## ELECTRIFIED MOTOR/GEARBOX SUB-ASSEMBLIES

**VibraTec presents a technical course in the form of 5 FREE webinars on electric motor + gearbox sub-assemblies (E-powertrain, electric actuators, gear drives, etc) **

**(given in French with English presentations)**

Their content will cover various industrial sectors / applications.

#### On the agenda:

**Webinar n°1****: Introduction and challenges (June 16 ^{th} 11h30-12h)**

This webinar presented:

**NVH issues associated with E-powertrains**and other electric motor + gearbox sub-assemblies,**Possible approaches in the design phase**(technical aspects and integration of NVH in the design process).- A state-of-the-art of
**existing tools and methods to control the NVH**performance of E-Powertrains and other electric motor + gearbox sub-assemblies.

Didn’t participate ? Webinar available on YouTube.

**Webinar n°2****: Electric machines (June 24 ^{th} 14h-15h)**

This webinar will detail the **physical phenomena that make an electric machine a source of noise and vibration**, from magnetic excitations to the vibration and noise generated by machines and their modal behavior. At each step of the process, important points will be highlighted and key indicators will be given.

**Webinar n°3****: Gearboxes and Gear Drives (June 25 ^{th} 14h-15h)**

This webinar will analyze the **physical phenomena that make a gear transmission a source of noise and vibration**, from the excitations to the vibration and noise generated by the gearbox to the dynamic phenomena of excitation amplification. Orders of magnitude will be indicated and the main technical nodes will be detailed for each step of the process.

**Webinar n°4****: Optimum multi-stress design (July 1 ^{st}, 14h-15h)**

This webinar will present the latest methods we use to very significantly reduce E-Powertrains excitations: sound power, forces transmitted to the receiving structure and torque pulsations. The **overall approach based on optimization algorithms** will be presented and then applied to gear systems and electric motors. A few industrial examples will illustrate the gains obtained with these methods.

** ****Webinar n°5****: Integration in the final application (July 2 ^{nd}, 14h-15h)**

The **successful acoustic integration of an « electric motor + gearbox » assembly in its final application**: our last webinar will address this topic, by recalling the principles of solid-state transfer and aerial transfer, with specificities related to E-Powertrains. The numerical tools and experimental methods used will also be presented.

## ELECTRIC MOTOR DESIGN OPTIMIZATION

Noise and vibrations generated by electric motors can be divided into three main contributions related to three distinct sources:

- Mechanical noise and vibration sources,
- Aerodynamic noise and vibration sources,
- Electromagnetic noise and vibration sources.

These electromagnetic excitations result in tonal contributions: the engine “whistles”. Even if the sound power radiated is lower than that of a combustion engine, the noise can be extremely annoying. Noise perception has to be taken into account.

#### Electromagnetic excitation

Maxwell pressure is taken as the main phenomenon responsible for stator vibration and stator acoustic radiation. Flux density and thus Maxwell pressure can be calculated using finite element electromagnetic solvers.

The Maxwell pressure is calculated for virtual sensors located inside the air gap. Since the rotor is moving, the electromagnetic solver is based on a time resolution so that the electromagnetic calculation results in a time evolution of the magnetic excitation at each virtual sensor. Thus, for each motor speed, the electromagnetic simulation provides two time-space excitation matrices (one related to the radial component, the second related to the tangential component). The influence of every relevant parameter is contained in this simulation: number of poles, number of stator slots, number of rotor slots, current shape, eccentricity and saturation of the magnetic core. These parameters affect the excitation content in the time domain as well as its spatial distribution.

The first step is to transform the time excitations into a sum of frequency excitations using the Fourier series. Then, the Maxwell pressure (expressed in N/m²) has to be projected onto the structural mesh and transformed into forces (in N). This mapping algorithm can be applied to any type of structural mesh.

Our methodology to simulate electric motor noise and vibration is based on the determination of the space-frequency content of the excitation applied on the stator. Thus, the Maxwell pressure is decomposed into elementary rotating forces characterized by their frequency f and their spatial order m. The spatial order corresponds to the spatial distribution waveform of the Maxwell pressure in the air gap. Space-frequency maps synthesize the harmonic content of the Maxwell pressure.

**Calculation of the Maxwell pressure inside the air gap**

#### Structure of the stator and modes

The structure of the stator of an electric motor can be approximated by a cylinder whose deformation modes are described using two integers (m, n) giving:

- m: the number of nodal diameters. It is called circumferential spatial order.
- n: the number of nodal circles in the length.

Each deformation mode is also characterized by a natural frequency f. The stator’s modal basis can be determined using either an experimental modal analysis or a finite element model.

To predict the resonances and the potential vibroacoustic issues related to electric motors, it is necessary to both analyze the content of the electromagnetic excitation applied to the stator and know its modal basis.

If an excitation contribution coincides frequentially as well as spatially with a stator mode, a resonance occurs, leading to very high vibration levels.

Stator breathing mode (0,0) ~5000 Hz |
Stator deformation mode (2,0) ~1000 Hz |

#### Dynamic response simulation

The basic principle of the calculation is to perform a weakly coupled electromagnetic-dynamic calculation. The electric motor is modelled using a finite element electromagnetic software program in order to calculate the electromagnetic excitations applied to the stator.

This excitation data is projected onto the structural mesh of the e-machine with the aid of a dedicated mapping tool; a dynamic calculation can be performed using a finite element method. As this kind of procedure is included in an acoustic scope, the output value of interest is the stator’s vibration velocity. The last step is the calculation of the vibrating structure radiation. An acoustic finite element method (FEM) is used (a boundary element method (BEM) can also be used). Vibration velocity is taken into account as a boundary condition. The output data is the sound power radiated by the machine.

**Basic principle of the calculation procedure**

#### Run-up calculations

With the projection of the electromagnetic forces described in the previous section, excitation data is available in the frequency domain for each motor speed. It is therefore possible to achieve a dynamic calculation for each motor speed, leading to the motor’s vibration response using a structural finite element solver.

By identifying the frequencies where the acceleration is maximal, it becomes possible to identify critical speeds and frequencies. In the majority of the Campbell diagram, the stator’s dynamic response is a forced response: there is no resonance and this spatial order of the deflection is due to the spatial order of the excitation. But in some cases, there are both a spatial and a frequency coincidence between the electromagnetic excitation and a stator mode. This leads to high vibration levels and possibly to a high sound power radiation, depending on the radiation efficiency related to the deflection shape.

**E-motor acoustic radiation Campbell diagram**

#### Optimization

VibraTec has developed a new method to optimize the electromagnetic design of electric motors in order to fulfil NVH specifications. This method is equally applicable to BEV, HEV or PHEV traction motors, as well as any other motor. It relies on an NVH performance calculation process for an electric motor submitted to electromagnetic excitations. This calculation process is integrated into an optimization loop minimizing noise and vibration levels by reducing electromagnetic excitations. Moreover, criteria such as the mechanical power delivered by the motor, torque ripple, efficiency or temperature are constrained by the optimization algorithm so that a multi-constraint optimization can be performed.

Indeed, electrical engineers and NVH engineers work to optimize the same raw quantity, i.e. the flux density in the air gap, to comply with their respective specifications. The innovative approach is to treat both topics at the same time in the project and with the same tools to achieve the best compromise.

Electric motors give the opportunity to be inherently quiet,as long as the origins of the dynamic excitation are known and integrated in the design process without compromises against global performance. Once the NVH constraints are integrated at the same level as the other constraints, the design process becomes safer and more efficient.

A major interest of these optimization methods is that they can be directly applied during the e-powertrain design process.

**Basic principle of the calculation of optimization objectives and constraints from a common electromagnetic simulation**

The optimization of electric machines in order to minimize their noise or vibration levels without deteriorating the aforementioned electro mechanical performance criteria requires the modelling of the related physical phenomena. As explained before, the estimation of a motor’s dynamic and acoustic behavior requires the determination of the flux density time evolution for different rotor positions, just as for the estimation of common electro mechanical performance criteria such as torque or efficiency.

In order to determine the Maxwell pressures which apply to the stator and represent the main phenomenon responsible for noise and vibrations, the radial and tangential flux densities are calculated along the air gap, using 2D electromagnetic finite element calculations.

The instantaneous torque calculation is straightforward after the electromagnetic finite element computation. It can be performed by integrating Maxwell tangential pressures around the surface of the air gap. The instantaneous torque can also be defined directly from the electromagnetic results using the virtual works method.

A loss model can also be implemented in order to include the motor’s efficiency in the optimization objectives or constraints, as well as thermal criteria via a further thermal model. The losses can be divided into copper losses due to the current flow in the windings, and iron losses composed of hysteretic losses and eddy current losses.

The iron losses can be computed from the electromagnetic simulation results using a loss model such as the Bertotti model.

Electric motor design optimization currently aims at minimizing a cost function corresponding to the machine’s acoustic power level, while respecting constraint functions defined using the previously mentioned criteria, so as to not deteriorate the motor’s overall performance.

The acoustic power level is minimized by reducing the Maxwell pressures’ harmonic contributions, which are responsible for high vibration and noise levels. These contributions’ amplitudes are very sensitive to electromagnetic design parameters such as the shape of the rotor poles or stator teeth. Consequently, this optimization technique’s great advantage is that the geometrical changes applied to the design are small from a mechanical point of view, and they do not involve any increase of the motor’s mass or any significant additional cost.

Moreover, although current use of the optimization methods aims at a noise decrease without deteriorating the overall performance (efficiency, torque, thermal, etc), they can be equally used to maximize the overall performance without increasing the noise level, or even to find a trade-off both improving the overall performance and decreasing the noise level.

#### Optimization example

The motor to be optimized was a 10-pole interior permanent magnet synchronous motor (IPMSM) dedicated to automotive traction. The objective of the optimization was to minimize the motor’s acoustic power level without decreasing its mean torque and without increasing its torque ripple.

**Initial IPMSM geometry**

Before running the optimization, the motor’s dynamic behavior was diagnosed. This was done using a previously validated workflow based on electromagnetic and structural finite element models.

This diagnosis lead to an optimization aiming at reducing the noise level at the speed where the noise and vibrations prevailed, i.e. at 4700 rpm. For this purpose, the cost function was defined as the acoustic power level when the engine speed is 4700 rpm.

Some inequality constraints were also defined, so that the mean torque produced by the motor was not reduced and torque ripple was not increased in comparison with the initial design.

The optimization results show a very significant 14 dB decrease of the Sound Power Level. This reduction was achieved by small changes on the shape of the rotor poles. They do not affect the motor’sprice or weight.

**Comparison of initial and optimized IPMSM design SWL**

## GEARBOX DESIGN OPTIMIZATION

#### Introduction

A gearbox can be one of the main sources of E-Powertrain noise and vibration. There are various internal gearbox excitation sources, which depend on the state of the gearbox. The excitation source is a parametric excitation that is transmitted to the gearbox housing via the crankshafts and bearings. The gearbox housing will then transmit the vibration and noise directly via the surrounding panels’ noise transparency, and indirectly via structure-borne vibration propagation.

**Generation and transmission of gearbox whining noise**

1: parametric excitation between teeth,

2: propagation in the gearbox, 3: housing vibration

The main source of excitation in gearboxes is generated by the meshing process. The excitation is divided in two phenomena: transmission error and mesh stiffness fluctuations. The transmission error is mainly due to voluntary (tooth modifications) and involuntary (manufacturing errors) geometrical deviations of the teeth, at a micrometric scale. The flexibility of the teeth, the pinion and the shafts result in additional transmission error fluctuations.

#### Transmission Error Computation

For geared system, the STE under load is one of the main noise sources. It corresponds to the difference between the actual position of the driven gear and its theoretical position for a very slow rotation velocity and for a given applied torque. Its characteristics depend on the instantaneous situations of the meshing tooth pairs. STE results from teeth deflections, teeth surface modifications and manufacturing errors. The calculation of STE is relatively classical. For each position of the driving gear, a kinematical analysis of the mesh allows the determination of the theoretical contact line on the gearing teeth mating surfaces within the plane of action.

One important remark on the STE computation concerns the global system, in which the gears are inserted. Indeed, when the gears are wide, thetransmitted torque is high, the bearing soft and the gears not centered on their shaft, the global static deflection can become a first order parameter.

**Static deflection and its potential impact on the STE**

#### Dynamic Response Calculation

This computation scheme requires a finite element model of the complete gearbox in order to obtain its modal basis. The contact between the gears is modeled with a stiffness matrix linking the degrees of freedom of each pair of meshing gears. The scheme then uses a powerful resolution algorithm in the frequency domain to solve the dynamic equations with an iterative procedure (ISIS).

The final output is thus the dynamic transmission error (DTE), the teeth dynamic loads and the housing vibration as a function of the frequency. The operating speeds corresponding to resonance peaks and the amplitude of the housing vibration characterize the whining noise’s severity. The process can be repeated for several applied torques.

Our computation scheme has been validated step by step by comparisons with extensive and complex measurements on different products such as automotive gearboxes. Several quantities have been measured and compared with the simulation: static transmission error fluctuation, dynamic transmission error, housing vibration and whining noise.

**Test vs. Measurements: STE and housing vibrations**

#### Gear Optimization

The first part of the computation scheme, i.e. STE computation, can be used to optimize the teeth geometry in order to minimize excitation. An optimization problem requires a correctly defined fitness function and an appropriate algorithm to be solved.

The criterion retained to estimate STE fluctuations is the peak-to-peak amplitude (STEpp). Considering that the modifications made have to reduce the STEpp for a given torque range, the fitness function is defined as the integral of STEpp over this torque range.

The algorithm retained is based on a Particle Swarm Optimization (PSO). This method is based on a population’s stigmergic behavior; being in constant communication and exchanging information about their location in a given space to determine the best location according to what is being searched. In this case, some informant particles are considered, which are located in an initial and random position in a hyper-space built according to the different optimization parameters. The best location is thus the combination of parameters which ensures the minimum value of the fitness function defined earlier.

Let’s say that a solution S0 is determined by the PSO. The robustness study is done using a Monte-Carlo simulation, i.e. 10000 other solutions are computed, chosen randomly in a hyperspace centered on the optimized solution parameters’ values, limited by the tolerance intervals of each parameter and considering possible lead and involute alignment deviations. These 10000 results allow the establishment of the density probability function of each selected optimized solution. They also make it possible to compute statistical variables such as mean value and standard deviation.

**Optimization results and robustness study**

VibraTec has developed and validated these tools to design and optimize parallel axis helical gears, parallel axis spur gears, plastic gears and also planetary gears.

#### Planetary gears

Planetary gear sets provide high gear ratios in a compact package. They are widely used in automatic gearbox transmissions of hybrid vehicles and in E-axles. Compared to cylindrical gears with fixed and parallel axes, predicting and controlling the whining noise emitted from a planetary gear set remains a difficult problem because of the coupling between the multiple gear meshes and the mobility of the planet axes.

In the case of planetary gears, contact equations are solved taking account of all the meshings simultaneously. In a first step, one planet gear is taken as a reference and the contact points for the other gears are deduced for each successive angular position of the reference gear. The knowledge of contact line localization between the sun and planets, geometrical construction makes it possible to deduce where the contacts between ring and planets occur.

**Planetary gear numerical model**

Comparisons between numerical simulations and measurements show the ability of the proposed numerical model to compute the overall STE and to accurately predict the housing’s vibratory state, responsible for the whining noise. The dynamic response order tracking shows a multi-harmonic behavior, mainly driven by internal parametric excitations.

**Simulation vs. test – Planetary gear**