Contact bounce is a common phenomenon in the closing process of electromagnetic relays, which seriously affects the electrical contact reliability of the system. The whole closing process of flexible spring components consists of the free movement stage and the contact bounce stage, which increases the complexity of analyzing the electrical contact behaviors of electromagnetic relays. In this study, a novel nonlinear dynamic model of the flexible spring components in electromagnetic relays is established to analyze the physical mechanism of the electrical contact bounce. Considering the electromagnetic effect, flexible deformation, contact collision, and welding force, the proposed mathematical model can capture the dynamic closing process accurately and collect bounce parameters efficiently. The Kelvin–Voigt viscoelastic contact model is employed to evaluate the nonlinear contact force between the static and movable contacts. Based on the Galerkin method, the nonlinear equations of motion for the movable spring within electromagnetic relays are formulated. A series of experimental work are designed to measure the typical waveforms of displacements of the movable contact and currents in the coil during the closing process of the movable spring. Furthermore, the theoretical results are validated by comparing with the experimental data. The dependences of some key factors, including the structural, material, and electrical contact parameters on the contact bounce characteristics, are investigated explicitly. The theoretical results attest that the proposed model can accurately predict the dynamic behaviors of the movable spring, and the contact bounce can be effectively suppressed by adjusting the parameters of the flexible spring component.