Lateral dynamics of articulated frame steer vehicles
Articulated frame steering is commonly used on wheeled construction machines. Instead of applying the steering angle to the wheels, the steering of an articulated frame vehicle is achieved by changing the relative yaw angles of the front and rear vehicle parts, which are connected by a centre joint. Normally hydraulic cylinders are used to accomplish this, although mechanical solutions exist for smaller vehicles. Articulated steering is said to have been first patented in the early twentieth century.
Articulated steering allows a smaller turning radius than traditional, Ackermann-type steering and is also a robust installation as no steering rods or similar components need to be integrated in the wheel-axle assembly. A known drawback is the reduced rollover stability at standstill, caused by lateral shifting of the centre of gravity. Besides its installation in construction machines, the steering type can be found on various agricultural and forestry machines, as well as specialised off-road transport vehicles.
The in-plane dynamics of an articulated vehicle can be described by the model seen in figure 2. The vehicle frames are represented by the two rigid bodies with masses and moments of inertia denoted by m1, I1 and m2, I2. The position vector (x1, y1) describes the motion of the front frame in an earth-fixed system, while the longitudinal and lateral velocities u and v quantify the front frame velocities in body-fixed coordinates. The torsional stiffness and damping KR and CR represent the equivalent stiffness of the hydraulic steering system, which is assumed to have some degree of flexibility due to oil compressibility and component elasticity. This torsional spring and damper element acts around an equilibrium position corresponding to the set steer angle. In straight line driving, this angle is zero.
The articulation angle φ is defined as the difference in yaw angle between the front and rear frames, i.e.:
ϕ =ψ 1 −ψ 2
The front and rear tyre forces Fy,1 and Fy,2 are assumed to be linear functions of the slip angle αi, so that:
Fy,i = Ci αi
A similar relation defines the front and rear aligning moments Mz,1 and Mz,2:
M z,i = CMiαi
The parameters C1, C2 and CM1, CM2 define the cornering stiffness and aligning moment coefficients of the front and rear tyres, respectively. Assuming small articulation angle and a constant longitudinal velocity, u, the equations of motion of the vehicle model in figure 1 can be expressed as a linear system of first order equations, in the form:
where the state vector x is given by:
The matrix A is dependent on the vehicle parameters, as well as the longitudinal velocity u, since this velocity implicitly defines the tyre forces. A full derivation has been presented by Azad, McPhee and Khajepour. The in-plane stability properties of the vehicle can be analysed on a fundamental level by studying the eigenvalues of the system matrix A. Any eigenvalue with a positive real part will indicate a divergent or unstable mode. Furthermore, a purely real eigenvalue will correspond to an exponentially divergent or decaying mode, whereas eigenvalues with nonzero imaginary parts signify an oscillatory motion. These two modes have been termed “folding” and “snaking”, in accordance with their physical interpretation. Examples of eigenvalue analysis can be seen in figure 2. Here the method described above has been applied to the model in figure 1, with parameters corresponding to those of an articulated tractor, and with a total mass of 3,000 kg and identical front and rear frames. The real part of each eigenvalue has been plotted against the vehicle velocity. In the graph, crosses symbolise purely real eigenvalues while dots indicate eigenvalues with an imaginary component. A total of four eigenvalues exist. As complex eigenvalues appear as conjugates, only one point appears in the graph for each pair of eigenvalues.
In figure 2 (a), two real eigenvalues can be seen at velocities below 6.0 m/s. These correspond to strongly damped modes dominated by the lateral velocity v, thus indicating a divergent lateral motion. The two complex eigenvalues seen at low velocities have small negative real parts, and therefore indicate a lightly damped oscillatory mode dominated by oscillations in the articulating angle φ. At higher velocities, only complex eigenvalues exist, appearing as two conjugated pairs. These two eigenvalues indicate that one lightly damped or slightly divergent oscillatory mode exists, as well as a similar, more strongly damped mode. Figure 2 (b) shows an analysis with similar parameters, but with the centre of gravity positions moved closer to the centre, so that the centres of gravity are located between the respective axles and the central joint. It can be seen that two divergent modes exist, one of which becomes unstable at 12.5 m/s, as the eigenvalues become positive at this point. The oscillatory mode is stable throughout the velocity range for this configuration. Using the method outlined above, Crolla and Horton (1983) investigated the influence of vehicle parameters on snaking and folding behaviour. Generally the most important parametric influences can be summed up as follows: - Mass and inertia: Eigenvalue analysis showed that increased inertia of the rear frame decreases the stability. Increased rear frame mass has a similar but less pronounced effect. - Centre of gravity positions: Moving the rear frame’s centre of gravity further aft deteriorates the snaking stability in the same manner as increasing the rear frame inertia. Moreover, moving the centres of gravity closer to the centre joint seemed to reduce the margin for folding instability. - Steering system properties: It was found that a lower equivalent stiffness (KR) resulted in highly deteriorated stability. In practice, decreased stiffness may occur because of air inclusion in the hydraulic system. Hence, this is highly undesirable with regard to snaking stability. It was also found that increased equivalent damping (CR) results in a more stable vehicle.
The effect of mass distribution is particularly important for a construction machine, since there may be a considerable difference between the loaded and unloaded condition. For a loader, it can be expected that the vehicle will be rear heavy when travelling unloaded, and hence it is most prone to snaking instability in this configuration. An articulated dump truck is loaded closer to the vehicle centre, and hence could be expected to display increased tendency towards folding when loaded.
Expanded linear analysis
A more refined study using the planar model was later published by Horton and Crolla (1986). Here a full model of the hydraulic steering system was included, replacing the simple equivalent torsion and damping seen in figure 11. Using this model, it was shown that, besides the snaking and folding modes, a third, “oversteering” mode also exists. This mode takes the form of a slow divergence in the articulation angle. It was hypothesised that this “oversteering” divergence may be the true cause of snaking oscillations, as an effect of the driver overcorrecting in order to maintain a straight path. Later full vehicle tests conducted with a wheel loader on a paved road (Lopatka and Muszynski, 2003) supports this to some extent, since the main frequency of snaking oscillation seemed there to occur at 0.1 Hz, which matches the primary frequency of driver inputs. Hence, the low frequency snaking behaviour seen in the tests could well be the effect of driver corrections, as suggested by the results from the expanded planar analysis.