https://hal.archives-ouvertes.fr/hal-01509017/document

A periodically forced piecewise linear oscillator

Chaos after period-doubling bifurcations in the resonance of an impact oscillator

A new closed-form solution based on the use of two square wave functions to model stiffness change is proposed for bilinear oscillators under low-frequency excitation. The solution procedure is extended to cases with bilinear forcing function, which usually occur in the dynamics of damaged structures. The proposed solution provides the spectral pattern and the magnitude of each harmonic component for a damaged rectangular beam according to the size and location of the crack. The predicted spectral pattern and the magnitude of each harmonic component for a simply supported beam containing a fatigue crack are validated by numerical simulation; experimental work has not been done. The results show that the proposed solution may be used to predict efficiently spectral patterns of damaged structures, while avoiding lengthy computation. Such a procedure may be developed into an attractive on-line diagnostic tool.

Analysis of forced bilinear oscillators and the application to cracked beam dynamics

Periodic forced vibration of unsymmetrical piecewise-linear systems by incremental harmonic balance method

Periodic response and stability of oscillators with symmetric trilinear restoring force

This paper discusses forced vibrations in an unsymmetric piecewiselinear system excited by general periodic force functions. The system has no damping. The superharmonic resonances up to the second order are analysed by means of a perfect Fourier series method. As the simplest example, the exciting forces are assumed to be of even function with the second higher harmonics only. The theoretical resonance curves are computed by a digital computer and are compared with the results obtained by an analog computer. They show a fairly good agreement.

Forced Vibrations in an Unsymmetric Piecewise-Linear System Excited by General Periodic Force Functions

This paper deals with perfect Fourier series solutions for superharmonic vibrations in various piecewise-linear systems with viscous damping, and gives a stability criterion for these solutions utilizing Hill's infinite determinant. Resonance curves for superharmonic vibrations of the 3rd order and stability charts for vibrations of Type I are constructed. The results using analog computer are shown to confirm these curves and charts.

Superharmonic Resonance in Piecewise-Linear System : Effect of Damping and Stability Problem

In this paper, steady, forced vibrations in an unsymmetrical piecewise-linear system with one degree of freedom, whose characteristics of the restoring and damping force are composed of two distinct linear regions, are analysed by means of a new method utilizing appropriate perfect Fourier series expansion. Main features of this method consist in the following procedures : 1) Linearizing the original nonlinear equation of motion by presumedly expanding the nonlinear part of the restoring and damping force into a Fourier series with the same period as of the given exciting force. 2) Obtaining the formal solution of the thus linearized equation by deeming the above-mentioned nonlinear part as if it were an exciting force from without. This solution contains certain unknown coefficients of the Fourier expansion above assumed. 3) Determining these unknown coefficients from the conditions that the above obtained formal solution satisfies the given piecewise-linear characteristics of the system. A certain process of convergency improvement by means of series transformation may prove highly effective in this step.

Steady, Forced Vibration of Unsymmetrical Piecewise-linear System : 1st Report, Explanation of Analytical Procedure

The bow force necessary to maintain steady vibration of a bowed string is greatly increased, and its fluctuations are severely intensified by the presence of inharmonicity due to flexural stiffness of the string. Timoshenko's beam theory gives a slightly smaller estimation of inharmonicity than the ordinary beam theory and decreases the necessary bow force considerably. Meanwhile, the internal friction of the string in bending and torsional motions and the compliance and damping of the bow mitigate the effect of inharmonicity that comes from the stiffness of the string. These factors are related with the configurations of the velocity resonance curve of the string, especially of its bottom parts in the higher frequency range. Along with some of our previous reports, the sound energy transmitted through the bridge to the body of an instrument is considered in this analysis, in which a Fourier series solution by means of series transformation for a piecewise-linear system is used.

Vibration of a Bowed String : Application of Timoshenko's Beam Theory and Transmission of Sound Energy Through the Bridge

By further developing the hitherto utilized dynamical model of the typical ultrasonic carving machine, we considered it as an unsymmetrical piecewise-linear system containing a solid horn as a continuous body And by applying the analytical method introduced by the author in the 1 st report to the solving of the forced vibrations of this system, we deduced theoretically various working characteristics of the ultrasonic carving machine Some of them explain the experimental results only qualitatively, but some do even quantitatively

Seiichiro Maezawa

Papers

Forced Vibrations in an Unsymmetric Piecewise-Linear System Excited by General Periodic Force Functions

Superharmonic Resonance in Piecewise-Linear System : Effect of Damping and Stability Problem

Steady, Forced Vibration of Unsymmetrical Piecewise-linear System : 1st Report, Explanation of Analytical Procedure

Vibration of a Bowed String : Application of Timoshenko's Beam Theory and Transmission of Sound Energy Through the Bridge

Steady, Forced Vibration of Unsymmetrical Piecewise-linear System : 3 rd Report, Application to the Ultrasonic Carving Machine