Duffing Oscillator: Could an Impact Load of Finite Duration be Substituted with an Instantaneous Impulse?

Abstract

We show that a single degree of freedom system with a rigid cubic characteristic of the restoring force (known in the physical literature on nonlinear vibrations as “Duffing oscillator”) is a suitable analytical (“mathematical”) model that can be used, when appreciable reactive in-plane (“membrane”) forces occur, to evaluate the dynamic response of a printed circuit board (PCB) subjected to a drop or a shock impact. When modeling such a response either of the PCB itself, or of a surface mounted device (SMD) package, including ball-grid-array (BGA) or pad-grid-array (PGA) structure, on a board level, there is an obvious incentive in trying to simplify the modeling by substituting an impact load of finite duration with an instantaneous impulse. On the other hand, when there is an intent to replace drop tests with shock tests, one has to properly “tune” the shock tester, so that to adequately mimic the drop test conditions. In this analysis we obtain exact solutions to the Duffing equation for the cases of an instantaneous impulse and for a suddenly applied and suddenly removed constant loading. We use these solutions to determine the error (in terms of the predicted amplitudes and accelerations) from substituting an impact load of finite duration with an instantaneous impulse. We consider an elongated PCB, which is currently employed in the Nokia accelerated test vehicle (experimental setup). The PCB’s short edges are simply supported, while its long edges are support-free. The in-plane reactive forces arise because the PCB’s short edges (supports) cannot get closer during its impact induced vibrations. We have determined, based on the obtained model, that the nonlinear system in question is, in general, less sensitive to the duration of the applied load than a linear system, and that this sensitivity decreases with an increase in the degree of the nonlinearity. Since the nonlinear frequency is strongly dependent on the magnitude of the applied load, we suggest that the nonlinear analysis be carried out prior to the assessment of the possible error in a modeling or a testing effort.