Lightweight structures tend to be susceptible to vibration, which can lead to fatigue and failure. Piezoelectric shunt damping is an efficient technology that converts mechanical energy into electrical energy and dissipates it through shunt circuits without requiring external power. This study models several piezoelectric patches coupled to different shunt circuits on a thin plate, utilizing Rayleigh-Ritz modal analysis to examine patch parameters including size, position, and distribution. Optimization demonstrates that effective patch-circuit topologies correspond with structural mode shapes, resulting in maximum vibration reduction. The findings introduce a fresh technique to enhance vibration control which improves structural reliability and performance.