The Influence of Pitch on the STD and BSTAM Layout

Even though it is derived under the assumption of an unsprung chassis, the mathemati- cal definition of the OPT BSTAM in terms of instantaneous center of steering axis inclination (cf. eq. (3.23)) and king-pin inclination angle (cf. eq. (3.19), (3.20), (3.24)) is universally true for every conventional front suspension / steering system with two steering bearings. Depending on the chassis layout, there may however be practical implications for the implementation of OPT BSTAM on a real sprung chassis.

One such implication arises, when a standard chassis with a telescopic fork front sus- pension is chosen as baseline, on which the effect of brake pitch causes significant variations in effective caster angle and trail. The mathematical description of the OPT BSTAM remains principally the same, only the caster angle τ has to be replaced in the already known equation set by:

(3.25)

with τ0 being the design value of the caster angle in static trim (23°55’ for the test mo-

Figure 3.5: Instantaneous center of steering axis inclination of OPT BSTAM under the influence of brake pitch with telescopic fork suspension

Figure 3.5 illustrates, how h2 (cf. eq. (3.23) and (3.26)) increases under the influence of

pitch and the new instantaneous center of steering axis inclination (Gν) moves lower

along the vertical connection line between front axle (M) and tire ground contact point:

(3.26)

For the chassis parameters of the test motorcycle and a pitch angle of ν = 10°, h2,ν ≈ 124.7 mm compared to the original h2 ≈ 74.0 mm in static trim.

In order to achieve the same geometric compensation ratio (gcr) with this lower instan- taneous center of steering axis inclination (Gν vs. G), the king-pin inclination angle needs to be increased as well. Also this effect is already included in the mathematical definition through replacing the original caster angle τ (in eq. (3.7), (3.19), (3.20), and (3.24)) with τν (from eq. (3.25)). It yields the dark grey lines in Figure 3.6 with much higher required inclination angles than for the reference with no pitch, cf. black lines. Besides implications for the construction space needed for larger displacements of the steering bearing adjustments, this also means, that the control algorithm of an ideal OPT BSTAM with a telescopic fork needs to take the pitch angle into account.

Even though the variable height of instantaneous center of steering axis inclination is theoretically feasible with two independently adjustable steering bearings (cf. Table 4.2, KC 4-6), it stands to question, if this effort is justified by functional superiority of the mathematically ideal to simpler solutions in terms of steering torque demand. In the following, this question is discussed for a “non ideal” OPT BSTAM with a fixed instan- taneous center of steering axis inclination and a king-pin inclination angle σ computed

 h2 h3= rr,ft t₀ t_  (plane of) steering axis vertical reference  w/o pitch vertical reference  with pitch

 ground reference w/o pitch  ground reference with pitch M

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

Figure 3.6: King-pin inclination angles in dependency of the roll angle for the OPT BSTAM at full or partial compensation under the influence of brake pitch (ν = 10°) for the mathematically ideal (black and dark grey lines) and non ideal case (light grey lines). Note that the diagram presents effective values in perpendicular frontal projection. The slightly smaller king-pin inclination angles σst in the steering axis plane can be computed using eq. (3.31).

STD of a Non Ideal OPT BSTAM under the Influence of Pitch

Looking back at Figure 3.5, for this configuration, the effective vertical position of the lower steering bearing in the frontal projection hardly changes under the influence of brake pitch:

(3.27)

For a pitch angle of ν = 10°, this yields h2,eff ≈ 72.9 mm (grey reference system), instead

of h2 ≈ 74.0 mm (black reference system). Since the frontal projection of the instantane-

ous center is placed slightly higher, also the projected king-pin inclination angle σ needs to be reduced to avoid over-compensation (i.e. ecr > tcr, but especially when ecr > 1). Using a twin-fold projection of the original king-pin inclination angle σ (according to equations (3.19), (3.20), and (3.24) and based on τ0) – from the vertical reference plane

(black) to the steering axis plane and back to the new vertical reference (grey) – yields the new effective king-pin inclination angle under brake pitch:

(3.28)

As desired and illustrated in Figure 3.6 (light grey lines), this leads to slightly lower values as in the ideal reference case with no pitch angle (black lines). Referring back to Figure 3.1, left (triangle D-E-G), the compensated portion of the tire scrub radius srcmp can be computed as follows:

Together with the roll angle dependent tire scrub radius srtir (eq. (3.4)), the effective compensation ratio ecr (cf. eq. (3.5)) is then given by:

(3.30)

yielding slightly lower effective compensation ratios as would be the target ecr ≤ tcr. In order to evaluate, what all these deviations from the ideal case mean in terms of steering torque demand, the already known equation set can be used to compute the lever arms and steering torque demand contributes of the front tire contact forces. The results are illustrated in Figure 3.7 for the same boundary conditions as before.

Figure 3.7: Steering Torque Demand generated by front tire forces for different chassis setups at ay = 6 m/s² (λ ≈ 35°) and ax = 0 – 7 m/s² for riding style “lean with” under variation of pitch angle (ν = 0 or 10°). Refer to the main text for a detailed description.

The black lines in the first graph show the STD and its three contributes for the standard chassis with lean with riding style and no pitch. The lowest two solid black lines indi- cate the STD of the ideal OPT BSTAM at partial (gcr = 0.65) and full compensation (gcr = 1) while the curved arrow describes the possible field of adjustment. The dashed grey line is the STD of the standard chassis at ν = 10° brake pitch (cf. second diagram in the figure) which is in the order of 5 to 10 Nm higher than without pitch, but still lower as with 10% lean out riding style and no pitch in the reference corner braking situation, as indicated by the dotted grey line (also cf. Figure 3.2).

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

under the influence of ν = 10° brake pitch. While the more direct transmission ratio of the brake force is to be slightly recognized in the gradient of Tx, the balance of Ty and Tz takes place at lower absolute values and gradients. This leads to an initial offset in STD in comparison with the no pitch reference as indicated by the solid grey line. The math- ematically ideal OPT BSTAM is exactly following this offset (of approximately 5.3 Nm in the example, cf. common root of black diagram curves at ax = 0 vs. grey reference). The third and fourth diagram in Figure 3.7 show the decomposition of steering torque demand contributes for the non ideal OPT BSTAM – with a fixed instantaneous center and pitch angle invariant actuation of steering axis inclination – for full and partial compensation. While the target compensation ratios are tcr = 1 and 0.65, the effective compensation ratios that are achieved according to eq. (3.27) through (3.30) are slightly lower, ecr ≈ 0.95 and 0.62, respectively. The STD of the standard chassis is repeated for reference, the dashed grey line indicating the situation with pitch and the solid grey line the one without.

While, for a given pitch angle, the balance of Ty and Tz remains identical for any com- pensation ratio 0 ≤ gcr ≤ 1 and the corresponding steering axis inclination angles on the ideal OPT BSTAM (cf. first and second diagram in the figure), this does not hold true in the non ideal case, where this balance changes slightly with varying compensation ratio. Therefore, the resulting STD TF of the non ideal OPT BSTAM starts with a gap towards the standard reference (solid black vs. dashed grey line) in free cornering (ax = 0). Its magnitude is biggest for full compensation and decreases with lower compensation ratios. In the example, it is 3.6 Nm for tcr = 1 and 2.3 Nm for tcr = 0.65, respectively. However, it has to be stated, that a pitch angle of 10° does not suddenly occur during free cornering. On the one hand, the absolute pitch angle depends on the deceleration level and on the other, pitching is a transient process that involves a certain time span, due to the motorcycle’s pitch inertia. Even for the standard chassis, the resulting steer- ing torque demand curve is therefore always a blend between the model calculations for the situation with pitch and without. Moreover, during free cornering at zero pitch angle (i.e. parallel suspension travel), the geometry of the OPT BSTAM is the same for both the ideal and non ideal case. Consequently, also the stationary steering torque demand is the same, and both curves start from the same origin as for the standard chassis (see first diagram in Figure 3.7).

In the presented example, the “blended” average increase rates from free cornering (ν = 0, ax = 0) to the end-point of the model calculation (ν = 10°, ax = 0.7g ≈ 6.87 m/s²)

are 8.36 Nm/m/s² for the standard chassis, 2.78 Nm/m/s² for the ideal OPT BSTAM at

gcr = 0.65, and 3.61 Nm/m/s² for the non ideal OPT BSTAM at tcr = 0.65 (ecr ≈ 0.62).

This is a reduction by a factor of 3.0 in the ideal case and by 2.3 in the non-ideal case, corresponding to an efficiency of roughly 77% for the non ideal solution, which even

In conclusion it can be stated, that – despite the strong influence of brake pitch on the steering geometry of a standard chassis with telescopic fork – even a mechanically more simple and mathematically non ideal implementation of the OPT BSTAM concept proves effective to mitigate the BST effect and to provide a stationary steering torque demand (from tire contact forces) very close or even identical to that of the baseline. Finally, the less sensitive the baseline chassis is to brake pitch and related changes in caster angle and trail, the closer will a simplified practical implementation of OPT BSTAM be to the ideal case. As already mentioned earlier, this can favorably be obtained on the basis of a hub-center or king-pin steering.

Since the practical functionality of a non ideal OPT BSTAM does not deviate too much from the mathematically ideal case and is even identical at zero pitch angle, a rigid chassis remains the simplified basis for further considerations, if not stated otherwise.