of linear springs and dry friction sliders (see figure 4.3 and 4.4).
A significant property of this model by Piotrowski is that the stiffnesses of the pa-rameters in the model are assumed to vary linearly with respect to the load that the suspension supports. Thus in this chapter we ultimately determine a set of normalized parameters, which can be scaled to correct values depending on how much load the suspension supports.
Another consequence of the model that we adopt is that we assume Coulomb’s law of friction holds for sliding in the joints. This has been argued for in [8], and a comparison of measurement with the theoretical curve for Coulomb’s friction law can be see in figure F.2 in the appendix. As can be seen, there seems to be an acceptable degree of similarity. Ultimately, this entails that we do not differentiate between static and kinetic coefficients of friction when it comes to sliding in the suspension joints.
The mathematical model is derived through a differential succesion of the dry friction element. A derivation1of this is found in appendix F and the result is summarized below.
F = −ky+T1 Lateral F = −ky+
X4 i=1
Ti Longitudinal where Ti are defined by
T˙i =
−kiy˙ if |Ti| < T0i
−[kiy]˙ + if Ti = T0i [−kiy]˙ + if Ti = −T0i
The parameters k, ki and T0 are determined through real experiments on the UIC suspension. The experiments and identification of the parameters is the topic of the following sections.
4.2 Experiment
The aim of this section is to produce a set of parameters for the lateral and longitudinal dynamics of the UIC suspension model. Actual experimental measurements were carried out at the Institute of Vehicles, Warsaw University of Technology, in the month of March, 2003, in cooperation with Artur Grzelak and under the guidance of Jerzy Piotrowski.
As inspired by [8], we focus mainly on the longitudinal and lateral dynamics of the UIC suspension here, and thus it is sufficient for us to create a setup involving only the actual linkages present in the UIC suspension, omitting the leaf spring. The linkages were delivered in a worn state as desired, but in that they were disassembled,
1This technique is known in non-smooth mechanics but this derivation is not shown in [8].
32 UIC Suspension Links
it was impossible to determine the exact original configuration of the linkages. Unable to assemble the linkages as they originally fit together, we had no other option but to sandblast and reprofile the linkages in an attempt to yield them as new in order to attempt meaningful experimentation with them.
Furthermore, in that the leaf spring is of no interest in our measurements, it is re-placed by a stiff beam upon which mass is re-placed in order to load the linkages. The actual construction of the suspension setup is peformed in an upside down fashion, in-spired by [8]. This setup is illustrated in figure 4.5. The mass of the beam and added masses total 378.2 kg. This value has remained constant throughout experimentation.
The length of the beam replacing the leaf spring is 1.22 m. The angle α= 25.9◦ repre-sents the angle the linkages form with respect to vertical. The UIC linkage dimensions correspond to those where the longitudinal pivot element has a diameter of 35 mm.
Figure 4.5: The suspension setup.
4.2.1 Measuring Equipment
The suspension setup was instrumented with linear displacement sensors in the longi-tudinal and lateral directions. They operate on the basis of translating physical dis-placement into a voltage value that is subsequently transmitted to an amplifier. The amplifier then feeds the analog signal to an analog to digital converter that interfaces with the computer through a PCI card device, and the digital signal is subsequently recorded by software. The displacement sensors can be seen in figure 4.5.
A brief overview of the measurement equipment:
4.2 Experiment 33
• 2 linear displacement sensors.
• Amplifier.
• Analog to digital converter.
• Computer with available PCI slot.
• Oscilloscope software for sampling data.
4.2.2 Procedure
We here outline the procedures taken to setup the oscilloscope software correctly for measurement. This must be done for both sensors.
1. Ensure that the suspension is at rest.
2. Select the channel for the sensor of interest.
3. Shift the sensor physically such that it registers a voltage corresponding to the middle of the range of voltage it is capable of generating (its own zero-point).
4. Set the oscilloscope software’s zero-point for the voltage given by the sensor of interest.
5. Utilizing a standardized block of metal with a known dimension, insert this be-tween the sensor and suspension, taking care not to displace the suspension.
6. Set the oscilloscope software to register the voltage now measured.
7. The oscilloscope software is calibrated by entering the displacement in mm corre-sponding to the standardized block of metal.
8. Repeat for the other sensor.
Once this is done, we can proceed to limit ourselves to measure only two channels of the sixteen that the digital to analog converter delivers. We also choose the total measurement time to record. In recording, the sampling was done at a rate of 1000 samples per second.
Recordings were performed in one direction at a time, since the mathematical model we strive to implement explicitly separates longitudinal and lateral mathematical ele-ments. Theoretically, they should be kinematically independent. However, in measure-ment, we are interested in preventing as much dissipation as possible other than that dissipation that will yield parameters for the mathematical model. This is because that low amplitude oscillations do not seem to be nearly as kinematically independent as
34 UIC Suspension Links
the larger amplitude oscillations. This leads to the low amplitude pure rolling oscilla-tions to die off too quickly for proper measurement if oscillation goes on laterally and longitudinally simultaneously.
When recording starts, the suspension is excited in the lateral direction, carefully preventing too much excitation in the longitudinal direction. This is done to an ampli-tude that takes advantage of the entire range of motion the corresponding sensor can register. Furthermore, excitation of the suspension ceases prior to the suspension reach-ing an extremum in motion. This is of importance when we later need to determine parameters from measurements. An analogous procedure was used for measurement in the longtitudinal direction.
Several measurements were recorded for excitations in both longtitudinal and lateral directions in order to limit the data from a poor measurement run polluting calculated parameters.
Once a recording is done, it can be saved in its raw format to disk, but in order to analyse results, the data is converted to ASCII format, and only every 5th data point is sampled. This reduces both the file sizes as well some high frequency noise in the data.