Strain proportional damping in Bernoulli-Euler beam theory

In structural dynamics, different damping models are used; however, due to modal decomposition, those models typically result in the use of the damping ratio as the modal damping parameter. If proportional viscous damping is used, the damping ratio can be related to the mass and stiffness parameters of a particular dynamic system, i.e. the damping is structure-specific. Lord Rayleigh introduced the idea of proportional damping based on the global kinetic and potential energies of a dynamic system. This global or system-wide approach becomes questionable at the local scale, i.e., at a particular location of the researched system: for a particular mode, the potential energy is related to the strain mode shape and the kinetic energy is related to the displacement mode shape. As the strain and displacement mode shapes have different spatial distributions, also the spatial distributions of the potential and kinetic energies differ. Based on the Bernoulli-Euler beam theory, this research proposes an extension to the proportional damping approach, which results in a material-specific damping parameter. It is shown that using this material damping parameter and the assumption of damping energy proportionality to the local modal strain energy, the modal damping ratio of each mode can be obtained theoretically. This finding was confirmed against several experimental test-cases. The proposed material-specific damping parameter opens up the possibility to obtain the structure-specific damping parameters using the theoretical/numerical mode shapes.