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43-507: Lowering the air pressure in a tire creates a larger contact patch between the tire and the ground, and makes driving on softer ground much easier. It also does less damage to the tire surface. This is important on work sites and in agricultural fields; by giving the driver direct control over the air pressure in each tire, maneuverability is greatly improved. Softer tires also cushion against rough terrain and road damage, such as washboarding , more effectively. Reducing tire pressure also reduces

86-412: A distributed load p ( x ) {\displaystyle p(x)} is applied to the surface instead, over the range a < x < b {\displaystyle a<x<b} . The principle of linear superposition can be applied to determine the resulting stress field as the solution to the integral equations: The same principle applies for loading on the surface in

129-416: A doubling of transmission and differential life. The technology is extensively used in many off-road transport operations. In many countries, especially Australia, New Zealand and South Africa, CTIS is used in logging, mining, and power line maintenance, as it significantly reduces environmental impact when transporting logs, or travelling on gravel or dirt roads. From 1984, General Motors offered CTIS for

172-504: A nonlinear function of the deformation. To simplify the solution procedure, a frame of reference is usually defined in which the objects (possibly in motion relative to one another) are static. They interact through surface tractions (or pressures/stresses) at their interface. As an example, consider two objects which meet at some surface S {\displaystyle S} in the ( x {\displaystyle x} , y {\displaystyle y} )-plane with

215-463: A railroad wheel is much smaller than for pneumatic rubber tires; it is only about the size of a dime (252 mm ). Contact mechanics Contact mechanics is the study of the deformation of solids that touch each other at one or more points. A central distinction in contact mechanics is between stresses acting perpendicular to the contacting bodies' surfaces (known as normal stress ) and frictional stresses acting tangentially between

258-466: Is The pressure distribution is given by The stress has a logarithmic singularity at the tip of the cone. In contact between two cylinders with parallel axes, the force is linearly proportional to the length of cylinders L and to the indentation depth d : The radii of curvature are entirely absent from this relationship. The contact radius is described through the usual relationship with as in contact between two spheres. The maximum pressure

301-480: Is a boundary value problem of linear elasticity subject to the traction boundary conditions : where δ ( x , z ) {\displaystyle \delta (x,z)} is the Dirac delta function . The boundary conditions state that there are no shear stresses on the surface and a singular normal force P is applied at (0, 0). Applying these conditions to the governing equations of elasticity produces

344-431: Is allowed to occur within the contact area, i.e., contacting bodies can be separated without adhesion forces. Several analytical and numerical approaches have been used to solve contact problems that satisfy the no-adhesion condition. Complex forces and moments are transmitted between the bodies where they touch, so problems in contact mechanics can become quite sophisticated. In addition, the contact stresses are usually

387-470: Is considered highly proprietary and, therefore, very little is published on the subject. Because pneumatic tires are flexible, the contact patch can be different when the vehicle is in motion than when it is static. Because it is so much easier to make observations of the contact patch without the tire in motion, it is more common to conduct studies of the static contact patch. Statically, the size, shape, and pressure distribution are functions of many things,

430-516: Is equal to The contact in the case of bearings is often a contact between a convex surface (male cylinder or sphere) and a concave surface (female cylinder or sphere: bore or hemispherical cup ). Some contact problems can be solved with the method of dimensionality reduction (MDR). In this method, the initial three-dimensional system is replaced with a contact of a body with a linear elastic or viscoelastic foundation (see fig.). The properties of one-dimensional systems coincide exactly with those of

473-400: Is in actual contact with the road surface . It is commonly used in the discussion of pneumatic (i.e. pressurized) tires, where the term is used strictly to describe the portion of the tire’s tread that touches the road surface. The term “footprint” is used almost synonymously. Solid wheels also exhibit a contact patch which is generally smaller than the pneumatic “footprint”. The contact patch

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516-470: Is often used as a starting point and then generalized to various shapes of the area of contact. The force and moment balances between the two bodies in contact act as additional constraints to the solution. A starting point for solving contact problems is to understand the effect of a "point-load" applied to an isotropic, homogeneous, and linear elastic half-plane, shown in the figure to the right. The problem may be either plane stress or plane strain . This

559-412: Is pressed into an elastic half-space, it creates a pressure distribution described by where R {\displaystyle R} is the radius of the cylinder and The relationship between the indentation depth and the normal force is given by In the case of indentation of an elastic half-space of Young's modulus E {\displaystyle E} using a rigid conical indenter,

602-466: Is small compared to the sizes of the objects and the stresses are highly concentrated in this area. Such a contact is called concentrated , otherwise it is called diversified . A common approach in linear elasticity is to superpose a number of solutions each of which corresponds to a point load acting over the area of contact. For example, in the case of loading of a half-plane , the Flamant solution

645-668: Is the maximum contact pressure given by The radius of the circle is related to the applied load F {\displaystyle F} by the equation The total deformation d {\displaystyle d} is related to the maximum contact pressure by The maximum shear stress occurs in the interior at z ≈ 0.49 a {\displaystyle z\approx 0.49a} for ν = 0.33 {\displaystyle \nu =0.33} . For contact between two spheres of radii R 1 {\displaystyle R_{1}} and R 2 {\displaystyle R_{2}} ,

688-402: Is the only connection between the road and the vehicle. The size and shape of the contact patch, as well as the pressure distribution within the contact patch, are important to the ride qualities and handling characteristics of a vehicle. Since the wear characteristics of tires is a highly competitive area between tire manufacturers , a lot of the research done concerning the contact patch

731-626: The z {\displaystyle z} -axis assumed normal to the surface. One of the bodies will experience a normally-directed pressure distribution p z = p ( x , y ) = q z ( x , y ) {\displaystyle p_{z}=p(x,y)=q_{z}(x,y)} and in-plane surface traction distributions q x = q x ( x , y ) {\displaystyle q_{x}=q_{x}(x,y)} and q y = q y ( x , y ) {\displaystyle q_{y}=q_{y}(x,y)} over

774-615: The Chevrolet Blazer and various pickups . There have been attempts at employing central tire inflation systems on aircraft landing wheels (notably on the Soviet Antonov An-22 military transport) to improve their preparedness for unpaved runways . CTIS was first used in production on the American DUKW amphibious truck , which was introduced in 1942. The Czech Tatra T813 's central inflation and deflation system

817-452: The elastic moduli and ν 1 {\displaystyle \nu _{1}} , ν 2 {\displaystyle \nu _{2}} the Poisson's ratios associated with each body. The distribution of normal pressure in the contact area as a function of distance from the center of the circle is where p 0 {\displaystyle p_{0}}

860-583: The Flamant solution for the 2D half-plane, fundamental solutions are known for the linearly elastic 3D half-space as well. These were found by Boussinesq for a concentrated normal load and by Cerruti for a tangential load. See the section on this in Linear elasticity . Distinctions between conforming and non-conforming contact do not have to be made when numerical solution schemes are employed to solve contact problems. These methods do not rely on further assumptions within

903-627: The Johnson et al. model came to be known as the Johnson–Kendall–Roberts (JKR) model for adhesive elastic contact. This rejection proved to be instrumental in the development of the David Tabor and later Daniel Maugis parameters that quantify which contact model (of the JKR and DMT models) represent adhesive contact better for specific materials. Further advancement in the field of contact mechanics in

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946-398: The area of contact is a circle of radius a {\displaystyle a} . The equations are the same as for a sphere in contact with a half plane except that the effective radius R {\displaystyle R} is defined as This is equivalent to contact between a sphere of radius R {\displaystyle R} and a plane . If a rigid cylinder

989-438: The contact area is approximately proportional to the normal force . Further important insights along these lines were provided by Jonh A. Greenwood and J. B. P. Williamson (1966), A. W. Bush (1975), and Bo N. J. Persson (2002). The main findings of these works were that the true contact surface in rough materials is generally proportional to the normal force, while the parameters of individual micro-contacts (pressure and size of

1032-433: The contact problem of two elastic bodies with curved surfaces. This still-relevant classical solution provides a foundation for modern problems in contact mechanics. For example, in mechanical engineering and tribology , Hertzian contact stress is a description of the stress within mating parts. The Hertzian contact stress usually refers to the stress close to the area of contact between two spheres of different radii. It

1075-420: The depth of the contact region ϵ {\displaystyle \epsilon } and contact radius a {\displaystyle a} are related by with θ {\displaystyle \theta } defined as the angle between the plane and the side surface of the cone. The total indentation depth d {\displaystyle d} is given by: The total force

1118-459: The extent to which the tires grind against loose surfaces, significantly reducing dust and silt. Another function of CTIS is to maintain tire pressure if there is a slow leak or puncture. In this case, the system controls inflation automatically based on the selected pressure the driver has set. CTIS also extends truck, tire, and drive train life, by significantly reducing vibration and shock loading. Feedback from Australian logging contractors show

1161-484: The field may include stress analysis of contact and coupling members and the influence of lubrication and material design on friction and wear . Applications of contact mechanics further extend into the micro - and nanotechnological realm. The original work in contact mechanics dates back to 1881 with the publication of the paper "On the contact of elastic solids" "Über die Berührung fester elastischer Körper" by Heinrich Hertz . Hertz attempted to understand how

1204-400: The geometry of the area of contact. A conforming contact is one in which the two bodies touch at multiple points before any deformation takes place (i.e., they just "fit together"). A non-conforming contact is one in which the shapes of the bodies are dissimilar enough that, under zero load, they only touch at a point (or possibly along a line). In the non-conforming case, the contact area

1247-400: The micro-contact) are only weakly dependent upon the load. The theory of contact between elastic bodies can be used to find contact areas and indentation depths for simple geometries. Some commonly used solutions are listed below. The theory used to compute these solutions is discussed later in the article. Solutions for multitude of other technically relevant shapes, e.g. the truncated cone,

1290-704: The mid-twentieth century may be attributed to names such as Frank Philip Bowden and Tabor. Bowden and Tabor were the first to emphasize the importance of surface roughness for bodies in contact. Through investigation of the surface roughness, the true contact area between friction partners is found to be less than the apparent contact area. Such understanding also drastically changed the direction of undertakings in tribology. The works of Bowden and Tabor yielded several theories in contact mechanics of rough surfaces. The contributions of J. F. Archard (1957) must also be mentioned in discussion of pioneering works in this field. Archard concluded that, even for rough elastic surfaces,

1333-438: The most important of which are the load on the tire and the inflation pressure : These two properties are not linearly proportional to the area of the contact. For example, a 10% change in load or inflation pressure usually does not result in a 10% change in the contact patch area because the load or pressure on a tire can be altered freely, and the contact patch area is affected by the tire geometry and stiffness. Further,

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1376-399: The normal contact force, the radii of curvature of both bodies and the modulus of elasticity of both bodies. Hertzian contact stress forms the foundation for the equations for load bearing capabilities and fatigue life in bearings, gears, and any other bodies where two surfaces are in contact. Classical contact mechanics is most notably associated with Heinrich Hertz. In 1882, Hertz solved

1419-403: The optical properties of multiple, stacked lenses might change with the force holding them together. Hertzian contact stress refers to the localized stresses that develop as two curved surfaces come in contact and deform slightly under the imposed loads. This amount of deformation is dependent on the modulus of elasticity of the material in contact. It gives the contact stress as a function of

1462-563: The original three-dimensional system, if the form of the bodies is modified and the elements of the foundation are defined according to the rules of the MDR. MDR is based on the solution to axisymmetric contact problems first obtained by Ludwig Föppl (1941) and Gerhard Schubert (1942) However, for exact analytical results, it is required that the contact problem is axisymmetric and the contacts are compact. The classical theory of contact focused primarily on non-adhesive contact where no tension force

1505-446: The plane of the surface. These kinds of tractions would tend to arise as a result of friction. The solution is similar the above (for both singular loads Q {\displaystyle Q} and distributed loads q ( x ) {\displaystyle q(x)} ) but altered slightly: These results may themselves be superposed onto those given above for normal loading to deal with more complex loads. Analogously to

1548-660: The region S {\displaystyle S} . In terms of a Newtonian force balance, the forces: must be equal and opposite to the forces established in the other body. The moments corresponding to these forces: are also required to cancel between bodies so that they are kinematically immobile. The following assumptions are made in determining the solutions of Hertzian contact problems: Additional complications arise when some or all these assumptions are violated and such contact problems are usually called non-Hertzian . Analytical solution methods for non-adhesive contact problem can be classified into two types based on

1591-418: The result for some point, ( x , y ) {\displaystyle (x,y)} , in the half-plane. The circle shown in the figure indicates a surface on which the maximum shear stress is constant. From this stress field, the strain components and thus the displacements of all material points may be determined. Suppose, rather than a point load P {\displaystyle P} ,

1634-410: The size of the contact patch cannot be simply calculated as load divided by inflation pressure, and the average contact pressure a tire has with the road surface is not equal to the inflation pressure. The contact patch size of solid materials is described by the equations of contact mechanics . It is mainly related to the stiffness of the material in terms of Young's modulus . The contact patch of

1677-777: The subject is built upon the mechanics of materials and continuum mechanics and focuses on computations involving elastic , viscoelastic , and plastic bodies in static or dynamic contact. Contact mechanics provides necessary information for the safe and energy efficient design of technical systems and for the study of tribology , contact stiffness , electrical contact resistance and indentation hardness . Principles of contacts mechanics are implemented towards applications such as locomotive wheel-rail contact, coupling devices, braking systems, tires , bearings , combustion engines , mechanical linkages , gasket seals, metalworking , metal forming, ultrasonic welding , electrical contacts , and many others. Current challenges faced in

1720-424: The surfaces ( shear stress ). Normal contact mechanics or frictionless contact mechanics focuses on normal stresses caused by applied normal forces and by the adhesion present on surfaces in close contact, even if they are clean and dry. Frictional contact mechanics emphasizes the effect of friction forces. Contact mechanics is part of mechanical engineering . The physical and mathematical formulation of

1763-626: The worn sphere, rough profiles, hollow cylinders, etc. can be found in An elastic sphere of radius R {\displaystyle R} indents an elastic half-space where total deformation is d {\displaystyle d} , causing a contact area of radius The applied force F {\displaystyle F} is related to the displacement d {\displaystyle d} by where and E 1 {\displaystyle E_{1}} , E 2 {\displaystyle E_{2}} are

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1806-496: Was designed to maintain pressure even after multiple bullet punctures. Military-spec Tatra trucks are equipped with CTIS as standard. Several trucks used by the U.S. military, such as the HMMWV , HEMTT , and FMTV , are equipped with CTIS. The feature is also common in Soviet and Russian military trucks. Contact patch The contact patch is the portion of a vehicle's tire that

1849-549: Was not until nearly one hundred years later that Kenneth L. Johnson , Kevin Kendall , and Alan D. Roberts found a similar solution for the case of adhesive contact. This theory was rejected by Boris Derjaguin and co-workers who proposed a different theory of adhesion in the 1970s. The Derjaguin model came to be known as the Derjaguin–Muller–Toporov (DMT) model (after Derjaguin, M. V. Muller and Yu. P. Toporov), and

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