Modelos para el Estudio del Crecimiento y Remodelación

As pointed out in the previous chapter, the deceleration of the front wheel against its spinning inertia is causing additional aligning steering torque components during the whole braking process for a BSTAM with lateral inclination of the steering axis.

148 _{Magiera (2011): Simulation Model, Bachelor-Thesis} 149 _{Vasylyev (2012): Multi Body Simulation, Bachelor-Thesis}

150 _{This was confirmed in orienting slalom tests with the prototype that required only very little steering}

While their quasi-stationary portion is anyway overridden by the misaligning BST (i.e. the Tx component) during the steady braking phase and could beneficially be considered through a reduction in compensation ratio, their disturbing influence is biggest in the initial phase of braking. In order to generate the brake force in the contact patch, the tire first needs to build up brake slip. This means, that the brakes are quickly reducing the spinning velocity of the wheel against its inertia and generate the said aligning (out- ward) steering disturbance just an instant before the brake force and misaligning BST (i.e. Tx) occur and override the effect. The significance of the effect for the layout and riding feel of a BSTAM chassis shall be illustrated by the following rough calculation. The brake reaction torque that decelerates the front wheel against its spinning inertia is:

(3.34)

with Iyy being the spinning inertia of the front wheel in kgm² and being the reduction in angular velocity in rad/s² of the same during the initial braking phase.

The initial angular velocity ω0 of the front wheel in rad/s is defined by the initial velocity

v0 in m/s and the current roll angle dependent tire rolling radius (see eq. (3.3)) in m:

(3.35)

The percental brake slip s leads to a reduction in wheel speed:

(3.36)

The time needed to generate the brake slip is limited by the build-up of brake pressure respectively brake torque. In the driving experiments conducted during this study, typi- cal front brake pressure rise times ranged between 0.1 < Δt < 0.3 s, with a higher per- centage of lower values close to 0.1 s. Taking these values as reference for a combina- tion of equations (3.35) and (3.36) leads to:

(3.37)

As already introduced in eq. (3.31), the relationship between the king-pin inclination angle in the frontal projection σ and that in the steering coordinate system σst is depend- ing on the caster angle τ:

(3.38)

The aligning steering torque disturbance resulting from the generation of initial brake slip is then given as:

3.3 Layout and STD of a BSTAM with Laterally Inclined Steering Axis (KPI)

Filling in from equations (3.3), (3.34), (3.37), and (3.38), it can finally be written as:

(3.40) Figure 3.10 shows the results of a parameter study under variation of initial velocity v0

and king-pin inclination angle σ, conducted under the assumption of a brake slip s = 5%, rise times of 0.1 s respectively 0.3 s, and a roll angle of λ = 35°, for the parameter data of the test motorcycle (cf. appendix A.4.2, with a front wheel inertia of Iyy = 0.48 kgm², caster angle τ = 23°55’, and tire geometry defined by rft = 295 mm and rc,ft = 64.6 mm).

Figure 3.10: Aligning steering disturbance in Nm caused by front wheel inertia while generating 5% brake slip at λ = 35° for a wheel inertia of 0.48 kgm² (average between new and worn tire)

The realized BSTAM prototype features less than 2° projected king-pin inclination angle. For a given initial velocity of 60 to 70 km/h, a steering disturbance in the order of only 0.5 to 1 Nm is to be expected from Figure 3.10, left. It was therefore only very rarely recognized and reported by the test rider and could only once be captured in a measurement (cf. chapter 5.2.2). On one hand, the effect duration is only a few tenths of a second and directly followed by the opposing BST effect. Thus, no real steering angle or even roll angle disturbances occur due to steering system and vehicle roll inertia. On the other hand, capturing of the effect in terms of steering torque measurement requires a relatively pre-tensioned rider. In these regards, the elevated stationary steering torque demand with active BSTAM setups can be seen as a small help, since the test rider typically was very much at ease and relaxed while doing the test rides.

Going back to Figure 3.10 for an OPT BSTAM design, with king-pin inclinations of 5°, 10°, or even more as well as higher speeds of up to 100 km/h not uncommon on rural roads, the effect can assume values of 5 Nm or even more. Even for the short duration,

an unexpected outward steering impulse of that dimension is potentially dangerous, especially when cornering close to the roll angle limits of the vehicle.

Remedy may be found in reducing the front brake pressure increase rates, as exemplari- ly illustrated in Figure 3.10, right. Choosing a rise time of 0.3 s instead of 0.1 s brings the disturbance to more acceptable levels of only 1 or 2 Nm. Theoretically, this is of cause compromising the minimal achievable braking distance. However, for most prac- tical cases, only partial decelerations are required (and requested by the rider’s inputs), so that the reductions in front brake force can be more than outbalanced by a rear-wheel oriented brake force distribution (cf. chapter 3.6), without compromising the braking distance. Moreover, on a real sprung chassis, a braking strategy that activates the rear brake slightly in advance of the front brake is regarded as beneficial. It is triggering a forward shift in wheel load and the pitch process, so that a small misaligning effect on the steering is generated (through Tz) just about the time of the occurrence of the inertia effect. The fact that such a strategy is already incorporated into the C-ABS brake system of the test motorcycle is seen as a further contribute to the rare recognition of the effect. In conclusion, the inertia effect of a BSTAM with inclined steering axis has the follow- ing three facets that need to be considered for the system layout. Firstly, its quasi- stationary portion can beneficially reduce the required compensation ratio – and with it ultimately the construction space. Secondly, the initial aligning disturbance can be mitigated to an acceptable level through limited front brake pressure increase rates and advanced rear brake activation. Finally, also tire wear needs to be considered, since it significantly affects the front wheel inertia and hence the magnitude of the inertia effect (cf. Table A.6).

3.4 Layout and STD of a BSTAM with Parallel