Tyre is the only connecting part between the vehicle and the road, which provides all the vehicle driving, steering, and braking force. Accuracy tyre dynamics models are increasingly needed. Because of the structure nonlinear and complexity of contact state, how to establish effective tyre model is always the difficulty of vehicle dynamics modeling. Based on the idea of modal parameters modeling, establish the quantitative static tyre model in the vertical direction and research the effects of modal root and tread model on this method. With the tyre vertical load on the road, research the pressure and deformation features in the tyre contact patch and the interrelation between the horizontal load distribution, contact length, and sinking amount. Use the iterative method to establish tyres static vertical model on the horizontal road. The theoretical calculation and experiment results indicate the relation curves of vertical force distribution and sinking amount, vertical load, and length of contact patch and show good agreement on the qualitative and quantitative aspects.
Tyre is one of the important components in the vehicle, and its mechanical properties and structure design directly affect the driving performance of vehicle. Tyre is the only support and force transmission element to connect vehicle and road. Tyre mechanics is the basis of vehicle dynamics and the further development of vehicle dynamics and control technology relies on the precise tyre modeling techniques [
For a long time, scholars have conducted extensive research and exploration on the tyre model. According to the modeling methods, tyre model can be divided into empirical models [
In the process of car driving, the tyre has a periodic force from the road, so the rolling characteristics are related to its frequency and vibration mode [
General experimental modal analysis system includes excitation systems, signal measurement acquisition, and analysis system. With modal analysis theory, input and output signals can be got through modal test methods; then frequency response function of the system can be solved [
Excitation input is got by hammer knocking tyre and then using charge amplifier to import data acquisition and processing system. By the time domain and frequency domain analysis to obtain data modal identification and verification, finally, the natural frequencies, damping ratios, and vibration modes of the tested tyre are obtained; experimental modal flow chart is shown in Figure
Experimental modal flow chart.
By using test method to obtain tyre steady vertical model of modal parameters, analyze the sinking amount, contact length, force deformation, and stress distribution caused by the vertical load [
During the rolling process of the tyre, the tyre and road surface are in contact with each other, and the interaction force between the horizontal and the vertical directions is generated. It is assumed that the tyre is a driven wheel and assuming that the axle is only subjected to horizontal driving force and vertical gravity and does not bear the driving and braking torque, it rolls at a constant speed on a horizontal rigid road surface. As the tread width is not included in the tyre rolling model, the longitudinal and vertical characteristics of the contact patch are the focus. The tyre needs to be uniformly dispersed in the circumferential direction, and the rolling trace should be divided into
The tyre is divided into two parts: the carcass and the tread, with connecting each other in series. The deformation and force of each part are deduced as follows.
Assume carcass deformation:
Then, derivate
According to the symmetry of the tyre, relationship between
Assume
With the tyre rolling on the horizontal smooth surface and its response being the model superposition results of all orders, then the response function of
Assuming undamped tread, tread deformation can be represented:
Since the tread comprises rubber damping material, use the relationship between shear modulus and elastic modulus to establish the expression of tread stiffness.
Define displacement admittance expression of tread elements
Then, expression of tread deformation is as follows:
Along the circumferential direction, the tyre is divided into discrete units; then, the whole analysis is carried on. As in Figure
Discrete units force condition within contact patch.
According to the horizontal force
With modal parameter model, tyre is divided into carcass and tread, which form series model [
Force and deformation of tyre.
After the deformation of tread and carcass,
After coordinate transformation, (
Further elucidate the relationship between tread element deformation and carcass element deformation:
Analyze the mechanical components of tread elements in different coordinates:
Analyze the relationship between carcass tangential force
Define vertical force and horizontal force in the contact patch as
With (
As the test object, tyre is placed on the soft sponge support, using mobile hammer method to get tyre excitation with 12 pick-up point around the tyre center; the test equipment is shown in Table
Test equipment.
Number | Tool type | Configuration instruction | Factory number |
---|---|---|---|
1 | DASP-V10 | Multichannel signal acquisition and real-time analysis software | 2240-2014 |
2 | Poly LSDF | Poly LSDF modal analysis software | |
3 | MIMO | Test and analysis software of multi-input and multioutput modals | |
4 | INV3060S | 24 bit intelligent collecting instrument of network type | A0B0A006 |
5 | MSC-3 (ICP/LEM0) | Medium-sized hammer | YD-5T-140309 |
6 | INV9832A | Three-dimensional acceleration vibration sensor | 140509 |
Modal frequency and damping ratio of each order.
Type | Number | Frequency (Hz) | Damping ratio (%) |
---|---|---|---|
SF-350 | 1 | 112.839 | 4.528 |
2 | 113.661 | 3.958 | |
3 | 125.580 | 5.220 | |
4 | 133.580 | 3.528 | |
5 | 135.528 | 3.666 | |
6 | 155.739 | 3.370 | |
7 | 156.657 | 2.766 | |
8 | 180.092 | 3.198 | |
9 | 182.454 | 3.972 | |
10 | 207.977 | 3.180 | |
11 | 210.787 | 3.920 | |
12 | 238.917 | 3.166 | |
13 | 241.519 | 2.780 | |
14 | 250.064 | 0.193 | |
15 | 274.594 | 3.417 | |
16 | 312.030 | 3.779 |
Modal test and analysis.
DASP test equipment
Test process
Steady-state diagram of frequency response function
Frequency and orthogonal mode of the multiple root
From the results of the modal test, it is known that the vibration mode of the tyre is characterized by harmonic wave. When performing function fitting with trigonometric series, use the least squares method to determine the undetermined coefficients; therefore,
There are
With least squares method for the undetermined coefficients, the result should be
The description for undetermined coefficients obtained by fitting function is as follows: with the model test results of tyre, modal shapes are all symmetric harmonic, so, in most cases with the specific order, some
With the above steady vertical model of modal parameters, solve contact patch length, pressure, deformation, and sinking amount of tyre on rigid horizontal surface under static load; the analysis results are shown in Figures
Vertical pressure distribution.
Deformation of contact patch.
Figure
Results for contact patch test.
Vertical load/N | Sinking amount mm | | | Contact patch |
---|---|---|---|---|
2131 | 8 | 104 | 86 | |
3261 | 12 | 123 | 112 | |
5478 | 20 | 142 | 117 | |
6502 | 23 | 146 | 126 | |
7322 | 26 | 147 | 132 | |
8321 | 29 | 149 | 135 | |
The self-developed test bench is shown in Figure Clean up dirt debris and other contaminants on the tyre surface. Put the tyre on the test bench. Mark the required test points on the tyre sidewall. According to GB/T521, measure the outer diameter and section width of tyre. Place the pressure-sensitive film between tread and simulation platform, and apply vertical radial load to tyre axle at a constant speed and keep pressure-sensitive film without wrinkle, holding more than 2 minutes after loading, and then measure the static load radius and section width. Process data by pressure-sensitive film.
Contact patch test.
In order to regulate the tyre contact patch measurement, the coordinate system is defined in Figure
Coordinate of contact patch.
After processing data, the original measurement data are shown in Table
Sinking amount with loads.
With the rolling theory based on modal parameters, the force and deformation model of tyre carcass and tread is established. By using test method to obtain tyre steady vertical model of modal parameters, analyze the sinking amount, contact of the imprinted length, force deformation, and other laws caused by the vertical load. Deformation of carcass is the main reason for sinking amount and is also the key factor for the overall elastic deformation of tyre, and its force deformation characteristics are embodied in the modal parameters. According to modal test results by DASP V10 of Dongfang Institute Research Institute, tyre mold shapes render harmonic characteristics, by using trigonometric series for modal fitting and the least squares method for solving undetermined coefficients. Finally, self-developed test bench and a pressure-sensitive film Prescale TM are used in contact patch test; it can be seen that the calculation results of modal parameters are well consistent with the test bench results on contact patch length and sinking amount.
The authors declare that they have no competing interests.
The research described in this paper was financially supported by National Natural Science Foundation of Youth Fund Project (61503163), Jiangsu Province “333 project” training funded project (BRA2015365), and Natural Science Foundation of the Higher Education Institutions of Jiangsu Province, China (Grant no. 16KJB130002).