The imbedding equations for the Timoshenko beam
Journal article, 1998
Wave reflection in a Timoshenko beam is treated, using wave splitting and the imbedding technique. The beam is assumed to be inhomogeneous and restrained by a viscoelastic suspension. The viscoelasticity is characterized by constitutive relations that involve the past history of deflection and rotation of the beam through memory functions of the suspension. By applying wave splitting, the propagating fields are decomposed into left- and right-moving parts. An integral representation of the split fields in impulse responses is presented. This representation gives the reflected and transmitted fields as convolutions of the incident field with the reflection and transmission kernels, respectively. The kernels are independent of the incident field and depend only on the material properties. Invariant imbedding is used to obtain equations for these kernels. In general, the kernels contain discontinuities for which transport equations are derived and solved. Some numerical solutions are presented for the reflection by a homogeneous beam suspended on two separated, semi-infinite layers of continuously distributed, viscoelastically damped, local acting springs.