13 Vehicle Driveline Benchmarking to Support Predictive CAE Modeling Development 143 Fig. 13.2 Free body diagram of simplified engine on frame reaction Fig. 13.3 Flow chart of lumped parameter model power flow The major objective of modal testing on the powerplant, propeller shaft, and differential was to come up with a damping estimate to support a CAE lumped parameter model of the vehicle. The modes of interest include rigid body modes of the powerplant and differential while evaluating the first torsional mode of the propshaft. During tip-in or tip-out events there is a change in driver requested torque which modifies the torque output of the engine. The engine torque output causes a transverse weight to shift due to reaction torques, shown in Fig. 13.2, known as roll which is associated with the rigid body mode about the X-axis. To model the roll modes of the powerplant and differential, a simple rack and pinion type model can be incorporated in parallel with the power flow that converts rotational degrees of freedom into translational degrees of freedom with a lever arm and spring/damper model to match the equivalent stiffness and damping of the roll mode estimates. This parallel modeling technique is depicted in Fig. 13.3 above. In Fig. 13.3, ellipses represent the torsional elements while rectangles represent the translational elements. Torque is passed back and forth between the elements and whenever the torque splits from one torsional element to a linear element it is incorporating an equivalent mass, stiffness and damping to represent the roll modes of the powerplant, differential and the fore-aft stiffness of the vehicle suspension. 13.3 Analysis As commonplace with any modal analysis testing method, one of the first things to consider when acquiring data is how much data is appropriate and necessary to capture the relevant mode shapes that are of interest. With respect to the powerplant and differential roll modes, these are low frequency modes and can be found <20 Hz and <200 Hz for the powerplant and differential respectively. However, the test vehicle utilizes an aluminum propeller shaft instead of steel. This requires an analytical estimate to find the shaft natural frequency and set an adequate sample rate for experimental testing. The torsional natural frequency for the propeller shaft can be calculated by assuming several material properties as well as a constant cross-sectional area for the length of the shaft. The physical shaft has a taper (reduction in diameter) at both the front and rear where the Cardan Joint’s connect to the transmission output and differential input respectively. These tapers are ignored when computing the solution to the closed form equation estimate the propshaft natural frequency. The closed form solution may be computed from Eq. (13.2) [3] wherefi is theith natural frequency, λi is the ith dimensionless parameter
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