Vibration analysis of the drive train of a tire testing machine under MATLAB / SIMULINK
An uneven pattern of wear occurs in an existing tire testing machine, which essentially serves to examine the wear behaviour during constant speed driving in the speed range of 80 to 120 km/h with adjustable camber, track, wheel load and torque. This leads to the conclusion that vibrations occur in the drive train of the tire testing machine.
These vibrations become noticeable with a simple rotation frequency of the tire. This can be seen on the one hand in the fact that the corresponding amplitude increases occur in the frequency spectrum of the load, moment and revolution, and on the other hand in the wear pattern of the tire tread. To clarify these vibration phenomena, the following investigations were carried out:
Development of a mathematical model for the description of the tire mounting subsystem with drive unit from the perspective of vibrational engineering and program-technical implementation.
Determination of system-describing parameters such as mass, mass moment of inertia, spring stiffness based on available drawings and other documents.
Simulation of the inherent behaviour of the subsystem and correlation with existing measurements.
Evaluation and documentation of the calculations.
This test does not take into account either the drive train of the drum or the contact between the tire and drum in the model. If the observed vibrations cannot be explained by the characteristic behaviour of the drive train for the tire mount with drive unit, the model would have to be extended by the drive train of the drum and the tire/drum contact. This would then have to be the subject of a further investigation.
Modelling of the powertrain
Figure 1: Principle diagram of the components in the powertrain
Figure 2: Principle diagram of the components in the powertrain
a) Determination of system parameters
Based on the construction drawings provided in the form of partial production drawings of the components for the drive train of the components shown in Fig. 1, a physical model of the drive train of the tire testing machine is first of all set up (Fig. 2). The mathematical model can then be developed and analyzed using the MATLAB/SIMULINK simulation and analysis program. The simulation of the model has the task of understanding the physical mechanisms of action that lead to uneven tread wear in continuous operation.
In order to investigate the natural vibration behaviour of the drive train, the torsional vibration model illustrated in Fig. 3 is constructed consisting of a bound nine-mass oscillator. Table 1 shows the physical parameters that were determined on the basis of the manufacturer's specifications for the individual components or with the help of the CAD program MECHANICAL DESKTOP from the supplied geometry data, as well as the calculated reduced mass moments of inertia and spring stiffness. Subgroup 1 consists of the adapter rotor-joint, the torque measuring shaft and a part of the adapter shaft-rotor. While subgroup 2 consists of a part of the adapter shaft-rotor, the wheel bearing shaft, rim flange and parts of the test piece.
b) Modeling under MATLAB/SIMULINK
The nine-mass oscillation model is in the following described as a linear oscillation system with constant parameters. The modeling and movement simulation are carried out using the MATLAB/SIMULINK program.
Shortening the mass moments of inertia under matrix J, the damping constants under K, the stiffnesses under C, the angular deflections under vector j and the drive torque under M (t) results in the following differential equation system to describe the system dynamics:
J j _ .. + K j _ . + C j _ = M(t) 
Figure 3: Discrete mass-spring-damper-system as a model for the powertrain
_ = [j1 j2 . . j8 j9 ]T ; M(t) = [Mantr 0 . . 0]T
To describe the transfer functions, the linear system of equations of motion is transferred to the state model. This produces the equation of state in the form:
x.(t) = A x (t) + b (t) ;
- x(t) = [ji ji . ]Ti=1..9 und b = [ b1 b2 ] ;
b1 = (9,1)-Nullvektor
b2 = J-1 M (t)
- A = [ A1,1 A1,2 A2,1 A2,2 ]
A1,1 = (9,9)-Nullmatrix
A1,2 = (9,9)-Einheitsmatrix
A2,1 = J-1 C
A2,2 = J-1 K
Figure 4.1: Measured frequency spectra at 90 km/h
Figure 4.2: Measured frequency spectra at 120 km/h
Figure 4.3:1. natural mode for f01 = 14.60 Hz
Figure 4.4:2. natural mode for f02 = 38.85 Hz
To determine the eigenvalues or eigenforms, the equation of state represented in equation  is used as a homogeneous equation without excitation term to solve the special eigenvalue problem. This allows both the natural frequencies and the eigenforms of the system to be determined. The natural frequencies of the system are system characteristics and depend only on the values of mass inertia, stiffness and damping. The investigation of the systems own behaviour provides important information about the dynamic behaviour of the drivetrain even before performing a simulation calculation in the time domain, so that an initial assessment of the system can be made. A comparison of the calculated frequencies with the frequency spectra determined from the measurements on the real machine, for a speed of 90 km/h and 120 km/h and a tire type (XXX/XX R XX) is qualitatively represented in Figures 4.1 and 4.2. With the exact values for the mass moment of inertia and torsional stiffness, it is now also confirmed by the natural frequency analysis that at a speed of 120 km/h you are in the resonance range, while at a speed of 90 km/h you are driving below the resonance range. Figure 4.3 and Figure 4.4 graphically illustrate the 1st and 2nd eigenmodes of oscillation of the 9-mass oscillator model for the drive train of the tire testing machine. The two lowest eigencircle frequencies of the system were determined for an example of the combinations of wheel load, tire size and rim type. The eigencircle frequency f01_test wheel rim is the lowest frequency at which the wheel support subsystem and the other rotating masses of the drive train oscillate substantially relative to the raceway drum, which is regarded as fixed, and f02_test wheel rims the second lowest eigencircuit frequency resulting mainly from the elasticity of the angular gear unit. The mass moments of inertia J9 and torsional stiffness c9 have been recalculated for different tire and wheel sizes as well as different tire/rim combinations.