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.
In the optimal configuration design of piezoelectric smart structures, it is favorable to use actuation elements with certain predefined geometries from the viewpoint of manufacturability of fragile piezoelectric ceramics in practical applications. However, preserving the exact shape of these embedded actuators and tracking their dynamic motions presents a more challenging research task than merely allowing them to take arbitrary shapes. This paper proposes an integrated topology optimization method for the systematic design of compliant smart structures with embedded movable PZT (lead zirconate titanate) actuators. Compared with most existing studies, which either optimize positions/sizes of the actuators in a given host structure or design the host structure with pre-determined actuator locations, the proposed method simultaneously optimizes the positions of the movable PZT actuators and the topology of the host structure, typically a compliant mechanism for amplifying the small strain stroke. A combined topological description model is employed in the optimization, where the level set model is used to track the movements of the PZT actuators and the independent point-wise density interpolation (iPDI) approach is utilized to search for the optimal topology of the host structure. Furthermore, we define an integral-type constraint function to prevent overlaps between the PZT actuators and between the actuators and the external boundaries of the design domain. Such a constraint provides a unified and explicit mathematical statement of the non-overlap condition for any number of arbitrarily shaped embedded actuators. Several numerical examples are used to demonstrate the effectiveness of the proposed optimization method.
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