A New Approach to Nonlinear Oscillations

2001 ◽  
Vol 68 (6) ◽  
pp. 951-952 ◽  
Author(s):  
B. Wu ◽  
P. Li
Keyword(s):  
Governing Equation ◽  
Nonlinear Oscillation ◽  
Harmonic Balance ◽  
Nonlinear Oscillations ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
New Approach ◽  
Single Degree ◽  
Method Of Harmonic Balance ◽  
Approximate Formulas

This paper deals with nonlinear oscillation of a general single-degree-of-freedom system. By combining the linearization of the governing equation with the method of harmonic balance, we establish two analytical approximate formulas for the period. These two formulas are valid for small as well as large amplitudes of oscillation.

A Harmonic Balance Approximation of Dynamic Snap-Through Boundaries in a Single-Degree-of-Freedom Structure

2013 ◽  
Author(s):  
Richard Wiebe ◽  
Lawrence N. Virgin
Keyword(s):  
Large Amplitude ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Nonlinear Phenomena ◽  
Amplitude Response ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Analytical Approximations

Under dynamic loading, systems with the requisite condition for snap-through buckling, that is co-existing equilibria, typically exhibit either small amplitude response about a single equilibrium configuration, or large amplitude response that transits between the static equilibria. Dynamic snap-through is the name given to the large amplitude response, which, in the context of structural systems, is obviously undesirable. Structures with underlying snap-through static behavior may exhibit highly nonlinear and unpredictable oscillations. Such systems rarely lend themselves to investigation by analytical means. This is not surprising as nonlinear phenomena such as chaos run counter to the predictability of an analytical closed form solution. However, many unexpected analytical approximations of global stability may be obtained for simple systems using the harmonic balance method. In this paper a simple single-degree-of-freedom arch is studied using the harmonic balance method. The equations developed with the harmonic balance approach are then solved using an arc-length method and an approximate snap-through boundary in forcing parameter space is obtained. The method is shown to exhibit excellent agreement with numerical results. Arches present an ideal avenue for the investigation of snap-through as they typically have multiple, often tunable, stable and unstable equilibria. They also have many applications in both civil engineering, where arches are a canonical structural element, and mechanical/aerospace engineering, where arches may be used to approximate the behavior of curved plates and panels such as those used on aircraft.

scholarly journals Investigation of high order stochastic differential equations using averaging method

2007 ◽  
Vol 29 (3) ◽  
pp. 249-255
Author(s):  
Nguyen Dong Anh ◽  
Ngo Thi Hong Hue
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Nonlinear Oscillation ◽  
Colored Noise ◽  
Averaging Method ◽  
Nonlinear Oscillations ◽  
High Order ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Stochastic Problems

The averaging method is a useful tool for investigating both deterministic and stochastic quasilinear system. In the stochastic problems, however, the method has often been developed only for mechanical systems subjected to white noise excitations.In the paper this method is applied to high order stochastic differential equations. The nonlinear oscillations in high order deterministic differential equations were investigated in the fundamental work of Prof. Nguyen Van Dao. As an application of high order stochastic differential equations the nonlinear oscillation of single degree of freedom systems subjected to the excitation of a class of colored noises is outlined. The results obtained show that the higher order averaging method can also be successfully extended to the cases of colored noise excitation.

Transients in Simple Undamped Oscillators Under Inertial Disturbances

1959 ◽  
Vol 26 (2) ◽  
pp. 217-223
Author(s):  
Antongiulio Dornig
Keyword(s):  
Exact Solution ◽  
Maximum Amplitude ◽  
Degree Of Freedom ◽  
Inertial Forces ◽  
Constant Acceleration ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Approximate Formulas

Abstract Single-degree-of-freedom systems acted upon inertial forces are often found in technical applications. In this paper we shall study the transients in the vibrations of the system due to a change in speed in the machine in which the inertial forces are generated. We shall state the problem in the most general case, and then study the starting and the stopping with constant acceleration. After giving the exact solution of the problem we shall derive very simple approximate formulas for the determination of the maximum amplitude reached in these transients.

Rational harmonic balance-based approximate solutions to nonlinear single-degree-of-freedom oscillator equations

2010 ◽  
Vol 81 (3) ◽  
pp. 035003 ◽  
Author(s):  
Serge Bruno Yamgoué ◽  
Jean Roger Bogning ◽  
Aurélien Kenfack Jiotsa ◽  
Timoléon Crépin Kofané
Keyword(s):  
Harmonic Balance ◽  
Approximate Solutions ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Oscillator Equations
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