This article presents a novel moving isosurface threshold (MIST) method for designing flexible structures using graded materials with multivolume fractions and constraints and viscous or hysteretic damping under harmonic loadings. By employing a unit dynamic load with the same frequency of an applied load, the displacement amplitude at chosen degrees-of-freedom (DOFs) can be expressed in an integral form in terms of mutual modal strain and kinetic energy densities over the entire design domain. Such integrals enable the introduction of novel physical response functions for solving a range of topology optimization problems, including single and multiple objectives with single and multiple volume fractions and/or constraints, e.g., single-input and single-output (SISO) and multi-input and multi-output (MIMO). Numerical examples are presented to validate the efficiency and capability of the present extended MIST method. Experiments are also conducted on rectangular plates with and without damping layer, fully and optimally covered, to demonstrate the benefits of the optimal damping layer design.

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