Modes for the coupled Timoshenko model with a restrained end

The modes of the second-order Timoshenko system for the displacement and rotation of a fixed beam with a restrained end at the left are formulated in terms of a fundamental spatial response. This is done without decoupling the system into fourth-order scalar equations. The restrained end leads to time–space boundary conditions which introduce the frequency as a parameter into the system of equations for determining the modes. These equations involve first-order derivatives and, consequently, the modes are determined by solving a non-conservative differential system. This modal differential equation is discussed in terms of a fundamental matrix response. It is determined by applying a closed formula that was obtained by the first author and involves the characteristic polynomial of the modal differential equation.