1 Introduction

In the general topic of reducing energy consumption and emission in the automobile industry, the development of more energy efficient transmissions is required. Thus, it is essential to comprehensively investigate transmission losses, and to have a detailed understanding of different loss contributions and their influential parameters. According to ISO/TR14179‑1 [1], analytical simulations and laboratory experiments are major methods for characterization of transmission losses. In terms of analytical simulations, power losses of complex transmissions with various configurations are characterized based on each individual loss model. Compared to this, laboratory experiments are applied to evaluate the total transmission losses and their influential operation conditions on the system level.

Regarding power loss evaluation for complex transmissions, most investigations use both methods to get a full insight of transmission loss distributions to discover possibilities for efficiency optimization. For example, Schaffner et al. [2], Vacca et al. [3] and Zhou et al. [4] present power loss investigations of different types of transmissions with both simulation and experiment. In their study, available analytical models are introduced and employed for the prediction of each loss mechanism at the system level. After this, experiments are conducted considering different operating conditions. A detailed analysis of transmission loss behaviors is then performed under different conditions. Wink et al. [5] present a hybrid analytical-experimental method to characterize the total transmission loss. Here, the data-based modeling strategy with polynomial equations is employed to fit the total loss data generated by the combination of measurement and simulation. The polynomial model is then compared with the measured total power losses for its verification. Concerning power loss measurements, almost all available studies use the traditional factorial design approach with predefined level for the determination of measurement points. Without any prior information, this approach could result in a more time-consuming experiment with oversized points, or lead to insufficient measurements that cannot fully characterize transmission losses within the whole input domain.

In this article, analytical and experimental investigations are both conducted for a complex seven-speed dual clutch transmission (DCT), which is originally published in the dissertation work of Liu [6]. First, most available analytical models are studied and filtered for further simulation. Following this, a sequential adaptive sampling strategy is applied to overcome the shortcoming of the one-shot factorial design approach in the experiment. A surrogate model is generated and optimized in each experiment cycle for the determination of new measuring candidates. Furthermore, a detailed discussion about the power loss and efficiency behavior of the seven-speed DCT is presented based on the analytical simulation and the surrogate modeling, which shows its practical value in further fuel consumption study and efficiency optimization. Relevant outcomes and perspectives are provided in the final section.

2 Research subject

As the study case, the seven-speed DCT (Fig. 1) has a dual wet clutch that induces hydrodynamic losses. These losses exist between the disengaged separate and friction discs because of the oil shearing caused by the different rotational speeds. Two clutches are separately connected to the solid and the hollow input shaft, in order to realize gear shifting without load interruption. For subsequent power transfer, the shafts are equipped with gear sets that further contribute to both load-dependent and no-load losses. Moreover, synchronizers are employed on the intermediate shafts to achieve gear engagement. Analogous to the disengaged clutch discs, oil drag losses are generated in the open synchronizer ring and cone due to their different rotational speed. Bearings are another fundamental mechanical component used to support shafts and idle gears in the transmission. Bearing losses can also be classified into load-dependent and no-load contributions. Radial shaft seals are utilized to prevent oil leakage. However, the relative motion between the rotating shaft and the stationary seal lip leads to no-load friction.

Fig. 1
figure 1

Schematic of the seven speed DCT [6]

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As described in the previous chapter, for the purpose of identification of above-described individual loss contributions and total loss behaviors under various operating conditions, analytical simulations and laboratory experiments are both applied in the power loss investigation. Their overall characteristics are discussed in the following sections.

3 Individual loss modeling approaches

Based on the introduction to the individual losses of mechanical components, all these losses can be divided into load-dependent and no-load categories. Figure 2 demonstrates most available analytical models that have been developed since the 1950s.

Fig. 2
figure 2

Available individual loss models (underlined models are used in this study)

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3.1 Gear load-dependent losses

Gear meshing losses indicate the friction between gear teeth during power transfer, and are mathematically described by averaging the total losses along the tooth contact line:

$$P_{m}=\frac{1}{L_{m}}\int\limits_{0}^{L_{m}}f\left(x\right)F_{n}\left(x\right)V_{s}\left(x\right)dx,$$
(1)

where Lm is the total length of tooth contact line, f is the friction coefficient, Fn is the meshing normal force, Vs is the sliding velocity and x is the coordinate along the contact line. The key in building these analytical models is the determination of friction coefficient f between tooth contact surfaces. A very simple model is developed by Naruse et al. [7]. In this model, only the tooth sliding velocity is considered. Misharin [8] and Benedict et al. [9] additionally include the tooth rolling velocity and the oil kinematic viscosity in their analysis. More complex models are constructed by Donoguhe et al. [10] and ISO/TR14179‑2 [11], taking into account the tooth surface roughness and geometric parameters. Based on these models, Xu [12] proposes an advanced friction coefficient model, treating the tooth contact as an elastohydrodynamic lubrication (EHL) contact. This model includes more features, such as slide-roll ratio and Hertzian pressure, and is mathematically expressed as:

$$f=e^{\sigma }\cdot {P}_{h}^{k_{1}}\cdot \left| SR\right| ^{{k_{2}}}\cdot {V}_{e}^{k_{3}}\cdot {\eta }_{\mathrm{oil}}^{k_{4}}\cdot \rho ^{{k_{5}}},$$
(2)
$$\begin{aligned}\sigma =k_{6}+k_{7}\cdot \left| SR\right| \cdot P_{h}\log \left(\eta _{\mathrm{oil}}\right)+k_{8}\cdot e^{-\left| SR\right| {\cdot P_{h}}\cdot \log \left(\eta _{\mathrm{oil}}\right)}+\\k_{9}\cdot e^{{R_{a}}},\end{aligned}$$
(3)

where k1−9 are empirical coefficients, Ph is the maximum Hertzian pressure, SR is the slide-roll ratio, Ve is the entraining velocity, ηoil is the oil dynamic viscosity, ρ is the oil density and Ra is the tooth surface roughness.

3.2 Gear no-load losses

Gear churning losses are defined as the resistance between gears and their surrounding oil during splash lubrication. Most analytical models are built based on the measurement with gears submerged in an oil sump. An early detailed model is constructed by Terekhov [13]. He introduces Reynolds number and Froude number in the calculation, to distinguish different oil flow regimes. Lauster et al. [14] extend this basic model for truck transmissions. Boness [15] additionally considers the transitional oil flow state in his simulation. A later study by Changenet et al. [16] introduces a new acceleration term to separately consider the influence of oil temperatures and rotational speeds in the calculation. This model is described as follows:

$$P_{c}=\frac{1}{2}\cdot \rho \cdot \omega ^{2}\cdot S_{m}\cdot {r_{p}}^{3}\cdot C_{m},$$
(4)

where ρ is the oil density, ω is the rotational speed, Sm is the gear submerged surface area in oil sump and rp is the gear pitch radius. The term Cm is a dimensionless coefficient that is defined under the form:

$$\begin{aligned}C_{m}=\psi _{1}\cdot \left(\frac{2\cdot h}{r_{p}}\right)^{{\psi _{2}}}\cdot \left(\frac{8\cdot V_{0}}{{r_{p}}^{3}}\right)^{{\psi _{3}}}\cdot \left(Fr\right)^{{\psi _{4}}}\cdot\\ \left(Re\right)^{{\psi _{5}}}\cdot \left(\frac{b}{r_{p}}\right)^{{\psi _{6}}},\end{aligned}$$
(5)

where ψ1−6 are coefficients that are determined based on Reynolds number and an acceleration term, h is the gear immersion depth, V0 is the oil volume, Fr is the Froude number, Re is the Reynolds number and b is the tooth face width.

Gear windage losses denote no-load resistances between rotating gears and the oil-air mixture. Normally, these losses hold small proportions of the total, and can be just influential in aero engine applications [17]. Townsend [18] outlines two analytical models considering rotational speeds and different geometric gear parameters. Compared to these, Dawson [19] weights the effect of the housing and the oil on windage losses in his model:

$$P_{w}=n^{2.9}\cdot (0.16\cdot {d_{p}}^{3.9}+{d_{p}}^{2.9}\cdot b^{0.75}\cdot m^{1.15})\cdot 10^{-20}\cdot \Upphi \cdot \lambda ,$$
(6)

where n is the rotational speed, dp is the gear pitch diameter, b is the tooth width, m is the gear module, Φ is the oil influence coefficient and λ is the housing effect coefficient.

3.3 Bearing load-dependent losses

Bearing load-dependent losses consist of the rolling resistance between rollers and bearing rings, and the sliding resistance during the relative motion of rollers, bearing cages and bearing rings. Palmgren [20] proposes an analytical model to separately describe the losses generated by axial and radial loads. A more recent model is developed by SKF [21]. Compared to Palmgren’s Model, SKF considers all friction surfaces and distinguishes the rolling and sliding loss contributions in the model:

$$T_{bl}=T_{rr}+T_{sl}.$$
(7)

Here, the rolling losses Trr are determined by:

$$T_{rr}=10^{-3}\cdot \phi _{\mathrm{ish}}\cdot \phi _{rs}\cdot G_{rr}\cdot \left(\nu \cdot n\right)^{0.6},$$
(8)

where ϕish is the coefficient related to the rolling friction reduction by reverse flow, ϕrs is the factor for kinematic starvation phenomenon at high speed, Grr is the rolling friction coefficient, ν is the oil kinematic viscosity and n is the rotational speed. The sliding losses are calculated with:

$$T_{sl}=10^{-3}\cdot G_{sl}\cdot [\phi _{bl}\cdot \mu _{bl}+(1-\phi _{bl})\cdot \mu _{\mathrm{EHL}}],$$
(9)

where Gsl is the sliding friction coefficient, ϕbl is the weighting factor for mixed lubrication and full EHL conditions, μbl is the frictional coefficient under mixed conditions and μEHL is the coefficient under full EHL conditions.

3.4 Bearing no-load losses

Analogous to gears, bearing no-load losses are induced by the rotating components immersed in the oil sump. A very early model for the characterization of these losses is developed by Palmgren [20]. In this model, the bearing dimension, the lubricant property and the rotating condition are taken into consideration. Based on this simple model, Eschmann [22] makes an extension by introducing a coefficient related to the lubricant viscosity and the rotating speed. Later, the mathematical model of SKF [21] evaluates bearing hydrodynamic losses and seal losses for ball bearings and roller bearings, respectively:

$$T_{bn}=T_{\mathrm{seal}}+T_{\mathrm{drag}},$$
(10)
$$T_{\mathrm{seal}}=10^{-3}\cdot \left(K_{S1}\cdot {{d_{s}}^{\beta }}+K_{S2}\right),$$
(11)
$$\begin{aligned}T_{\mathrm{drag},\mathrm{ball}}=4\times 10^{-4}\cdot V_{M}\cdot K_{\mathrm{ball}}\cdot {d_{m}}^{5}\cdot n^{2}+1.093\times\\ 10^{-10}\cdot n^{2}\cdot {d_{m}}^{3}\cdot \left(n\cdot {d_{m}}^{2}\cdot f_{t}/\upsilon \right)^{-1.379}\cdot R_{s}\end{aligned}$$
(12)
$$\begin{aligned}T_{\mathrm{drag},\text{roller}}=4\times 10^{-3}\cdot V_{M}\cdot K_{\mathrm{roll}}\cdot C_{w}\cdot B\cdot {d_{m}}^{4}\cdot n^{2}+\\1.093\times 10^{-10}\cdot n^{2}\cdot {d_{m}}^{3}\cdot \left(n\cdot {d_{m}}^{2}\cdot f_{t}/\upsilon \right)^{-1.379}\cdot R_{s},\end{aligned}$$
(13)

where Tseal are the seal losses and KS1, β and KS1 are the coefficients determined by seal and bearing types. ds is the seal diameter, Tdrag,ball and Tdrag,roller are the ball and roller bearing drag losses respectively, VM is the drag loss factor, Kball and Kroll are the factors related to rolling element dimensions. Cw is the coefficient for roller bearings, B is the bearing width, dm is the bearing mean diameter, n is the bearing rotational speed, υ is the oil kinematic viscosity, ft and Rs are the coefficients depending on the bearing immersion depth and its dimensions.

3.5 Other losses in transmissions

In order to determine the hydrodynamic losses between open clutch discs, all available models first analyze the oil flow behavior in disc gaps based on Navier-Stokes-equations (NS-equations) [23]. Under the assumption of steady, incompressible, laminar and symmetrical oil, the theoretical oil flow rate in the disc gap can be described along the radial direction. Li et al. [24] assume that there is always a full oil film between disc gaps, even when the actual oil flow rate is slower than the theoretical value. In this case, a factor named equivalent radius is employed in the model to indicate the full oil film area. With this, final hydrodynamic losses are calculated based on Newtonian friction theorem. Similar to Li’s assumption, Yuan et al. [25] also consider a full film in the disc gap. However, they take the surface tension with discs into account in the form of flow resistance. Iqbal et al. [26] analyze the oil flow behavior and consider that it consists of a full and a ruptured oil film. The ruptured region composes of a full film part and a mist film part that are identified with the indicator ϕc:

$$\phi _{c}=\frac{6\cdot \eta _{\mathrm{oil}}\cdot Q_{ac}}{\rho _{\mathrm{oil}}\cdot \pi \cdot r^{2}\cdot {h}_{i}^{3}}\cdot \left(\frac{1}{{n}_{c1}^{2}+n_{c1}\cdot \Updelta n+0.3\cdot \Updelta n^{2}}\right),$$
(14)

where ϕc is the area ratio of full film part in the ruptured region, ηoil is the oil dynamic viscosity, Qac is the actual oil flow rate, ρoil is the oil density, r is the radius, hi is the oil gap length, nc1 is the lower disc speed and Δn is the disc differential speed. With this, total clutch losses Tc can be determined as follows:

$$\begin{aligned}T_{c}&=\int\limits_{r_{i}}^{r_{c}}\int\limits_{0}^{2\pi }N_{c}\cdot r^{3}\cdot \frac{\eta _{\mathrm{oil}}\cdot \Updelta n}{h_{i}}drd\theta +\\&\int\limits_{r_{c}}^{r_{o}}\int\limits_{0}^{2\pi }N_{c}\cdot \phi _{c}\cdot r^{3}\cdot \frac{\eta _{\mathrm{oil}}\cdot \Updelta n}{h_{i}}drd\theta +\\&\int\limits_{r_{c}}^{r_{o}}\int\limits_{0}^{2\pi }N_{c}\cdot \left(1-\phi _{c}\right)\cdot r^{3}\cdot \frac{\eta _{cm}\cdot \Updelta n}{h_{i}}drd\theta ,\end{aligned}$$
(15)

where ri is the clutch inner radius, rc is the boundary radius of the full and the ruptured oil film, ro is the clutch outer radius and ηcm is the dynamic viscosity of mist film.

In terms of synchronizer losses, the oil shearing exists in the gap between open synchronizer ring and cone. Weider [27] builds a simple analytical model which considers the oil gap as being completely meshed with the full film. Wirth [28] conducts extensive measurement for different types of synchronizers and generates an empirical model, in which synchronizer geometries are not considered. Liu et al. [29] propose a synchronizer loss model by the investigation on oil flow behavior with NS-equations. Based on their investigation on oil dynamics, the overall oil region is classified into a full film and a mist film. Their boundary is determined by the comparison of the theoretical oil flow rate and the actual flow rate. The total synchronizer losses Tsyn are then defined with Newtonian friction theorem under all possible flow states:

$$\begin{aligned}&T_{\mathrm{csyn}}=\int\limits_{0}^{z_{0}}\int\limits_{0}^{2\pi }\left(r_{\mathrm{syn}}-0.5\cdot h_{s}\right)^{2}\cdot \frac{\eta _{\mathrm{oil}}}{h_{s}}\cdot\\ &\left[\left(n_{s2}+\Updelta n\right)\cdot \left(r_{\mathrm{syn}}-h_{s}\right)-n_{s2}\cdot r_{\mathrm{syn}}\right]dzd\theta +\\&\int\limits_{z_{0}}^{b_{s}\cdot cos\alpha _{s}}\int\limits_{0}^{2\pi }\left(r_{\mathrm{syn}}-0.5\cdot h_{s}\right)^{2}\cdot \frac{\eta _{sm}}{h_{s}}\cdot\\ &\left[\left(n_{s2}+\Updelta n\right)\cdot \left(r_{\mathrm{syn}}-h_{s}\right)-n_{s2}\cdot r_{\mathrm{syn}}\right]dzd\theta ,\end{aligned}$$
(16)

where z0 is the boundary of the two flow states, rsyn is the synchronizer ring radius, hs is the gap width, ηoil is the oil dynamic viscosity, ns2 is the rotational speed of synchronizer ring, Δn is the differential speed, bs is the ring length, αs is the cone angle and ηsm is the mist film viscosity.

For describing shaft seal losses, Simrit [30] proposes a simple equation based on measurements. In this model, only shaft diameters and rotational speeds are considered. A more detailed model is introduced by Jelaska [31]. The oil temperature and its viscosity are additionally included, and the mathematical expression is:

$$\begin{aligned}P_{se}=[145-1.6\cdot t_{\mathrm{oil}}+350\cdot \log (\log (\nu _{\mathrm{oil}40}+0.8))]\cdot\\ {d}_{sh}^{2}\cdot n\cdot 10^{-7},\end{aligned}$$
(17)

where toil is the oil temperature, νoil40 is the oil viscosity at 40 °C, dsh is the shaft diameter and n is the shaft speed.

4 Experiment strategy for transmission total loss measurement

Generally, the commonly used factorial design in transmission loss measurements cannot effectively determine the measurement points in the input region, due to a lack of prior information. Against this one-shot sampling strategy, Liu et al. [32] select the sequential adaptive design category and develop a surrogate modeling based adaptive sampling strategy. The key of this strategy is to constantly evaluate the surrogate model after each measurement and modeling iteration, and to position new measurement candidates in the area with high interest.

4.1 Gaussian Process Regression Subset Simulation (GPRSS)

As comprehensively compared in the previous research [32], GPR can be properly applied for high-dimensional and non-linear applications, with a relatively uncomplicated hyperparameter optimization in the model learning process. Besides, due to a probabilistic modeling feature [33], GPR can additionally deliver the estimation variance for further evaluation. On the basis of multivariate Gaussian distribution [33], GPR is mathematically expressed with a mean and a covariance function:

$$\hat{y}\left(x\right)\sim \mathcal{GP}\left(m\left(x\right),c\left(x,x^\prime\right)\right)$$
(18)

where \(\hat{y}\left(x\right)\) is the non-parametric model output, m(x) is the mean value and \(c(x,x^\prime)\) is the covariance matrix of various input x and \(x^\prime\). Without any prior knowledge, the mean value m(x) is normally set with 0, and the model structure is defined with the covariance matrix \(c(x,x^\prime)\). The commonly employed squared exponential covariance function is used in this study and numerically described as:

$$c\left(x,x^\prime\right)={\sigma }_{f}^{2}\exp \left(-\frac{1}{2l^{2}}\left(x-x^\prime\right)^{2}\right)$$
(19)

where \({\sigma }_{f}^{2}\) is the signal variance and l is the characteristic length scale. Finally, the prediction of the unobserved point \(x_{*}\) can be formulated with:

$$\left[\begin{array}{c} \hat{y}\\ \hat{y}_{*} \end{array}\right]\sim \mathcal{N}\left(0,\begin{bmatrix} K & k_{*}\\ {k}_{*}^{\top } & k_{**} \end{bmatrix}\right)$$
(20)

where \(\hat{y}_{*}\) is the unobserved target, K is the covariance function \(c\left(x,x^\prime\right)\), \(k_{*}=c(x,x_{*})\) and \(k_{**}=c(x_{*},x_{*})\). The prediction of mean value \(m_{*}\) and variance value \(\mathrm{var}_{*}\) are expressed with:

$$m_{*}={k}_{*}^{\top }\left(K+{\sigma }_{n}^{2}I\right)^{-1}y$$
(21)
$$\mathrm{var}_{*}={k_{**}-k}_{*}^{\top }\left(K+{\sigma }_{n}^{2}I\right)^{-1}k_{*}$$
(22)

where \({\sigma }_{n}^{2}\) denotes the noise variance. With the structured GPR model, the new measurement candidates are selected based on a stochastic simulation algorithm Subset Simulation. It is originally introduced to efficiently search the event with small probabilities [34]. The key feature of Subset Simulation is to replace the rare event F with a product of conditional probabilities of more common events:

$$\begin{aligned}p\left(F\right)&=p\left(F_{k}|F_{k-1}\right)p\left(F_{k-1}\right)\\&=p\left(F_{1}\right){\prod }_{i=2}^{k}p\left(F_{i}| F_{i-1}\right)\end{aligned}$$
(23)

where p(F) is the small probabilities, k is the number of total search events and F1 is a common event that can be relatively easily discovered.

The determination of new measurement candidates xc are conducted by maximizing the minimum distance of all received green samples, to avoid insufficient measurements:

$$x_{c}=\{x\colon d\left(x_{i},x_{j}\right)> D_{\min }\}$$
(24)

where d(xi,xj) is the distance of the corresponding points xi, xj and Dmin is the minimum acceptable distance in the input region. This combination of Subset Simulation and candidate selection provides both a local exploitation of desired region and a global exploration of sufficient measurement candidates. A detailed analysis and verification of the strategy can be found in previous studies [32, 35].

4.2 Test bench introduction and preliminary measurement

Figure 3a demonstrates the transmission test bench which has a modular structure to install various transmission configurations. The synchronous drive motor has a maximum power of 240 kW for the simulation of engine. Two asynchronous motors are employed on the transmission output to simulate the road load. Additionally, an oil conditioning system is equipped to realize the oil temperature regulation in the transmission.

Fig. 3
figure 3

Transmission test bench at IMS, TU Darmstadt

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In terms of the measurement devices, three optical encoders are employed on the drive and load motors for the speed quantification. Besides, three digital transducers HBM TP12 (accuracy level 0.02%) are installed on the drive and load side for the measurement of shaft torques. The oil temperature at the transmission outlet bore is observed with a TP100 sensor (class A according to DIN 60751) and further regulated with the oil conditioning. After the oil temperature is stabilized within the set tolerance of ± 1 °C in an observation time window of 90 s, all the signals in that time window for each operating point (with a constant set speed and torque) are applied to calculate the transmission losses Tl with:

$$T_{l}=\frac{\overline{T}_{i}\cdot \overline{n}_{i}-\overline{T}_{o}\cdot \overline{n}_{o}}{\overline{n}_{i}}-T_{r}$$
(25)

where \(\overline{n}_{i},\overline{n}_{o}\) are the mean rotational speed of the input and output shafts, respectively, \(\overline{T}_{i},\overline{T}_{o}\) indicate the mean torque on the input and output shafts and Tr is the not transmission related torque loss. As shown in Fig. 3b, in addition to the transmission losses, the measured total losses also include losses from two drive shafts (left shaft joint angle: 0.17°, right: 0.06°) and two bearings (SKF 6010, 50 × 80 × 16). To determine these “not transmission related” losses, the power loss measurement protocols from drive shaft manufacturer and the SKF bearing model [21] are used in the evaluation process. According to a preliminary analysis with 102 operating points under 3nd gear and 50 °C oil temperature, the proportion of “not transmission related” losses \(T_{r}/(T_{r}+T_{l})\) is within 0.3–2.6% referring to the total measured torque losses.

Regarding the operating conditions for the torque loss evaluation of the DCT, all operating ranges are listed in Table 1. Due to the limited cooling capacity of the oil conditioning, 50 and 70 °C are adjusted in the torque loss mapping. Besides, the max. input speed under 6th and 7th gear is set with 4000 rpm, for reaching a steady oil temperature under the corresponding stable speed and torque state.

Table 1 Operating conditions for the torque loss and efficiency analysis in seven-speed DCT
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4.3 Adaptive transmission loss measurement

As introduced in Sect. 4.1, the whole strategy GPRSS combines the surrogate modeling and the stochastic simulation for the determination of new measurement candidates. Figure 4 demonstrates a typical GPRSS progression under 1st gear, 50 °C oil temperature. The first GPR model is built with 20 measured points based on Latin Hypercube Sampling (LHS) [36]. All measurement points are conducted on the CONNECT test bench, and all required signals are recorded under the stable speed, torque and oil temperature. With this initial GPR model, new measurement candidates are explored in multiple areas with relatively large estimation variances. It should be mentioned that the goal function FG in the Subset Simulation is defined with a variance threshold aG:

$$F_{G}=\{x\colon a_{G}-f_{\mathrm{var}}\left(y\left(x\right)\right)< 0\}$$
(26)

where x is the input region, y(x) is the GPR estimation value and fvar indicates the estimation variance. The critical variance threshold in each Subset Simulation process (grey lines in Fig. 4) is determined with the cumulative distribution function (CDF) gcdf of the evaluation data with GPR model:

$$a_{G}=\{f_{\mathrm{var}}\left(y_{e}\right)\colon g_{\mathrm{cdf}}\left(f_{\mathrm{var}}\left(y_{e}\right)\right)=v_{\mathrm{cri}}\}$$
(27)

where ye is the evaluation data and vcri is the set critical CDF value. In this research, it is set with 0.6. Based on this, GPRSS finally terminates with total 48 measurement points after five iterations.

Fig. 4
figure 4

GPRSS progression under 1st gear and 50 °C oil temperature

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5 Detailed investigations on the efficiency and loss behavior of seven-speed DCT

5.1 Total loss and efficiency analysis

The adaptive strategy GPRSS is employed for various operating conditions for characterizing the toque loss behavior of the complex DCT. Since GPRSS uses data-based GPR algorithm to describe transmission losses in the whole input region, its validity is first investigated with the traditional factorial design approach. Figure 5a, c, e and g demonstrate the estimated torque loss surfaces with GPR models and 102 measured validation points for 1st, 2nd, 3rd and 5th gear conditions under 50 °C oil temperature. In general, the generated final GPR model (with final 48, 44, 54 and 50 training data for each gear condition) can deliver a good torque loss prediction for all conditions, with a MAPE of 3.75%, 3.21% 3.75% and 6.00%, respectively.

Fig. 5
figure 5

Analysis of transmission total losses and efficiency under various conditions

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Input torques have a straightforward influence on total losses, and they increase almost linearly with input torques under all speed and gear conditions. Compared to this, input speeds show more complex effects on total losses. At lower input torques, total losses have a rising characteristic with increasing input speeds under all gear conditions, while at higher input torque states, the loss map exhibits a slight reduction of the total loss in the lower input speed range under 1st, 2nd and 3rd gear condition. At the higher 5th gear, this trend disappears and the total loss gradually increases. A more detailed explanation will be conducted later with analytical simulations.

Regarding the efficiency behavior, the seven-speed DCT has a rising total efficiency with increasing input torques, and stabilizes in the range of 95–96% at the max. input torque under all gear conditions. As the gear increases, a clear drop of the efficiency in the lower input torque range can be detected, especially at high input speeds, which is related to the large increase of no-load losses in the system.

5.2 Analysis of the influences of operating conditions

5.2.1 Influences of input speeds

With the help of analytical simulation, individual loss distributions and influences of various parameters can be discussed in detail. Figure 6a, c, e and g illustrate the influence of input speeds under no-load conditions. Without load transfer, gear churning, bearing no-load, clutch and synchronizer losses are major loss contributors. With increasing input speeds, the total loss displays an almost linear growth. Under a higher gear condition, the total loss has a larger contribution and growth rate at a same input speed condition. This result is related to a larger amount of gear churning and bearing no-load losses, due to higher speeds of the corresponding rotating components on the intermediate and output shafts under higher gear conditions.

Fig. 6
figure 6

Influences of input speeds under various conditions

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As mentioned earlier, input speeds show a different effect on the total loss behavior for different gear conditions, especially at higher load states. A detailed inside view is shown in Fig. 6b, d, f and h. Under 1st gear for example, the gear meshing and bearing load-dependent losses dominate the total contributions and their significant reduction in the low speed range induces the drop of the total loss. In the higher speed range, the reduction of load-dependent losses slows down and the change of total losses is more dependent on the increase of no-load contributions. However, under the higher 5th gear, no-load losses hold a relatively larger proportion. Thus, their increase has a higher impact and leads to a continuous growth of the total loss.

5.2.2 Influences of input torques

Figure 7 displaces DCT loss behaviors with increasing input torque under different gear conditions at the input speed of 500 rpm (in a, c, e and g) and 5000 rpm (in b, d and f, h for 4000 rpm due to test bench limit), respectively. Based on the loss mechanisms of all components, no-load losses have constant contributions in the entire torque range. The almost linear correlation between the total loss and the input torque is caused by the gear meshing and bearing load-dependent losses. It is also clear that at a lower input speed of 500 rpm, the higher 6th gear holds a smaller total loss than 1st gear at the max. input torque. This reflects that a higher gear ratio induced load-dependent loss decrease is stronger than the increase of no-load losses. Compared to this, no-load losses are more influential to the total loss at the input speed of 5000 rpm, and a higher total loss value can be noticed at the max. input torque under 6th gear, against the lower gear conditions.

Fig. 7
figure 7

Influences of input torques under various conditions

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5.2.3 Influence of oil temperature

As mentioned in Table 1, besides the torque loss measurement under the oil temperature of 50 °C, the GPRSS experiment design strategy is also employed for mapping the DCT losses under 70 °C. The influence of the oil temperature in oil sump are thus studied with both experiments and analytical simulations here.

Figure 8 shows the comparison of the DCT no-load loss behavior at the oil temperature of 50 and 70 °C under different gear conditions. According to analytical simulations, a reduced oil viscosity at higher temperature leads to a lower hydrodynamic oil shear resistance and a lower drag torque. Consequently, the estimated no-load losses are lower at a higher oil temperature of 70 °C. In contrast, no clear differences of the total losses from the measurement-based curves can be observed at different temperatures, especially under higher gear conditions. A similar phenomenon is found in the churning loss simulation for a two-speed transmission with Computational Fluid Dynamics (CFD) by Liu et al. [37]. In their study, larger amount of oil can be observed around gear teeth in the circumferential direction at higher oil temperature, which induces a larger pressure on the teeth surface and a higher drag torque. The analytical loss model does not consider the interaction between oil distribution and oil temperature in complex systems. Thus, a large deviation between simulations and experiments can be observed, especially in the speed region dominated by churning losses.

Fig. 8
figure 8

Influences of oil temperatures under various gear conditions

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6 Conclusion

In this article, the loss and efficiency behaviour of a seven-speed DCT is investigated with both analytical simulations and laboratorial experiments. For the estimation of individual loss mechanisms, most available modeling approaches are analyzed and screened for the power loss simulation in the complex DCT. In terms of the measurement, the total loss sensitivity to the oil temperature stabilization tolerance is studied, before a novel sequential adaptive sampling strategy is implemented for mapping the total loss behaviour. This strategy continually establishes and evaluates the data-based GPR model, in order to search and determine new measurement candidates for the next sequence. In general, the adaptive sampling strategy can deliver a good mapping quality while reducing the total number of measurement points by about half.

With both methods, a detailed investigation on the DCT loss and efficiency is performed for various operating conditions. Under different gear conditions, the investigated DCT can have a high efficiency up to 95% in a wide range of input speeds and torques. Regarding the influences of various conditions, no-load losses demonstrate a rising trend with increasing input speeds, while the load-dependent losses decrease. Input torques only affect the load-dependent losses and they exhibit almost a linear relationship. Moreover, the total loss variation and individual loss contributions are highly sensitive to the gear condition, since an increasing gear ratio leads to different speed and load distributions of all individual components in the system. More specifically, no-load losses have more proportions with rising gear and gradually dominate total losses in high input speed area, while load-dependent loss proportions shrink.

Compared to analytical simulations, the measured total loss at different oil temperatures has no clear deviations. Therefore, there is potential and need to advance the analytical models for their usage on the transmission system level, and to include the correlation of oil temperatures and oil distribution for the prediction of transmission total losses.