Relaxation length

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Plot showing lateral force building up as a bicycle tire rolls forward at a 2.4o slip angle. The results from three separate test runs are superimposed. Bicycle tire lateral force vs distance rolled.png
Plot showing lateral force building up as a bicycle tire rolls forward at a 2.4º slip angle. The results from three separate test runs are superimposed.

Relaxation length is a property of pneumatic tires that describes the delay between when a slip angle is introduced and when the cornering force reaches its steady-state value. [1] It is also described as the distance that a tire rolls before the lateral force builds up to 63% of its steady-state value. [2] It can be calculated as the ratio of cornering stiffness over the lateral stiffness, where cornering stiffness is the ratio of cornering force over slip angle, and lateral stiffness is the ratio of lateral force over lateral displacement. [1]

Slip angle

In vehicle dynamics, slip angle or sideslip angle is the angle between the direction in which a wheel is pointing and the direction in which it is actually traveling. This slip angle results in a force, the cornering force, which is in the plane of the contact patch and perpendicular to the intersection of the contact patch and the midplane of the wheel. This cornering force increases approximately linearly for the first few degrees of slip angle, then increases non-linearly to a maximum before beginning to decrease.

Cornering force

Cornering force or side force is the lateral force produced by a vehicle tire during cornering.

In physics and engineering, the time constant, usually denoted by the Greek letter τ (tau), is the parameter characterizing the response to a step input of a first-order, linear time-invariant (LTI) system. The time constant is the main characteristic unit of a first-order LTI system.

Contents

Values

Pacejka gives a rule of thumb that "at nominal vertical load the relaxation length is of the order of magnitude of the wheel radius". [1] Relaxations lengths have been found to be between 0.12 and 0.45 meters, with higher values corresponding to higher velocities and heavier loads. [2] Tests on motorcycle tires have found that the ratio of cornering stiffness over lateral stiffness produces values 20-25% higher than those calculated as 63% of the steady state-value. [3] The relaxation length associated with camber thrust has been found to be nearly zero. [2]

Camber thrust

Camber thrust and camber force are terms used to describe the force generated perpendicular to the direction of travel of a rolling tire due to its camber angle and finite contact patch. Camber thrust is generated when a point on the outer surface of a leaned and rotating tire, that would normally follow a path that is elliptical when projected onto the ground, is forced to follow a straight path while coming in contact with the ground, due to friction. This deviation towards the direction of the lean causes a deformation in the tire tread and carcass that is transmitted to the vehicle as a force in the direction of the lean.

Importance

A tire's relaxation length controls how much the tire will contribute to speed wobble. [4]

Wobble, shimmy, tank-slapper, speed wobble, and even death wobble are all words and phrases used to describe a quick (4–10 Hz) oscillation of primarily just the steerable wheel(s) of a vehicle. Initially, the rest of the vehicle remains mostly unaffected, until translated into a vehicle yaw oscillation of increasing amplitude producing loss of control. Vehicles that can experience this oscillation include motorcycles and bicycles, skateboards, and in theory any vehicle with a single steering pivot point and a sufficient amount of freedom of the steered wheel, including that which exists on some light aircraft with tricycle gear where instability can occur at speeds of less than 80 kmh; this does not include most automobiles. The initial instability occurs mostly at high speed and is similar to that experienced by shopping cart wheels and aircraft landing gear.

See also

Bicycle and motorcycle dynamics

Bicycle and motorcycle dynamics is the science of the motion of bicycles and motorcycles and their components, due to the forces acting on them. Dynamics falls under a branch of physics known as classical mechanics. Bike motions of interest include balancing, steering, braking, accelerating, suspension activation, and vibration. The study of these motions began in the late 19th century and continues today.

Pneumatic trail

Pneumatic trail or trail of the tire is a trail-like effect generated by compliant tires rolling on a hard surface and subject to side loads, as in a turn. More technically, it is the distance that the resultant force of side-slip occurs behind the geometric center of the contact patch.

For vehicles such as cars, vehicle dynamics is the study of how the vehicle will react to driver inputs on a given solid surface.

Related Research Articles

Tire Ring-shaped covering that fits around a wheels rim

A tire or tyre is a ring-shaped component that surrounds a wheel's rim to transfer a vehicle's load from the axle through the wheel to the ground and to provide traction on the surface traveled over. Most tires, such as those for automobiles and bicycles, are pneumatically inflated structures, which also provide a flexible cushion that absorbs shock as the tire rolls over rough features on the surface. Tires provide a footprint that is designed to match the weight of the vehicle with the bearing strength of the surface that it rolls over by providing a bearing pressure that will not deform the surface excessively.

Circle of forces

The circle of forces, traction circle, friction circle, or friction ellipse is a useful way to think about the dynamic interaction between a vehicle's tire and the road surface. The diagram below shows the tire from above, so that the road surface lies in the x-y plane. The vehicle to which the tire is attached is moving in the positive y direction.

Understeer and oversteer

Understeer and oversteer are vehicle dynamics terms used to describe the sensitivity of a vehicle to steering. Oversteer is what occurs when a car turns (steers) by more than the amount commanded by the driver. Conversely, understeer is what occurs when a car steers less than the amount commanded by the driver.

The lowsider or lowside is a type of motorcycle or bicycle crash usually occurring in a turn. It is caused when either the front or rear wheel slides out as a result of either too much braking into the corner, too much acceleration through or out of the corner, or too much speed carried into or through the corner for the available grip. It may also be caused by unexpected slippery or loose material on the road surface.

Weight transfer

Weight transfer and load transfer are two expressions used somewhat confusingly to describe two distinct effects: the change in load borne by different wheels of even perfectly rigid vehicles during acceleration, and the change in center of mass (CoM) location relative to the wheels because of suspension compliance or cargo shifting or sloshing. In the automobile industry, weight transfer customarily refers to the change in load borne by different wheels during acceleration. This is more properly referred to as load transfer, and that is the expression used in the motorcycle industry, while weight transfer on motorcycles, to a lesser extent on automobiles, and cargo movement on either is due to a change in the CoM location relative to the wheels. This article uses this latter pair of definitions.

Rolling resistance rolling friction or rolling drag, is the force resisting the motion when a body (such as a ball, tire, or wheel) rolls on a surface

Rolling resistance, sometimes called rolling friction or rolling drag, is the force resisting the motion when a body rolls on a surface. It is mainly caused by non-elastic effects; that is, not all the energy needed for deformation of the wheel, roadbed, etc. is recovered when the pressure is removed. Two forms of this are hysteresis losses, and permanent (plastic) deformation of the object or the surface. Another cause of rolling resistance lies in the slippage between the wheel and the surface, which dissipates energy. Note that only the last of these effects involves friction, therefore the name "rolling friction" is to an extent a misnomer.

Steering ratio refers to the ratio between the turn of the steering wheel or handlebars and the turn of the wheels.

A Bundorf analysis is a measure of the characteristics of a vehicle that govern its understeer balance. The understeer is measured in units of degrees of additional yaw per g of lateral acceleration.

Tire Uniformity refers to the dynamic mechanical properties of pneumatic tires as strictly defined by a set of measurement standards and test conditions accepted by global tire and car makers. These standards include the parameters of radial force variation, lateral force variation, conicity, plysteer, radial run-out, lateral run-out, and sidewall bulge. Tire makers worldwide employ tire uniformity measurement as a way to identify poorly performing tires so they are not sold to the marketplace. Both tire and vehicle manufacturers seek to improve tire uniformity in order to improve vehicle ride comfort.

Hans Bastiaan Pacejka was an expert in vehicle system dynamics and particularly in tire dynamics, fields in which his works are now standard references. He was Professor emeritus at Delft University of Technology in Delft, Netherlands.

In (automotive) vehicle dynamics, slip is the relative motion between a tire and the road surface it is moving on. This slip can be generated either by the tire's rotational speed being greater or less than the free-rolling speed, or by the tire's plane of rotation being at an angle to its direction of motion.

Self aligning torque

Self aligning torque, also known as aligning torque, aligning moment, SAT, or Mz, is the torque that a tire creates as it rolls along, which tends to steer it, i.e. rotate it around its vertical axis. In the presence of a non-zero slip angle, this torque tends to steer the tire toward the direction in which it is traveling, hence its name.

Motorcycle tyre

Motorcycle tyres are the outer part of motorcycle wheels, attached to the rims, providing traction, resisting wear, absorbing surface irregularities, and allowing the motorcycle to turn via countersteering. The two tyres' contact patches are the motorcycle's connection to the ground, and so are fundamental to the motorcycle's suspension behaviour, and critically affect safety, braking, fuel economy, noise, and rider comfort.

Outline of tires Overview of and topical guide to tires

The following outline is provided as an overview of and topical guide to tires:

References

  1. 1 2 3 Pacejka, Hans B. (2006). Tyre and vehicle dynamics (2nd ed.). SAE International. p. 22. ISBN   978-0-7680-1702-1. The relaxation length ... is an important parameter that controls the lag of the response of the side force to the input slip angle.
  2. 1 2 3 Cossalter, Vittore (2006). Motorcycle Dynamics (Second ed.). Lulu.com. p. 58. ISBN   978-1-4303-0861-4. The relaxation length represents the distance the wheel has to cover in order for the lateral force to reach 63% of the steady state force.
  3. R.T. Uil (June 2006). "Non-lagging effect of motorcycle tyres: An experimental study with the Flat Plank Tyre Tester" (PDF). Eindhoven University of Technology, Department of Mechanical Engineering. p. 14. Retrieved 2010-10-31. For the relaxation behaviour of motorcycle tyres, it can be concluded that the relaxation length for side slip defined by Pacejka is higher than the relaxation length determined from the steady-state side force.
  4. Peter Willem Anton Zegelaar (March 1998). "The Dynamic Response of Tyres to Brake Torque Variations and Road Unevennesses" (PDF). Delft University of Technology. p. 16. Archived from the original (PDF) on 2013-10-04. Retrieved 2013-10-02. the relaxation length plays an important role in the ‘shimmy’ phenomenon.