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A mathematical model for a cubic nonlinear packaging system with a cantilever beam type critical component with concentrated tip mass is proposed. The finite element method and the implicit finite difference method together with the Rung-Kutta method are applied to study the dropping impact dynamics of the critical component and the effect of system parameters, such as the value of the concentrated tip mass and the frequency of the main component, is discussed. The results show that the relative displacement and acceleration change remarkably with the length of the cantilever beam, and the maximum internal stress occurs at the joint end of the critical component. With the increase of the value of the concentrated tip mass and/or a higher frequency of the main component, the amplitudes of the responses increase obviously.