Excerpt

## Contents

1 Introduction

2 Laboratory setup

3 Signal analysis

4 Mode extraction

5 Results

6 Appendix

List of Figures

1 Test structure

2 Grid points for the measurements

3 Grid points marked on the frame

4 Schematised laboratory setup^{1}

5 Signal processing in a FFT analyzer^{1}

6 Typical FRFs and coherence functions

7 Typical FRFs and coherence functions

8 Typical curve fit of a resonance peak

9 Summed FRF magnitudes

10 Mode shapes 1 - 6

11 Mode shapes 7 - 10

12 Side views of various mode shapes

13 Measurement grid

## 1 Introduction

Within the framework of this project, it is the aim to investigate the dynamic properties of an automotive sub-structure provided by GKN Driveline. The modal parameters are extracted experimentally and these results are used to animate the structure’s eigenmodes and to hereby get a better understanding of its dynamic behaviour.

This section shall give a short introduction to the test object and the given problem. It is followed by a listing of the laboratory setup. Section 3 will present an explanation of the signal analysis and hereafter the principal theory behind mode extraction from the obtained frequency response functions (FRFs) is given. In the end, we will present our gathered results and use a self-designed MATLAB program to animate the eigenmodes of the structure. In our last section, we will discuss our findings, prove consistency and try to explain particularities.

Figure 1: Test structure.

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Figure 1 depicts the given automotive subframe of GKN Driveline, which is the object of investigation. It features a basically rectangular metal hollow profile with a rectangular cross section. At its corners, it offers mounting points including an elastomer decoupling of its envi- ronment. In its centre, it carries a power train’s transmission block, which is mounted by using soft bushings. Formerly used hard bushings caused significant noise in the passenger cell, but GKN assumes that the new soft bushings solved the problem. It is now the aim to understand how the soft bushings changed the overall dynamic behaviour of the structure. Therefore, the overall experimental setup is marked by the three following versions, which then have to be be compared.

- Assembled structure with soft bushings

- Assembled structure with hard bushings

- Frame without gearbox

It is the task of this project group to investigate the first configuration. The frequency range of interest is from 80 to 500 Hertz.

## 2 Laboratory setup

It is of vital importance to avoid unwanted interference of support loads when investigating the eigenmodes of a test object. Hence, we suspend the frame from the ceiling with soft elastic springs. Since our frequency range of interest is 80 - 500 Hz, it is recommended to make sure that the rigid body modes of the supported structure are at maximum 30% of 80 Hz to make sure that the support forces do not interfere with the elastic modes^{1}. A short measurement of the period time of the oscillating up and down movement reveals a rigid body motion at 20 Hz. Later we will see that the elastic modes will begin above 100 Hz, so that we can neglect the support loads in our investigation. We need to overlay the test structure with a number of

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Figure 2: Grid points for the measurements.

measurement points. One have to find a good trade-off between accuracy and number of points. We pursue several strategies to extract the mode shape details that we would like to highlight hereafter. To detect relative motion of the gear box, we choose the connection points of the gear box and frame as well as the outer mounting points (black crosses in Fig. 2). Several points are located on the gear box to detect pitching and dipping. We expect an overall better visualisation of the bending modes by adding an array of points at the bottom of the frame as it is more flat. Additionally, it adds a backup set of points to filter for possible outliers. The deformation of the cross section requires us to put at least two points in transverse direction. For one side we even decided to add a third row (cf. Fig. 2 left side and 3). The overall amount of grid points adds up to 134 nodes. Additionally, we defined a triangulation table similar to the mesh generation

Figure 3: Grid points marked on the frame.

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in Finite Element Methods. In our case, it helps us to better visualise the structure’s surfaces (point clouds do not give a good plastic impression of the object’s movements). A comprehensive listing of coordinate values and triangulation tables can be found in our Appendix. We decided to mount the accelerometers to the long and the short side to minimise the risk of positioning them on nodal lines. The positions are marked with circles in Fig. 2. Namely, our channel setup is:

- Channel 1: Force

- Channel 2: Accelerometer Point 12 (left in Fig. 2)

- Channel 3: Accelerometer Point 32 (centre in Fig. 2)

- Channel 4: Accelerometer Point 75 (right in Fig. 2)

We set up our laboratory equipment similar to the abstract sketch as seen in Figure 4. We excite our test object using hammer excitation. It is cost effective and provides easy handling considering the number of measurement points. A soft rubber tip provides enough excitation energy and avoids the risk of double hits. The signal of the force transducer and the three ICP accelerometers are transferred to a sufficient energy level by a simple amplifier. The signal is then passed to an A/D interface with included Fast Fourier Transform (FFT) analyser. The electronic system is managed by a proper MATLAB program called SIGLAB. A comprehensive list of lab components can be found in the Appendix.

We want to cover a required frequency range of 80 - 500 Hz. The upper frequency limit determines our sampling frequency fs. To sample the signal adequately, we have to follow the Shannon sampling theorem^{1}, stating that the sampling frequency has to be at least twice the upper frequency limit fu

(1)

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and the built-in anti-aliasing filters will cut off signal components larger than fu to avoid information mixing in the frequency domain of the FFT. As no filters can be ideal (infinitely steep cut off), the practical sampling rate is defined as

(2)

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We add an additional buffer and set fu = 1000 Hz. The frequency resolution in the frequency domain should be at least Δf = 1 Hz. This will influence our time record length to

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(3)

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Figure 4: Schematised laboratory setup^{1}

In the measuring software, we have to reverse engineer these equations, as we can only set the number of recorded samples per record. We use the maximum N = 8192 and in combination with the sampling frequency we get

(4)

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(5)

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As we like to make use of the noise-reduction in signals, we use the built-in averaging function. The measurement is automatically triggered with the pulse in the force transducer’s signal. The input channels’ sensitivity were tuned to fit to the different signal levels to obtain the best possible signal to noise ratio. A more detailed insight in the signal treatment will be given in the paragraphs hereafter.

## 3 Signal analysis

The signal processing is guided by the diagram which is depicted in Fig 5. First of all, the time signals of the impulse hammer’s force and all acceleration signals are filtered in order to truncate frequencies which are larger than half of the sampling frequency. This procedure avoids aliasing, which would occur when digitalising the signals by using the A/D converter.

Measured signals in experiments are commonly non-periodic and have a finite signal length. For such signals, algorithms of the FFT would cause leakage errors, which lead to wrong frequency spectra. In order to reduce the influence of non-periodic and finite signals, weighting windows are normally used. The main characteristic of all windows used for FFT is that they start and end with zero. Since we use a hammer excitation, we assume our signals to be zero before the excitation and zero at the end of the measurement periods. Therefore, weighting windows were not used in our experiments. Next, the fast fourier transform is performed for the accelerations and excitation force. Based on the FFT, the averaged auto spectrum is computed for all signals, as well as the averaged cross spectrum between excitation force and all accelerations. Since we perform five measurements per excitation position, in order to lower the influence of noise and

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Figure 5: Signal processing in a FFT analyzer^{1}

possible disturbances, the auto and cross spectra of all measurements are averaged. Then, the frequency response functions

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(6)

between the acceleration x at position n and force F at position m are estimated by dividing the cross spectra by the auto spectra of the excitation force. This approach is established since it provides the best FRF estimation. Finally, a calibration of all FRFs is performed. To this end, a known mass was excited with a hammer hit. The constant value for low frequencies of the calibration FRF as well as the mass are then used to calibrate the FRFs of the structure. Furthermore, the coherence function is determined. It is an indicator for linear relation between two signals.

Fig. 6 shows a small selection of various FRFs and the corresponding coherence in the fre- quency range of interest. One can see that some FRFs have common resonance peaks which indicates eigenfrequencies. Moreover, the coherence functions are close to one. That indicates linearity between input and output signal and good signal quality, respectively. However, the coherence drops if the corresponding FRF has an anti resonance. The measurement position 125 is located on the the gear while position 12 is on the top of the frame. It can clearly be seen, that the corresponding FRF h12,125 has a low magnitude for higher frequencies. This means, that the impact of the hammer to position 12 is lower, if the forces are transferred through the soft bushings. A well isolating behaviour of the bushings can be concluded. However, the coherence within the high frequencies becomes worse as it shows more noise. Therefore, the signal should not be interpreted for very high frequencies, say above 600 Hz, which are not shown in the figure.

Another interesting fact is visible in Fig. 6. All FRFs have small disturbance which are noticeable as small local peaks. For instance, two of them occur within the frequency range from 150 Hz to 200 Hz. These disturbances emerge in the entire range and have a distance to each other of approx. 20 Hz. Because of this regularity, they cannot be caused by the investigated structure itself. The reason for the appearance must be in the in the setup of the measurement instruments. We assume that the disturbances are leakage errors, which are caused by omitting weighting windows.

**[...]**

- Quote paper
- Martin Rakowitsch (Author)Maximilian Scheid (Author), 2016, Experimental modal analysis of an automotive drivetrain subframe, Munich, GRIN Verlag, https://www.grin.com/document/342046

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