Parking lock integration for electric axle drives by multi-objective design optimization

As already mentioned in Sect. 1, various mounting positions of the parking lock in the e‑drive system are eligible. In the following, the focus is put on e‑drives with a single-speed, two-stage helical gearbox with an offset between EM axis (gearbox input shaft axis) and axle (gearbox output shaft axis), which is also schematically depicted in Fig. 1. Furthermore, the rotor of the electric machine is directly mounted on the gearbox input shaft and this shaft is supported by three bearings. This architecture will later be used in the case study presented in Sect. 3. For such an e‑drive architecture, a total of five different parking lock mounting positions on the input and intermediate shaft of the gearbox appear reasonable, considering aspects from series production as well as assembly. Positions on the output shaft are in general considered suboptimal as the highest acting torque on all shafts inside the gearbox is present at the output shaft. For the designs shown in Sect. 3, the torque at the intermediate shaft is around 3.3 times lower than on the output shaft due to the transmission ratio of the second gear pair. This high torque level at the output shaft would require a large and costly design of the parking lock. The mentioned mounting positions are schematically depicted in Fig. 4, where two positions on the input shaft (denoted as “INPA” and “INPB”) and three positions on the intermediate shaft (denoted as “INTA”, “INTB” and “INTC”) are visualized. When designing the e‑drive system, a decision for one of the mounting positions needs to be made, which means the mounting position represents a design parameter c_PLMount that maps to an element in the set composed of the described positions:

In the following, a given architecture of an electrically actuated parking lock is considered as shown in Fig. 5, which will later be used in the case study presented in Sect. 3. For such a parking lock—independent of the mounting position c_PLMount—further degrees of freedom exist when integrating the parking lock. These include the axial placement of the parking lock on the shaft y_PL, the mounting angle of the parking lock on the shaft Θ_PL, the mounting angle of the actuator Θ_a and two flipping parameters f_PL and f_a, which allow to rotate the parking lock and the actuator by 180 ° around the z‑axis respectively. A visualization of all possible combinations of the parameters f_PL and f_a as well as the definition of the parameters y_PL, Θ_PL and Θ_a (Θ_a is measured from the closest possible position to the shaft) are depicted in Fig. 5. The origin of the shown coordinate system (x, y, z) is defined by the position parameter c_PLMount.

That way, a total of six design parameters (c_PLMount, y_PL, Θ_PL, Θ_a, f_PL, f_a) are used to define the geometrical integration of the parking lock in the e‑drive system. In order to extend the established design method depicted in Fig. 3 and described in [3, 6, 7] by the capability to integrate the parking lock, these six parameters are defined as additional optimization parameters for the optimization algorithm.

Moreover, certain restrictions apply when geometrically integrating the parking lock. In particular, the body of the parking lock must not clash with any other components in the e‑drive system (e.g. with shafts, bearings, gear wheels, rotor and stator of the electric machine). As the e‑drive system and the parking lock itself in general show a 3D shape, a clash detection in 3D needs to be performed. This is done by the e‑drive analysis, which uses 3D-CAD methods to automatically check for any clashes [13]. That way, it is determined if the integration suggested by the optimization algorithm is valid or not. In case clashes are identified, the design variant is considered invalid and automatically removed from the optimization loop.

Aside from geometrically integrating the parking lock and checking for clashes, further constraints need to be satisfied. Specifically, the load cases induced by the parking lock, which are described in Sect. 1, must not lead to a failure of the mechanical components inside the e‑drive system. In order to determine if a sufficient load capacity of the mechanical components is present or not, a suitable model of the drivetrain and vehicle to approximately calculate the parking-lock induced loads on the components is required. Based on the loads determined by the model, further criteria need to be applied to determine the load capacity of the mechanical components. For that purpose, the minimal information required is the torque transmitted by both gear pairs and the acting torque on the parking lock.

To calculate these torques, a simple multibody simulation (MBS) model is presented in the following, which requires low computational effort and is thus well suited for the application in the design optimization loop shown in Fig. 3. The mechanical model for mounting positions “INPA”, “INTA” and “INTC” is shown in Fig. 6 and represents a rotational system composed of four masses, which are connected by rotational spring-damper combinations. For reasons of simplicity, all kinematic and kinetic quantities are referred to the level of the gearbox input shaft. The actual torques and velocities at the intermediate and output shaft can be calculated by applying the transmission ratios of the two gear pairs. The torques acting in the springs and dampers are given byrespectively, where c is the spring rate, d the damping coefficient, Δφ the relative angle between the two end points of the spring and \(\Updelta \dot{\varphi}\) the relative angular velocity between the two end points of the damper. The e‑drive with its moment of inertia J_ed around the pitch axis of the vehicle is mounted in a suspension with spring rate c_S and damping coefficient d_S. The parking lock is connected to J_ed by its spring rate c_PL and damping coefficient d_PL. Following the torque paths visualized in Fig. 6, a certain spring rate c₁ and damping coefficient d₁ accounts for the elasticity and damping of the path between parking lock and the nearest gear wheel. The rotor of the electric machine with moment of inertia J_EM is connected to this gear wheel by c₂ and d₂. The elasticity and damping of the remaining drivetrain connecting the single wheels with moment of inertia J_w and the described gear wheel are considered by c_rDT and d_rDT. The gear wheel itself represents an intersection of the three spring-damper combinations and is modelled with moment of inertia J_gw.

Analogously, the model for mounting positions “INPB” and “INTB” is defined and shown in Fig. 7. The model is slightly simpler due to the missing spring-damper combination defined by c₁ and d₁. The intersection of the three spring-damper combinations is now at the parking lock wheel, which is modelled with its moment of inertia J_PL.

The connection to the longitudinal dynamic of the vehicle is made by the torque r_w F_tire acting on the wheels, where r_w is the effective rolling radius and F_tire is the longitudinal force at the tire-road contact modelled by the Magic Formula given in [11] (equation (4.49), page 173).

The mechanical model to describe the longitudinal vehicle dynamic is schematically shown in Fig. 8 for a vehicle with a parking lock mounted on the front axle (for a parking lock mounted on the rear axle, the force F_tire would act on the wheels of the rear axle). The vehicle with body mass m_b is modelled on a slope with angle β. At the center of gravity (CoG) the gravitational force m_bg is acting. The single wheels with unsprung mass m_w are connected to the body by spring-damper combinations representing the longitudinal wheel suspension. The spring rates and damping coefficients for the front axle are given by c_wsf and d_wsf, the ones for the rear axle by c_wsr and d_wsr, all of them being inputs to the design optimization (possibly taken from similar existing vehicles if this information is unknown in the early design phases). In analogy to Eq. 2 and 3, the spring and damper forces are determined bywhere Δx is the spring deflection and \(\Updelta \dot{x}\) the relative velocity between the two end points of the damper. That way, the force F_tire couples the longitudinal dynamic of the vehicle and wheels with the rotational dynamic of the drivetrain.

Applying the conservation of momentum to the described models yields equations of motion for all masses, which are numerically solved to obtain the torques acting on the parking lock, gear pair 1 (connecting input and intermediate shaft) and gear pair 2 (connecting intermediate and output shaft). For an exemplary e‑drive with parking lock mounting position “INTA”, the described model is compared to a reference MBS model, which is implemented in Simcenter Amesim [14]. The reference model is more sophisticated than the suggested model and considers the masses of all gear wheels, shafts (including the axle shafts) and the differential as well as a backlash between parking lock wheel and pawl. Additionally, the vertical movement of the vehicle is modeled and the wheel suspensions are considered by their full compliance in longitudinal and vertical direction. That way, the reference model considers 76 state variables, needs the specification of significantly more parameters and requires a higher computational effort for solving than the suggested model, which only uses 12 state variables. Figure 9 shows the obtained torque curves for an initial vehicle velocity of 3.5 km/h and slope β = 0° from the suggested and the reference MBS model. Mainly due to the modelled backlash, the torques from the reference model tend to oscillate more and phase shifts compared to the curves from the suggested model are noticeable. However, the general torque levels are predicted well by the suggested model, which means it is well-suited for determining the load capacity of the mechanical components with low computational effort and a small number of required model parameters—both properties are of importance for an early-stage design optimization as shown in Fig. 3.

In order to calculate the load capacity of the mechanical components based on the obtained torque curves, the quasi-stationary forces acting on the gear wheels and the bearing reaction forces are determined from start of the simulation until the vehicle speed reaches zero for the first time. After that, the following criteria are applied to determine whether or not a sufficient load capacity for the parking-lock-induced load case is present:

These conditions are checked by the e‑drive analysis and in case an insufficient load capacity is determined, the design variant is considered invalid and removed from the optimization loop.

That way, the established design method described in [3, 6, 7] for e‑drives is extended to be capable of holistically integrating the parking lock by adding the parameters regarding the geometrical integration (c_PLMount,  y_PL, Θ_PL, Θ_a, f_PL, f_a) as optimization parameters and checking the constraints regarding clashes and load capacity in the parking-lock-induced load cases.

Figure 10 shows a simplified flow chart of the major steps that are performed by the gearbox analysis: The design parameters of the gearbox, including the parameters regarding the geometrical integration of the parking lock, serve as input. With these parameters, the gear geometry is calculated and the bearings for all shafts are selected first. After that, the parking lock is positioned according to the values of (c_PLMount, y_PL, Θ_PL, Θ_a, f_PL, f_a) and the detailed geometry of the shafts is determined. Finally, the gearbox design variant is investigated regarding internal clashes of components and the load capacity of the mechanical components is evaluated (including a fatigue analysis based on a load spectrum and the described investigation of parking-lock-induced load cases).