Vibration of a Viscoelastic Timoshenko Beam

Coupled equations for vibrations of a linear viscoelastic Timoshenko beam of an incompressible material are obtained by replacing the elastic moduli by viscoelastic operators. Time-variables in these equations are separated out by means of a dual eigenfunction expansion in terms of the corresponding elastic modes. Specific problems on free and forced vibration for a beam of either Kelvin or Maxwell material are treated in detail. The problem of moving load and the problem of a time-dependent boundary condition are also included in the treatment. It is shown that the foregoing solutions obtained for different specific problems retain their forms if the viscoelastic material is changed from an incompressible one to one of identical behavior in bulk and in shear.