The dynamic analysis of nonuniformly pretwisted Timoshenko beams with elastic boundary conditions

Abstract The coupled governing differential equations and the general elastic boundary conditions for the coupled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beam are derived by Hamilton's principle. The closed-form static solution for the general system is obtained. The relation between the static solution and the field transfer matrix is derived. Further, a simple and accurate modified transfer matrix method for studying the dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relation between the steady solution and the frequency equation is revealed. The systems of Rayleigh and Bernoulli–Euler beams can be easily examined by taking the corresponding limiting procedures. The results are compared with those in the literature. Finally, the effects of the shear deformation, the rotary inertia, the ratio of bending rigidities, and the pretwist angle on the natural frequencies are investigated.

[1]  Z. Celep,et al.  On the influence of pretwisting on the vibration of beams including the shear and rotatory inertia effects , 1986 .

[2]  W. Carnegie,et al.  The Coupled Bending—Bending Vibration of Pre-Twisted Tapered Blading , 1972 .

[3]  B. Dawson,et al.  Coupled Bending-Bending Vibrations of Pre-Twisted Cantilever Blading Treated by the Rayleigh-Ritz Energy Method , 1968 .

[4]  S. M. Lin,et al.  Exact solutions for extensible circular curved timoshenko beams with nonhomogeneous elastic boundary conditions , 1998 .

[5]  William Carnegie,et al.  Vibrations of Pre-Twisted Cantilever Blading , 1959 .

[6]  Aviv Rosen,et al.  Structural and Dynamic Behavior of Pretwisted Rods and Beams , 1991 .

[7]  M. Sabuncu.  Coupled vibration analysis of blades with angular pretwist of cubic distribution , 1985 .

[8]  B. Dawson,et al.  Modal Curves of Pre-Twisted Beams of Rectangular Cross-Section , 1969 .

[9]  Shueei Muh Lin.  Vibrations of Elastically Restrained Nonuniform Beams with Arbitrary Pretwist , 1997 .

[10]  J. S. Rao.  Advanced theory of vibration , 1992 .

[11]  H. A. Slyper.  Coupled Bending Vibrations of Pretwisted Cantilever Beams , 1962 .

[12]  J. S. Rao,et al.  Coupled bending-bending vibrations of pretwisted tapered cantilever beams treated by the Reissner method , 1982 .

[13]  R. S. Gupta,et al.  Finite element eigenvalue analysis of tapered and twisted Timoshenko beams , 1978 .

[14]  J. S. Rao,et al.  Coupled bending-bending vibrations of pre-twisted cantilever blading allowing for shear deflection and rotary inertia by the Reissner method , 1981 .

[15]  B. Dawson,et al.  Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section , 1971 .

[16]  J. S. Rao.  Flexural Vibration of Pretwisted Tapered Cantilever Blades , 1972 .

[17]  Shueei Muh Lin.  Pretwisted Nonuniform Beams with Time-Dependent Elastic Boundary Conditions , 1998 .