Exact Frequency Analysis of a Rotating Cantilever Beam With Tip Mass Subjected to Torsional-Bending Vibrations

Abstract

An exact frequency analysis of a rotating beam with an attached tip mass is addressed in this paper while the beam undergoes coupled torsional-bending vibrations. The governing coupled equations of motion and the corresponding boundary condition are derived in detail using the extended Hamilton principle. It has been shown that the source of coupling in the equations of motion is the rotation and that the equations are linked through the angular velocity of the base. Since the beam-tip-mass system at hand serves as the building block of many vibrating gyroscopic systems, which require high precision, a closed-form frequency equation of the system should be derived to determine its natural frequencies. The frequency analysis is the basis of the time domain analysis, and hence, the exact frequency derivation would lead to accurate time domain results, too. Control strategies of the aforementioned gyroscopic systems are mostly based on their resonant condition, and hence, acquiring knowledge about their exact natural frequencies could lead to a better control of the system. The parameter sensitivity analysis has been carried out to determine the effects of various system parameters on the natural frequencies. It has been shown that even the undamped systems undergoing base rotation will have complex eigenvalues, which demonstrate a damping-type behavior.

Vibration of Spinning Cantilever Beams With an Attached Rigid Body Undergoing Bending-Bending-Torsional-Axial Motions

“…They derived the governing equations for the coupled bending-torsional vibration of a spinning elastic beam with a tip mass. Ansari et al [2] determined closed-form expressions for the rotating bending-torsion beam model in Ref. [1].…”

“…This study focuses on spinning beam gyroscopes that experience bending-bending-torsional-axial motions [1,2], although the derived model applies to other systems.…”

“…Spinning elastic beams where the rotation axis is perpendicular to the beam represent diverse engineered structures such as gyroscopes [1,2], rotating turbine and fan blades [3][4][5], and helicopter blades [6], among others. This study focuses on spinning beam gyroscopes that experience bending-bending-torsional-axial motions [1,2], although the derived model applies to other systems.…”

Nonlinear Dynamics of a High-Dimensional Model of a Rotating Euler–Bernoulli Beam Under the Gravity Load

“…Oh and Librescu [3] developed a linear dynamic theory of rotating blades with a constant angular velocity, which are made of fibrous composite materials and modeled as anisotropic thin-walled beams; trans verse shear, pretwist, and rotary inertia are included in the dynamic model. Ansari et al [4] derived a closed-form frequency equation of coupled torsional and bending vibrations of a rotating beam with a tip mass to determine its natural frequencies.…”

Identification of equivalent mechanical properties for unreinforced masonry walls

“…Low (1999Low ( , 2000 focused on the frequency analysis of an Euler-Bernoulli beam bearing a concentrated mass at an arbitrary location and developed a modified Dunkerley formula to approximate the exact solution with shorter computational time. Because of its increasing use as robot arms, machines or structures, Bruch and Mitchell (1987), Oguamanam (2003), Salarieh and Ghorashi (2006) and Ansari et al (2011) investigated the free vibrations of an Euler-Bernoulli or a Timoshenko cantilever beam with a rigid tip in the perspective of studying the behaviour of a flexible member. More recent contributions dealt with non-homogeneous or cracked beams (Li et al, 2013a;Khaji et al, 2009).…”