
System Of Differential Equations ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Hardening Behavior ◽  
Thin Walled

Some significant behaviors on strongly nonlinear vibrations are examined for a thin-walled cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a single radial harmonic excitation at the inner surface. First, with the aid of Donnell’s nonlinear shallow-shell theory, Lagrange’s equations and the assumption of small strains, a nonlinear system of differential equations for the large deflection vibration of a thin-walled shell is obtained. Second, based on the condensation method, the nonlinear system of differential equations is reduced to a strongly nonlinear Duffing equation with a large parameter. Finally, by the appropriate parameter transformation and modified Lindstedt–Poincar[Formula: see text] method, the response curves for the amplitude-frequency and phase-frequency relations are presented. Numerical results demonstrate that the geometrically nonlinear characteristic of the shell undergoing large vibrations shows a hardening behavior, while the nonlinearity of the hyperelastic material should weak the hardening behavior to some extent.

A Technique for Estimating Linear Parameters Using Nonlinear Restoring Force Extraction in the Absence of an Input Measurement

2005 ◽  
Vol 127 (5) ◽  
pp. 483-492 ◽  
Author(s):  
Muhammad Haroon ◽  
Douglas E. Adams ◽  
Yiu Wah Luk
Keyword(s):  
System Identification ◽  
Nonlinear System ◽  
Nonlinear System Identification ◽  
Restoring Force ◽  
Nonlinear Parameters ◽  
System Parameters ◽  
System Data ◽  
Vehicle Suspension System ◽  
Two Stages ◽  
Parameter Estimation Techniques

Conventional nonlinear system identification procedures estimate the system parameters in two stages. First, the nominally linear system parameters are estimated by exciting the system at an amplitude (usually low) where the behavior is nominally linear. Second, the nominally linear parameters are used to estimate the nonlinear parameters of the system at other arbitrary amplitudes. This approach is not suitable for many mechanical systems, which are not nominally linear over a broad frequency range for any operating amplitude. A method for nonlinear system identification, in the absence of an input measurement, is presented that uses information about the nonlinear elements of the system to estimate the underlying linear parameters. Restoring force, boundary perturbation, and direct parameter estimation techniques are combined to develop this approach. The approach is applied to experimental tire-vehicle suspension system data.

Dynamic Response of Nonlinear Oscillators With Hysteresis

2015 ◽  
Author(s):  
Biagio Carboni ◽  
Walter Lacarbonara
Keyword(s):  
Hysteresis Loops ◽  
Base Excitation ◽  
Response Curves ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Wire Ropes ◽  
Dynamical Behaviors ◽  
Nonlinear Features ◽  
Restoring Forces ◽  
Force Displacement

The nonlinear features of the steady-state periodic response of hysteretic oscillators are investigated. Frequency-response curves of base-excited single-degree-of-freedom (SDOF) systems possessing different hysteretic restoring forces are numerically obtained employing a continuation procedure based on the Jacobian of the Poincaré map. The memory-dependent restoring forces are expressed as a direct summation of linear and cubic elastic components and a hysteretic part described by a modified version of the Bouc-Wen law. The resulting force-displacement curves feature a pinching around the origin. Depending on the hysteresis material parameters (which regulate the shapes of the hysteresis loops), the oscillator exhibits hardening, softening and softening-hardening behaviors in which the switching from softening to hardening takes place above certain base excitation amplitudes. A comprehensive analysis in the parameters space is performed to identify the thresholds of these different behaviors. The restoring force features here considered have been experimentally obtained by means of an original rheological device comprising assemblies of steel and shape memory wire ropes. This study is carried out also with the aim of designing the restoring forces which give rise to dynamical behaviors useful for a variety of applications.

Asymmetric Solutions of SDOF System with Wire Rope Vibration Isolator Subjected to Harmonic Excitation

2015 ◽  
Vol 15 (06) ◽  
pp. 1450089 ◽  
Author(s):  
Keguan Zou ◽  
Satish Nagarajaiah ◽  
Andrew Dick
Keyword(s):  
Harmonic Excitation ◽  
Analytical Approximation ◽  
Wire Rope ◽  
Vibration Isolator ◽  
Response Curves ◽  
Restoring Force ◽  
Sdof System ◽  
Analytical Approximations ◽  
Homotopy Analysis ◽  
Isolation Systems

A new modification of homotopy analysis method (HAM) is proposed for capturing asymmetric solutions of wire rope isolation systems. Analytical expressions of asymmetric solutions to wire rope isolation systems are obtained. A dynamic system with quadratic polynomial restoring force is investigated specifically. Then the analytical results are applied to a single-degree-of-freedom (SDOF) system with wire rope vibration isolator to investigate the response curve and other dynamic characteristics. The analytical approximations match satisfactorily with the numerical results. The presented analytical approximation is a useful method to derive the response curves and examine limit cycles without resorting to numerical simulations.

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.

scholarly journals Validation of Two Nonlinear System Identification Techniques Using an Experimental Testbed

2004 ◽  
Vol 11 (3-4) ◽  
pp. 365-375 ◽  
Author(s):  
V. Lenaerts ◽  
G. Kerschen ◽  
J.-C. Golinval ◽  
M. Ruzzene ◽  
E. Giorcelli
Keyword(s):  
Experimental Data ◽  
Nonlinear System ◽  
Nonlinear System Identification ◽  
Physical Parameters ◽  
Free Response ◽  
Restoring Force ◽  
Surface Method ◽  
Experimental Testbed ◽  
Linear And Nonlinear ◽  
Identification Techniques

The identification of a nonlinear system is performed using experimental data and two different techniques, i.e. a method based on the Wavelet transform and the Restoring Force Surface method. Both techniques exploit the system free response and result in the estimation of linear and nonlinear physical parameters.

Sign in / Sign up

Export Citation Format

Share Document