Resonance-induced excessive vibrations can degrade performance, reduce reliability, and shorten the service life of engineering structures. To reduce the risk of resonance, structural natural frequencies are generally engineered to be shifted away from excitation frequencies. To achieve these frequency shifts, topology optimization provides a systematic approach to tailor structural stiffness and mass as an effective inverse design approach. Multi-material topology optimization exploits the high-dimensional design space of spatially varying material constituents by simultaneously optimizing material selection and spatial distribution over the design domain, offering great flexibility in controlling natural frequencies. This study proposes a multi-material topology optimization that attenuates excessive structural dynamic vibrations in free-vibration scenarios via a novel multivariate weight-based interpolation method. Additionally, to address non-smoothness due to mode switching amid optimization, we employ a bound formulation with an auxiliary scalar design variable to replace the max-min eigenvalue objective with constraints on individual eigenvalues. Furthermore, we study and compare three optimization formulations for multi-material design to attenuate vibrations, including maximizing fundamental eigenfrequency, maximizing the gap between adjacent eigenfrequencies, and excluding frequencies from targeted frequency bands, which are smoothly approximated by a proposed rational polynomial formulation. Numerical examples are presented to demonstrate the effectiveness and applicability of our proposed method.

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