Complex modal analysis of rods with viscous damping devices

•Rods with viscous damping devices are studied by the theory of generalized functions.•Exact closed-form characteristic equations and eigenfunctions are derived.•Eigenfunctions fulfill the required conditions at damping devices locations.•Forced response is built by modal superposition using orthogonality of eigenfunctions.•Exact closed-form dynamic stiffness matrix is derived for truss modal analysis.

The complex modal analysis of rods equipped with an arbitrary number of viscous damping devices is addressed. The following types of damping devices are considered: external (grounded) spring-damper, attached mass-spring-damper and internal spring-damper. Within a standard 1D formulation of the vibration problem, the theory of generalized functions is used to model axial stress and displacement discontinuities at the locations of the damping devices. By using the separate variable approach, a simple solution procedure of the motion equation leads to exact closed-form expressions of the characteristic equation and eigenfunctions, which inherently fulfill the required matching conditions at the locations of the damping devices. Based on the characteristic equation, a closed-form sensitivity analysis of the eigensolution is implemented. The displacement eigenfunctions exhibit orthogonality conditions. They can be used with the complex mode superposition principle to tackle forced vibration problems and, in conjunction with the stress eigenfunctions, to build the exact dynamic stiffness matrix of the rod for complex modal analysis of truss structures. Numerical results are discussed for a variety of parameters.