This paper investigates the optimal distribution of damping material in vibrating structures subject to harmonic excitations by using topology optimization method. Therein, the design objective is to minimize the structural vibration level at specified positions by distributing a given amount of damping material. An artificial damping material model that has a similar form as in the SIMP approach is suggested and the relative densities of the damping material are taken as design variables. The vibration equation of the structure has a non-proportional damping matrix. A system reduction procedure is first performed by using the eigenmodes of the undamped system. The complex mode superposition method in the state space, which can deal with the non-proportional damping, is then employed to calculate the steady-state response of the vibrating structure. In this context, an adjoint variable scheme for the response sensitivity analysis is developed. Numerical examples are presented for illustrating validity and efficiency of this approach. Impacts of the excitation frequency as well as the damping coefficients on topology optimization results are also discussed.
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