Comparative analysis of stiffness measurement methods for lamination stacks in electric machines

With the increasingly strict global regulations on vehicle emissions, the demand for electromobility has risen due to its potential to reduce local emissions. However, designing an electric motor for use in passenger cars poses significant challenges. The e-motor should operate reliably in various climates with good mechanical and electrical characteristics, be lightweight, withstand repetitive thermal and structural loads, be cost-effective, and - last but not least - exhibit good noise and vibration properties. To meet these requirements, virtual testing through simulations has become the most efficient and economical approach, enabling the identification of weaknesses and unwanted behaviors before physical prototyping. This paper focuses on the testing of lamination stacks, a critical component of electric motors. Two methods of stiffness measurement for such parts are compared: static stiffness determination and dynamic analysis. The former involve compressing the specimen and measuring the force-displacement response, while dynamic method uses the restoring force surface method to obtain the stiffness and damping characteristics. The study highlights the importance of considering nonlinearity in stiffness measurements. The stiffness of lamination stacks varies depending on the pretension state, and a significant hysteresis exists between the loading and unloading curves. The paper discusses the experimental procedures for each method. The findings emphasize the necessity of accurate stiffness characterization for different applications, such as structural strength, modal analysis, dynamic analysis, and noise-vibration-harshness (NVH) studies. The research contributes to the development of electric machines by providing insights into effective stiffness measurement techniques for lamination stacks.


Introduction
The automotive industry is experiencing a major transformation nowadays: instead of developing larger and more powerful internal combustion engines, electric drive systems are seen as the main products of the future.Although electric motors have been used for centuries in every possible application, the usage in a car that meets the up to date expectations of customers poses a great challenge.
Various operating conditions, mass-, power-, efficiency-and acoustic requirements induce high costs in the design process of such a motor, which increases the overall price of the end product.In order to fulfill these requirements and reduce costs, a number of virtual experiments and optimizations are performed before building prototypes.Virtual prototyping requires sophisticated modeling techniques and high quality material models.
The lamination stack is a key component of every electric motor, used in the rotor and the stator.In order to achieve high efficiency, the formation of Eddy-currents parallel to the axle has to be avoided.This is achieved by stacking varnish coated steel plates, which are pressed on one another.The result -from the structural mechanics point of view -is an orthotropic stiffness behavior.[1] For purely radial loads, like rotation around the axle, (Figure 1 a) the behavior is similar to a steel block.For axial compression and shear loads (Figure 1 b and c) the resulting stiffness values are highly nonlinear.[2] The causes for this nonlinearity are the cut surfaces of the sheet edges, the nonlinear stiffness of the varnish, the surface roughness, and the imperfect planar shape of the individual sheets.[3].As the sheets are pressed together, the contact area between them is steadily increasing, resulting in an increase in the overall contact force.The focus in this research is on axial compression (Figure 1   In order to obtain material stiffness data, the required parameters can either be measured directly by applying force and measuring the displacement, or they can be derived from the results of a modal analysis in a prestressed state.The static approach is able to characterize the whole nonlinear stiffness spectrum of the lamination stack, including hysteretic effects and cyclic loading [3], but excludes strain rate dependent effects.The dynamic approach yields equivalent stiffness values at a single pretension state, required to reproduce the modal behavior of the specimen. For NVH Finite Element Analyses (FEA) linear material models can be used by linearizing the material model at a predefined pretension state.For special cases of static strength analyses where 3D pretension effects and plasticity have to be taken into account, full nonlinear stiffness material models are required.
Various studies are available about the usage of dynamically obtained stiffness values, and a few examining static loading.[4], [5], [6], [7] No publications about performing the study on the same specimen after cyclic static loading has been found by the authors.In this research, static and dynamic experiments are performed on the same specimen -a real rotor lamination stack used in an automotive electric drive, to avoid altering the nonlinear behavior of the assembly.Results are compared, in order to evaluate the usage of static stiffness material properties in dynamic simulations, or dynamic stiffness for static applications.

The quasi static experiment
For measuring static stiffness properties there were basically two types of procedures found by the authors.Examples for the first type of procedure can be found in [4] and [5].It consists of two rigid steel plates, connected through bolts, with the specimen between them.As the bolts are tightened, their strain and the displacement of the steel plates is measured relative to each other.Based on these data a stress-strain curve can be calculated.The method is relatively slow.The bolts have to be tightened gradually and simultaneously, and excessive loading has to be avoided, in order to avoid asymmetric loading [8], overloading of the bolts and / or the bending of the plates.Due to these reasons it is not optimal for performing cyclic load experiments.
Figure 2: Bolted pretension method to measure static stiffness of a lamination stack An advantage of this procedure is, however, that it does not require a hydraulic press or expensive machinery and the setup is relatively simple.Its layout makes it possible to perform dynamic tests in a prestressed state: this type of testing was used in the dynamic experiment described in the next chapter.
The second type of procedure, which was used to perform the quasi static measurements in this research, involves direct compression and displacement measurement of the specimen, and is described in detail in [2].
Figure 3 illustrates the experiment setup.The specimen (diameter: 140 mm, uncompressed height: 19 mm), a part of a lamination stack of an automotive rotor, was put between a steel plate and a steel base support.The steel plate was loaded using a hydraulic cylinder at a rate of 1 kN/s, slow enough to avoid/minimize viscoelastic effects.A spherical joint was used between the hydraulic cylinder and the support frame, so that the angle of the contact surface could automatically be adjusted.This feature was important in order to provide even pressure on the whole surface of the specimen.The deformation of the specimen was measured by monitoring the change of the distance of the plates at 4 locations, using displacement sensors.The displacement data was additionally used to secure an even and parallel deformation of the specimen.The data of the 4 sensors was averaged during postprocessing.The specimen was loaded and unloaded 5 times using an F test reference load, resulting in a loading-unloading curve visible in Figure 4.During the load cycles a hysteresis was observed.While loading, the force required to induce a given deformation is higher than during the unload phase.Additionally it is to be observed, that when subjecting the specimen to multiple loadingunloading cycles, loading and unloading curves approximate the unloading curve of the first cycle.This phenomenon, and the possibility of using a single approximating curve has been studied in [3].In the current paper, however, a specimen with a different geometry has been used, with similar results.This result provides additional support for the theory of using the unload curve of the first cycle as an approximation, in order to represent the stiffness of the whole structure after multiple loading-unloading cycles.In a modal dynamic FEM analysis, a single E-modulus is needed to describe the stiffness.In case of a lamination stack, however, as described in the introduction, the behaviour of the stack is steel-like in the radial direction, and corresponds to the curve in Figure 4 in the axial direction.To this curve applies, that σ = f (ϵ).In order to compare the results to the dynamic ones, this curve needed to be linearized: The linearization was done at a single pretension state (Load Case -LC in this experiment).This load corresponds to the pretension state of the stack where the dynamic analysis described in the next chapter is performed.As there are different curves for loading and unloading, the E-moduli for loading and unloading will be different.Figure 5 illustrates graphically, how this difference is to be interpreted.E l and E u are compared to dynamic results in chapter "Results and conclusion".

The dynamic experiment
The dynamic experimental setup consisted of two robust 30mm thick steel push plates positioned on either side of the rotor segment, and was described in [9].The plates had positioning pins to significantly reduce asymmetric axial loads set by 4 bolted joints (Figure 2, Figure 6) in the 4 corners of the push plates.This setup drew inspiration from a prior study on the stiffness and damping characteristics of an electric rotor's lamella package [10].However, in the current study, the focus was on comparing dynamic properties to static ones.To conduct the study, the rotor specimen was positioned between the two push plates, and impedance and acceleration sensors were used to measure both the input force and corresponding accelerations.This particular measurement technique is known as a dilatation test.For data acquisition, the Simcenter SCADAS Mobile Frontend and Simcenter Testlab software from Siemens were employed.The analysis of the results involved identifying material properties, and this relied on the geometry and masses of the test setup.These parameters define the dilatational resonance of the push plates where the nodal line is in the center of the lamella segment.Initially, an EMA (Experimental Modal Analysis) measurement was conducted with one excitation and six response points, specifically examining the axial compression setup.The data was processed using MIMO (Multiple-Input, Multiple-Output) FRFs (Frequency Response Functions), which are standard measures of linearity.Various checks were performed on the reciprocity and magnitudes of FRFs at different force excitation.However, some noise was observed in the energy transfer, and reciprocity was not satisfactory.For further investigation, a stepped sine excitation method was employed to gain better control over the input force and energy.Different excitation forces were tested, including higher force amplitudes.To ensure a detailed response of the laminated structure, a sampling rate of 102.4 kHz was set during the data acquisition process.Accelerometers that were installed on each of the push plates, which were considered to be infinitely rigid compared to the lamination stacks, simplifies the calculation of relative acceleration of the lamella package (Figure 7).
Calculation of the relative displacement by the means of a double integration of the acceleration data often contains error due to the static drift and low frequency component.This was eliminated by a high pass filter and zero phase filtering technique.The filtering parameters were selected by care, in order to minimize any possible phase distortion.The analysis needed to be done where the first dilatational resonance frequency lies.This frequency determines the corresponding time segment.The dynamic contact force of the lamination was derived from the weight of push plates multiplied by the measured acceleration.This calculation leads to the estimation of the restoring force, that allows the visualization of the dynamic hysteresis.The hysteresis curve is an ellipse, where the angle of the main axis represents the stiffness and the area of the ellipse proportional with the cyclic energy dissipation.The static stiffness results measured while loading and unloading the specimen, and the stiffness derived from the dynamic experiment are compared in Figure 8.The specimen examined was the same automotive rotor lamination stack, and all the results were extracted at the LC pretension state after performing 5 cycles of loading and unloading.There is a significant difference between the loading and unloading stiffness (visually shown in Figure 1 on the right hand side).The difference is even larger between the static and the dynamic results.The static experiment excludes effects which depend on the deformation rate, which can be an explanation to this difference.The resulting dynamic stiffness is 3-4 times larger, than the statically measured one.The conclusion is, that using static stiffness data for dynamic analyses, or using dynamic stiffness material data for static analyses can lead to significant deterioration in the quality of the results in cases, where the specific longitudinal (Figure 1 b) stiffness plays a significant role.Intermixing static and dynamic stiffnesses is therefore not recommended.

Figure 3 :
Figure 3: Schematic view of the quasi static experiment

Figure 4 :
Figure 4: The resulting stiffness curves of 5 loading cycles, and the location of the load case LC, used as pretension for the dynamic experiment

Figure 5 :
Figure 5: Schematic visualisation of the E-Moduli for loading (E l ) and unloading (E u ) at LC

Figure 8 :
Figure 8: Comparison of the stiffness values at LC