Topology optimization of free-layer damping material on a thin panel for maximizing modal loss factors expressed by only real eigenvalues

Damping material is usually applied to steel panels of vehicles to reduce vibration levels. On the other hand, the weight of a vehicle must be reduced to improve the rate of fuel consumption. Therefore, the modal loss factors caused by the treatment of damping material on the steel panels of a vehicle body structure must be maximized within a given volume. In this paper, we propose a practical design method to maximize modal loss factors by optimizing the layout of damping material under a volume constraint. The modal loss factor for an eigenmode can be obtained conventionally by the modal strain energy method as the material loss factor multiplied by the ratio of the strain energy stored in the damping material over the total strain energy in the system under consideration. In the proposed method, we assume that the eigenvectors with damping material are almost identical with the eigenvectors without damping material. The modal loss factor can then be expressed approximately by using a corresponding real eigenvalue, for which the stiffness of the damping material is taken into account but its mass density is set to zero and ignored. Several numerical examples are provided to demonstrate that the proposed method obtains optimal layouts of damping material applied to a flat rectangular panel. Our results indicate that the damping material is mainly distributed in areas where strain energy is stored, which agrees well with the results obtained using conventional design methodologies. Moreover, by applying a design sensitivity filter that was improved recently, the layout of damping material can be unified into a single domain to meet practical requirements for manufacturing.