Theoretical and Experimental Free Vibration Analysis of a Loaded Curved Beam With Moving Boundaries: Vibration Analysis of Beam With Moving Boundaries

Theoretical and Experimental Free Vibration Analysis of a Loaded Curved Beam With Moving Boundaries: Vibration Analysis of Beam With Moving Boundaries

Sushanta Ghuku (Jadavpur University, Kolkata, India) and Kashi Nath Saha (Jadavpur University, Kolkata, India)
DOI: 10.4018/IJMMME.2019100103

Abstract

The present article theoretically and experimentally investigates free vibration characteristics of generalized curved beams with moving boundaries. The dynamic behavior is characterized about deformed configuration, attained under different concentrated loads, and rigidly connected to the midpoint of the beam. The coupled static and dynamic analysis of the geometric nonlinear problem is decomposed into two parts: the static problem dealing with large deformed configuration and the dynamic problem dealing with small amplitude free vibration of the deformed configuration beam. The analysis is carried out incrementally in embedded curvilinear coordinate frames using variational principle. The governing equation of the static problem is derived for a combined effect of bending and center line extension. The governing equation for free vibration is derived at the particular configuration of the updated beam geometry, using Hamilton's principle. The comparison between the numerical and experimental results successfully validates the proposed semi-analytical model and leads toward some meaningful observations.
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Introduction

Beam like flexible structures are being used in variety of real-world engineering applications, from ancient structures to modern day’s micro and nano-level electromechanical systems. Hence, proper prediction of mechanical behavior of such one-dimensional continuum has been of great interest to researchers since ancient times. Over the centuries several classical beam theories have been established, among which the most celebrated one in engineering application is the Euler-Bernoulli beam theory (Timoshenko, 1953). Design of beam like flexible structures based on such classical beam theories results failure due to inherent nonlinear system characteristics and hence such design demands high factor of safety leading to compromise on their operating range. Finite limit of natural resource and revolution in the field of computational mechanics motivated researchers to predict inherent nonlinearity in mechanical behavior of beam structures. Hence, the research area becomes alive once again for the last few decades and several classical nonlinear beam models are developed (Fertis, 2006). Two major sources of nonlinearity of a beam problem are associated with geometry of deformation and material behavior of the beam. The nonlinearity associated with the material of construction of the beam has not been addressed in this paper. The present paper deals with geometric nonlinearity involved in free vibration problem of initially curved beam around large deformed configuration with lumped mass. Such free vibration analysis of large deformed and pre-stressed curved beam around the deformed configuration is very essential not only for the purpose of completeness of analysis in their design but also for their health monitoring as well. As the present coupled problem involves free vibration of curved beam in association with large deformation static problem, some relevant research papers related to geometric nonlinear static and free vibration of curved beam are presented in the following paragraphs.

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