Laplace transform is utilized to solve the coupled equation of plate and acoustic cavity and analysis sensitivity of the vibro-acoustic system. The sensitivity of the displacement amplitude with respect to size and shape design parameters for the coupled system is derived. Research on vibro-acoustic coupling system includes computation of the coupling frequencies and coupling modes, and sensitivities of the coupling system to design variables. The reducing displacement amplitude of the specified nodes on the plate is the target of depressing sound pressure of the measured point. The distribution of the damping material on the plate is then optimized by penalized density topology optimization theory. Filter method by Sigmund is developed to suppress the numerical instabilities, such as checkerboards. Numerical examples are given to show the validity and the efficiency of the sensitivity analysis and design optimization methods.