Longitudinal Dynamics of Driven Tire

Longitudinal Dynamics of Driven Tire

• Torque applied to a tire results in longitudinal force within the contact patch.
• Low stiffness of tire components leads to deformation within the contact patch.
• Driving torque creates compressive strain in the tread as it enters the contact patch.
• The tread length within the contact patch of a driven tire is less than that of a free-rolling tire.
• The effective rolling radius of a driven tire is less than the rolling radius of a free-rolling tire.

Tire Radii Definitions

• The effective radius of a tire depends on the circumstances.
• Geometric/unloaded radius ((r_u)) can be estimated from the tire size code.
• Free-rolling/loaded radius (r) is affected by tire deformation due to load, which reduces the effective radius.
• Driving/braking radius ((r_e)) experiences strain in the contact patch, altering the effective perimeter and, thus, the effective radius.
• (r_u > r > r_e)

Tire Slip (Driven Tire)

• Slip (i = (1 - \frac{V}{r\omega}) \times 100%), which can also be expressed as (i = (1 - \frac{r_e}{r}) \times 100%).
• As rotational speed increases for a given vehicle speed, slip increases.
• Slip approaches 100% if (\omega > 0) and (V = 0).
• A spinning tire has (\omega > 0) and either (V = 0) or (V \neq f(\omega)), where (f(\omega)) is some function of (\omega).

Effect of Tire Slip on Driving Force

• Slip is required to generate longitudinal force.

Julien's Theory for Driven Tire Longitudinal Dynamics

• The tire tread is treated as a 2-D elastic band.
• Sidewall is ignored.
• Normal force is approximated as uniformly distributed within the contact patch.
• The contact patch is divided into two zones: adhesion and sliding.
• Sliding within the contact patch is not the same as tire slip; a tire can slip even without sliding in the contact patch if (r_e < r).

Simplified Julien's Theory Results: Driven Tire

• Slip ratio i = (1 - \frac{V}{r\omega} \cdot 100%)
• The contact patch is purely in adhesion when (F_x = C_i \cdot i) for (i < i_{crit}), with (i_{crit} = \frac{\mu_p W}{2C_i}) and (C_i = \frac{k_t L_t^2}{2})
• There are mixed adhesion/sliding zones in the contact patch when (F_x = \mu_p W (1 - \frac{\mu_p W}{4 C_i i})) for (i_{crit} < i < i_{peak}).
• The peak tractive force, when exactly 100% of the contact patch is sliding, is (F_x = \mu_p W) at (i = i_{peak}).
• Beyond the peak slip ratio, (F_x \rightarrow \mu_s W) for (i > i_{peak}).
• Simplified Julien's Theory does not predict (i_{peak}).

Tire Skid (Braked Tire)

• Skid definition for a braked tire: (i_s = (1 - \frac{r\omega}{V}) \cdot 100%), which can also be written as (i_s = (1 - \frac{r}{r_e}) \cdot 100%).
• Skid increases as the rotational speed of the tire decreases for a given vehicle speed.
• Skid approaches 100% if (\omega = 0) and (V > 0).

Simplified Julien's Theory Results: Braked Tire

• Skid ratio, (i_s = (1 - \frac{r\omega}{V}) \times 100%), also expressed as (i = \frac{-i_s}{1-i_s}) and (i_s = \frac{-i}{1-i}).
• The contact patch is purely in adhesion when (F_x = C_s \frac{i_s}{1-i_s}) for (i_s < i_{s,crit}), where (i_{s,crit} = \frac{\mu_p W}{2C_s + \mu_p W}), and (\frac{dF_x}{di_s}|_{i_s=0})
• There are mixed adhesion/sliding zones when (F_x = \mu_p W (1 - \frac{\mu_p W (1 - i_s)}{4 C_s i_s})) for (i_{s,crit} < i < i_{s,peak}).
• Peak braking force (when exactly 100% of C.P. is sliding) is (F_x = \mu_p W) at (i = i_{s,peak})
• The braking force beyond the peak skid ratio is (F_x \rightarrow \mu_s W) for (i > i_{s,peak})
• Simplified Julien's Theory does not predict (i_{s,peak}).

Effect of Construction & Surface on Braking

• Braking effort coefficient is defined as Fx/Fz.

Variation of Peak & Sliding Friction

• Typical values for road adhesion coefficients:
  • Asphalt & Concrete (Dry): Peak Value ((\mu_p)) = 0.8-0.9, Sliding Value ((\mu_s)) = 0.75
  • Asphalt (Wet): Peak Value = 0.5-0.7, Sliding Value = 0.45-0.6
  • Concrete (Wet): Peak Value = 0.8, Sliding Value = 0.7
  • Gravel: Peak Value = 0.6, Sliding Value = 0.55
  • Earth Road (Dry): Peak Value = 0.68, Sliding Value = 0.65
  • Earth Road (Wet): Peak Value = 0.55, Sliding Value = 0.4-0.5
  • Snow (Hard-packed): Peak Value = 0.2, Sliding Value = 0.15
  • Ice: Peak Value = 0.1, Sliding Value = 0.07
• The values of (\mu_p) & (\mu_s) depend more on the surface and rubber composition rather than tire construction style and inflation pressure.
• The peak value is possible to be greater than 1.

Additional Effects: Threshold Speed Due to Hysteresis

• Standing waves can occur in the tread surface at high speeds because the deformation from the contact patch does not relax quickly enough.
• Standing waves result in repeated cycles of compression and relaxation, creating substantial heat through hysteresis.
• Rolling resistance is increased greatly and rapidly because of standing waves through hysteresis.
• Threshold speed is given by (V_{thresh} \approx \sqrt{F_t/\rho_t}), where (F_t) is the tension in the tread and (\rho_t) is the tread density per unit area.

Hydroplaning Mechanism

• Hydroplaning happens when the water film volume flux is greater than the tread evacuation capacity, thus creating a hydrodynamic bearing film between the tire and road.
• Hydroplaning happens when the film has enough volume to support the load on the tire approximated by the inflation pressure and contact patch area.
• Hydrodynamic force is proportional to the contact area and the square of its velocity for slippery or smooth tires.

Additional Effects - Hydroplaning

• Peak and sliding friction coefficients are lowered on wet surfaces.

Hydroplaning Resistance

• Tread ribs (longitudinal grooves) reduce the lateral distance water must be displaced, increasing hydroplaning resistance.
• Tread sipes (lateral grooves) increase lateral flow capacity to improve hydroplaning resistance.
• The onset of hydroplaning for grooved or grooved & siped tires primarily depends on inflation pressure; greater pressure helps displace water from under the contact patch.
• Hydroplaning speed:
  • (V_{p(km/h)} \approx 6.34\sqrt{p_i (kPa)})
  • (V_{p(mph)} \approx 10.35\sqrt{p_i (psi)})