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TIRES
F = F (x, k, ϑ, ft, alignment ...)
- pure empirical
- semiempirical
- simple physical
- complex physical
BRUSH MODEL
UNCONDITIONAL SLIPPAGE
k => dFx
dFx/dk > 0 Rake Gauge
dFx/dk < 0 Mutee Value
dFx/dk = kL Sliding Tolerance
State of the system
deviation in the contact patch (nL-v)/v
k << 0 braking k >> 0 driving
A adhesion zone B slippage zone
- A - Force provided by elastic deformation τ = μfₛq₁
- B - Force provided by sliding friction τ = μdqf₇
k ff => A I, B I
BRUSH MODEL
=> reliability ↑, tire temp ↓, complexity ↓, material temp ↓
TPS:
- Tread made by ceramic elements several brushes, it is valid the Hooke's Law F = α main
- Brushes cannot be deformed radially
- Neglected axial dimension
- Pressure distribution assumed parabolic (+ Hertz that defines it as uni - elliptical) => no rolling resistance.
biuve is an elastic element (brief) with a nucleus cp
it is considered isotropic = > there properties depend x and y
∑max = μg ≈ friction limit f(x) because F(x)
shear stress ∅
9: Ax2 + Bx + C + BC @ x = ± a; Ft = ∅
= > Ft = ∫CPx(x)dx = > 3 unknowns, 3 conditions
= > ∂x(x) = 3∑z/a (1 - (x/a)2)
pure rolling condition |Vx = Ωle|
(lo > le > lc)
Vt = Ωlelinear rolling speed
vsx = Vx - Rle = Vx - Vr; relative speed = > Vi = Vx - Vsx
Δt = a - x/Vi
θ = -Vsx Δt (definition in equilibrium condition)
= -Vx(a - x)/Viproportional to x in the contact area
= > Θ = -Vsx/Vx
u = (a - x) ∂x
pianoce slip Φx = -Vsx/Vi = Vr - Vx/Vr
= -Vsx + Vx = k
-Vsx + Vx
k + 1
= > c~ = cpμ = -DFx∫aa(x)dx
-a
= cpVx∑asub - a(a - x)x dx
= l 2a2∑xcp = > no friction limit
kl ≈ 2a2cp considered
indeed |k| = > Vx| = > DFx' = > it is not true
we can introduce the friction limit as μd = μs
∑(x) = | C it is able to take into account goods
Δk = k_front - k_rear => if Δk < Δk̅ (threshold) => k_rear is reduced compensating the rear brake pressure.
The ideal value is reached again k_rear ↑, Δ = Δk̅ ↔ Δk > Δk̅
if the required braking increases it must k_rear ↑
The tire is not incompressible; it has its own dynamics. For this reason, we model it as a simple system P/T = 1 / ∑i-22c2i+1.
The ABS is very important from a safety point of view, mainly for two reasons linked with the fact that tire spin is able and required to prevent wheel locking:
- You brake in the μwhe_max. => Fx ≅ Fx max => easier
- You can develop lateral force Fy, because k does not become too high.
where Eom => J·ω̇ = T_m - T_b - f_R - M_y, f_x = f_x(f(u,v))
mining T_m,T_b,M_ul,V_x ≅ can => k = ωR - v_x / V_x => ω̇ = kV_x/R, k-k_0 = k
∫k Vx2/R3 = - df_x |f_x
F_R => ∫k = - df_x | k_o
=>⇒
=⇒ = - R2/Vx3 dF_O dF_O/k|f_x => model if df_x/k|f_x
because v_x ≅ can ⇒ λ ≪ 1
where f_x = f_x|kps dkx +dFk/dω|vxu̇ + dFx/dx k/sub>u̇
xL = | x θ | xv = | x1 xL |
M = | M 0 | | 0 J |
K = | k1 + k2 -k1 + k2 -ku -k2 | | (k1 + k2) e2 ke -k2 e | | + k1 0 k2 |
M ¨ + R ¨ + K ¨ = 0 RL xL xv xL xv xL xv xL xL xv = f(t)
if β = kL n → found and knew where we are from → so much before the 1 β found we have in more, we have only we work
if β = 1/2 kL n → found and knew where we have been → only ml is exist, we don’t have heavy water
2β/λ = 2n - 1
is associated to the travelled distance of the suspension, you want to maintain it low to avoid bumps etc., that can give fatigue and mechanical problems.
For commercial cars you tend to improve first bounce frequency, which is mostly related to heave motion of car body. It is because it is the most comfortable for the human body. Vice-versa for race cars you increase where it is better for load handling performance.
NB the model with sprung and unsprung mass, is valid up to ≅20-25 Hz. @ higher frequencies you must consider tire's model. If you want it is important to consider also the air inside the tire, you are entering in rigid field. Also, if l≈2 cm (few cm) you need more complex models because tire's deformability becomes important.
The unsprung mass act as a low-low filter, lowering speed and acceleration amplitudes. The unsprung mass follows the road to generate contact force between tire and road. It moves to leave the sprung mass walking in kinematic-like loop.
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