Simulative Investigation and Optimization of a Rolling Moment Compensation in a Range-Extender Powertrain

Abstract

Battery electric vehicles (BEVs) are gaining market share, yet range anxiety and sparse charging still create demand for hybrids with combustion-engine range extenders. Range-extender vehicles face high customer expectations for noise, vibration, and harshness (NVH) due to their direct comparability with fully electric vehicles. Key challenges include the vibrations of the internal combustion engine, especially from vehicle-induced starts, and the discontinuous operating principle. A technological concept to reduce vibrations in the drivetrain and on the engine mounts, called “FEVcom,” relies on rolling moment compensation. In this concept, a counter-rotating electric machine is coupled to the internal combustion engine via a gear stage to minimize external mount forces. However, due to high speed fluctuations of the crankshaft, the gear drive tends to rattle, which is perceived as disturbing and must be avoided. As part of this work, the rolling moment compensation system was examined regarding its vibration excitation, and an extension to prevent gear rattling was simulated and optimized. For the simulation, the extension, based on a chain or belt drive, was set up as a multi-body simulation model in combination with the range extender and examined dynamically at different speeds. Variations of the extended system were simulated, and recommendations for an optimized layout were derived. This work demonstrates the feasibility of successful rattling avoidance in a range-extender drivetrain with full utilization of the rolling moment compensation. It also provides a solid foundation for further detailed investigations and for developing a prototype for experimental validation based on the understanding gained of the system.

1. Introduction

The global automotive industry is undergoing a transformation process toward electrified propulsion systems. While purely battery electric vehicles (BEVs) are increasingly gaining market share, limited range and inadequate charging infrastructure continue to present challenges. Hybrid vehicles with combustion engines as range extenders offer a bridging technology that combines electric driving with extended range.

Range-extender vehicles are subject to special NVH (noise, vibration, and harshness) requirements, as they are directly compared to purely electric vehicles. Particular challenges include the vibrations generated by the combustion engine, especially during vehicle-initiated engine start and during discontinuous operation of the range extender.

The “FEVcom” concept developed by FEV uses rolling moment compensation to minimize vibrations transmitted to the vehicle body. Here, a counter-rotating electric machine is coupled to the combustion engine via a gear stage. However, the inevitable speed fluctuations of the combustion engine can lead to acoustically disturbing gear rattling, especially on engines with a low number of cylinders, which poses NVH and durability concerns and thus requires mitigation. This creates a fundamental design conflict in range-extender systems with rolling moment compensation. Whereas gear rattling in vehicle transmissions is typically suppressed by decoupling the crankshaft side from the gears, often with the use of a dual-mass flywheel, systems with rolling moment compensation require rigid connections, which makes effective decoupling impossible. Traditional systems force a choice between these requirements—either accepting gear rattling with optimal compensation or reducing rattling at the cost of rolling moment compensation.

This paper investigates a solution for avoiding gear rattling in a serial hybrid powertrain using a pretensioned belt or chain drive while preserving the full potential from rolling moment compensation. The main objective of this work is the analysis of the functioning of the system using a multi-body simulation (MBS) approach. For a successful simulative verification of the solution, gear rattling is effectively avoided in the typical operating range of a range extender up to approximately 4000 1/min while not negatively affecting the rolling moment compensation. Furthermore, optimal system parameters for different operating conditions are derived by performing variant simulations. In particular, the minimum chain pretension and optimal stiffness values that prevent gear backlash traversal are determined while minimizing parasitic power losses and system complexity. Finally, quantitative design guidelines for prototypical implementation are derived.

2. Low-Vibration Serial Hybrid Engines

Serial powertrains represent a special architecture of electrified propulsion systems, where the mechanical decoupling between the internal combustion engine (ICE) and drive wheels enables an optimized operating strategy. In this configuration, the ICE drives an electric machine (EM) that produces electrical energy, which in turn feeds one or more electric motors that propel the vehicle and can also be used to charge the battery. Through this separation of energy generation and propulsion, the combustion engine can be operated in a narrow, map-optimized range, which both increases thermodynamic efficiency and significantly reduces vibration levels. This is particularly relevant for NVH criteria, as reduced vibration excitations lead to lower acoustic load and higher driving comfort. Additionally, the serial structure facilitates the use of active or passive measures for vibration decoupling, for example, through elastic mounting or active vibration absorbers. In the overall system assessment, there is thus considerable potential for optimizing both powertrain excitations and drive comfort. However, the purely electric operation of a vehicle with a serial powertrain also leads to a direct comparison with battery electric vehicles (BEVs), which is why customers have high expectations for NVH behavior. These high requirements concern both airborne noise and structure-borne noise transferred through the body. Airborne sound directly emitted from the powertrain can often be reduced via encapsulation and insulation measures, and structure-borne sound can also be reduced to a certain extent through optimized structure-borne sound paths. However, particularly for structure-borne sound, a greater reduction in vibrations perceptible to vehicle occupants is possible if as few vibrations as possible are introduced into the body. For this purpose, the primary vibrations of the serial powertrain must be reduced. One possibility is to provide for rolling moment compensation in the powertrain.

Rolling moments occur due to rotating mass inertias that experience acceleration. According to the principle of angular momentum, the rolling moment $M_{rot}$ is proportional to the angular accelerations α, with the rotating mass inertia $J_{rot}$ as the proportionality factor:

$$M_{rot}=J_{rot}·\alpha$$

(1)

In combustion engines, the crankshaft together with the flywheel has the greatest rotating mass inertia and experiences non-uniform angular acceleration due to, among other things, intermittent combustion in the cylinders. In addition to the work principle-induced angular accelerations during stationary engine operation, transient processes such as engine start or engine acceleration also cause angular accelerations of the crankshaft. In a serial powertrain, the engine start is particularly critical with respect to NVH, as the start of the combustion engine is not driver- but vehicle-induced, and the resulting vibrations can therefore increasingly lead to irritation for the occupants. The angular accelerations result in rolling moments of the combustion engine, which must be supported by the engine mounts and thus excite the vehicle chassis, leading to structure-borne sound [1,2].

One possibility to compensate for the rolling moments within a serial powertrain is shown in Figure 1. The rotating mass inertia of the crank train is directly coupled with a counter-rotating mass inertia. The counter-rotating mass inertia can be represented, for example, by an electric machine, an oil pump, or another auxiliary unit. For a complete rolling moment compensation, it is important that the ratio of the two mass inertias and the ratio of the two rotational speeds are equal. Additionally, the two mass inertias must rotate in opposite directions on parallel rotational axes. A range-extender system with rolling moment compensation has already been successfully built up as a prototype and has been described in [1,3].

The basic equation for the rolling moment compensation is thus according to [1]:

$$J_2=i·J_1\to M_{Roll}=0 with i={\displaystyle \frac{z_2}{z_1}} as gear ratio$$

(2)

$$Combining Equations \left(1\right) and \left(2\right): {\displaystyle \frac{M_{rot,2}}{M_{rot,1}}}=i·{\displaystyle \frac{{\alpha }_2}{{\alpha }_1}}$$

(3)

With a gear ratio of one, the rotational mass moment of inertia of the electric machine $J_2$ must correspond to the rotational mass moment of inertia of the crank train $J_1$ for a complete rolling moment compensation. From preliminary investigations, it is known that for optimal rolling moment compensation, the connection between the two mass inertias should be as direct and rigid as possible to prevent a phase shift between angular accelerations. A phase shift leads to reduced effectiveness of the rolling moment compensation, as the counteracting rolling moments only fully compensate without phase shift, as described in [4]. Due to these requirements, coupling via gears is particularly desirable. However, the advantage of a direct, rigid coupling in a gear stage can lead to gear rattling due to the high rotational fluctuations of crankshafts in combustion engines. The tendency toward gear rattling for this system layout can also be derived from (3). Based on this equation, it follows that the ratio of rotational accelerations is directly coupled to the ratio of rotational torques via the transmission ratio. Since the dynamic excitations of the combustion engine and the electric machine are fundamentally different, and in particular the dynamics of the combustion engine are very high, this also results in fundamentally different rotational accelerations on the combustion engine side compared to the electric machine side. This deviation in rotational accelerations can lead to gear rattling.

Gear rattling occurs due to changes in the dynamic tooth flank with uneven power transmission in the gear pair. Figure 2 shows the different phases for gear rattling in a vehicle gearbox [5].

Phase 1 represents the desired nominal state. The crankshaft drives the electric machine. The power is transferred from the crankshaft to the electric machine. In phase 2, the tooth contact is lost due to the rotational non-uniformity of the crankshaft. The crankshaft decelerates significantly, whereas the electric machine remains at a nearly constant speed due to its mass inertia. In this phase, no power is transmitted through the gear stage. After the power-free state, there is an impact on the opposite flank side (phase 3). This impact causes an impulsive excitation in the gear stage and is emitted as airborne and structure-borne sound. Phases 2 and 3 are triggered by a deceleration of the crankshaft’s rotational speed and lead to a reversal of the power flow in phase 4. With contact on the opposite flank side, the electric machine drives the crankshaft. When the crankshaft accelerates again, phases 5 and 6 are passed through. First, the tooth contact from the opposite flank side is lost (phase 5), and subsequently, the teeth of the two gears impulsively meet on the initial flank (phase 6). This again causes an emission of airborne and structure-borne sound [5]. At the same time, gear rattling results not only in NVH issues but can also have severe effects on the long-term durability of gears, as shown in [6].

Gear rattling is a known problem in vehicle powertrains, and various measures are known to reduce or avoid rattling. Since gear rattling in powertrains, including combustion engines, is primarily caused by the rotational speed fluctuations induced by the combustion engine, the most common solution is to reduce these fluctuations before they can reach a gear mesh. Therefore, flywheels in various designs are used between the combustion engine and the gearbox. A single-mass flywheel already provides smoothing of the rotational fluctuations due to its rotational mass inertia. However, the integration of a single-mass flywheel cannot entirely eliminate gear rattling in a range-extender engine with a low number of cylinders with rolling moment compensation. Due to the direct coupling of the ICE-side and EM-side inertia according to Equation (2), the single-mass flywheel inertia cannot be increased to a point at which no gear rattling occurs without extensively increasing the EM-side inertia. Dual-mass flywheels, which provide strong decoupling between the combustion engine and gearbox via torsional stiffness, are particularly effective in reducing fluctuations, as shown in [7]. However, decoupling the mass inertias is disadvantageous for rolling moment compensation, as rolling moment compensation works best without phase shift of the rolling moments and therefore relies on a very rigid connection between the mass inertias [4,8].

Another possibility to reduce excitations at the gear mesh is the use of highly dynamic control of the electric machine. However, the disadvantage is the high switching frequency that is necessary to significantly change the torque at the electric machine several times within a combustion cycle. This leads to high control effort and can adversely affect the efficiency and thermal load of the inverter. Several investigations [9,10,11,12] have been performed to investigate dynamic controlling.

If the excitation acting on the gear stage cannot be reduced, there are further possibilities to reduce gear rattling directly at the gear stage. Particularly at low torques, pre-loaded split gears have proven effective. However, the friction in the system increases due to the preload, so a compromise for a maximum preload must usually be found. Gear rattling can thus only be avoided to a certain degree. Especially for applications with high torques, as in range-extender applications, the effect of split gears for rattle avoidance is therefore limited [11,13].

Another possibility to improve the NVH of gear rattling is to make parts of gears or entire gears from plastics. Through the targeted use of plastic, sound transmission paths can be damped and vibrations can be reduced. Gear rattling is not prevented, but the acoustic effects can be reduced. This solution is known, for example, in mass balancing shafts, where plastic gears are sometimes used. However, this does not represent a solution for range-extender applications, as the power transmitted between the crankshaft and the rotor of the electric machine is significantly higher than in mass balancing applications. In particular, due to the temperature and wear behavior of plastic gears, their use with high torques in automotive applications is not advisable [14,15,16,17].

Further modifications can be made directly to the gears to reduce rattling. These include targeted tooth flank and production error optimization, reduction in gear backlash, and optimization of the oil film in tooth contact. However, these measures only reduce the acoustic effects of gear rattling and cannot prevent the rattling itself [5,7,18,19,20,21].

The different solutions for gear rattling reduction described above are summed up and evaluated in

Table 1. Gear rattling minimization solutions in rolling moment compensation systems.

Solution	Complexity	Speed Capability	Torque Capability	Gear Rattle Avoidance	Rolling Moment Compensation
Single-mass flywheel	Low	High	High	Low	High
Dual-mass flywheel	Low	High	High	Medium	Low
Dynamic EM control	High	Medium	Medium	Medium	High
Preloaded gear	Medium	Medium	Medium	Medium	High
Plastic gears	Low	High	Low	Low	High
Gear optimization	Medium	High	High	Low	High
Chain solution [22]	Medium	High	High	High	High

regarding their specific performance as gear rattle reduction systems in powertrains with rolling moment compensation. Single-mass and dual-mass flywheels reduce fluctuations at the gear stage but offer only limited gear rattle avoidance capability. Dynamic control of the electric machine and the use of preloaded gear offer only mitigation of gear rattling up to medium engine speeds and medium torque levels. Plastic gears and the optimization of gear geometry do not mitigate gear rattling but only dampen the transmission. Furthermore, plastic gears offer only low torque capability. The chain solution investigated in this paper combines high speed and high torque capabilities. Due to its layout, the rolling moment compensation is not affected. The effectiveness of the gear rattle avoidance up to high engine speeds is investigated in this paper.

3. Description of Solution

A way to combine rolling moment compensation with a system for avoiding gear rattling is described in [22]. Two possible layouts for the combination of gear stage and belt or chain drive are outlined. Both layouts contain a belt or chain drive mounted parallel to the power-transmitting gear stage. One of the layouts is shown in Figure 3.

The power between the crankshaft and the electric machine is still mainly transferred via the gear stage, and the additional belt or chain drive acts exclusively against gear rattling. For this purpose, the belt or chain drive is pretensioned via a tensioning device. Two of the spans are located directly at the tensioning device, while a third span acts between the gear stages. The technical innovation of this system is based on the principle of asymmetric tension conditions in the belt or the chain, whereby a significantly greater tension is generated on the two free belt sides at the tensioning device than on the independent belt side directly between the crankshaft and EM sprockets. By means of this controlled tension difference, tooth flank changes between the gears themselves are effectively prevented during the compression phase of the combustion engine and even during transient operating conditions. The achieved matching of the transmission ratios between the gears and toothed belt pulleys also guarantees a harmonized power flow with minimal mechanical wear. This drive configuration offers significant NVH advantages for hybrid range-extender applications while simultaneously reducing space and component requirements [22].

The belt or chain drive aims to add pretension into the system, and the pretension acts as additional torque at the sprockets resulting from the span forces. Both span forces at the tensioner sprocket $F_{span,ten,1}$ and $F_{span,ten,2}$ can be assumed to be equal with low friction. Therefore, two span force amplitudes influence the drive system: the span force at the tensioner $F_{span,ten}$ and the force in the span between the sprocket of the electric machine and the crankshaft sprocket $F_{span,EM,ICE}$. The additional torque $M_{add,EM}$ at the sprocket of the electric machine with the radius $R_{sprocket,EM}$ is thus

$$M_{add.EM}=(F_{span,ten}-F_{span,EM,ICE})· R_{sprocket,EM}$$

(4)

The additional torque $M_{add,ICE}$ at the sprocket of the crankshaft with the radius $R_{sprocket,ICE}$ is accordingly

$$M_{add.ICE}=(F_{span,ten}-F_{span,EM,ICE})·R_{sprocket,ICE}$$

(5)

For $M_{add.ICE}>0$ and $M_{add.EM}>0$, the belt or chain system amplifies the force that acts at the gear contact. This leads directly to the conclusion that the span force at the tensioner must always be bigger than the span force between the sprocket of the electric machine and the crankshaft sprocket to support the gear contact with additional pretensioning:

$$F_{span,ten}>F_{span,EM,ICE}$$

(6)

4. Simulation Model

To investigate the functionality of the system, two simulation models were set up in a multi-body simulation environment: a base model, which represents the rolling moment compensation with a gear stage and without an additional system for rattle avoidance, and an extended model, which additionally contains the system for rattle avoidance described in the patent application.

The base model consists of submodels of a two-cylinder four-stroke in-line engine, a gear stage, and an electric machine. The internal combustion engine model consists of rigid bodies for pistons, piston pins, connecting rods, and the crankshaft. All components are reflected by their respective mass and inertia properties, and they are linked by joints offering a certain degree of freedom between the bodies. Engine excitations are considered through a crank angle-dependent application of cylinder pressures. Cylinder pressures are multiplied with the effective piston surface area, and the resulting gas forces act in the simulation model between the piston and housing structure. This type of modeling reflects the standard approach for multi-body simulations of crank trains and is used for accurate modeling of stresses and vibrations for combustion engines. Rotational fluctuations of the crankshaft are particularly important for this investigation, and their realistic representation using this modeling approach has already been demonstrated, for instance, in [23].

The excitations by the electric machine are modeled via the rotational mass inertia of the rotor and a rotation angle- and speed-dependent torque. A feedforward controller gives back an estimation for a torque level based on the current operating point. This torque estimation is supplemented by a torque value from a speed-dependent proportional controller, which compares the target speed to the actual speed of the electric machine. The P-controller is specifically tuned to minimize the torque fluctuations induced by the electric machine while accurately maintaining the target speed. These excitations result in torque fluctuations for both the combustion engine and the electric machine that are introduced into the system. Both excitations are designed to reflect a typical behavior of a gasoline engine and an electric machine. The excitation integration in the simulation model is shown in Figure 4.

A single-mass flywheel in the form of a rotational inertia and a flexplate, which is modeled as a flexible body, are considered at the crankshaft. The connection between the rotational axes of the crankshaft and electric machine is via a gear stage. The parameters of the gear drive can be found in

Table 2. Parameters of the gear drive.

Parameter	Value	Unit
Number of teeth (electric machine gear)	43	-
Number of teeth (crankshaft gear)	50	-
Gear contact stiffness	8.36 × 105	N/mm
Gear contact damping	7.67	Ns/mm
Oil film thickness	0.04	mm
Gear lash at working pitch diameter	0.08	mm

. The tooth contact is modeled with constant tooth stiffness, as effects resulting from fluctuating stiffness, such as gear whine, are not relevant for this investigation.

Gear contact is modelled using a three phases approach embedded in the multi-body simulation software using rigid gear body elements. In this approach the teeth contact is separated in the three contact phases: force free phase, oil compression phase and structure phase as shown in Figure 5.

The resulting gear force for each phase is separated into a stiffness force and a damping force. During the force-free phase, no teeth are in contact, and subsequently neither damping nor stiffness force is active. The teeth are in the area of the gear lash $L$ without interacting with the oil film thickness $f$. This force-free phase correlates to phases 2 and 5 in Figure 2. Considering a realistic impact behavior at the gear contact, there is the oil compression phase, during which the damping force is defined by a quadratic function, while the stiffness force is modeled with a cubic function. The first contacts at phases 3 and 6 in Figure 2 are dominated by the effects of the oil compression phase followed by the solid contact of the structure phase. The structure phase is the preferred driving state for teeth contacts. This phase is reflected by phases 1 and 4 in Figure 2. Considering all three phases from [24], the stiffness force $F_c$ can be written as

$$F_c=\left\{\begin{array}{c} 0, x<\left(L-f\right)\\ c·{\displaystyle \frac{4}{27}}·{\displaystyle \frac{{\left(x-L+f\right)}^3}{f^2}}, \left(L-f\right)\le x\le \left(L+{\displaystyle \frac{f}{2}}\right)\\ c·\left(x-L\right), x>\left(L+{\displaystyle \frac{f}{2}}\right)\end{array}\right.$$

(7)

The damping force $F_k$ is, according to [24], defined as

$$F_k=\left\{\begin{array}{c} 0, x<\left(L-f\right)\\ k·\dot{x}·{\displaystyle \frac{{\left(x-L+f\right)}^2}{f^2}}, \left(L-f\right)\le x\le L\\ k·\dot{x}, x>L\end{array}\right.$$

(8)

Stiffness and damping forces are calculated at the working pitch of the gear contact for all teeth contacts, and at the center of each gear, the appropriate resulting forces and moments are applied. This modeling approach is based on the linear Kelvin–Voigt contact model for the solid contact as described in [25]. The additional nonlinear transition phase for the oil film thickness improves the base equation by reducing the high impact velocities at the solid contact, aiming for better modeling of impacts [26].

Both simulation models represent complete rolling moment compensation. Therefore, the mass inertia on the combustion engine side (primary side) and the electric machine side (secondary side) of the gear stage must be matched as precisely as possible. The gear ratio of the gear stage must also be considered in relation to the mass inertias on the primary and secondary sides. In Figure 6, it can be seen that the sum of all mass inertias on the secondary side is 0.86 times smaller than the sum of the mass inertias on the primary side. This factor corresponds exactly to the gear ratio, thus ensuring a complete rolling moment compensation according to Equation (2).

For the rattle avoidance system, a chain drive is selected. Timing belt systems were considered but rejected due to limited power transmission capability. Furthermore, chain systems are less sensitive regarding environmental conditions like temperature. To investigate potential influences of a belt system, a variant with reduced stiffness is analyzed in Section 5.2. The geometry of the chain elements and the wheels can be easily modeled through standardization in ISO 606:2015 [27]. In contrast to a belt, the chain drive has significantly higher longitudinal stiffness and damping and can also endure higher loads. To evaluate a possible effect when using a belt drive on the system performance, an additional stiffness variation is carried out. The polygon effect, which arises from the use of chain wheels, plays only a subordinate role in this application, as all chain wheels significantly exceed the minimum number of teeth of 17 required in the literature [28].

The data for the chain drive can be found in

Table 3. Parameters of the chain drive.

Parameter	Value	Unit
Stiffness (translational)	90,000	Nmm/mm
Damping (translational)	0.9	Nmms/mm
Damping (rotational)	0.1	Nmms/deg
Profile	ISO 606: 05B	-
Pitch	8.0	mm
Number of teeth (electric machine sprocket)	43	-
Number of teeth (crankshaft sprocket)	50	-
Number of teeth (tensioner sprocket)	43	-

.

The chain and the sprockets are modeled using the linear chain model from [29]. In this approach, the chain consists of links and bushings. The links represent the mass and inertia properties of the single chain elements, whereas the bushings are used to model the stiffness between the links. The contact to sprockets is ensured by general forces, which take the geometry into account to calculate resulting forces. For each bushing, the stiffness matrix ${\mathit{C}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}$ and damping matrix ${\mathit{K}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}$ are solved in each time step of the simulation using the six-dimensional displacement vector ${\mathit{x}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}$ and the 6-dimensional velocity vector ${\dot{\mathit{x}}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}$ to calculate the resulting force and torque vector ${\mathit{f}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}$:

$${\mathit{f}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}=-{\mathit{C}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}·{\mathit{x}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}-{\mathit{K}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}·{\dot{\mathit{x}}}_{\mathit{b}\mathit{u}\mathit{s}\mathit{h}\mathit{i}\mathit{n}\mathit{g}}$$

(9)

In this model, only the values specified in Table 3 are individually specified, while the other values of the stiffness and damping matrices are determined automatically. For more information on additional values, please refer to [29]. The chain property values in Table 3 are derived from previous development projects at FEV Europe GmbH, showing good correlation to measurement data.

The extended simulation model is created by combining the original base model and the rattle avoidance system. Through the modular extension, direct comparisons between the system behavior with and without chain drive are possible. Both systems are schematically shown in Figure 7.

This investigation represents a concept study focused on verifying the working principle of the proposed gear rattling suppression system and establishing quantitative design guidelines for prototype development. The simulation model employs well-established multi-body dynamics components and proven modeling approaches to ensure reliable representation of the fundamental system behavior. The crank train modeling methodology has been validated for rotational fluctuation prediction in previous investigations [23], which is the critical excitation source for gear rattling in this application. Gear contact forces are modeled using the standard Kelvin–Voigt approach with oil film consideration [24], capturing all relevant effects for gear rattling analysis, including contact stiffness, damping, and impact dynamics. Chain system parameters are derived from FEV’s validated component database, ensuring realistic representation of stiffness and damping characteristics.

The modeling approach incorporates several simplifications appropriate for this concept study. Key limitations include simplified gear impact dynamics using linear contact models rather than detailed finite element approaches, neglect of friction effects in both gear and chain contacts, and restriction to steady-state operating conditions without transient effects, such as engine start–stop events. Additionally, thermal effects, manufacturing tolerances, and structural flexibility are not considered, and lubrication modeling employs simplified oil film assumptions. These limitations are deliberately accepted to maintain focus on the fundamental working principle and avoid model complexity that could obscure the essential parameter relationships.

Rather than pursuing high-fidelity modeling that could lead to overfitting in concept investigations, this study deliberately employs simplified approaches to identify the key system parameters and their effective operating ranges—specifically the critical span force difference criterion and its speed-dependent threshold values. This methodology aligns with the stated research objectives of deriving optimal system parameters and quantitative design guidelines for prototypical implementation. The simulation framework provides the essential foundation for focused experimental validation studies, where the identified parameter ranges can be verified under real operating conditions, the working principle can be demonstrated in hardware, and the effects of the modeling simplifications can be quantified.

The simulations are carried out as steady-state simulations at constant engine speed in a range from 1000 1/min to 4500 1/min. First, the base simulation model is investigated as a baseline for the evaluation of the effectiveness of the gear rattle avoidance system in the extended simulation model. This extended simulation model is firstly investigated using parameter variation of the tensioner pretension and the chain stiffness at a constant engine speed of 4000 1/min. Subsequently, the simulations are carried out for the same operating points between 1000 1/min and 4500 1/min as the base simulation model to enable a direct comparison. All results are absolute values unless otherwise stated.

5. Results

5.1. Analysis of Gear Rattling in the Base Model

As a foundation, the mechanism of gear rattling is first investigated using the base model. Based on the simulation results in Figure 8, it is clearly recognizable that a sign change of the tangential gear force occurs for most simulated engine speeds. According to the definition in the simulation, the tooth forces are negative when the gear pair is engaged in a way that the combustion engine delivers power and the electric machine absorbs power. Accordingly, the power flow is reversed with a positive sign. For the operation of a serial hybrid powertrain, only operation with negative tooth forces is necessary and desired.

When simultaneously considering the cylinder pressures, the influence of the resulting rotational non-uniformity on the gear forces can be directly derived. For this purpose, Figure 9 shows the absolute cylinder pressures and the absolute tangential force of the gear mesh in one diagram for a speed of 4000 1/min over crank angle.

During combustion in cylinder 1, which starts at TDC at 0 deg crank angle (degCA), the gear forces increase abruptly. This rapid increase in tooth forces results from the positive acceleration of the crankshaft, which is transmitted via the gear stage to the mass inertia of the electric machine rotating at almost constant speed. In the subsequent working strokes of cylinder 1 (exhaust) and cylinder 2 (compression), the crankshaft experiences a negative acceleration, as less work is expended for both working strokes. Due to the almost-constant speed of the rotor of the electric machine, this subsequently leads to a relief of the tooth contact. As a result of the different speeds of the crankshaft and rotor of the electric machine, the gear backlash (complete contact force loss) is first traversed, and then a contact on the corresponding opposite flanks is established.

5.2. Analysis in the Extended System

In the next step, the system behavior with the additional system for avoiding gear rattling is investigated. In particular, the tensioner forms the core element of the adapted system and should therefore be considered in variant investigations at the beginning. For this purpose, a high speed of 4000 1/min is considered, at which clear gear rattling is visible in the gear forces in the base investigation. First, the preload of the tensioner is considered and varied from 2 kN to 5 kN in four steps. As a base parameterization, a stiffness of 50 Nm/deg and a damping of 0.1 Nms/deg are set. Based on the results in Figure 10, it is clearly visible that a higher preload leads to better system behavior.

In particular, the area around 270 degCA becomes uncritical from 4 kN onwards. With a preload of 2 kN, the preload in the system is not yet sufficient to prevent a tooth flank change at 4000 1/min, and 3 kN also still shows an area without power transmission (phase 2 according to Figure 2), which must also be prevented due to the subsequent impulsive impact of the tooth flanks (phase 6).

Even though high preloads are advantageous for system dynamics, the resulting chain forces should be mentioned as a limiting factor, as these cannot be increased arbitrarily due to drawbacks in friction losses and chain load restrictions. This leads to the consideration of the chain forces, as shown in Figure 11.

The highest chain forces occur in the spans around the tensioner (span 2 and span 4). For a preload of 4 kN, the chain forces reach 2.4 kN and thus lie in a range between 2.0 kN and 2.5 kN, which is acceptable for the chain used here with a pitch of 8.0 mm. For comparability, an acceptable range for a chain with a pitch of 9.525 mm is also shown. This direct comparison, however, only serves for the estimation of possible limiting values. By using a chain with a different pitch, the system dynamics change, among other things, due to different chain stiffness, so the calculation with an 8.0 mm chain cannot be directly transferred to a 9.525 mm chain. The analysis of the chain forces shows that none of the spans is completely relieved, and the chain thus does not run the risk of being excited to strong oscillations.

Further variations with the tensioner under a preload of 4 kN regarding stiffness and damping lead to these parameters playing only a subordinate role in the simulation. Both the static movement of 0.5 deg and the dynamic movement, which is more than a factor of 10 smaller than the static movement, are negligibly small at a speed of 4000 1/min. The static movement arises due to the chain elongation at the beginning of the simulation under the influence of the preload. The dynamic movement, on the other hand, results from the rotational fluctuations of the crankshaft.

The second important element for the functionality of the system is the properties of the chain. Therefore, in a further investigation, the chain stiffness is varied over wide ranges to investigate the influence of stiffness on the overall system behavior. The nominal chain stiffness of 90,000 N/mm is reduced by a factor of 10 and increased by a factor of 10. This already provides an outlook on the use of a double-sided belt in the lower range, and chains with significantly higher stiffness, such as duplex chains, can be considered in the upper range. The results of the stiffness variation are shown based on the tangential force of the gear stage in Figure 12.

Both the chain with increased stiffness and the chain with reduced stiffness show disadvantages regarding gear rattling. In particular, with increased chain stiffness, gear rattling is clearly visible in all six phases around 270 degCA. The reasons for a poorer suppression of gear rattling are different with increased and reduced chain stiffness. A reduced chain stiffness leads to increased initial chain elongation with the same preload, which in turn proportionally reduces the tensioner force due to the resulting relaxation of the tensioner. Thus, the chain with reduced chain stiffness starts with a tensioner force reduced by 1.3 kN compared to the nominal variant. This disadvantage can be compensated again via a corresponding increase in the preload. Regarding the dynamic behavior, the chain with the reduced stiffness does not show any special features compared to the original chain. This result shows that a system with significantly lower stiffness can also successfully be utilized to prevent gear rattling. Therefore, a belt system might be a suitable solution as well, although the strength and durability of the belt would still need to be investigated, since the forces in the system exceed typical timing belt loads.

Increased chain stiffness also leads to gear rattling, with the corresponding mechanism becoming clear from the chain forces in Figure 13. The span most relevant for the functionality of the rattle avoidance lies between the chain sprocket of the electric machine and the chain sprocket of the crankshaft. As in the other areas of free spans (see also Figure 11, numbers 2 and 4), the chain force in this short span (see also Figure 11, number 6) forms a horizontal plateau of constant force. With the chain with increased stiffness, the force in the span between the electric machine and the crankshaft is at a very high level compared to the original chain. Meanwhile, the force level in the other free spans remains comparable for the two chain variants.

This shows, particularly for the chain with increased stiffness, that the working principle of the rattle avoidance depends on the difference between the chain force in the span between the electric machine and the crankshaft and the chain force in the free spans at the tensioner. For the chain with standard stiffness, this critical difference is 1570 N, while the difference for the chain with increased stiffness is only 630 N. The chain with low stiffness also has a smaller difference of 1240 N, although the maximum span force at the tensioner is already significantly reduced due to the initial preload force loss. The difference between the two spans must therefore be above a certain threshold for the system to successfully prevent gear rattling. The results in Figure 10 on the variation of the preload force also confirm a lower limit of the span force difference. With a preload force of 3 kN, the span force difference is 1080 N and thus below the value of the chain with standard stiffness at 4 kN. At the same time, the system already shows first gear rattling with a preload of 3 kN. An overview of the span force differences for different variants is shown in Figure 14. Here, the trend of the rattling tendency decreasing with increasing span force difference is particularly noticeable. For an engine speed of 4000 1/min, the limit of the span force difference to avoid gear rattling lies between 1240 N and 1570 N. At the same time, it also becomes clear that the necessary minimum span force difference depends on the speed. In Figure 14, the two engine speeds of 4000 1/min and 4500 1/min are directly compared, each with 4 kN preload force and the same chain stiffness. Although the variants with a speed of 4500 1/min have a span force difference of 1623 N and are thus above the variant at 4000 1/min, gear rattling still occurs in the simulation. For this reason, a minimum limit for the span force difference for successful avoidance of gear rattling can only be specified depending on the maximum engine speed.

In the final step, a fixed system layout is considered over the entire speed range of the engine. As boundary conditions, a preload force of 4000 N, a stiffness of the tensioner of 50 Nm/deg, and a damping of the tensioner of 0.1 Nms/deg are set. This results in significantly improved behavior of the overall system regarding gear rattling, as shown in Figure 15.

Gear rattling is reliably avoided up to speeds of 4000 1/min. Only from a very high speed for a range-extender application of 4500 1/min do tooth flank changes occur again in the system. It is noticeable that the amplitudes of the gear force at 4500 1/min increase significantly compared to the investigation with the base model, especially after the tooth flank change. This can be explained by the fact that the extended system is pretensioned by the chain drive when changing the tooth contact flank and thus has significantly more energy than the base variant when changing the tooth flank again. From this observation, it can be directly derived that the extended system must always be operated with sufficient distance to the speed limit for gear rattling in order not to be acoustically more conspicuous than the base system. For an engine speed of 1000 1/min, low-amplitude oscillations are visible in the gear tangential force that were absent in the base investigation (Figure 8). Those vibrations can also be seen in the chain force in the span between the crankshaft and the electric machine pulley (position 6 in Figure 11), and the frequency matches the 50th order in the system, correlating to the number of teeth of the crankshaft gear and the crankshaft sprocket. This all indicates that a system eigenfrequency is excited at this certain operating point because the pretensioned chain introduces an additional flexibility into the system. This needs to be considered for future system designs to avoid introducing additional oscillations into the system.

To verify the remaining effectiveness of the rolling moment compensation in the system while utilizing the chain system to suppress gear rattling, the transfer function between the crankshaft side and the electric machine side is investigated. To do this, the torque between the gear wheel and shaft is evaluated for the crankshaft side and the electric machine side, and the phase difference between the two torques is evaluated at the frequency of the first order of the combustion engine. The first order is the firing order of the two-cylinder inline engine and therefore the main excitation order in the system. The results can be derived from

Table 4. Phase shift between gear torques for base and advanced variants.

Engine Speed/(1/min)	Frequency of 1st Order/Hz	Base Variant Phase Shift/°	Extended Model Phase Shift/°
1000	16.7	−7.1	−1.3
2000	33.3	−2.4	+0.7
3000	50.0	−0.4	+1.3
4000	66.7	−0.9	+1.2
4500	75.0	−2.9	+0.7

.

The phase shift between the crankshaft gear and the electric machine gear attachment torque is low for all engine speeds for both the base system and the extended model with the gear rattling suppression mechanism. Only the lowest engine speed of 1000 1/min for the base variant shows a deviation. This can be explained by the high-speed fluctuations that occur at low engine speeds in combustion engines. These high fluctuations already lead to gear rattling at low speeds, as shown in Figure 8, and result in an increased phase shift. In the extended model, gear rattling is not only avoided for all engine speeds up to 4000 1/min, but the phase shift between the torques is also on a very low level, enabling the full potential of the rolling moment compensation.

6. Conclusions

This investigation successfully addresses a fundamental NVH challenge that limits the use of rolling moment compensation in range-extender systems needed for the automotive industry’s electrification transition. The introduction identified a critical problem, namely that range-extender vehicles face exceptionally high customer expectations for noise and vibration performance due to direct comparison with battery electric vehicles, yet the use of rolling moment compensation in range-extender powertrains suffers from acoustically disturbing gear rattling.

In this work, a system for avoiding gear rattling in a range-extender internal combustion powertrain with rolling moment compensation was investigated using dynamic multi-body simulations. The concept based on patent application DE102020007360A1 uses a chain drive parallel to the power-transmitting gear stage to suppress tooth flank changes.

The simulation studies conducted prove the effectiveness of the chosen concept, which can reliably prevent gear rattling up to a speed of 4000 1/min. The fundamental working principle of the system is based on the controlled difference between the chain forces in the short span between the gear stage and the free spans at the tensioner. This span force difference represents the decisive design parameter and must exceed a speed-dependent threshold value to effectively prevent gear rattling. At higher speeds above 4000 1/min, additional measures would be required, as the required threshold value of the span force difference continues to increase. At the same time, it is proven that the rolling moment compensation is not negatively affected by the chain system. The comparison of the phase shift between crankshaft torque and the torque at the electric machine shaft shows that even with the chain drive, there are no significant phase shifts in first-order torque, which could reduce the rolling moment compensation. Previous solutions were either limited in maximum torque and speed, like plastic gears or dynamic electric machines, or reduced the rolling moment compensation effect. The investigated concept not only resolves the tradeoff between gear rattling and rolling moment compensation but also enables speeds up to 4000 1/min and torque levels of up to 120 Nm.

As part of the parameter variation, it is demonstrated that a design with a medium chain stiffness of about 90,000 N/mm and a sufficient preload force of at least 4 kN represents the best compromise between effective rattle avoidance and mechanical load on the chain elements. Special attention should be paid to the fact that, when exceeding the critical speed, the system can lead to increased amplitudes during tooth flank change, which is why an adequate safety margin to the speed limit must be maintained in practical application. The research establishes a quantitative design framework through the span force difference criterion, specifically demonstrating that values exceeding 1570 N at 4000 1/min govern successful gear rattling suppression. This provides the first quantitative relationship between chain system parameters and gear rattling suppression effectiveness in range-extender applications.

The findings gained form a solid basis for the development of a prototype and the experimental validation of the system under real conditions. Further research is needed to investigate the long-term stability, as effects of chain elongation can lead to a reduction in the preload force. In addition, the system behavior under transient conditions, especially during engine start, should be analyzed in more depth. For further refinement of the simulation model and its dynamic behavior, the influence of structural elasticity can be considered. The results obtained from the multi-body simulations already allow for a targeted design of the system and thus significantly reduce the experimental effort in the prototype phase.

From a simulation methodology perspective, this work demonstrates effective multi-body modeling approaches for coupled gear-chain systems with discontinuous contact dynamics. The systematic parameter variation studies covering stiffness, damping, and pretension effects provide modeling techniques for similar complex mechanical systems.

In summary, it has been demonstrated that the investigated system can make an important contribution to improving the NVH behavior of range-extender vehicles. The successful suppression of gear rattling while maintaining the advantages of rolling moment compensation supports the acceptance of this bridging technology in the transformation process toward climate-neutral mobility.

The simulation-based foundation established in this work enables several promising research directions for automotive engineering. The most important research need is the development of prototype systems for empirical verification of simulation predictions, particularly for real-world performance validation under dynamic operating conditions. Advanced multi-objective optimization studies balancing NVH performance, efficiency, durability, and cost considerations would support commercial implementation optimization and enable broader market adoption of the technology. Detailed investigation of system performance during engine start events, load transitions, and other transient conditions remains essential for customer acceptance in real-world operation. These transient behaviors significantly influence consumer perception of vehicle refinement and require comprehensive characterization. This research demonstrates that innovative mechanical design can resolve seemingly fundamental engineering contradictions, supporting the automotive industry’s successful navigation of the electrification transition through improved bridging technologies.