This paper presents to determine the contact stress of large diameter ball bearings using analytical and numerical methods. The hertzian model of contact does not consider adhesion possible. This is a pdf file of an unedited manuscript that has been accepted for publication. Simplified computation of edgeloaded contact stresses could. To engineer a contact condition like that the cylinder needs a profile with end reliefs. Hertzian contact practical adhesion science prof steven.
As a preparatory step, we will summarize a few results of the theory of elasticity that have a direct application to contact mechanics. Hertz law has been used beyond the limits of its validity on the. This is true for tires, needle bearings and other roller type of devices. Assessment of the applicability of the hertzian contact. Hertzian contact pressure is applied in a fatigue life model to describe the damage of wheels in 4. Contact stresses and deformations university of utah. In mechanical engineering and tribology, hertzian contact stress is a description.
The theory of contact between elastic bodies can be used to find contact areas and. Hertz in terms of the solution of a transcendental equation involving elliptic integrals and used by him to describe the deformation of bodies subjected to contact stresses. Contact of bodies occurs in all interfaces transmitting force or motion, or both. Examples are a railway wheel and rail, or a pair of gear wheel teeth. Hertzian contact stress theory university of arizona. Theoretical background and applications of dem simon lo. The hertzian theory has previously been used to describe the contact mechanics of the congruent hip couple 4, but not the edgeloaded couple. Present study utilizes the application of hertz contact theory to the wheelrail contact problem to study the effect of variation of wheel.
For example, the hct is applied in 2, 3 to investigate the growth of shelltype defects in the head of a railroad rail. An approximate analysis for hertzian elliptical wheelrail. Hertzian theory has previously been used to describe the contact mechanics of the congruent hip couple 4, but not the edgeloaded couple. When applicable, the hertz theory can be used to accurately model contact dimensions, normal pressures, and stress fields. In 1882, hertz solved the contact problem of two elastic bodies with curved surfaces. Me 383s bryant february 17, 2006 1 contact mechanical interaction of bodies via surfaces surfaces must touch forces press bodies together size area of contact dependent on forces, materials, geometry, temperature, etc.
Maximum stress stress occurs at 45 degree to the normal stress, and it can cause deformation by movement of dislocation. In 1880 heinrich hertz developed his theory for contact stress after studying newtons rings with two glass lenses. Assessment of the applicability of hertzian contact theory. Tutorial report on hertzian contact stress theory, opti 521. Introduction when there are two nonconforming surfaces that come into contact the point or line of contact can be a location of very high stress values. Hertzian loaddisplacement relation holds for spherical. Jkr adhesion theory starts with measurements of little rubber balls being pressed into contact with a flat surface.
Spherical contact is a special case of the hertz contact, which is between two spheres, or between a sphere and the surface of a half space. Failure analysis of rolling bearings based on explicit. Pdf hertz contact stress analysis and validation using finite. Lecture 28 antennas and radiation and the hertzian dipole.
Stress concentration in periodically rough hertzian contact. Hertz contact stress mechanical engineering general. Hertz developed a theory to calculate the contact area and pressure between the two surfaces and predict the resulting compression and stress induced in the. E lr a, where r is the radius of curvature of the probing sfm tip, and e is the modulus of the sample only, if the sample material stiffness is much smaller than the. You could divide the hertz stresses by the same factors before comparing them to your allowables. Dowson, d fisher, j jin, z m auger, d d and jobbins, b 1991. The finite element analysis also performed to predict contact pressure in. It compares experiment results with predictions based on hertzs original theories. This stillrelevant classical solution provides a foundation for modern problems in contact mechanics. It applies to normal contact between two elastic solids that are smooth and can be described locally with orthogonal radii of curvature such as a toroid.
The result was the first satisfactory theory for contact mechanics and is still in use today. General theory a general b geometry of the unstressed surface in the region of contact c equations for area of contact, pressure distribution and compression iii. Another contact case entails friction that causes shear stresses on the contact zone. Hertz contact stress involutes gear tooth contact stress analysis one of the main gear tooth failure is pitting which is a surface fatigue failure due to many repetition of high contact stresses occurring in the gear tooth. In dynamic wheelrail models, the hertzian contact stiffness is an important parameter. Tutorial on hertz contact stress university of arizona. Below contacting surface, sigma x, sigma z, and shear stress varies with depth. Relation between hertz stresslife exponent, ballrace conformity, and ball bearing life. These elevated stresses are beyond the hertzian field. Fischer summary railwheel contact problems have been analyzed by applying threedimensional. Ai palmgren revisited a basis for bearing life prediction erwin v.
Cone cracks and the auerbach relationship in diamond. With the application of the smallest load, elastic deformation enlarges these into contact areas across which the loads are distributed as. Hertz contact stress of deep groove ball bearing sciencedirect. The company was founded in 2011 and is majority owned by fouriertransform ab. The following assumptions are made in determining the solutions of hertzian contact problems. Therefore, it is very imporant to calculate the actual stress at the point of contact, the socalled contact stress by theory of elasticity, developed by hertz. Oct 16, 2019 the smaller difference between simulated results obtained by edm and theoretical results calculated by hertzian point contact theory validates the reliability of explicit dynamic model established in this paper. Vicura ab is an engineering company that delivers products which are used in different mechanical transmission systems and electrical drivelines. Elastic compression of spheres and cylinders at point and. Lecture 28 antennas and radiation and the hertzian dipole in this lecture you will learn. Jan 21, 2014 2014w engr380 lecture18 spur gear design lewis equation for tooth bending stress, part i duration.
When adhesion between the two bodies in contact is considered, several modifications from the hertz theory have been proposed. Hertz mindlin no slip this contact model is the default model used in edem due to its accurate and efficient force calculation. Although hertzian contact theory has been applied in the case of concentric ceramiconceramic hip contact. Jones 10 relates the hertz contact theory for the stresses of nonconforming bodies in contact for both ball and roller. This paper presents a study by the finite element method fem of the contact tensions between two bodies and the results obtained are compared with the calculations obtained on basis of the relations of the theory of h. In this model the normal force component is based on hertzian contact theory hertz 1882. C contact mechanicsi the theory developed by hertz in 1880 remains the foundation for most contact problems encountered in engineering. Pdf rigorous treatment of contact problems hertzian contact. Based on these models, the applicability of the hertz contact theory hct to railwheel contact problems is veri. The limits normally adopted for hertz s contact stresses are 4 times the allowable stress for linear contact and 5.
However, in the late 1960s, several contradictions were observed when the hertz theory was compared with experiments involving contact between rubber and glass spheres. Stresses surface tractions and sliding contact cylinder. Special cases a two spheres in contact b sphere in contact with a plane c sphere in contact with an internal sphere d equal cylinders. If determined to be applicable to edgeloaded bearings, the hertz. Assessment of the applicability of the hertzian contact theory to edgeloaded prosthetic hip bearings. Hertzian elastic contact and jkr adhesiveelastic contact as function of load. Archive of applied mechanics 70 2000 255268 o springer.
When any two curved bodies of different radii of curvature are brought into contact they will initially touch at either a point or along a line. It was observed that, though hertz theory applied at. Morning coffee jazz relaxing instrumental bossa nova jazz playlist have a nice day. Palmgren revisited a basis for bearing life prediction. Pdf hertz contact stress analysis and validation using. Nov 10, 2011 although hertzian contact theory has been applied in the case of concentric ceramiconceramic hip contact mak and jin, 2002, its merits and shortcomings have not been assessed in abnormal states and multiple material couples. The displacement of the points on the surface in the contact area between an originally even surface and a rigid sphere of radius r is equal to uz d. If determined to be applicable to edgeloaded bearings, the hertz theory would offer the benefit of an analytical solution to a complicated problem. Step by step procedure of how to do static structural analysis of hertzian contact analysis ansys workbench. Tables of hertzian contact stress coefficients duane h. The hertz contact theory consists of a series of mathematical equations to calculate normal, hoop, radial, and shear stress components on and within a soft flat surface or halfspace, known as the specimen, when pressed by a rigid hemisphere called the indentor appendix figure 1 1.
The finite element analysis also performed to predict contact pressure in ball and raceway. Hertz 1882 contact model is both elastic and quasistatic in nature. Equations 7 and 8 can be used to determine the contact parameters between a cylinder and an internal spherical surface or a flat surface. Design considerations for cushion form bearings in artificial hip joints. Nov 15, 2008 an example of hertz contact fem simulation. He became concerned about the effect of contact pressure between the two lenses and set out to analyze the effects. Tutorial hertz contact stress brian taylor opti521 fall 2016, sid. Roller bearing contact stress causes deformation, plastic or elastic. Cooper abstract formulations are given for the coefficients x, u, u defined by h.
Experimental study of the force pulse and resulting stress waves gregory c. The unit of frequency, cycle per second, was named the hertz in his honor. This treatise is concerned with the stresses and deformation of solid bodies in contact with each other, along curved surfaces which touch initially at a point or along a line. Hertz is well known for his contributions to the field of electrodynamics see below. Hertzian contact theory is a classical theory of contact mechanics and is a very useful tool for engineers and researchers. Normal contact of elastic solids hertz theory chapter 4.
If we determine the size and shape of this contact area and the distribution of normal pressure, then the interval stresses and deformation can be calculated. Applicability of the hertz contact theory to railwheel contact problems w. In analytical method the contact stress is found out using the hertzian elliptical contact theory. We can also do wear analysis due to frictional rubbing between the contact. Hertzian contact theory applied to edgeloaded ceramicon. A theoretical background to the hertz contact theory can be found here. Therefore, it is very imporant to calculate the actual stress at the point of contact, the socalled contact stress.
The hertzian contact apparatus is a selfcontained and easytouse unit that shows the nature of contact between two surfaces. Dem interaction hertz mindlin contact this is a noslip contact model used for modeling contact between particles the hertz mindlin contact model is a variant of the nonlinear springdashpot contact model based on the hertz mindlin contact theory the forces. The hertzian theory of elastic circular point contact for a planar surface and an assumed spherically caped tip, fig. Pure hertzian model only describes normal deformation of spheres. Glaser department of civil and environmental engineering, university of california, berkeley, 621a sutardja dai. To understand the effects of adhesion we first have to see what happens when theres no adhesion, which is the hertz case. Hertzian theory of nonadhesive elastic contact in hertzs classical theory of contact, he focused primarily on nonadhesive contact where no tension force is allowed to occur within the contact area. The simplest modification is to crown the cylinder that the contact is of double radius type, just as hertzian theory handles. Even though the derivation of the theory is relatively difficult, the final solution is a set of simple analytical equations relating the properties of the system to the developed stress.
In a static contact, the interface is the principal plane no shear stress. Next he discusses the influence of friction and the topographical roughness of surfaces, and this is incorporated into the theory of contact mechanics. A track structure and wheel set connected by means of a hertzian spring is used to describe high frequency. For an electronic copy pdf of the relevant report, click here consider a simple problem that illustrates a comparison between the analytical and numerical results. Generation of radiation by oscillating charges and currents hertzian dipole antenna ece 303 fall 2005 farhan rana cornell university maxwells equations and radiation maxwells equation predict outgoing radiation from sinusoidally. Hertzian definition of hertzian by the free dictionary. Later extensions of the model add viscosity, shear response including friction and adhesion.
This helps engineers to predict contact areas between common machined surfaces and materials, for example different types of bearings. Hertzian contact equations for elliptical, spherical and. The tangential force model is based on mindlinderesiewicz work mindlin 1949 mindlin and deresiewicz 1953. Force modulation sfm and hertzian theory the hertzian theory of elastic circular point contact for a planar surface and an assumed spherically caped tip, fig.
Equations 7 and 8 can be used to determine the contact parameters between a cylinder and an internal spherical surface or a. E lr a, where r is the radius of curvature of the probing sfm tip, and e is the modulus of the sample only, if the sample material stiffness is much smaller than the modulus of the cantilever material. Such as the hertzian theory, the jkr solution is also restricted to elastic sphere contacts. An improvement over the hertzian theory was provided by johnson et al. This paper discusses the hertz contact theory validation using finite element analysis. Contact mechanics is the study of the deformation of solids that touch each other at one or more. Keywords hertz contact stress, contact fe analysis, stresses within rollers, roller contact analysis and.
After hertzs work, people do a lot of study on the stresses arising from the contact between two elastic bodies. On the compression of a cylinder in contact with a plane. Contact mechanics in 1881 1882, hertz published two articles on what was to become known as the field of contact mechanics. The hertz contact theory the hertz contact theory consists of a series of mathematical equations to calculate normal, hoop, radial, and shear stress components on and within a soft flat surface or halfspace, known as the specimen, when pressed by a rigid hemisphere called the indentor appendix figure 1 1. Contact stresses and deformations me en 7960 precision machine design topic 7 me en 7960 precision machine design contact stresses and deformations 72 curved surfaces in contact the theoretical contact area of two spheres is a point 0dimensional the theoretical contact area of two parallel cylinders is a line 1. Assessment of the applicability of hertzian contact theory to. Hertzian contact stress refers to the localized stresses that develop as two curved. Palmgren, as will be discussed, distrusted hertz theory 12 and depended on a loadlife relation.
Relation between hertz stresslife exponent, ballrace. Zaretsky national aeronautics and space administration lewis research center cleveland, ohio 445 summary bearing technology, as well as the bearing industry, began to develop with the invention of the bicycle in. The contact area will change depending on the magnitude of the contact stress. The second missing element was a comprehensive life theory that would fit the observations of palmgren. Photoelasticity analysis of hertz contact problem shear stresses. When the load was applied there is some elastic deformation that takes place to provide a larger area so that the stresses are not in nite. The hertzian contact hertz 1881, 1882 is referred to the frictionless contact between two bodies. Ball bearing thrust bearing roller bearing contact stress causes deformation, plastic or elastic. Assessment of the applicability of hertzian contact theory to edgeloaded prosthetic hip bearings.
1522 1079 68 1210 874 642 690 170 1121 1608 849 1041 1482 1548 978 952 139 1234 1033 1205 403 698 188 1573 1034 552 1614 1620 495 895 717 1284 162 673 1055 1324 1121 1416 1284 1173 498 1414 926 91 415