In this thesis, the further investigation and improvement on synchronized switch shunt damping on inductor technique (SSDI technique) are described. The optimization of the nonlinear shunt network based on eigenvalues analysis of discrete dynamic equations is introduced. A novel switch shunt damping circuit for improved damping performance is introduced. It is a combination of the SSDI shunt and a negative capacitance. It combines the adaptive nature of the SSDI technique and the improved performance through a negative capacitance. The switching control laws for handling broadband multimodal vibration are theoretically studied, an enhanced switching law using a modal observer is presented. Additionally, the approach of synthesizing the nonlinear switch shunt damping network is presented.