I don't know how to deal with the complex damping E*. I am trying to study the dynamics of a cantilever subjected to bending and torsion, the beam is made of a viscoelastic material. After setting up the equations of motion and using Galerkin's method to convert the equations to odes in matrix form, I have a difficulty in the implementation of the viscoelastic property ( what to do with E* & G*).
I have seen the equations in the attached file. Even if the exact meaning of all the terms is not evident, I understand that these are two coupled PDEs describing the dynamics of a beam in bending and torsion. The problem looks rather complicated, moreover if you want to include the effects of viscosity. I'm not sure that the problem can be correctly formulated by a simple replacement of the real elastic moduli, E and G, with their complex counterparts, E* and G*, in the equations for the elastic beam. You should consider that when you write sigma = E* epsilon for the material, both sigma and epsilon may be complex quantities. As a consequence, you should consider that the generalised displacements, w and theta, as well as the internal forces, in the equations for the beam may be complex quantities!
My suggestion would be to start from a simpler case, for instance a beam subjected solely to bending or torsion. In this case, assume that some viscoelastic model (Maxwell, Kelvin etc.) describes the behaviour of the material at the microscale (i.e. for the longitudinal fibres of the beam). Then, introduce the common assumptions of beam theory (i.e. plane sections remain plane etc.) and derive the constitutive laws for the beam at the cross-section level. At this point, you should be able to write the dynamic equilibrium equations for the beam (for instance, by imposing the equilibrium of an elementary beam segment). Boundary and initial conditions should complete the formulation of your problem.
Once you have studied bending and torsion separately, you may try to consider the coupled problem.
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