Reliability analysis for product package via probability density function of acceleration random response

Analytical probability density function of acceleration random response for cubic nonlinear product package has seldomly been investigated due to the difficulty of solving nonlinear random problem and less attention paid to the acceleration response. In this paper, the analytical solution for the probability density function of linear package acceleration response under random vibration is obtained, and on this basis, the equivalent linearization method is introduced to further explore the approximate analytical solution of acceleration response probability density function for cubic nonlinear product package under random vibration. Some suggestions on packaging optimal design are given by analyzing the sensitivity of acceleration random response to the external excitation and system parameters, and the first-passage failure probability for the acceleration response of cubic nonlinear package under random vibration is analyzed. The results show that the equivalent linearization method can precisely predict the acceleration response probability density function for the cubic nonlinear package under certain conditions, and the applicable range of the method is given by the equivalent stiffness formula. Nonlinear characteristic parameter ? of the cubic package activates acceleration response a completely different trend from the quasi-linear system. Properly increasing the system stiffness and decreasing the damping ratio can effectively reduce the acceleration response strength of the product package. As the vibration time increases and the product fragility decreases, the first-passage failure probability of the product package increases. The analysis provides theoretical foundation and guidance for packaging optimization design.