These functions, together with the functions in 3.4.3, enable the driver to operate the vehicle longitudi- nally with precision and in an intuitive and consequent way.
3.4.4 Brake Proportioning
For brake performance it is important that both axles are used as much as possible during braking. But one also should consider that, in most driving situations, it is preferred that the front wheels lock first, because:
• A vehicle with locked front wheels ( = 0) tends to be yaw stable. However, steering ability is lost, so vehicle continues straight, incapable of curving its path.
• A vehicle with locked rear wheels ( = 0) tends to be yaw unstable. It turns around and ends up sliding with the rear first.
Hence, there are trade-offs when designing the wheel torque distribution. Same reasoning works for propulsion, if “locked“ is replaced by “spin”, meaning large positive longitudinal tyre slip. Spin at front makes vehicle more yaw stable than spin rear. The yaw stability then has a trade-off with acceleration performance.
Wheel torques is influenced simultaneously by both propulsion system and (friction) brake system, especially if regenerative braking with electric propulsion system. So, coordination of brake and pro- pulsion systems might be needed.
The basic function of a brake system is that brake pressure (hydraulic on passenger cars and pneu- matic on trucks) is activated so that it applies brake pads towards brake discs or drums. In a first ap- proximation, the pressure is distributed with a certain fraction on each axle. For passenger cars this is typically 60..70% of axle torque front. In heavy trucks, the proportioning varies a lot, e.g. 90% for a solo tractor and 30% for heavy off-road construction rigid truck. The intention is to utilize road fric- tion in proportion to the normal load, but not brake too much rear to avoid yaw instability.
If neglecting air resistance and road grade in Eq [3.13], the vertical axle loads can be calculated as function of deceleration (− ). An ideal brake distribution would be if each axle always utilize same fraction of available friction: 𝑓𝑥
𝑓∙ 𝑓𝑧
=
𝑟𝑥 𝑟∙ 𝑟𝑧⇒ {
𝐴𝑠𝑠𝑢 𝑒
=
;} ⇒
𝑓𝑥 𝑟𝑥=
𝑓𝑧 𝑟𝑧⇒
𝑓𝑥 𝑟𝑥=
∙(𝑔∙𝑙𝑟𝐿−𝑎𝑥∙𝐿) ∙(𝑔∙𝑙𝑓L+𝑎𝑥∙L)=
𝑔∙𝑙𝑟−𝑎𝑥∙ℎ𝑔∙𝑙𝑓+𝑎𝑥∙ℎ
;
Combining with + = ∙ ; gives the optimal and := ∙ ∙ ( 𝑙𝑟 𝐿− 𝑎𝑥 𝑔 ℎ 𝐿) ; and = ∙ ∙ ( 𝑙𝑓 𝐿 + 𝑎𝑥 𝑔 ℎ 𝐿) ; or, if eliminating : = 𝑙𝑟 2 − 𝑓𝑥− 2 √𝑙𝑟 2− 4 𝑓𝑥 ; or: ⁄ = 𝑙𝑟 𝐿− ℎ 𝑔 𝐿 ; and ⁄ = 𝑙𝑓 𝐿 + ℎ 𝑔 𝐿 ; [3.17]
If including non-zero 𝑎𝑖 and non-zero (but small) 𝑦: ⁄ = 𝑙 ⁄ − ∙ ( ∙ )⁄ + 𝑎𝑖 ∙
( − 𝑎𝑖 ) ( ∙ )⁄ ; and ⁄ = 𝑙 ⁄ + ∙ ( ∙ )⁄ − 𝑎𝑖 ∙ ( − 𝑎𝑖 ) ( ∙ )⁄ ;. Air re-
sistance 𝑎𝑖 and road gradient 𝑦, of course, influences so that we need to adjust to reach a certain . However, road gradient does not influence the distribution of longitudinal tyre force be-
tween axles and air resistance only if ≠
𝑎𝑖.
Conceptual purpose with pressure limiting valve or EBD
−
𝑁
⁄
−
⁄
𝑁
Figure 3-24: Brake Proportioning diagram. The curved curves mark optimal distribution for some variation in position of centre of gravity.
The proportioning is done by selecting pressure areas for brake calipers, so the base proportioning will be a straight line, marked as “Hydrostatic brake proportioning”. For passenger cars, one typically designs this so that front axle locks first for friction below 0.8 for lightest vehicle load and worst vari- ant. For heavier braking than 0.8 , or higher (or front-biased) centre of gravity, rear axle will lock first, if only designing with hydrostatic proportioning.
To avoid rear axle lock-up, one restricts the brake pressure to the rear axle. This is done by pressure limiting valve, brake pads with pressure dependent friction coefficient or Electronic Brake Distribu- tion (EBD). In principle, it bends down the straight line as shown in Figure 3-24. With pressure de- pendent values one gets a piece-wise linear curve, while pressure dependent friction coefficient gives a continuously curved curve. EBD is an active control using same mechatronic actuation as ABS. EBD is the design used in today’s passenger cars, since it comes with ABS, which is now a legal requirement on most markets.
On heavy vehicles with EBS (Electronic Brake System) and vertical axle load sensing, the brake pres- sure for each axle can be tailored. For modest braking (corresponding to deceleration ≤ 2 𝑠⁄ 2) all
axles are braked with same brake torque, to equal the brake pad wear which is importance for vehicle maintenance. When braking more, the brake pressure is distributed more in proportion to each axle’s vertical load.
3.4.5 Heave and Pitch
So far, in 3.3.5 and 3.4.4, we modelled transfer of vertical forces between axles, but neither heave and pitch motion nor displacement. This will be added in 3.4.5. In 3.4.5.1, the load transfer is steady state and the linkage “trivial”. In 3.4.5.2, the load transfer is transient and the linkage “non-trivial”.
3.4.5.1 Steady State Load Transfer and Trivial Linkage
Additional to that the axle vertical loads change due to acceleration , there are also change in out-of- road-plane motion (heave and pitch). In the following section, we study constant acceleration, e.g. when mild braking for a long time. We propose the steady state model in Figure 3-26. The model dif- fers between the “unsprung mass” (wheels and the part of the suspension that does not heave) and the “sprung mass” or “body” (parts that heaves and pitches as one rigid body). The wheels are assumed to be linked to the body through “trivial linkage” as in Figure 2-54.
ECE regulation limits to this region
Ideal curve
[data from BMW 320i E46]
Figure 3-25: Brake Proportioning. From (Boerboom, 2012). If looking carefully, the “HydroStatic” curve is weakly degressive, thanks to brake pad material with pressure dependent friction coefficient.