Investigation of the suspension’s influence
With suspended axles, it can be expected that the suspended mass roll will interact with snaking oscillations, possibly influencing the snaking stability. This effect is not covered by the planar model in figure 1 and therefore requires a model with more degrees of freedom.
To simulate suspended mass roll, the model uses an equivalent roll stiffness approximation of the wheel suspension, modelled using rigid axles pivoting around the centre of each axle. Thus, the suspended mass has the freedom to roll, but not to pitch or bounce except on the tyres. The suspension roll stiffness is represented by an equivalent torsion stiffness acting around the roll centres, located in the middle of each axle, while the steering system flexibility is simulated by a centre torsion stiffness in the same way as in the planar model (figure 1). The masses of the front and rear parts are assumed to be 7,200 kg and 12,000 kg, respectively. The wheelbase is 3.6 m and the track width is 2.4 m.
Analysis of the snaking stability is performed by simulating the vehicle going straight forward on a level surface at a constant velocity, and initiating a snaking oscillation by a steering torque applied at the centre joint. The baseline suspension is set up so that the undamped suspended mass roll frequency is 1.00 Hz, while the relative roll damping is approximately 0.10. An unsuspended configuration is also analysed for comparison. In this configuration, the front axle is locked to the front frame while the rear axle is allowed to pivot freely about the roll centre. Figure 4 shows a comparison between the responses of the suspended and unsuspended models after an initial steering disturbance at 6 s, when evaluated at a velocity of 10 m/s. It is seen that the articulation angle slowly diverges for the suspended vehicle, as compared to the near-neutral stable unsuspended configuration. In addition, the frequency of the snaking oscillations is slightly decreased. This seems to indicate that the snaking and rolling motions of the suspended vehicle behave in a coupled manner, so that the combined yaw and roll inertias lead to a lower frequency compared to the pure yaw oscillations of the unsuspended vehicle.
Further investigation of the snaking and rolling motions shows that body roll and snaking motion are closely connected, as seen in figure 5. Both angles exhibit a harmonic oscillation at almost the same frequency, with the roll angle showing some time delay. Changing the roll stiffness does not affect the roll frequency during snaking to any greater extent. Hence, the snaking motion of the suspended vehicle can be characterised as a combination of yawing and rolling motions with closely matched frequencies.
As the rolling and snaking motions are closely coupled, it can be expected that the stability will be most affected when the eigenfrequency in roll coincides with the snaking frequency. It is found that minimum stability occurs when the suspended mass roll resonance is set to a value similar to the snaking frequency, but that the effect is generally small and can be offset by small changes in other parameters, such as mass and inertia or steering system stiffness.
Scale model tests
Scale model tests of articulated vehicle stability have been performed with the remotecontrolled test vehicle “Hjulius”. The vehicle is an approximate 1:10 scale model of an articulated wheel loader, with a total length of about 100 cm and a mass of around 20 kg, depending on the load configuration. The vehicle uses an electromechanical steering system that uses coil spring to simulate the flexibility of a hydraulic steering system. The articulation angle is measured by a potentiometer and the vehicle is also equipped with accelerometers and gyros to allow the study of other vehicle dynamic states.
In the basic setup, the vehicle has a fixed front axle and a freely pivoting rear axle, and is configured for all-wheel drive. Various combinations of mass and inertia properties have been analysed with respect to snaking and folding stability. Two examples of results are seen in figure 6. Here the vehicle has been subjected to a steering input while travelling in a straight line at a constant velocity of 1.5 m/s. Figure 6 (a) shows a case where the vehicle is in the original configuration, with no added weights. It is seen that the response is highly stable. Figure 6 (b) shows a case where extra weight has been added to the rear part. This rendered the rear yaw inertia higher in relation to the yaw inertia of the front part, resulting in an unstable configuration.
The corresponding stability predictions using the eigenvalue method are seen in figure 7. As seen in figure 7 (a), the original configuration is stable at the velocity considered (1.5 m/s), while the rear-heavy setup has an unstable snaking mode as shown in figure 7 (b). This is in agreement with the experimental results. Generally, the linearised analysis is able to predict the stability properties of the vehicle with reasonable accuracy, although some tendency to under-predict the stability margins in comparison with the test results is noted. This is likely to be caused by various damping effects occurring in the model but not included in the linear analysis, such as the tyre aligning moment and the presence of friction in the centre joint.