Changing the suspension kinematics in the early development phase requires FE models to be rebuilt which includes generating new component FE models. Our method aims at simplifying and automating this process allowing two main goals. On the one hand, we want to predict NVH results from kinematics changes. On the other hand, we want to use the method to optimize suspension kinematics regarding NVH performance.
Our method uses the first available FE model of the axle under investigation and modifies it automatically in order to investigate kinematics influences on interior road noise. To fit existing models to new kinematics, we use two different approaches depending on the complexity of the changed parts. Simple parts like suspension links for instance are modeled in
Section 2.2.1 using simplified geometries such as FE beam elements. Complex parts like the knuckle adapt to new kinematics by being morphed in
Section 2.3.1. This method allows us to change the simulation model parametrically without knowing changes beforehand, as Helm et al. suggest [
23] (p. 27). Additionally, we are not dependent on manual, time consuming, and fault-prone modeling of suspension kinematics as described by Niersmann [
19] (p. 2).
2.1. Simulation Model
In this paper, we investigate the rear axle of a front driven car. The complete transfer chain from excitation to the passenger’s ears is visualized in
Figure 1. The excitation on the tire patch creates structure-borne vibrations that are transmitted through the tire and the wheel via the wheel hub into the axle. From there, the suspension’s links transmit them to the subframe attachment points as well as the damper and spring attachment points. The car body transmits them into the passenger cabin where the structure-borne noise becomes airborne noise.
We investigate only the axle subsystem to identify potential modifications to reduce noise transfer through the axle. The FE simulation model consists of the complete axle from both wheel hubs to all the attachment points between axle and car body. We will use the coordinate system given in
Figure 2. The axle in this figure differs from the axle under investigation. As the actual axle is still under development, no images of it can be shared. The coordinates’ origin lies in the center between both wheel hubs. With the driving direction indicated by the large arrow, the
x-axis points backwards. The
y-axis points to the right when looking in the driving direction and
z to the top.
We calculate the Frequency Response Functions (FRF) that characterize the transfer behavior of the axle. We investigate the axle subsystem from the wheel hub to the car body connection points. The attachment points are fixed to the inertial system which is known as blocked force investigation [
25] (p. 225). The simulation consists of 10 separate load cases. For each side, left and right, we excite the wheel hub with three translational force spectra and two torque spectra. In
Figure 2, the yellow arrows mark force and torque excitations. The forces attack in
x,
y and
z directions. The excitation amplitudes are 1 N for the complete spectrum from 1 Hz to 200 Hz. Torque is applied around the
x- and
z-axis at 1 Nm for the same spectrum. The torque in the
y-direction would simply turn the wheel hub. As we do not consider friction in the wheel hub, there would be no excitation to the axle. Using these excitations, we calculate the transfer functions for the axle subsystem that can characterize the transmission of road disturbances that occur not only in the vertical direction [
26] (p. 395).
For each of these 10 load cases, we get resulting forces at all the attachment points. In
Figure 2, the blue circles highlight the eight attachment points: the spring and damper as well as the front and back of the subframe for both the left and the right part of the axle. For each of these eight points, we receive three force spectra—respectively in the
x-,
y- and
z-directions—per load case. In
Figure 2, the small red coordinate systems indicate these force spectra.
All mentioned calculations add up to
result spectra per simulation. In order to reduce complexity, we add up all the individual load case result spectra. For all eight attachment points
i, we name the number of the load case
and the result spectrum of this load case for example for the
x-direction
. We calculate the resulting spectrum for one attachment point and one resulting force direction by energetically adding up all
j spectra
leaving 24 root mean square (RMS) spectra. This is done following van der Linden et al. [
27] (p. 3) and Sell who recommends adding energetically for high frequencies [
28] (p. 2). As we are analyzing up to 200 Hz and wanting to get transfer functions without preliminary phase assumptions, we decided to apply this method. It is possible to further reduce the number of spectra, as the left part of the axle is symmetrical to the right part. This omits 12 spectra leaving 12 force spectra for each simulation that represents energetically the added transfer functions from the wheel hub to each attachment point direction.
Using these transmitted forces, we are able to rate the axle’s transmission behavior directly by interpreting transferred forces into the car body. In order to optimize the sound pressure levels at the passenger seats, we can include the simulated axle subsystem back into the complete transfer path from
Figure 1 to calculate interior road noise [
2,
27].
2.2. Modeling Parts for Changed Kinematics
Changing the kinematics implies modifying components of the suspension. Shifting the hard point between the knuckle and track rod requires the modification of these two parts and the relocation of the bushing. All the other parts stay unchanged.
The modification of FE parts is split into two steps. First, simple parts such as the track rod are automatically rebuilt from scratch fitting the new kinematics, as described in
Section 2.2.1. Secondly, complex parts like the knuckle are automatically morphed as explained in
Section 2.3.1 in a way they fit the new kinematics without corrupting unchanged connections. Morphing includes the disadvantage of changing the part’s stiffness, but, on the other hand, it is what happens when the geometry of a part is changed in reality. This problem is more thoroughly discussed by Schlecht [
10] (p. 89). Elastomer bushings automatically rotate into the new kinematics. According to Fang and Tan and Schlecht, their properties remain the same, as stiffness and damping cannot be specified definitely in the early development phase [
10,
22].
2.2.1. Modifying the Track Rod
The track rod is a simple part of the suspension. It is a long, thin rod that connects two hinges mainly by tension or compression [
5] (p. 550). The original part in
Figure 3a is completely modeled using FE shell elements for the rod part, the bushing ring, and the bushing clamp. On the left, we display the back view—on the right, the top view. In the middle, the cross section is displayed. The track rod is hollow because it is made from a bent sheet metal. The FE model was created by meshing a Computer Aided Design (CAD) part in a preprocessor.
For the investigation of numerous kinematics variants, this process is too time-consuming. Therefore, we simplified the structure and automated its creation. The first simplification in
Figure 3b, which we will call
version 1, is to replace the shell meshed rod part in the middle of the track rod by FE beam elements. These elements describe a beam analytically [
21] (p. 98). All of the input parameters are cross section locations and cross section dimensions. The bevel of the cross section is omitted. The amount of beam elements is determined by the bending mode needed to model the axle behavior. Additionally, we need individual beam sections to model changing cross sections. This requires more elements than the bending modes. The ring part and the clamp part are replaced by parametrically defined FE meshes. This way, we are able to create a complete track rod with some sampling points, dimensions for the cross section and dimensions for ring and clamp. This allows us to easily modify geometric parameters such as coordinates or cross sections.
The next simplification in
Figure 3c,
version 2 is to connect the ring and clamp by omitting all the sampling points. Errors introduced into the simulation by these simplifications will be investigated in
Section 2.3. Our Python generator enables us to automatically create an array of simplified track rod variations to systematically identify significant geometric parameters which can be used for NVH optimization.
2.3. Validation of a Simplified Model
The effects resulting from the changes made to the track rod in
Section 2.2.1 must be well known. Using that knowledge, we can assess whether changes in the transfer function result from model simplifications or kinematics changes. We investigate the simplified track rod on three levels. First, we check changes in the component properties itself. Then, we look at the eigenmodes of the complete axle using the original track rod as well as the simplified versions. Finally, we investigate changes in the FRFs resulting from simplifying the track rod.
The validation on a component level compares the original model (
Figure 3a) with the simplified
version 1 of the track rod (
Figure 3b).
Version 2 is not only a simplification but also a geometry change and therefore another component, which is why we investigate
version 2 only in the complete axle and not on the component level.
The relative error for the mass of
version 1 with respect to the original mass
is below 1%. The absolute error of the center of gravity differs by less than 0.5 mm in each direction
x,
y, and
z.
The averaged relative error of the inertia in principal axes lies within 2%. The inertia of the rotation around the connection axis between both connection points is much smaller than the other two inertias. The relative error for the small inertia is greater than the other relative errors.
The next criterion to validate is the eigenmodes of the complete axle. We compare them using Modal Assurance Criterion (MAC) values. Using the eigenvectors
of model 1 for mode number
m and
of model 2 for mode number
n, the MAC value
calculates the orthogonality between both eigenvectors. Its values range from 0 to 1. Matching modes have higher MAC values. For example, 0.8 could be a threshold value [
29] (p. 132).
Figure 4 shows the MAC plots for the complete axle using the original track rod and the simplified track rod
version 2.
Figure 4a shows the comparison,
Figure 4b,c the auto MAC values for the original and
version 2 model, respectively. For the auto MAC value, the same model is used twice in Equation (
4). It is commonly used to identify similar looking modes. In the MAC plots, mode numbers start from 3 since modes 1 and 2 are the rigid body modes of the two break disks rotating around the wheel hub at 0 Hz.
Overall,
Figure 4a shows no noticeable difference between the original and
version 2 of the track rod up to mode 47 at 160 Hz. For modes 6 to 8, the auto MAC from
Figure 4b,c indicates similar looking modes at different frequencies. For modes 48 and 49, there is a mode flip. This indicates a frequency shift of at least one of these two modes. Above mode 52 at 180
, MAC values decrease with mode 53 disappearing.
The MAC plots from
Figure 4 don’t give any information regarding frequency shifts. We can only identify mode shifts if the frequencies of single modes are shifted so that the order of the modes changes. Therefore, we plot all the high MAC values next to the diagonal line in
Figure 4a into a new
Figure 5. We see the modes of the original model on the top line, the ones of
version 1 on the middle line and
version 2 on the baseline. Arrows indicate where mode frequencies lie for each model version using the MAC value colors from
Figure 4. Straight arrows from top to bottom indicate no change in mode frequencies.
It seems clear that there is no change in mode frequencies up to 130
. For modes 38 and 39, the two track rods are bending in phase and in opposite phase respectively. The simplified
version 1 shifts these two modes down by
and
version 2 up again with mode 39 ending in the same frequency as it was originally and mode 38 shifted down by 1
. For mode 48, there is a shift by 4
with no other shifts up to 180
. Above that, we see mode 53 disappearing in
version 2 as already seen in
Figure 4a.
After checking the component parameters and axle eigenmodes, we now investigate influences on the FRFs. We use the simulation model from
Section 2.1 for the original model, simplified model
version 1 and
version 2.
Figure 6 shows FRF for all three models for the left attachment points. We see all three directions for the attachment points spring, damper, subframe front, and subframe rear. The color code in the background indicates relative differences between the original model and the simplified models. It indicates the maximum relative deviation from the original model, to either simplified model
version 1 or
2. The scale of the color indicator from −30% to +30% will be relevant and discussed in
Section 3. All three FRF lines are almost indistinguishable, which leads us to the conclusion that the simplification of the modeling as well as the geometry itself is possible without affecting the FRF function in a relevant way. The darker colors to the right indicate some larger relative deviations where the FRF values themselves are below 1. This results in bigger relative errors for minimal deviations. For frequencies above 200
, changes are even smaller.
2.3.1. Knuckle
The knuckle needs a different approach to fit the mesh to the new suspension kinematics. Because of the complex shape of the part, it is not possible to simplify it using beam elements as it was done for the track rod. Therefore, we use the existing mesh and morph it in order to fit the new kinematics.
Each part of the suspension has got
k specific connection points that either have to stay unchanged or follow an imposed displacement. The knuckle is connected to all suspension links including the track rod, the brake caliper, and the wheel bearing. For all these attachment points, we impose translations. This is represented in
Figure 7 in the top view. In white, we see the original knuckle geometry. The black dots and red squares indicate some of the points that receive an imposed displacement. The red connection between the knuckle and the track rod is moved in the
x-direction and all the other black points shall stay unchanged. This results in the blue part. Ideally, only the FE nodes near the shifted point will be moved.
In a first, and later omitted approach, we used Lagrange polynomials to interpolate directly onto FE nodes. The sum of all Lagrange polynomials in one point
which describes its displacement
depending on all imposed displacements
. The left part of the product is the basic Lagrange polynomial [
30] (p. 188) expanded to three dimensions. Because of the function’s nonlinearity, small imposed displacements far away from the interpolation point could lead to huge displacements at the interpolation point. Even an additional factor
considering the distance to the imposed points didn’t fulfill the requirements because of varying distances. Therefore, we omitted this approach.
Instead of Lagrange polynomials, using Discrete Sibson interpolation was found to be a working approach. We used the Python implementation by Stevens [
31], which is released under MIT-License [
32]. Discrete Sibson interpolation, also called natural neighbor interpolation, uses a three-dimensional Voronoi diagram that defines cells around each imposed point. A regular grid specifies points, where function values from the imposed points are interpolated. The overlapping area of a new Voronoi cell around the grid point with the already existing Voronoi cells is used as weighting factor. This method ensures that each grid point is only affected by given displacements adjacent to the said grid point [
33].
Because the implementation by Stevens can only interpolate onto a regular grid, we need to attach a second interpolation. Using the displacement values on the regular grid, we can interpolate from the grid onto each individual FE node using Python’s RegularGridInterpolator library [
34]. This function uses the regular grid data received from the Discrete Sibson interpolation and linearly interpolates the data onto each individual FE node.
We use this tool chain consisting of two separate interpolations three times, once for each translational direction
x,
y, and
z. As we keep the original meshing and only shift FE nodes, we distort some of the FE elements. Because FE elements are sensitive towards distortion [
21] (p. 307), this method is limited to small displacements. In the future, for bigger displacements, a remeshing procedure should be attached to the interpolation process in order to ensure sufficient element quality.