Longitudinal vibration of a damped rod. Part I: Complex natural frequencies and mode shapes

A study of the mechanical behaviour of a damped rod is presented in this paper. The damper is characterized by a complex stiffness, whose properties are dependent on frequency and modellized by an exponential model. The paper is divided in two parts: the first analyses the complex eigenvalues and mode shapes, and the second investigates on forced response when the rod is subjected to a harmonic excitation. Influence that stiffness and loss factor of the damper have on modal properties is analysed in the first part. Firstly, undamped case is considered. Later, a damper whose properties are constant with frequency is taken into account, and finally, frequency-dependent properties are contemplated, characterized by an exponential model. The complex eigenproblem is derived from the field equation and is solved making use of optimization Nelder–Mead method.