Design sensitivity analysis of acoustical damping and its application to design of musical bells

Damping characteristics of a musical bell plays an important role in characterizing the musical sound. The total damping consists of acoustical damping and internal damping. Acoustical damping depends upon resonating frequencies and vibration patterns while internal damping is a material property. The acoustical damping of a vibrating structure is formulated via boundary element method and finite element method using eigenmode decomposition. The design sensitivity of acoustical damping is derived using an adjoint variable method of the eigenvalue problem. Design optimization of a musical bell is then performed in terms of acoustical parameters. The goal of the optimization problem is to design a harmonically tuned bell with given acoustical damping values. The proposed automated design process integrates finite element analysis, boundary element analysis, design sensitivity analysis, mode-tracking algorithm and optimization module, seamlessly. It is demonstrated by numerical examples to show practical applications.