Free and Forced Wave Vibration Analysis of a Timoshenko Beam/Frame Carrying a Two-Degree-of-Freedom Spring-Mass System

2021 ◽  
pp. 1-25
Author(s):  
Carole Mei
Keyword(s):  
Timoshenko Beam ◽  
Discrete Systems ◽  
Degree Of Freedom ◽  
Combined System ◽  
Bending Vibrations ◽  
Mass System ◽  
Short Beam ◽  
Beam Frame ◽  
Classical Vibration ◽  
Two Degree Of Freedom

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.

A Wave-Based Analytical Solution to Free Vibrations in a Combined Euler–Bernoulli Beam/Frame and a Two-Degree-of-Freedom Spring–Mass System

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
C. Mei
Keyword(s):  
Analytical Solution ◽  
Coupling Effect ◽  
Free Vibrations ◽  
Degree Of Freedom ◽  
Transverse Motion ◽  
Bernoulli Beam ◽  
Combined System ◽  
Mass System ◽  
Euler Bernoulli Beam ◽  
Two Degree Of Freedom

In this paper, natural frequencies and modeshapes of a transversely vibrating Euler–Bernoulli beam carrying a discrete two-degree-of-freedom (2DOF) spring–mass system are obtained from a wave vibration point of view 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 forces applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam and 2DOF spring–mass system, the vibrating discrete spring–mass system injects waves into the distributed beam through the spring forces at the two spring attached points. Assembling these wave relations in the beam provides an analytical solution to vibrations of the combined system. Accuracy of the proposed wave analysis approach is validated through comparisons to available results. This wave-based approach is further extended to analyze vibrations in a planar portal frame that carries a discrete 2DOF spring–mass system, where in addition to the transverse motion, the axial motion must be included due to the coupling effect at the angled joint of the frame. The wave vibration approach is seen to provide a systematic and concise technique for solving vibration problems in combined distributed and discrete systems.

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

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.

Dynamic Stiffness Formulation and Its Application for a Combined Beam and a Two Degree-of-Freedom System

2003 ◽  
Vol 125 (3) ◽  
pp. 351-358 ◽  
Author(s):  
J. R. Banerjee
Keyword(s):  
Stiffness Matrix ◽  
Vibration Analysis ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Degree Of Freedom ◽  
Equivalent Stiffness ◽  
Euler Beam ◽  
Mass System ◽  
Stiffness Coefficients ◽  
Two Degree Of Freedom

This paper is concerned with the dynamic stiffness formulation and its application for a Bernoulli-Euler beam carrying a two degree-of-freedom spring-mass system. The effect of a two degree-of-freedom system kinematically connected to the beam is represented exactly by replacing it with equivalent stiffness coefficients, which are added to the appropriate stiffness coefficients of the bare beam. Numerical examples whose results are obtained by applying the Wittrick-Williams algorithm to the total dynamic stiffness matrix are given and compared with published results. Applications of the theory include the free vibration analysis of frameworks carrying two degree-of-freedom spring-mass systems.

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