Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

Abstract In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.

[1]  P.A.A. Laura,et al.  Vibrations of an elastically restrained cantilever beam of varying cross section with tip mass of finite length , 1986 .

[2]  Stephen Gates,et al.  Transverse vibration and buckling of a cantilevered beam with tip body under axial acceleration , 1985 .

[3]  Glenn R. Heppler,et al.  VIBRATION OF ARBITRARILY ORIENTED TWO-MEMBER OPEN FRAMES WITH TIP MASS , 1998 .

[4]  P.A.A. Laura,et al.  A note on the vibrations of a clamped-free beam with a mass at the free end , 1974 .

[5]  K. H. Low.  A note on the effect of hub inertia and payload on the vibration of a flexible slewing link , 1997 .

[6]  Colin L. Kirk,et al.  NATURAL FREQUENCIES AND MODE SHAPES OF A FREE–FREE BEAM WITH LARGE END MASSES , 2002 .

[7]  K. H. Low,et al.  On the methods to derive frequency equations of beams carrying multiple masses , 2001 .

[8]  D.C.D. Oguamanam,et al.  Free vibration of beams with finite mass rigid tip load and flexural-torsional coupling , 2003 .

[9]  K. H. Low.  Eigen-Analysis of a Tip-Loaded Beam Attached to a Rotating Joint , 1990 .

[10]  Hans Wagner,et al.  Natural frequencies of a uniform cantilever with a tip mass slender in the axial direction , 1976 .

[11]  G. R. Heppler,et al.  Vibration Modes and Frequencies of Timoshenko Beams With Attached Rigid Bodies , 1995 .

[12]  C.W.S. To,et al.  Vibration of a cantilever beam with a base excitation and tip mass , 1982 .

[13]  R. P. Goel.  Vibrations of a Beam Carrying a Concentrated Mass , 1973 .

[14]  J. C. Bruch,et al.  Vibrations of a mass-loaded clamped-free Timoshenko beam , 1987 .

[15]  Leonard Meirovitch,et al.  Elements Of Vibration Analysis , 1986 .