120 Steady Collision Vibration in Spring-Mass-System for Two-Degree-of-Freedom Excited by Periodic Force with Arbitrary Functions : The Response Analysis for Harmonic, Superharmonic and Subharmonic Resonance

Abstract In this paper, free and forced vibrations of a transversely vibrating Timoshenko beam/frame carrying a discrete two-degree-of-freedom spring-mass system are analyzed using the wave vibration approach, in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities. From the wave vibration standpoint, external excitations applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam/frame and two-degree-of-freedom spring-mass system, the vibrating discrete spring-mass system injects waves into the distributed beam/frame through the spring forces at the two spring attached points. Assembling the propagation, reflection, transmission, and external force injected wave relations in the beam/frame provides an analytical solution to vibrations of the combined system. In this study, the effects of rotary inertia and shear deformation on bending vibrations are taken into account, which is important when the combined structure involves short beam element or when higher frequency modes are of interest. Numerical examples are given, with comparisons to available results based on classical vibration theories. The wave vibration approach is seen to provide a systematic and concise solution to both free and forced vibration problems in hybrid distributed and discrete systems.

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

Volume 7: 32nd Conference on Mechanical Vibration and Noise (VIB) ◽

10.1115/detc2020-22289 ◽

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.

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Nonlinear Normal Modes and Localization in Elastic Vibro-Impact Systems With Multiple Constraints

Volume 5: 6th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C ◽

10.1115/detc2007-34641 ◽

2007 ◽

Author(s):

David Wagg

Keyword(s):

Mathematical Formulation ◽

Normal Modes ◽

Nonlinear Normal Modes ◽

Degree Of Freedom ◽

Lumped Mass ◽

Localized Modes ◽

Impact Oscillator ◽

Mass System ◽

Nonlinear Normal ◽

Two Degree Of Freedom

In this paper we consider the dynamics of compliant mechanical systems subject to combined vibration and impact forcing. Two specific systems are considered; a two degree of freedom impact oscillator and a clamped-clamped beam. Both systems are subject to multiple motion limiting constraints. A mathematical formulation for modelling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom lumped mass system are considered. We then consider sticking motions which occur when a single mass in the system becomes stuck to an impact stop, which is a form of periodic localization. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. A numerical example of a sticking orbit for this system is shown and we discuss identifying a nonlinear normal modal basis for the system. This is achieved by extending the normal modal basis to include localized modes. Finally preliminary experimental results from a clamped-clamped vibroimpacting beam are considered and a simplified model discussed which uses an extended modal basis including localized modes.

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