On the Nonlinear Response of a Flexible Connecting Rod

In two recent papers (Chen, J.-S., and Chen, K.-L., 2001, “The Role of Lagrangian Strain in the Dynamic Response of a Flexible Connecting Rod,” ASME Journal of Mechanical Design, 123, pp. 542–548; Chen, J.-S., and Huang, C.-L., 2001, “Dynamic Analysis of Flexible Slider-Crank Mechanisms With Nonlinear Finite Element Method,” Journal of Sound and Vibration, 246, pp. 389–402) we reported that previous researches of others on the dynamic response of a flexible connecting rod may have overestimated the deflections by ten folds when the crank rotates near the bending natural frequency of the connecting rod because terms of significant order of magnitude were ignored inadequately. While the findings in (Chen, J.-S., and Chen, K.-L., 2001, “The Role of Lagrangian Strain in the Dynamic Response of a Flexible Connecting Rod,” ASME Journal of Mechanical Design, 123, pp. 542–548; Chen, J.-S., and Huang, C.-L., 2001, “Dynamic Analysis of Flexible Slider-Crank Mechanisms With Nonlinear Finite Element Method,” Journal of Sound and Vibration, 246, pp. 389–402.) were obtained via numerical simulations, the present paper emphasizes the analytical approach with an aim to exploring the physical insights behind these numerical results. The equations of motion are first derived by applying Hamilton’s principle with all high order terms in the strain energy function being retained. After careful examination of the order of magnitude of each term, the coupled equations are simplified to a single one in terms of the transverse deflection, which turns out to be a Duffing equation under parametric and external excitations simultaneously. Closed-form approximations of the dynamic response are then derived by using multiple scale method. It is found that the combined effects of parametric and external excitations dominate the response when Ω is close to 0.5 and 1. Away from these two speed ranges, on the other hand, the response is dominated by the external excitation alone.