Modelling and Parametrisation Approach for an Electric Powertrain in a Hardware-in-the-Loop Environment

Abstract

A device under test, when applied to the test rig, often does not come with much information about its mechanical properties to the user. There are different applications in which specific properties of the device under test are of interest to the user. Therefore, a suitable model approach and a parameterisation method are required. If there is a torsional model of the plant, including the device under test and the load machines, it can, for example, be used in a model predictive control architecture. The focus of the publication is on the frequency range of driveability ($f<$ 30 Hz) and, in particular, on the phenomenon of the vehicle shuffle mode, which is important for driving comfort. The model approach has to map these characteristics. To make this possible, the publication presents a suitable, simplified modelling approach for electric powertrains in the hardware-in-the-loop environment and the possibility of indirect parameterisation for the moment of inertia and stiffness. The investigations demonstrate that the model possesses the essential eigenmodes and frequencies observed in the measurements on the test rig. Taking into account extensions, the model enables the incorporation of the properties of an open differential, including delta speeds. The natural frequency matches the measured one with deviations less than 1%. The results also show that the parameters are smaller than assumed. The authors will revise the developed method on this basis to achieve higher information value and a better confidence interval. This further work will discuss the influence of the confidence interval on the resulting parameters.

1. Introduction

The following presents the motivation and background of the paper. The problem is then defined, and the practically relevant constraints are marked. In the end, the authors state the research gap and the paper’s structure.

1.1. Background

Investigating powertrains as device under test in a hardware-in-the-loop, environment enables consideration of many properties at an early stage in the vehicle development process.

One phenomenon of the vehicle as a whole is longitudinal vehicle shuffle, which is due to the powertrain’s first natural frequency. During this state, drive components oscillate against the wheels, with the side shafts representing the vibration node. Both the driver and the passengers can perceive this vibration and rate it as unpleasant depending on its frequency [1].

Maier [2] discusses possibilities for objectifying powertrain-induced vibrations primarily using the example of conventional drives and real driving tests. Cai [3] provides detailed information on an approach to objectifying the vibrational perception of a driver of a complete vehicle on a roller test rig.

The relevant frequency range of driveability as a comfort-relevant property is at frequencies below 30 Hz [4]. A comparison with the frequencies humans perceive particularly strongly (below 10 Hz) shows that these are within the range of driveability [5]. The early investigation of vehicle shuffling and the powertrain’s vibration modes is therefore significant.

Hao et al. [6] show that an original vehicle model with multiple degrees of freedom can be reduced to three degrees of freedom, while retaining the essential properties of the longitudinal shuffle. The moments of inertia of the drive, the combined rims, and the combined belt inertias, along with the reduced inertia of the vehicle body, remain in the model. The side shafts and tires remain the main sources of stiffness and damping. This model describes a simplified real vehicle for longitudinal manoeuvres only.

Fan [1], Figel et al. [7], Hülsmann [8], Lukac et al. [9], Pillas [10], and Sorniotti [11] describe the influence of the side shafts on the frequency and the influence of the properties of the tire or the slip on the damping of the full vehicle vibration.

For investigating the electric vehicle driveability properties on a subsystem rig, there are hardly any existing results [12]. For this purpose, the test rig in Figure 1 is used. It consists of an electrical drive unit with gear stages and side shafts (device under test), two dynamometers (permanent-excited synchronous machines), a frequency converter, a battery emulator, and related components. The modelling of the entire hardware-in-the-loop is therefore both mechanical and electronic.

1.2. Problem Definition and Challenges/Practical Constraints

Previous investigations of driveability on the test rig used show:

-

Due to the adaptation of the device under test in the hardware-in-the-loop, additional adapters are required. Furthermore, the load machines replace the vehicle’s wheels. The vehicle’s mass is not taken into account. The addition of these adapters and the lack of body mass significantly changes the oscillation behaviour of the torsional system [13].

-

The excitation of the system by step functions is suitable for evaluating the natural frequency. An evaluation of these tests with coupled vehicle simulation, cf. Figure 1, shows that dead times and the existing control architecture (PID) in particular strongly influence and falsify the results [12,14].

To overcome these deficits, the authors will pursue a model-predictive control approach. The main idea is to describe the given physical system on the test rig and its vibrational behaviour. If the plant’s properties are known, it is possible to determine the necessary adjustments to a coupled virtual vehicle and tire model. The plant consists of the device under test and the load machines, including the needed adapters. This cyberphysical approach enables road matching, thereby shifting the real driving test for driveability-relevant phenomena to the rig.

Therefore, this idea requires a model of the entire physical system in the first step. The user should regard the device under test as a black box. No information on the mechanical and electrical parameters is available from the respective manufacturer, or the manufacturer cannot disclose it for confidentiality reasons.

Thus, a model with few degrees of freedom or low complexity is appropriate. This model has to satisfy both: mapping the frequency range ($f<$ 30 Hz) and the eigenmodes needed, and being simple enough for parametrisation.

Additionally, it should be real-time-capable in the second step for use in model predictive control, but this investigation will not be part of this paper.

1.2.1. Electronical Domain

For modelling the electrical domain of the test rig, only permanently excited synchronous machines are considered. According to Kellner [15], the following parameters are needed to describe the motor equations:

-

number of pole pairs p

-

stator resistance $R_S$

-

inductances $L_d$ and $L_q$

-

permanent magnet flux ${\psi }_p$

The parameters are known from the data sheets for describing the test rig machines.

If the parameters are unknown, there are already many identification methods; Rafaq and Jung [16] provide a comprehensive overview.

The description of the electronic systems mentioned above is less significant for the free vibration behaviour required; therefore, this investigation primarily requires a model of the mechanical components and, in the first instance, neglects the electronic domain.

1.2.2. Mechanical Domain

For modelling the mechanical domain of the test rig, this approach considers only components of the torsional system. Parameters like the moments of inertia, stiffness, and damping can describe the components.

The detailed description of the electrical drive unit under test, therefore, requires the data of the following parts, among others:

-

rotor shaft,

-

gear stage 1,

-

countershaft,

-

gear stage 2 to the cage of the differential,

-

differential with cage, planets, bevel gears,

-

…

Modelling the following parts requires the data of the side shafts:

-

bell and shaft with multi-tooth profile of the tripod on the gearbox side,

-

tripod with star and rollers,

-

profile shaft,

-

wide angle joint,

-

bell and shaft with multi-tooth profile on wheel side,

-

depending on the side shaft, the wheel bearing and a pressed-on and bolted flange are also part of the rotating masses.

The system therefore consists of a large number of parts and corresponding parameters to describe the moments of inertia, stiffness and damping. Here, too, the disadvantage is that the manufacturer provides no information about the electrical drive unit or the supplied side shafts.

1.2.3. Challenges and Practical Constraints

In conclusion, there are the following challenges and practical constraints:

-

It is necessary to identify the significant properties of the test rig when creating a model of the plant with a focus on driveability. Regarding the described characteristics of real vehicles, the findings have to be transferred to the test rig, as the load machines replace the wheels on it. The rotors replace the moment of inertia of the rim, but there is no physical influence of the tire belt inertia and the coupling stiffness. The influence of tire properties is calculated virtually and accounted for in the specified torque of the load machines.

For modelling the powertrain as a device under test, the model could include the following components: the drive and output moments of inertia, and the stiffness of the side shafts. This paper has to prove this claim. The test rig considers cornering; therefore, the model must map the separate consideration of left and right outputs.

-

Because the electrical drive unit and the side shafts of the device under test are a black-box for the user, the determination of the necessary parameters from data sheets or CAD models is not possible. The model’s parametrisation has to be experimental.

-

As there is no suitable component test rig at the chair of automobile engineering, the side shafts cannot be measured individually. Parameterising the assembled subsystem has the additional advantage that the measurements describe the installed position. Therefore, for example, the influence of tooth stiffness between the side shaft and bevel gears on the resulting shaft stiffness remains in this approach.

-

The electrical drive units under test usually do not have sufficiently accurate measurement technology that is suitable for parameterisation. For example, the device under test used here does not have its own internal torque sensor. The device under test has a speed sensor installed. The communication frequency to the motor itself is only 100 Hz, which means that the resolution of the speed signal is unsuitable for frequency observations. The method must be adapted so that the high-precision sensors of the load machines serve as the basis for measurements.

Therefore, the main problem definition is:

→

What must the underlying mechanical model of the plant of the powertrain test rig look like for use in that rig? The model should cover the essential eigenmodes and the necessary frequency range ($f<$ 30 Hz), and be parameterisable by the test rig itself.

1.3. Research Gap and Existing Methods

The scientific novelty value of the presented method for the combined determination of moment of inertia and stiffness in the real system should be briefly classified.

To determine the moment of inertia, Dresig and Holzweißig [17] cite, for example, the torsion-bar suspension, multi-strand suspension, and rolling pendulum methods. These methods all require the removal of the bodies and are therefore unsuitable in this application. Schmidt [18] also describes a method for determining the moment of inertia of electric motor adapters using ramp tests and back-calculation from measured and friction torque. Due to the unsuitable sensors in the test motor, this method is not applicable. Schmidt [18] describes the use of empirical values to determine the missing parameters in this case.

To determine the shaft stiffness, Schmidt [18] describes the use of an optimisation algorithm. This calculation determines the stiffness until the algorithm reaches the termination criterion; this method requires the knowledge of the moment of inertia and the natural frequency. This method is therefore highly dependent on the selected inertia and, when poorly chosen, can lead to large deviations.

Given the existing setup, none of the presented methods are considered suitable; therefore, the authors sought an algorithm to determine stiffness and moment of inertia simultaneously. The main advantage in industry is in engineering service providers, benchmarking, or across all areas where a way to model a black-box system as closely as possible is needed, based on the principle of being as good as possible and as accurate as necessary in terms of abstraction. Knowledge of the plant on the test rig allows for predicting target trajectories so that properties dependent on the system under investigation can be examined, in the best case, without disturbance variables.

The authors divide the publication into the following steps. In Section 2, the procedure for modelling torsional systems, the reduction of degrees of freedom and the derivation of the model used for parameterisation are presented. Additionally, the paper presents the model expansion for use on the rig and the impact of static bending. In doing so, Section 2.1 and Section 2.2 draw on existing methods. In particular, Dresig and Fidlin [19] and Rosenlöcher [20] are used to demonstrate the torsional modelling of rotationally symmetric bodies and the reduction of degrees of freedom. In addition, the authors take up the idea of indirect parameterisation from Dresig and Holzweißig [17]. They show the indirect mass-determination method for a single-mass oscillator. In this investigation, the approach is extended and applied to a 15-mass oscillator with a branched drive, resulting in a 29-mass oscillator with the additional stiffnesses.

Section 3 shows the results from the measurements, and the algorithm for determining the parameters. Furthermore, verification shows that the model with the determined parameters meets the necessary natural frequencies and eigenmodes.

Section 4 critically discusses the presented method and the results before summarising all the key findings in Section 5.

2. Materials and Methods

All the methods used and developed are explained below.

2.1. Modelling of a Torsional System

For modelling the torsional dynamics of the test rig, the following boundary conditions are defined at the beginning:

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All considered parts are ideally rotationally symmetrical.

-

This approach considers only rigid (partial) bodies, and interprets all rotational masses of a stepped shaft as discrete bodies.

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Only the pre-tensioned state of the system is considered for the description of the parameterisation. There are no load changes around M = 0 Nm, so that the assumption of a series oscillator is permissible. This assumption neglects gear backlash in the first step.

-

The mechanical parameters of the motor (moment of inertia and stiffness) and the drive shafts are unknown, as explained in Section 1. Estimating these parameters is necessary for the initial description of the methodology. For the motor, the total moment of inertia, reduced to the output, is taken from comparable applications. The stiffness of the drive components is much higher than the output stiffnesses of the left and right branches due to the reduction. Therefore, the approach will neglect it. The moment of inertia and the stiffness of the side shafts are estimated from Equations (1) and (2).

To obtain a model of a torsional system from a series of rotationally symmetrical components, the method of Dresig and Fidlin [19] and Rosenlöcher [20] is used, briefly presented below using the example in Figure 2.

The example includes an adapter (H-flange, dark grey), screwed to components on both the left and right sides. In total, this results in a stepped shaft, describable as a set of cylinders, consisting of five discrete sub-segments. The polar moment of inertia $J_i$ and the partial stiffness $c_i$ can be calculated for each partial segment i. With the shear modulus G, the density of steel $\rho$, the length $l_i$ and the diameter $d_i$ of the segments. Here for a hollow cylinder with outer diameter $d_1$ and inner diameter $d_2$.

$$\begin{array}{cc}\hfill J_i& ={\displaystyle \frac{\rho ·l_i·\pi ·\left(d_{1i}^4-d_{2i}^4\right)}{32}}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill c_i& ={\displaystyle \frac{G·\left(d_{1i}^4-d_{2i}^4\right)}{32·l_i}}\hfill \end{array}$$

(2)

To determine the coupling stiffnesses $c_T$ ($c_{T1}$ to $c_{T4}$) between the moments of inertia $J_i$ ($J_1$ to $J_5$), only half the lengths $l_i$ are taken into account or the double stiffnesses $c_i$ as a series connection; see Equation (3). If the forces or moments are not applied directly through the respective end faces, the method neglects the half-stiffnesses at the ends of the shaft [20].

$$\begin{array}{c}\hfill c_T={\displaystyle \frac{2·c_i·c_{i+1}}{c_i+c_{i+1}}}\end{array}$$

(3)

However, the stiffnesses calculated in this way overestimate the real stiffnesses, as the force flow cannot follow a jump in diameter abruptly. This results in force-free flow areas, described by additional lengths $\Delta l$ [19].

This correction changes two neighbouring lengths, for example, for Section 3 and Section 4 in Figure 2 $\to l_4^{\ast }=l_4-\Delta l$ and $l_3^{\ast }=l_3+\Delta l$. This exemplary calculation needs the rounding radius and the diameter of the smaller segment:

$$\begin{array}{c}\hfill \overline{r}={\displaystyle \frac{2·r_i}{d_i}}\end{array}$$

(4)

Sähn [21] states the needed additional length $\Delta l/d_i$ in dependence of the ratio $d_i/d_{i+1}$ for given $\overline{r}$. For $\overline{r}$ = 0, according to Dresig and Fidlin [19], the following estimation applies:

$${\displaystyle \frac{\Delta l}{d_i}}=\left\{\begin{array}{cc}0.155\hfill & 0<{\displaystyle \frac{d_i}{d_{i+1}}}\le 0.3\hfill \\ -0.115{\displaystyle \frac{d_i}{d_{i+1}}}+0.1895\hfill & 0.3<{\displaystyle \frac{d_i}{d_{i+1}}}\le 0.5\hfill \\ 0.264·\left(1-{\displaystyle \frac{d_i}{d_{i+1}}}\right)\hfill & 0.5<{\displaystyle \frac{d_i}{d_{i+1}}}\le 1\hfill \end{array}\right.$$

(5)

The diameter $d_{i+1}$ represents the diameter of the bigger segment. With these new partial stiffnesses, it is necessary to recalculate the coupling stiffnesses $c_{Ti}$ from Equation (3).

For a detailed consideration of the calculation rules, please refer to Dresig and Fidlin [19]. The stiffnesses used below include this correction.

Table A1. Moment of inertia $J_i$ and stiffness $c_i$ used for modelling of torsional system.

Moment of Inertia	Stiffness	Remarks
in 10−3 kgm2	in 108 Nm/rad	-
$J_1$ = 850	$c_1$ = 0.0872	rotor shaft dynamometer M2
$J_2$ = 38	$c_2$ = 0.0460	torque sensor
$J_3$ = 18.8	$c_3$ = 7.2907	adapter segment 1
$J_4$ = 0.7	$c_4$ = 0.0169	adapter segment 2
$J_5$ = 3.7	$c_5$ = 2.7171	adapter segment 3
The additional stiffness (Table A2) is applied here
$J_6$ = 3.4	$c_6$ = 0.1408	flange of side shaft
$J_7$ = 0.3	$c_7$ = 0.0001	side shaft (short)
$J_8$ = -	$c_8$ = -	integrated in i = 9
$J_9$ = 3500	$c_9$ = 0.7266	reduced inertias and stiffnesses to load axle, including rotor shaft, gear stages, lay shaft, differential
$J_{10}$ = -	$c_{10}$ = -	integrated in i = 9
$J_{11}$ = 0.4	$c_{11}$ = 0.0001	side shaft (long)
$J_{12}$ = 3.4	$c_{12}$ = 0.1408	flange of side shaft
The additional stiffness (Table A2) is applied here
$J_{13}$ = 3.7	$c_{13}$ = 2.7171	adapter segment 3
$J_{14}$ = 0.7	$c_{14}$ = 0.0169	adapter segment 2
$J_{15}$ = 18.8	$c_{15}$ = 7.2907	adapter segment 1
$J_{16}$ = 38	$c_{16}$ = 0.0460	torque sensor
$J_{17}$ = 850	$c_{17}$ = 0.0872	rotor shaft dynamometer M3

lists the partial stiffnesses $c_i$ and moments of inertia $J_i$.

2.2. Reduction in Degrees of Freedom of the Torsional System

The model of the entire test rig derived in Section 2.1 comprises 15 moments of inertia $J_i$ and 14 coupling stiffnesses $c_{Ti}$ and Figure 3 shows the model. Table A1 lists the component names.

As explained at the beginning, a pre-tensioned state is always assumed below. In addition, a detailed resolution of the device under test and the side shafts into their subcomponents is not possible, as already explained in Section 1. Accordingly, the inertias $J_8,J_9,J_{10}$ already contain several individual components. In Figure 3, these are labelled as individual parts to indicate the existing differential. However, the sum of the three is used for further calculation, as it is not possible to parameterise the individual parts. Due to the large number of components, the figure shows only the left-hand branch and the drive machine. The drive machine can be understood as a plane of symmetry so that the left and right branches are approximately equal.

A modal analysis yields the natural frequencies; Figure 4 shows all 15 natural frequencies of the undamped system and the first four normalised eigenvectors. The representation omits the rigid body mode. Only the first two eigenvalues greater than 0 Hz ($f_2$ = 17.59 Hz and $f_3$ = 22.47 Hz) describe driveability phenomena and lie within the defined frequency range of 30 Hz. The second eigenmode (vector 2) shows an oscillation between the outputs. The third mode (vector 3) shows the phenomenon of vehicle shuffle, with the drive components oscillating against the output components, and the side shafts representing the vibration node. The subsequent modes show oscillations with local maxima. These modes are acoustic phenomena with frequencies above 1000 Hz and are not relevant for further consideration.

The aim below is to simplify the model, as described in Section 1, by removing irrelevant modes while retaining the significant mode shapes that contribute to driveability. One possible method is the reduction according to Rivin [22] and Di [23], which Dresig and Fidlin [19] explain in detail and is summarised briefly here. The torsional system is broken down into subsystems of types A and B, as shown in Figure 5. For an initial system with n moments of inertia, there are n subsystems of type A and $n-1$ subsystems of type B. The natural frequencies of all $2n-1$ systems are determined, and the position of the system with the highest natural frequency is determined. For the non-existent stiffnesses at $i=0$ and $i=n+1$, a stiffness $c_{T0,n+1}$ = 0 Nm/rad must be entered.

$$\begin{array}{c}\hfill {\omega }_{Ai}^2={\displaystyle \frac{c_{Ti-1}+c_{Ti}}{J_i}}\mathrm{or}{\omega }_{Bi}^2={\displaystyle \frac{c_{Ti}·\left(J_i+J_{i+1}\right)}{J_i·J_{i+1}}}\end{array}$$

(6)

Depending on whether this natural frequency occurs in type A (Equations (7) to (9)) or B (Equations (10) to (12)), the following reduction steps are carried out, with the new parameters marked with a *. The system’s degree of freedom decreases by 1. The reduction continues until the model has reached the structure in Figure 3b).

$$\begin{array}{cc}\hfill J_{i+1}^{\ast }& =J_{i+1}+{\displaystyle \frac{c_{Ti}·J_i}{c_{Ti-1}+c_{Ti}}}\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill J_{i-1}^{\ast }& =J_{i-1}+{\displaystyle \frac{c_{Ti-1}·J_i}{c_{Ti-1}+c_{Ti}}}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill c_{Ti-1}^{\ast }& ={\displaystyle \frac{c_{Ti-1}·c_{Ti}}{c_{Ti-1}+c_T}}\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill J_i^{\ast }& =J_i+J_{i+1}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{c_{Ti-1}^{\ast }}}& ={\displaystyle \frac{1}{c_{Ti-1}}}+{\displaystyle \frac{J_{i+1}}{\left(J_i+J_{i+1}\right)·c_{Ti}}}\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{c_{Ti+1}^{\ast }}}& ={\displaystyle \frac{1}{c_{Ti+1}}}+{\displaystyle \frac{J_i}{\left(J_i+J_{i+1}\right)·c_{Ti}}}\hfill \end{array}$$

(12)

Both the overall stiffness and the sum of the moments of inertia remain unchanged by the reduction measures. A reduction made at one end of the shaft, type B in Figure 5, creates a free spring as a stub shaft. Dresig and Fidlin [19] suggest neglecting this spring. The reduction ends when the three-mass oscillator from Figure 3 has been reached.

Table 1. Parameter of the reduced torsional model.

Parameter	Value	Remarks
$J_1^{\ast }$	0.9148 kgm2	Sum of $J_1$ to $J_6$ → 0.9146 kgm2
$J_9^{\ast }$	3.5003 kgm2	-
$J_{17}^{\ast }$	0.9148 kgm2	Sum of $J_{12}$ to $J_{17}$ → 0.9146 kgm2
$c_{T1}^{\ast }$	13,093 Nm/rad	Stiffness of the shaft $c_7$ = 13,254 Nm/rad
$c_{T14}^{\ast }$	10,211 Nm/rad	Stiffness of the shaft $c_{11}$ = 10,308 Nm/rad

summarises the resulting variables for the considered example.

The reduction summarises the system to the left and right of the side shafts. The reduced inertia of the outputs ($J_1^{\ast },J_{17}^{\ast }$) corresponds with a deviation of 0.02% to the sum of the moments of inertia in front of the side shaft. In a vehicle, these inertias represent the wheels’ moments of inertia.

The moment of inertia of the side shafts splits to the left and right, but as this is very small, the effect is minimal. The influence on the moment of inertia of the drive machine ($J_9^{\ast }$) is less than 0.01%.

The resulting stiffnesses ($c_{T1}^{\ast },c_{T14}^{\ast }$) of the left and right branches are the original stiffnesses of the side shafts with a change of 1%.

The natural frequencies of the undamped system shown here are $f_2$ = 17.59 Hz and $f_3$ = 22.46 Hz.

2.3. Parametrisation of the Torsional System

The torsional system model derived in Section 2.2, (Figure 3b), is described by the moments of inertia $J_1^{\ast },J_9^{\ast },J_{17}^{\ast }$ and the stiffnesses $c_{T1}^{\ast },c_{T14}^{\ast }$. Due to the boundary conditions mentioned and since $J_9^{\ast }\approx J_9$ and $c_{T1}^{\ast }\approx c_7,c_{T14}^{\ast }\approx c_{11}$ apply, $J_9,c_7\mathrm{and}{c}_{11}$ are used below.

Since the side shafts differ in length but are otherwise identical in construction, Equation (13) is introduced, taking into account Equation (2). The stiffness $c_{Ts}$ describes the stiffness of a side shaft.

$$\begin{array}{c}\hfill c_{11}={\displaystyle \frac{7}{9}}·c_7={\displaystyle \frac{7}{9}}·c_{Ts}\end{array}$$

(13)

Equation (14) describes the free and undamped system.

$$\underset{\underset{=1}{\mathrm{J}}}{\underbrace{\left[\begin{array}{ccc}{J}_1^{\ast }& 0& 0\\ 0& J_9& 0\\ 0& 0& J_{17}^{\ast }\end{array}\right]}}\left[\begin{array}{c}{\ddot{\phi }}_1\\ {\ddot{\phi }}_9\\ {\ddot{\phi }}_{17}\end{array}\right]+\underset{\underset{=1}{\mathrm{C}}}{\underbrace{\left[\begin{array}{ccc}{c}_{Ts}& -c_{Ts}& 0\\ -c_{Ts}& \left(1+{\displaystyle \frac{7}{9}}\right)·c_{Ts}& -{\displaystyle \frac{7}{9}}·c_{Ts}\\ 0& -{\displaystyle \frac{7}{9}}·c_{Ts}& {\displaystyle \frac{7}{9}}·c_{Ts}\end{array}\right]}}\left[\begin{array}{c}{\phi }_1\\ {\phi }_9\\ {\phi }_{17}\end{array}\right]=\underset{=}{0}$$

(14)

The eigenvalues ${\lambda }_1$ of the system can be calculated using Equation (15) and are complex numbers. Since the system is undamped, the real part is zero and the imaginary part is ${\omega }_1$.

$$det\left(\left[\begin{array}{cc}\underset{=}{0}& \underset{=}{\mathrm{E}}\\ \underset{=1}{{-\mathrm{J}}^{-1}}\underset{=1}{\mathrm{C}}& \underset{=}{0}\end{array}\right]-{\lambda }_1\underset{=}{\mathrm{E}}\right)=0$$

(15)

In contrast to the explanation of the example presented, only the moments of inertia $J_1^{\ast }$ and $J_{17}^{\ast }$ of the application are directly known on the test rig. In relation to Figure 1, the two variables represent the inertias of the dynamometers M2 and M3 plus the inertias of the needed adapters.

The natural frequency of the system can be measured on the test rig, for example, via a sine sweep. Since $c_{Ts}$ and $J_9$ are unknown, Equation (15) is underdetermined.

Dresig and Holzweißig [17] propose the introduction of an additional mass or additional spring for indirect mass determination. The natural frequency changes due to the change in the defined parameter.

The following uses this principle to parameterise the system. This approach needs two identical and additional adapters, constructed with a defined stiffness $c_{T\Delta }\approx$ 64,330 Nm/rad. The stiffness lies in the dimension of the side shaft, taking into account the assumptions from Section 2.2. Additional design restrictions for the extra spring result from the maximum possible travel of the load machines and the boundaries of the in-house manufacturing process. The added stiffness especially influences the overall stiffness and thus the measurable natural frequency of the test rig structure. The adapter consists of a stepped shaft with seven sections; see

Table A2. Moment of inertia $J_i$ and stiffness $c_i$ used for the additional stiffness.

Moment of Inertia	Stiffness	Remarks
in 10−3 kgm2	in 108 Nm/rad	-
$J_1$ = 3.9	$c_1$ = 3.0530	adapter segment 1
$J_2$ = 0.1	$c_2$ = 0.0035	adapter segment 2
$J_3$ = 0.4	$c_3$ = 0.0010	adapter segment 3
$J_4$ = 0.1	$c_4$ = 0.0036	adapter segment 4
$J_5$ = 0.4	$c_5$ = 0.4165	adapter segment 5
$J_6$ = 0.3	$c_6$ = 0.0880	adapter segment 6
$J_7$ = 3.7	$c_7$ = 2.7187	adapter segment 7

.

The algorithm can thus resolve the previously underdetermined equation system. By introducing the additional stiffness described earlier, the sum of the moments of inertia also changes, as indicated by the index Δ. The steps from Section 2.1 and Section 2.2 were repeated. The summarised moments of inertia of the new system are $J_{1\Delta }^{\ast },J_{9\Delta }^{\ast },J_{17\Delta }^{\ast }$. For the application example from Section 2.2, this was shown as an example in

Table 2. Parameter of the reduced torsional model with applied additional stiffness for the shown example in Section 2.2.

Parameter	Value	Remarks
$J_{1\Delta }^{\ast }$	0.9222 kgm2	Sum of $J_i$ before shaft → 0.9201 kgm2
$J_{9\Delta }^{\ast }$	3.5030 kgm2	-
$J_{17\Delta }^{\ast }$	0.9225 kgm2	Sum of $J_i$ before shaft → 0.9201 kgm2
$c_{T1\Delta }^{\ast }$	10,861 Nm/rad	Stiffness of series connection of the shaft $c_7$ and $c_{T\Delta }$ → 10,990 Nm/rad
$c_{T14\Delta }^{\ast }$	8801 Nm/rad	Stiffness of series connection of the shaft $c_{11}$ and $c_{T\Delta }$ → 8884 Nm/rad

. The natural frequencies of the system are $f_{2\Delta }$ = 16.18 Hz and $f_{3\Delta }$ = 20.50 Hz.

The results from Table 1 and Table 2 show that $J_9\approx J_9^{\ast }\approx J_{9\Delta }^{\ast }$. It can also be seen that the resulting stiffness of the branches is determined by a series connection of the two springs per branch, where:

$$\begin{array}{cc}\hfill c_{T\Delta 1}& ={\displaystyle \frac{c_{Ts}·c_{T\Delta }}{c_{Ts}+c_{T\Delta }}}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill c_{T\Delta 2}& ={\displaystyle \frac{7·c_{Ts}·c_{T\Delta }}{9·\left({\displaystyle \frac{7·c_{Ts}}{9}}+c_{T\Delta }\right)}}\hfill \end{array}$$

(17)

The following then applies to the customised system:

$$\underset{\underset{=2}{\mathrm{J}}}{\underbrace{\left[\begin{array}{ccc}{J}_{1\Delta }^{\ast }& 0& 0\\ 0& J_9& 0\\ 0& 0& J_{17\Delta }^{\ast }\end{array}\right]}}\left[\begin{array}{c}{\ddot{\phi }}_1\\ {\ddot{\phi }}_9\\ {\ddot{\phi }}_{17}\end{array}\right]+\underset{\underset{=2}{\mathrm{C}}}{\underbrace{\left[\begin{array}{ccc}{c}_{T\Delta 1}& -c_{T\Delta 1}& 0\\ -c_{T\Delta 1}& c_{T\Delta 1}+c_{T\Delta 2}& -c_{T\Delta 2}\\ 0& -c_{T\Delta 2}& c_{T\Delta 2}\end{array}\right]}}\left[\begin{array}{c}{\phi }_1\\ {\phi }_9\\ {\phi }_{17}\end{array}\right]=\underset{=}{0}$$

(18)

The eigenvalue problem can thus be defined:

$$det\left(\left[\begin{array}{cc}\underset{=}{0}& \underset{=}{\mathrm{E}}\\ \underset{=2}{{-\mathrm{J}}^{-1}}\underset{=2}{\mathrm{C}}& \underset{=}{0}\end{array}\right]-{\lambda }_2\underset{=}{\mathrm{E}}\right)=0$$

(19)

After converting Equation (15) and inserting it into Equation (19), the stiffness $c_{Ts}$ can be calculated by converting it again. Reinserting it yields the drive’s moment of inertia $J_9$. The system described in Equation (14) is thus fully parameterised.

2.4. Modelling of a Simplified Torsional System for Use on the Test Rig

The system shown in Figure 6 is described by the vector of generalised coordinates $q={\left[{\phi }_{m2}{\phi }_{d2}{\phi }_{drive}{\phi }_{d3}{\phi }_{m3}\right]}^T$. It differs from the model in Section 2.3 by accounting for the differential. The simplified approach neglected the properties. This simplification was permissible because, in the special case with the tensioned system, the speeds of the left and right branches were equal.

The following redefines the nomenclature used in the parameterisation:

$$\begin{array}{c}\hfill J_{m2}=J_1{J}_{d2}=J_8{J}_{drive}=J_9{J}_{d3}=J_{10}{J}_{m3}=J_{17}{c}_{T2}=c_{Ts}{c}_{T3}={\displaystyle \frac{7·c_{Ts}}{9}}\end{array}$$

(20)

To determine the equations of motion of the model, the inertia, stiffness and damping matrices are defined, using the Lagrange equation of the second kind; see Equations (21) to (24) [17].

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial \dot{q}}}\right)-{\displaystyle \frac{\partial L}{\partial q}}& =Q\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill W_{kin}-W_{pot}& =L\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{J}_i·{\dot{\phi }}_i^2& =W_{kin}\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{c}_{Ti}·{\left({\phi }_i-{\phi }_{i-1}\right)}^2& =W_{pot}\hfill \end{array}$$

(24)

Taking into account the vector of the generalised coordinate q (which contains the individual rotation angles ${\phi }_i$), the Lagrange function L, the generalised force Q, as well as the kinetic energy $W_{kin}$ and the potential energy $W_{pot}$ in the notation for rotational systems.

The kinetic (Equation (25)) and potential energy (Equation (26)) enable the calculation of the elements of the inertia and stiffness matrices using partial derivatives [17].

$$\begin{array}{cc}\hfill j_{mn}=j_{nm}& ={\displaystyle \frac{{\partial }^2{W}_{kin}}{\partial {\dot{q}}_n\partial {\dot{q}}_m}}\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill c_{Tmn}=c_{Tnm}& ={\displaystyle \frac{{\partial }^2{W}_{pot}}{\partial q_n\partial q_m}}\hfill \end{array}$$

(26)

To describe the system’s properties regarding the open differential, Equation (27) introduces differential speed between the two outputs and thus cornering. The final drive ratio occurs before the cage, and the reduction to the output already integrates it.

$$\begin{array}{c}\hfill {\phi }_{drive}={\displaystyle \frac{{\phi }_{d2}+{\phi }_{d3}}{2}}\to {\phi }_{d2}=2·{\phi }_{drive}-{\phi }_{d3}\end{array}$$

(27)

This results in the following matrices for the modified system:

$$\begin{array}{cc}\hfill J& =\left[\begin{array}{cccc}{J}_{m2}& 0& 0& 0\\ 0& J_{drive}+4·J_{d2}& 2·J_{d2}& 0\\ 0& 2·J_{d2}& J_{d2}+J_{d3}& 0\\ 0& 0& 0& J_{m3}\end{array}\right]\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill V& =\left[\begin{array}{cccc}{c}_{T2}& -2·c_{T2}& c_{T2}& 0\\ -2·c_{T2}& 4·c_{T2}& -2·c_{T2}& 0\\ c_{T2}& -2·c_{T2}& c_{T2}+c_{T3}& -c_{T3}\\ 0& 0& -c_{T3}& c_{T3}\end{array}\right]\hfill \end{array}$$

(29)

As explained at the beginning, the device under test to be analysed is a black box. The resolution of the inertias into the inertia $J_{drive}$ reduced to the output and the moments of inertia of the bevel gears $J_{d2}$ or $J_{d3}$ can therefore not be parameterised or represented. The assumption is made that: $J_{d2}=J_{d3}<<J_{drive}$. The inertia matrix J reduces to a 3 × 3 matrix. For the coordinate ${\phi }_{d3}$, Equation (30) can be generated from the expression $J\ddot{q}+Vq=0$, with $J_{d2}=J_{d3}=0$.

$$\begin{array}{c}\hfill {\phi }_{d3}={\displaystyle \frac{2·{\phi }_{drive}·c_{T2}-{\phi }_{m2}·c_{T2}+{\phi }_{m3}·c_{T3}}{c_{T2}+c_{T3}}}\end{array}$$

(30)

This results in the following notation for the potential energy.

$$\begin{array}{cc}\hfill W_{pot}& =W_{pot2}+W_{pot3}\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill W_{pot2}& ={\displaystyle \frac{c_{T2}}{2}}·{\left({\phi }_{m2}·\left(1-{\displaystyle \frac{c_{T2}}{c_{T2}+c_{T3}}}\right)+2·{\phi }_{drive}·\left({\displaystyle \frac{c_{T2}}{c_{T2}+c_{T3}}}-1\right)+{\displaystyle \frac{{\phi }_{m3}·c_{T3}}{c_{T2}+c_{T3}}}\right)}^2\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill W_{pot3}& ={\displaystyle \frac{c_{T3}}{2}}·{\left({\phi }_{m3}·\left(1-{\displaystyle \frac{c_{T3}}{c_{T2}+c_{T3}}}\right)-2·{\displaystyle \frac{{\phi }_{drive}·c_{T2}}{c_{T2}+c_{T3}}}+{\displaystyle \frac{{\phi }_{m2}·c_{T2}}{c_{T2}+c_{T3}}}\right)}^2\hfill \end{array}$$

(33)

The matrices from Equations (28) and (29) change after partial differentiation of the kinetic and potential energies to Equation (34):

$$\begin{array}{cc}\hfill J& =\left[\begin{array}{ccc}{J}_{m2}& 0& 0\\ 0& J_{drive}& 0\\ 0& 0& J_{m3}\end{array}\right]\hfill \end{array}$$

(34)

The stiffness matrix V is noted in the appendix (Equation (A1)) due to its size. The damping matrix K results analogously to the matrix V. The determined matrices obtain the state-space model.

$$\begin{array}{cc}\hfill \dot{x}& =A·x+B·u\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill y& =C·x+D·u\hfill \end{array}$$

(36)

With the vectors $q={\left[{\phi }_{m2}{\phi }_{drive}{\phi }_{m3}\right]}^T$, $x={\left[q^T{\dot{q}}^T\right]}^T$ and $u={\left[M_{m2}{M}_{drive}{M}_{m3}\right]}^T$ and the matrices:

$$\begin{array}{ccc}\hfill A& =\hfill & \left[\begin{array}{cc}\underset{=}{0}& \underset{=}{\mathrm{E}}\\ -J^{-1}V& -J^{-1}K\end{array}\right]\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill C& =\hfill & \left[\begin{array}{cc}\underset{=}{\mathrm{E}}& \underset{=}{0}\end{array}\right]\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill B& =\hfill & \left[\begin{array}{c}\underset{=}{0}\\ J^{-1}\underset{=}{\mathrm{E}}\end{array}\right]\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill D& =\hfill & \underset{=}{0}\hfill \end{array}$$

(40)

The form shown here shows the reduced variables of the drive related to the output. To obtain the actual parameters of the device under test, the torque $M_{drive}$ and the torsion angle ${\phi }_{drive}$ must be recalculated taking into account the total transmission ratio.

The model of the open differential used here is simplified. For more detailed modelling approaches of different types of differentials, please refer to Deur et al. [24] or Deur et al. [25]. This investigation neglects gear backlash. Figel [26] provides an overview of various models.

2.5. Influence of Static Bending

The addition of an extra torsion spring changes both the sum of the individual masses and moments of inertia and the resulting stiffness. This change is necessary to determine the model’s required parameters. On the other hand, it also results in a longer bending beam. Before use, it is therefore necessary to check whether the additional part leads to excessive bending and thus to a significant imbalance. In addition, the analyses exclude the possibility that the system applies excessive bending torque to the torque sensor. If the displacement of the critical point is large, creating an additional bearing is necessary, which increases costs to ensure safe operation.

The modified test rig set-up, containing the parts of Table A1 with the additional stiffness, is a special application that does not have to cover the entire engine map. The requirements for torque transmission and speed are therefore lower. Using the Castigliano theorem evaluates the influence of bending on a structure. The side shaft stores itself in the device under test. The wide-angle joint is assumed to be the critical point. The joint cannot transmit bending moments. To determine the displacement $v_i$, the auxiliary force $F_C$ is introduced here.

The device under test contains a tripod joint on every side of the gearbox output. Therefore, the device under test does not introduce normal forces during operation. The Figure 7 shows the free section of the system under consideration. Equation (41) shows the balance of transverse forces, and Equation (42) shows the moment balance at the location of force $F_A$. The system is underdetermined, whereby the bearing forces $F_A$ and $F_B$ are rearranged.

$$\begin{array}{cc}\hfill F_A+F_B+F_C& ={\displaystyle \sum _{i=1}^n}{q}_i·l_i\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill F_B·l_1+F_C·{\displaystyle \sum _{i=1}^n}{l}_i& ={\displaystyle \sum _{i=1}^n}\left(q_i·l_i·\left(0.5·l_i+{\displaystyle \sum _{i=1}^{i-1}}{l}_i\right)\right)\hfill \end{array}$$

(42)

To determine the bending line, the internal forces, bending moment $M_i$ and shear force $F_i$, are defined for each of the 13 segments as a function of a running variable $z_i$. Table A1 determines the number of segments by the mentioned parts. The wide-angle joint is located between the flange and the side shaft itself, accounting for the additional stiffness provided by these parts. The application of Castigliano’s theorem considers only the influence of the bending moment. Because of this simplification, the formulation is:

$$v_i={\displaystyle \sum _{i=1}^n}{\int }_0^{l_i}{\displaystyle \frac{M_i}{E·I_i}}·{\displaystyle \frac{\partial M_i}{\partial F_C}}d{z}_i$$

(43)

With the modulus of elasticity E and the axial moment of inertia $I_i$ of the respective segment. After partial differentiation, $F_C$ = 0 is applied. As boundary conditions for the integration constants, it is assumed that the displacement in the fixed and loose bearings v = 0. Otherwise, the following applies to the free cantilever beam: $v_{i-1}\left(z_{i-1}=l_{i-1}\right)=v_i\left(z_i=0\right)$. Figure 8 shows the result of the bending line. The greatest deflection occurs in the area of the additional spring. The required low stiffness explains the high length-to-diameter ratio. The maximum displacement $v_{max}$ is 61 μm and is therefore very small. For this reason, there is no need for an additional intermediate bearing.

The conclusions rely on an ideal component. In mechanical production, manufacturing tolerances, as well as shape and position tolerances, must be considered. These can result in deviations in the component geometry so that the actual maximum deviation of the additional spring from the axis of rotation is greater.

3. Results

To demonstrate the function of the developed method, the test rig is excited by a sine sweep to determine the natural frequency of the set-up. For this purpose, the motor under test operates at constant torque, and the load machines reach an approximately constant speed. The load machines are inactive (idle), allowing free oscillation to take place. When the system reaches a steady state, the motor applies an oscillating torque signal with an amplitude of 0.8 Nm and a frequency of 1 Hz around the torque set point. This frequency increases linearly up to 30 Hz; see

Table 3. Set point for test runs of all four variants.

Parameter	Value	Parameter	Value
initial set point	4 Nm	Amplitude	0.8 Nm
Frequency	1 Hz → 30 Hz	Number of repetitions	5
nominal Power of DuT	184 kW	Speed sensor dynamometer	ECN 1313

.

The device under test is operating in open-loop control. Because of the transmission gear ratio of ca. 10, the applied torque to the output is 40 Nm. Through the sine sweep, the peak-to-peak torque difference is therefore 16 Nm, which splits between the branches. The torque variation causes a change in rotor speed, which the speed sensor measures. The sensor is on the back of the rotors and was neglected in the analysis of the torsional system because its moment of inertia is 2.6 × 10⁻⁶ kgm².

This experiment is exemplary for the output system (V1) shown in Figure 9. The top diagram shows the torque applied by the drive motor. The middle diagram shows the power spectrum of the speed signal of machine M2 over time. The bottom diagram shows the speed signals of the two load machines. The speed increase due to excitation, visible between t = [60 s, 80 s], indicates the presence of a natural frequency. The power spectrum also shows a local maximum in this range.

In addition to the two variants from Section 2 (original system and original system with additional springs on the left and right), systems with unilateral changes to the structure are also tested. The four variants differ as follows and are shown in Figure 10.

V1

This variant considers the original test rig set-up.

V2

This variant considers the original test rig set-up, including the additional spring on both sides.

V3

This variant considers the original test rig set-up, including the additional spring on the right side.

V4

This variant considers the original test rig set-up, including the additional spring on the left side.

Since the additional moment of inertia of the extra spring is very small, all variants use the same default values. For variants V3 and V4, Equation (44) and Equation (45) apply, respectively. The eigenvalue problems arise analogously to the previous considerations.

$$\underset{\underset{=3}{\mathrm{J}}}{\underbrace{\left[\begin{array}{ccc}{J}_1^{\ast }& 0& 0\\ 0& J_9& 0\\ 0& 0& J_{17\Delta }^{\ast }\end{array}\right]}}\left[\begin{array}{c}{\ddot{\phi }}_1\\ {\ddot{\phi }}_9\\ {\ddot{\phi }}_{17}\end{array}\right]+\underset{\underset{=3}{\mathrm{C}}}{\underbrace{\left[\begin{array}{ccc}{c}_{Ts}& -c_{Ts}& 0\\ -c_{Ts}& c_{Ts}+c_{T\Delta 2}& -c_{T\Delta 2}\\ 0& -c_{T\Delta 2}& c_{T\Delta 2}\end{array}\right]}}\left[\begin{array}{c}{\phi }_1\\ {\phi }_9\\ {\phi }_{17}\end{array}\right]=\underset{=}{0}$$

(44)

$$\underset{\underset{=4}{\mathrm{J}}}{\underbrace{\left[\begin{array}{ccc}{J}_{1\Delta }^{\ast }& 0& 0\\ 0& J_9& 0\\ 0& 0& J_{17}^{\ast }\end{array}\right]}}\left[\begin{array}{c}{\ddot{\phi }}_1\\ {\ddot{\phi }}_9\\ {\ddot{\phi }}_{17}\end{array}\right]+\underset{\underset{=4}{\mathrm{C}}}{\underbrace{\left[\begin{array}{ccc}{c}_{T\Delta 1}& -c_{T\Delta 1}& 0\\ -c_{T\Delta 1}& c_{T\Delta 1}+{\displaystyle \frac{7·c_{Ts}}{9}}& -{\displaystyle \frac{7·c_{Ts}}{9}}\\ 0& -{\displaystyle \frac{7·c_{Ts}}{9}}& {\displaystyle \frac{7·c_{Ts}}{9}}\end{array}\right]}}\left[\begin{array}{c}{\phi }_1\\ {\phi }_9\\ {\phi }_{17}\end{array}\right]=\underset{=}{0}$$

(45)

To determine the position of the speed overshoot of the load machines, see Figure 9, and thus the natural frequency of the test rig, the following procedure applies for all four variants:

-

A high-pass filter (cut-off frequency = 1 Hz) corrects the raw speed signal for any fundamental oscillations and offset. Figure 9 includes this correction.

-

The power spectrum of the speed signal is determined. The frequency range is limited to frequencies between 1 Hz and 30 Hz.

-

The matrix of the power spectrum is used to determine the maximum value in the range of the overshoot.

-

To be more robust against fluctuations, the algorithm determines the range in which the power spectrum corresponds to at least 80% of the maximum value. The outer limits of the range serve as interval limits for the position of the natural frequency.

-

The centre of the interval is determined, which indicates the position of the natural frequency. This position determines the frequency driven from the time axis.

Figure 11 shows all determined frequencies and their standard deviations for the four variants. Here, the natural frequency of the output system V1 is the highest, as this structure is the most rigid. Adding the additional springs to V2 softens the system the most, and the resulting natural frequency is the smallest of the four. The natural frequencies of variants V3 and V4 lie between V1 and V2, as they also become softer due to the one-sided addition of the spring.

In addition, it is already noticeable here that the expected frequency shift is much smaller than discussed in the preliminary considerations. The frequency intervals overlap significantly. Nevertheless, observing the predicted trends is possible.

Two variants are required to solve the system of equations and determine the test rig model’s parameters. Therefore, there are six possible combinations for solving the system.

Table 4. Possible combinations of $J_9$ and $c_{Ts}$ for solving the systems of equations.

	Variant 1	Variant 2	Variant 3	Variant 4
Variant 1	X	-	-	-
Variant 2	$J_9$ = 0.529 kgm2 $c_{Ts}$ = 1479 Nm/rad	X	-	-
Variant 3	$J_9$ = $c_{Ts}$ = []	$J_9$ = $c_{Ts}$ = []	X	-
Variant 4	$J_9$ = 0.736; 7.570 kgm2 $c_{Ts}$ = 1888; 7073 Nm/rad	$J_9$ = $c_{Ts}$ = []	$J_9$ =$c_{Ts}$ = []	X
The combination of Variant 1 and Variant 4 results in two solutions for the moment of inertia and the stiffness. A semicolon separates them. These two solutions are labelled by V1/V4-1 and V1/V4-2 in the following.

shows the resulting parameters. The solutions consider only values with no imaginary component and a positive real part. Using Equation (13) calculates the second stiffness of the long shaft.

As an additional condition for the determined parameters, after reinserting them into Equations (14), (18), (44) and (45), the respective eigenvalues must contain the measured natural frequencies from Figure 11. The corresponding eigenvector must represent the shuffle mode that occurs at $f_3$ in each case.

Table 5. Measured natural frequencies of the variants and calculated natural frequencies of parametrised models.

Property	Variant 1	Variant 2	Variant 3	Variant 4
Measured System	f = 12.77 Hz	f = 12.62 Hz	f = 12.67 Hz	f = 12.65 Hz
Parameter set V1/V2	$f_2$ = 5.97 Hz $f_3$ = 12.77 Hz	$f_2$ = 5.89 Hz $f_3$ = 12.62 Hz	$f_2$ = 5.93 Hz $f_3$ = 12.72 Hz	$f_2$ = 5.93 Hz $f_3$ = 12.67 Hz
Parameter set V1/V4-1	$f_2$ = 6.74 Hz $f_3$ = 12.77 Hz	$f_2$ = 6.63 Hz $f_3$ = 12.59 Hz	$f_2$ = 6.68 Hz $f_3$ = 12.71 Hz	$f_2$ = 6.69 Hz $f_3$ = 12.65 Hz
Parameter set V1/V4-2	$f_2$ = 12.77 Hz $f_3$ = 15.07 Hz	$f_2$ = 12.19 Hz $f_3$ = 14.28 Hz	$f_2$ = 12.26 Hz $f_3$ = 15.01 Hz	$f_2$ = 12.65 Hz $f_3$ = 14.39 Hz

shows the proof of this condition. The results show that the parameter set V1/V4-2 does not meet these conditions.

4. Discussion

Some aspects of Section 2 and Section 3 will be discussed in more detail here.

4.1. Significance of the Results

The results from Section 3 show that the parameters determined for the test rig are significantly smaller than those discussed in the preliminary considerations. This approach demonstrates that estimating shaft stiffness solely based on length and diameter results in a significant overestimation. As explained, the additional spring with stiffness $c_{T\Delta }$ was developed based on the estimated shaft stiffnesses. Under the given boundary conditions (maximum travel of the load machines, static deflection, etc.), this extra spring is intended to cause a frequency shift of several Hz. However, this spring is significantly stiffer than the actual side shaft, and the overall stiffness is therefore hardly affected.

This small effect ultimately also results in a strong overlap of the frequency intervals from Figure 11. An assessment of the error range in the calculation that accounts for the minima and maxima of the respective intervals is therefore ineffective. In this consideration, only two pairings showed physically interpretable solutions. The solutions where the target frequencies match the required eigenvectors are of the same order of magnitude.

In a later study, the authors plan to revise the additional stiffness $c_{T\Delta }$ to account for the findings obtained here and to repeat the measurements accordingly. The aim is to ensure that the resulting frequency ranges differ more significantly. It is then necessary to check again whether all six possible combinations yield physically plausible solutions that are similar to one another and meet the requirements for natural frequencies and eigenmodes.

Using the appropriate method to determine the drive unit’s moment of inertia and the side shaft’s stiffness, validating the results remains a challenge. Since dismantling the motor is not possible, the side shafts must be measured externally, which causes additional costs. With the stiffness values, it is possible to determine the moment of inertia of the drive unit. Taking into account the measured natural frequency. These results then serve as a basis for determining whether the proposed parameterisation method yields valid results.

4.2. Assumptions in the Derivation of the Methodology

When deriving the method for parameterising the test rig model in Section 2.3, estimates were made for the unknown variables. The results in Section 3 show that the parameters of Table 4 determined from the measurement data deviate from the assumptions made.

A robustness analysis assesses whether the assumptions underlying the parameterisation are based solely on the selected sample values or retain broader validity when the previous estimate deviates significantly from reality.

For this purpose, the estimated stiffnesses and moments of inertia of the side shafts and the drive machine in Table A1 varied by a factor of 10. All other variables and assumptions remain constant.

Table 6. Parameter of the reduced torsional model with sensitivity analysis.

Parameter	Value	Remarks
variation of stiffness, $d_{7,11}·\sqrt[4]{10}$
$J_1^{\ast }$	0.9154 kgm2	Sum of $J_1$ to $J_6$ → 0.9146 kgm2
$J_9^{\ast }$	3.5048 kgm2	-
$J_{17}^{\ast }$	0.9159 kgm2	Sum of $J_{12}$ to $J_{17}$ → 0.9146 kgm2
$c_{T1}^{\ast }$	117,940 Nm/rad	Stiffness of the shaft $c_7$ = 132,540 Nm/rad
$c_{T14}^{\ast }$	94,030 Nm/rad	Stiffness of the shaft $c_{11}$ = 103,080 Nm/rad
variation of stiffness, $d_{7,11}/\sqrt[4]{10}$
$J_1^{\ast }$	0.9147 kgm2	Sum of $J_1$ to $J_6$ → 0.9146 kgm2
$J_9^{\ast }$	3.5 kgm2	-
$J_{17}^{\ast }$	0.9147 kgm2	Sum of $J_{12}$ to $J_{17}$ → 0.9146 kgm2
$c_{T1}^{\ast }$	1324 Nm/rad	Stiffness of the shaft $c_7$ = 1325 Nm/rad
$c_{T14}^{\ast }$	1030 Nm/rad	Stiffness of the shaft $c_{11}$ = 1031 Nm/rad
variation of moment of inertia, $J_9·10$
$J_1^{\ast }$	0.9148 kgm2	Sum of $J_1$ to $J_6$ → 0.9146 kgm2
$J_9^{\ast }$	35.0003 kgm2	-
$J_{17}^{\ast }$	0.9148 kgm2	Sum of $J_{12}$ to $J_{17}$ → 0.9146 kgm2
$c_{T1}^{\ast }$	13,093 Nm/rad	Stiffness of the shaft $c_7$ = 13,254 Nm/rad
$c_{T14}^{\ast }$	10,211 Nm/rad	Stiffness of the shaft $c_{11}$ = 10,308 Nm/rad
variation of moment of inertia, $J_9/10$
$J_1^{\ast }$	0.9148 kgm2	Sum of $J_1$ to $J_6$ → 0.9146 kgm2
$J_9^{\ast }$	0.3503 kgm2	-
$J_{17}^{\ast }$	0.9148 kgm2	Sum of $J_{12}$ to $J_{17}$ → 0.9146 kgm2
$c_{T1}^{\ast }$	13,093 Nm/rad	Stiffness of the shaft $c_7$ = 13,254 Nm/rad
$c_{T14}^{\ast }$	10,211 Nm/rad	Stiffness of the shaft $c_{11}$ = 10,308 Nm/rad

shows the results.

Both a significant increase and a decrease in the drive’s moment of inertia indicate that the derived assumptions remain valid. The diameters of the corresponding segments change to achieve the required change in stiffness. The change in diameter explains the increase in total inertia as stiffness increases.

At 10 times the stiffness, the side shaft stiffness is no longer far from the other torsional system stiffnesses. The first natural frequencies also increase significantly, as does the resulting stiffness after the reduction in degrees of freedom. A significantly lower stiffness compared to the initial state does not cause any change, as the side shaft is still much smaller than the others.

For the expected dimensions of shaft stiffnesses and moments of inertia of the motor, the assumptions:

-

$J_9^{\ast }\approx J_9$, respectively, $J_{drive}$;

-

$c_{T1}^{\ast }\approx c_7$, respectively, $c_{Ts}$ (short);

-

$c_{T14}^{\ast }\approx c_{11}$.

are still valid. It is therefore reasonable to assume that the resulting stiffness of the overall system approximates that of the side shaft. The drive’s moment of inertia is unaffected; using the parameterisation method is valid.

4.3. Impact of Damping

The damping of the system was neglected throughout the analysis, which is evident from the real parts of the eigenvalues of the systems.

Dresig and Holzweißig [17] state that a specific consideration of damping is not necessary if the focus is on the lowest natural frequencies. Rosenlöcher [20] also notes that considering the damping of each component or segment in a system is ineffective, as the entire system influences damping.

The main damping influence of the drive in a real vehicle comes from the tire–road contact or tire slip. This influence is simulated on the test rig by the load machines. The torque applied by the load machines is the main damping influence.

However, for use in model predictive control, a slight generic damping of the side shafts may be beneficial. The eigenvalues $\pm {\lambda }_i$ for the undamped natural frequencies ${\omega }_0$ of the undamped system are all on the imaginary axis due to the ’missing’ real part. The system is therefore marginally stable. Taking a small amount of damping into account (in the form of the damping coefficient D) shifts the eigenvalues of the damped natural frequencies ${\omega }_d$ to the real axis. It can thus lead to a stable system, as shown in Figure 12. For a detailed explanation, please refer to Lunze [27].

4.4. Consideration of the Relevant Modes

Section 1 emphasises the significant influence of the first natural frequency (shuffle mode) of the powertrain on driveability. The analyses in Section 2 show that this mode only occurs at the second natural frequency and that the first two natural frequencies are relatively close together.

Here, the discussion is about which eigenmode is actually measured on the test rig and whether the calculation based on this natural frequency is permissible. The eigenvalue problem shown in Section 2.3 has a solution for each eigenform that occurs. Thus, using the eigenvalue of the first or second eigenform is essentially irrelevant; the equations represent the physics of the underlying system. It is therefore permissible to conclude on the stiffness and moment of inertia.

Figure 13 shows a detailed excerpt from the experiment in Figure 9. The upper diagram shows the unfiltered speed signal of load machine 2. The speed offset at time t = 0 s has been removed from this figure. The fundamental oscillation of the signal is clearly visible here, which influences the frequency evaluation. At the beginning of the test, the speed signal fluctuates and then stabilises. The filter applied in Section 3 distorts this range. Figure 9 seems to indicate a natural frequency of the system at the beginning, but looking at the unfiltered signal refutes this impression. Since the measurement start time is far from the natural frequency region, the distortion caused by the high-pass filter is acceptable.

The lower diagram shows the detailed course of the speed signals from Figure 9 in the range of the natural frequency that occurs. The rotors of the load machines oscillate in phase during the tests. Therefore, the record shows the shuffle mode and not the out-of-phase oscillation of the outputs.

4.5. Modes of the Torsional System for Use on the Test Rig

The assumptions in Section 2.4 integrate the properties of the open differential as an additional extension of the derived test rig model. Additionally, the model substitutes the moments of inertia of the differential’s bevel gears. The stiffness matrix determined (Equation (A1) differs significantly from the original matrix in Equation (14). The following explanation must verify that the resulting natural frequency and vibration mode meet the specified conditions. Figure 14 uses the parameter set V1/V2.

The upper part of the figure shows the model’s properties, including the open differential’s extensions. The left-hand side shows the natural frequencies, with both the rigid body mode and the antiphase oscillation mode of the outputs occurring at 0 Hz. The third natural frequency occurs at 12.64 Hz. The corresponding eigenvectors on the right-hand side show that the shuffle mode is correctly represented, with a frequency deviation of approximately 1% relative to the measurement result. This theoretical consideration reinforces the finding that the shuffle mode is observable on the test rig.

The lower part of the figure shows the properties of the torsion model as a smooth strand. Here, the rigid body mode at 0 Hz, the antiphase vibration of the outputs at 5.97 Hz and the shuffle mode at 12.77 Hz are visible.

4.6. Influence of the Load Machines on the Free Oscillation

The load machines in this test are permanently excited synchronous machines. The frequency converter is idle, assuming that the motor side generates no voltage. However, the rotating rotors induce a voltage due to the permanent excitation applied to the frequency converter. The inverter rectifies this voltage and can charge the DC intermediate circuit. Due to the rotors’ comparatively low angular speed, the induced voltage is also low. This voltage is lower than the nominal intermediate circuit voltage, so no current flows, and the load machine generates no corresponding torque. The assumption that the rotors merely act as freely oscillating inertias is hence permissible.

5. Conclusions

The publication presents a modelling method for electric vehicle drives applied to a hardware-in-the-loop environment. The device under test is often a black box to the user, as no information is available about essential parameters.

In the investigations presented here, the paper identifies the driveability-relevant components and uses them to analyse the related properties. The model reduction retains the first eigenmodes and natural frequencies of the original system to be analysed. It demonstrates that the resulting stiffness in the reduced powertrain model equals the stiffness of both side shafts. The moment of inertia of the drive systems remains unchanged.

An indirect parameterisation option was also presented for the derived model, allowing the two unknown variables, the electric motor’s moment of inertia $J_{drive}$ and the stiffness of the side shafts $c_{Ts}$, to be determined. For this purpose, the introduction of a defined additional spring shows that the natural frequency of the test rig alters. The results showed that the original system is significantly softer than previously estimated. The moment of inertia of the device under test is also smaller than assumed based on known systems. Accordingly, the expected frequency shift is smaller, and the areas of standard deviation partly overlap. An assessment of the error range in the calculation that accounts for the minima and maxima of the respective intervals is therefore ineffective.

The developed and parameterised model has the essential eigenmodes for the driveability frequency range at the test rig’s measured natural frequencies. This condition was required at the outset to correctly map the system behaviour. Generally speaking, the outcome satisfies the main problem addressed in the introduction.

For use on the rig itself, additional boundary conditions for the model arose, including an open differential. The extensions enable differential speeds between the two outputs, allowing cornering. It was possible to show that the significant shuffle mode and frequency remain within deviations of less than 1% of the measured signals.

In future studies, the authors will revise the additional spring and enhance the method’s validity by using a substantially higher frequency shift. The investigations will also include an in-depth analysis of the error range in the frequency measurement on the determined parameters. In the end, the side shafts must be measured externally on a component test rig to validate the results of the shown method.