Torsional oscillations of an encased hollow cylinder of finite length

A hollow elastic cylinder of finite length is encased in a thin elastic shell. The analysis of free torsional vibrations of the elastic system yields a transcendental frequency equation that is solved numerically. The modes of free torsional motion are discussed and the proper relation establishing orthogonality of the principal modes is determined. The elementary mode of a free circular cylinder, when each transverse section rotates as a whole, does not occur for an encased cylinder. The exact frequencies are compared with estimates based on the assumption that the material of the core is very compliant as compared to the material of the shell. Fourier-Bessel analysis is used for the problem of forced torsional motion of the encased elastic core, for arbitrary dependence on radial coordinate and time of the prescribed displacements or stresses at the end-sections. For time-harmonic forcing functions the analysis is extended to forced torsional motion of an encased viscoelastic cylinder. The viscoelastic solutions are derived in terms of the complex shear modulus. The analysis of this paper is of interest for applications in solid-rocket technology.