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

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.

Free vibration analysis of large deformed curved beam incorporating rotary inertia and shear deformation effects and its experimental validation

“…The problem is already addressed in a previously reported paper, where effects of rotary inertia and shear deformation are not considered in the analysis. 25 Hence, the existing model fails to capture the vibration characteristics of thick curved beams. Moreover, the paper 25 concentrates on loaded natural frequency results of first mode only, although the experimental study shows some traces of complications associated with second vibration mode.…”

“…25 Hence, the existing model fails to capture the vibration characteristics of thick curved beams. Moreover, the paper 25 concentrates on loaded natural frequency results of first mode only, although the experimental study shows some traces of complications associated with second vibration mode. Such complicating effects are greatly influenced by deformation degrees of freedom and the associated problem parameters.…”

“…Kinematic and kinetic modeling of the moving roller supports are adopted from the paper 26 in association with Ghuku and Saha. 25 Effects of rotary inertia and shear deformation are incorporated in the mathematical formulation following the kinematic descriptions of generalized curved beam undergoing large deformation as presented in the paper. 27 Hence, some major mathematical relations are reproduced here from the papers 25–27 for ready reference.…”

“…The three coordinate systems are shown in Figure 1(b) for curved beam configuration Ci. Transformation relations between the frames are given below 25–27 …”

“…Such numerical solution technique converts the governing equation into linear eigenvalue problem. 25 The required sets of orthogonal functions for the approximated dynamic displacement fields are generated through Gram–Schmidt scheme, taking the corresponding known static displacement fields as start functions. Finally, with the generated orthogonal function sets, the linearized problem is solved for eigenvalues and eigenvectors, yielding the natural frequencies and mode shapes of the deformed vibrating beam.…”