Forced Nonlinear Oscillations of an Autoparametric System—Part 1: Periodic Responses

1983 ◽  
Vol 50 (3) ◽  
pp. 657-662 ◽  
Author(s):  
H. Hatwal ◽  
A. K. Mallik ◽  
A. Ghosh
Keyword(s):  
Numerical Integration ◽  
Harmonic Balance ◽  
Nonlinear Oscillations ◽  
Parametric Instability ◽  
Forced Oscillations ◽  
Instability Zone ◽  
Method Of Harmonic Balance ◽  
Numerical Integrations ◽  
System Part ◽  
Two Degree Of Freedom

Forced oscillations of a two degree-of-freedom autoparametric system are studied with moderately high excitations. The approximate results obtained by the method of harmonic balance are found to be satisfactory by comparing with those obtained by numerical integration. In the primary parametric instability zone, separate regions of stable and unstable harmonic solutions are obtained. In the regions of unstable harmonic solutions, depending on the forcing amplitude and frequency, the solutions may be amplitude modulated or completely nonperiodic. In the latter case the numerical integrations do not converge.

Forced Nonlinear Oscillations of an Autoparametric System—Part 2: Chaotic Responses

1983 ◽  
Vol 50 (3) ◽  
pp. 663-668 ◽  
Author(s):  
H. Hatwal ◽  
A. K. Mallik ◽  
A. Ghosh
Keyword(s):  
Statistical Analysis ◽  
Nonlinear Oscillations ◽  
Degree Of Freedom ◽  
Experimental Results ◽  
Forced Oscillations ◽  
Mean Square ◽  
Chaotic Oscillations ◽  
System Part ◽  
The Mean ◽  
Two Degree Of Freedom

Chaotic oscillations arising in forced oscillations of a two degree-of-freedom autoparametric system are studied. Statistical analysis of the numerically integrated nonperiodic responses is shown to be a meaningful description of the mean square values and the frequency contents of the responses. Some qualitative experimental results are presented to substantiate the necessity of performing the statistical analysis of the responses even though the system and the input are deterministic.

Forced Oscillations of Non-Linear Two-Degree-of-Freedom Systems

1966 ◽  
Vol 8 (3) ◽  
pp. 252-258 ◽  
Author(s):  
G. N. Bycroft
Keyword(s):  
Numerical Integration ◽  
Transient Response ◽  
Internal Resonance ◽  
Perturbation Technique ◽  
Degree Of Freedom ◽  
Forced Oscillations ◽  
Oscillatory Systems ◽  
Non Linear ◽  
Two Degree Of Freedom ◽  
Good Agreement

This paper shows how the Lighthill-Poincaré perturbation technique may be used to determine the transient response of ‘lightly coupled’ non-linear multi-degree-of-freedom oscillatory systems subject to arbitrary forcing functions. The results in general are complex but simplify in many important cases. A comparison is made between the analytical results and results obtained by a numerical integration of the equations on a computer. Good agreement is noted. The method fails under conditions of ‘internal resonance’ of the system.

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.

scholarly journals Functional analysis and the method of harmonic balance

1975 ◽  
Vol 32 (4) ◽  
pp. 457-464 ◽  
Author(s):  
C. A. Borges ◽  
L. Cesari ◽  
D. A. Sánchez
Keyword(s):  
Functional Analysis ◽  
Harmonic Balance ◽  
Method Of Harmonic Balance

Analysis of Rotating Blade Forced Vibration Due to Torsional Excitation Using the Method of Harmonic Balance

2002 ◽  
Author(s):  
B. O. Al-Bedoor ◽  
A. A. Al-Qaisia
Keyword(s):  
Forced Vibration ◽  
Harmonic Balance ◽  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Blade Vibration ◽  
Varying Coefficients ◽  
Rotating Blade ◽  
Method Of Harmonic Balance ◽  
Torsional Excitation ◽  
Truncated Fourier Series

This paper presents an analysis of the forced vibration of rotating blade due to torsional excitation. The model analyzed is a multi-modal forced second order ordinary differential equation with multiple harmonically varying coefficients. The method of Harmonic Balance (HB) is employed to find approximate solutions for each of the blade modes in the form of truncated Fourier series. The solutions have shown multi resonance response for the first blade vibration mode. The examination of the determinant of the harmonic balance solution coefficient matrix for stability purposes has shown that the region between the two resonance points is an unstable vibration region. Numerical integration of the equations is conducted at different frequency ratio points and the results are discussed. This solution provides a very critical operation and design guidance for rotating blade with torsional vibration excitation.

scholarly journals The Use of Integrals in Numerical Integrations of theN-Body Problem

1971 ◽  
Vol 10 ◽  
pp. 40-51
Author(s):  
Paul E. Nacozy
Keyword(s):  
Angular Momentum ◽  
Numerical Integration ◽  
Degrees Of Freedom ◽  
Body Problem ◽  
Center Of Mass ◽  
Integration Step ◽  
Gravitational System ◽  
Systems Of Differential Equations ◽  
Numerical Integrations ◽  
Original Number

AbstractThe numerical integration of systems of differential equations that possess integrals is often approached by using the integrals to reduce the number of degrees of freedom or by using the integrals as a partial check on the resulting solution, retaining the original number of degrees of freedom.Another use of the integrals is presented here. If the integrals have not been used to reduce the system, the solution of a numerical integration may be constrained to remain on the integral surfaces by a method that applies corrections to the solution at each integration step. The corrections are determined by using linearized forms of the integrals in a least-squares procedure.The results of an application of the method to numerical integrations of a gravitational system of 25-bodies are given. It is shown that by using the method to satisfy exactly the integrals of energy, angular momentum, and center of mass, a solution is obtained that is more accurate while using less time of calculation than if the integrals are not satisfied exactly. The relative accuracy is ascertained by forward and backward integrations of both the corrected and uncorrected solutions and by comparison with more accurate integrations using reduced step-sizes.

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