A note on the foreshortening effect of a flexible beam under oblique excitation

In Li et al. (Multibody Syst. Dyn. 21:249–260, 2009), we described the geometric nonlinear effects of a pivoted beam under point-surface impact. We used the floating frame of reference (FFR) formulation and the ABAQUS/Explicit model for comparative studies. The numerical solutions showed that the two approaches agree well with each other only if the FFR formulation includes proper geometric nonlinear terms. We were, however, not able to do convergence studies and critically find out the obscure stress stiffness term that reflects the negative bending stiffness of the flexible beam during oblique impact. In this paper, we rederive the foreshortening formulation with oblique excitation on a more original foundation, and extract the stress stiffness term right from the generalized active force vector. The stress stiffness effect is shown by a simple example of a cantilever beam under oblique excitation. As a partial comment to Li et al. (Multibody Syst. Dyn. 21:249–260, 2009), no extra effort is needed to account for the stress stiffness effect in the oblique impact case of the pivoted beam while using the foreshortening formulation, because it is a part of the foreshortening effect. Convergence studies are carried out to confirm the numerical solutions obtained using different approaches. As a result, the foreshortening formulation solution obtained using a sufficient number of modes can mostly agree with the benchmark in ABAQUS/Explicit, no matter if different sets of modes corresponding to different reference conditions are used.