A theorem on the representation of the field of forced vibrations of a composite elastic system

A theorem on the representation of a vibration field of an elastic system comprising two subsystems is proved in a general form. It is assumed that the system is linear and the subsystems are rigidly connected and interact along a continuous surface S. According to the theorem, forced vibrations of the system can be represented in the form of the sum of two components, which are the solutions of two simpler auxiliary boundary-value problems. The first component is the field of vibrations of the isolated (separated or blocked along S) subsystems under the effect of preset external forces. The second component represents the forced vibrations of the junction of the subsystems, where the external forces are taken equal to zero and only the reaction forces obtained in solving the first auxiliary problem act at the surface S. The theorem is applied to the problem on the reflection and transmission of elastic waves through a junction of two media. It is demonstrated that the theorem utilization reduces the amount of calculations. Other applications are discussed.