Closed-form dynamic response of damped mass–spring cascades

Closed-form responses due to step forces of finite and semi-infinite damped mass–spring cascades with variable boundary mass are derived. By convolution arbitrary responses may be obtained. Eigenfrequencies, eigenmodes and critical damping of nonuniform and uniform finite cascades are found, showing a decreasing critical damping with increasing eigenfrequency. Further the frequency-response functions of the cascades are given for three cases.