Laboratory experiments are broadly applied in transmission loss investigations. These measurement results are further used to validate and optimize mathematical models or directly employed in powertrain fuel consumption evaluations. In this article, an adaptive measurement strategy is proposed to circumvents the methodological limitations of the commonly used one-shot factorial design approach.
Figure1 IMS CONNECT at IMS, TU Darmstadt
To characterize transmission losses under various operating conditions, laboratory measurements demand a real full-scale test specimen and a dedicated test bench. As one of the essential aspects for drivetrain development, an effective measurement process can reduce the overall time and energy cost for development and application.
IMS (Institute for Mechatronic Systems, TU Darmstadt) has the test bench called IMS CONNECT (Advanced transmission and Powertrain Test Bench with Oil Conditioning). This test bench (in Figure 1) can be classified as the “back to back motor” concept. This concept locates electric motors on both of the input and output shafts of the test transmission. The transmission loss is then described as the difference of the input and output power. Based on this modular concept, various transmission configurations with different power ranges can be installed and tested.
Although an effective measurement is demanded in the overall development process, most of current power loss studies employ the traditional factorial design strategy, such as in [1, 2]. This offline factorial design strategy only samples once, by defining set levels for operating conditions within their variation range. However, without prior knowledge, the defined sample size can increase total time consumption.
In this article, an adaptive measurement strategy is employed, investigated and compared with the factorial design strategy. The adaptive strategy belongs to the sequential design category. Compared to offline strategies, sequential design strategies evaluate the acquired knowledge from each measurement to determine the next measurement candidates.
To achieve this, a goal function is defined based on surrogate modeling. Gaussian Process Regression (GPR) is one of the most common surrogate modeling approaches and is established based on Bayes’ Rule and multivariate Gaussian distribution. Different from other surrogate modeling approaches, GPR delivers not only prediction values, but also the predictive uncertainties with prediction variances. This predictive uncertainty is used to construct the goal function in the adaptive sampling process. To observe the region with high predictive variances for new measurement candidates, a stochastic simulation algorithm called Subset Simulation iteratively evaluates the constructed goal function. The core idea of Subset Simulation is to convert the searching event with small probabilities into a product of a sequence of large conditional probabilities.
Figure 2 demonstrates the workflow of the applied adaptive strategy for transmission loss measurements. The measurement begins with the initial samples generated with Latin Hypercube Sampling (LHS). The measured results are then applied to generate a data-based GPR model. Its prediction variances in the whole input domain are evaluated by Subset Simulation. After multiple regions with high variances are explored, new measurement candidates are then positioned within these regions. This whole adaptive process is conducted until a predefined stopping criterion is reached.
Figure 2 Workflow of adaptive measurement strategy
Figure 3 (a) illustrates a generated loss map for a seven-speed transmission under 1st gear and 50 °C oil temperature. The adaptive strategy is performed twice with different measurement points. The overlapped predicted loss maps indicate a good repeatability of the measurement strategy. 100 factorial designed measurement points are employed to validate the GPR model. Figure 3 (b) shows that the predicted contour corresponds well to the validation data with a mean absolute percentage error (MAPE) of 3.75 %.
Figure 3 Repeatability and validity analysis of the adaptive strategy
An adaptive strategy is applied in the transmission loss measurement to improve the overall measurement efficiency and to reduce costs. Compared with the commonly used factorial design approach, this strategy displays its technical advantages in the following aspects:
Literatures
[1]Wink, C. H., Marson, L. u. Goyal, S.: Hybrid analytical-experimental method to map power losses of automotive transmissions over their operating range. Tribology International 143 (2020), S.106070
[2]Schaffner, T., Allmaier, H., Girstmair, J., Reich, F. M. u. Tangasawi, O.: Investigating the efficiency of automotive manual gearboxes by experiment and simulation. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 228 (2014) 4, S.341–354
Zhihong Liu, M.Sc., research assistant, Institute for Mechatronic Systems, TU Darmstadt
Prof. Dr.-Ing. Stepan Rinderknecht, head of institute, Institute for Mechatronic Systems, TU Darmstadt