Uniformly Bounded Input Gives Ultimately Bounded Output: On the Dynamics of a Piecewise Linear Beam System

1999 ◽  
Author(s):  
Marcel F. Heertjes ◽  
Marinus J. G. van de Molengraft ◽  
Jan J. Kok
Keyword(s):  
Periodic Solution ◽  
Piecewise Linear ◽  
Wave Force ◽  
Degree Of Freedom ◽  
Harmonic Force ◽  
Periodic Force ◽  
Beam System ◽  
Bounded Input ◽  
Uniformly Bounded

Abstract A periodically excited piecewise linear beam system is studied. The beam system consists of a supported multi-degree-of-freedom beam with one-sided spring. This system is proved to have a 1-periodic solution to any uniformly bounded periodic force applied along the beam. The existence of a 1-periodic solution will be shown numerically and experimentally for both a harmonic force and a block-wave force.

The Dynamics of a Nonharmonically Excited System Having Rigid Amplitude Constraints

1992 ◽  
Vol 59 (3) ◽  
pp. 693-695 ◽  
Author(s):  
Pi-Cheng Tung
Keyword(s):  
Dynamic Response ◽  
Bifurcation Analysis ◽  
Piecewise Linear ◽  
Coefficient Of Restitution ◽  
Degree Of Freedom ◽  
Linear Oscillator ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Excited System

We consider the dynamic response of a single-degree-of-freedom system having two-sided amplitude constraints. The model consists of a piecewise-linear oscillator subjected to nonharmonic excitation. A simple impact rule employing a coefficient of restitution is used to characterize the almost instantaneous behavior of impact at the constraints. In this paper periodic and chaotic motions are found. The amplitude and stability of the periodic responses are determined and bifurcation analysis for these motions is carried out. Chaotic motions are found to exist over ranges of forcing periods.

Experimental Forced Response Analysis of Two-Degree-of-Freedom Piecewise-Linear Systems With a Gap

2020 ◽  
Author(s):  
Akira Saito ◽  
Junta Umemoto ◽  
Kohei Noguchi ◽  
Meng-Hsuan Tien ◽  
Kiran D’Souza
Keyword(s):  
Piecewise Linear ◽  
Excitation Frequency ◽  
Forced Response ◽  
Degree Of Freedom ◽  
Linear Oscillator ◽  
Phase Portraits ◽  
Response Analysis ◽  
Experimental Setup ◽  
The Masses ◽  
Two Degree Of Freedom

Abstract In this paper, an experimental forced response analysis for a two degree of freedom piecewise-linear oscillator is discussed. First, a mathematical model of the piecewise linear oscillator is presented. Second, the experimental setup developed for the forced response study is presented. The experimental setup is capable of investigating a two degree of freedom piecewise linear oscillator model. The piecewise linearity is achieved by attaching mechanical stops between two masses that move along common shafts. Forced response tests have been conducted, and the results are presented. Discussion of characteristics of the oscillators are provided based on frequency response, spectrogram, time histories, phase portraits, and Poincaré sections. Period doubling bifurcation has been observed when the excitation frequency changes from a frequency with multiple contacts between the masses to a frequency with single contact between the masses occurs.

EFFECT OF FRACTIONAL DERIVATIVE PROPERTIES ON THE PERIODIC SOLUTION OF THE NONLINEAR OSCILLATIONS

Fractals ◽  
2020 ◽  
Vol 28 (07) ◽  
pp. 2050095 ◽  
Author(s):  
YUSRY O. EL-DIB ◽  
NASSER S. ELGAZERY
Keyword(s):  
Periodic Solution ◽  
Fractional Derivative ◽  
Perturbation Method ◽  
Nonlinear Oscillations ◽  
Duffing Oscillator ◽  
Solution Process ◽  
Auxiliary Parameter ◽  
Periodic Force ◽  
Definition Of ◽  
Perturbed Equation

A periodic solution of the time-fractional nonlinear oscillator is derived based on the Riemann–Liouville definition of the fractional derivative. In this approach, the particular integral to the fractional perturbed equation is found out. An enhanced perturbation method is developed to study the forced nonlinear Duffing oscillator. The modified homotopy equation with two expanded parameters and an additional auxiliary parameter is applied in this proposal. The basic idea of the enhanced method is to apply the annihilator operator to construct a simplified equation freeness of the periodic force. This method makes the solution process for the forced problem much simpler. The resulting equation is valid for studying all types of possible resonance states. The outcome shows that this alteration method overcomes all shortcomings of the perturbation method and leads to obtain a periodic solution.

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