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In order to ensure the safety of a structure, adequate strength for structural elements must be provided. Moreover, catastrophic deformations such as buckling must be prevented. Using the linear finite element method, deterministic buckling analysis is completed in two main steps. First, a static analysis is performed using an arbitrary ordinate applied loading pattern. Using the obtained element axial forces, the geometric stiffness of the structure is assembled. Second, an eigenvalue problem is performed between structure's elastic and geometric stiffness matrices, yielding the structure's critical buckling loads. However, these deterministic approaches do not consider uncertainty the structure's material and geometric properties. In this work, a new method for finite element based buckling analysis of a structure with uncertainty is developed. An imprecise probability formulation is used to quantify the uncertainty present in the mechanical characteristics of the structure.

Reducing the high cost of hardware testing with analytical methods has been highly accelerated in the automotive industry. This paper discusses an analytical model to simulate the driveline bending integrity test for the longitudinal 4WD-driveline configuration. The dynamic stresses produced in the adapter/transfer case and propeller shaft can be predicted analytically using this model. Particularly, when the 4WD powertrain experiences its structural bending during the operation speed and the propeller shaft experiences the critical whirl motion and its structural bending due to the inherent imbalance. For a 4WD-Powertrain application, the dynamic coupling effect of a flexible powertrain with a flexible propeller shaft is significant and demonstrated in this paper. Three major subsystems are modeled in this analytical model, namely the powertrain, the final rear drive, and the propeller shafts.