[{"data":1,"prerenderedAt":43},["ShallowReactive",2],{"topic-en-strength-of-materials\u002Fcontact-stresses\u002Fcontact-of-cylinders-and-rollers":3},{"topic":4,"trail":28,"children":38,"tasks":39,"alternates":40},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},130,"Contact of Cylinders and Rollers","en","strength-of-materials\u002Fcontact-stresses\u002Fcontact-of-cylinders-and-rollers","Contact of Cylinders and Rollers — Hertz Line Contact","Hertz line contact of cylinders and rollers: contact-strip width, maximum pressure, reduced properties, edge effects, and engineering applications.","This topic explains the Hertz line-contact model for cylinders and rollers. It covers load per unit length, reduced elastic modulus and curvature, contact-strip half-width, maximum pressure, a numerical example, edge effects, and applications to rollers and rolling bearings.","\u003Cp>\u003Cstrong>Contact of cylinders and rollers\u003C\u002Fstrong> is a typical model of initially line contact. For long parallel cylinders under a normal force $F$, the initial contact line expands into a narrow strip.\u003C\u002Fp>\u003Ch2>Load per unit length\u003C\u002Fh2>\u003Cp>If total normal force $F$ is transmitted approximately uniformly over effective contact length $L$, define:\u003C\u002Fp>\u003Cp>$$F'=\\frac{F}{L},$$\u003C\u002Fp>\u003Cp>where $F'$ has units of force per unit length.\u003C\u002Fp>\u003Ch2>Reduced properties\u003C\u002Fh2>\u003Cdiv class=\"formula-chunk\">\u003Cp>The reduced elastic modulus of two isotropic contacting bodies is:\u003C\u002Fp>\u003Cp>$$\\frac{1}{E^*}=\\frac{1-\\nu_1^2}{E_1}+\\frac{1-\\nu_2^2}{E_2}.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$E^*$\u003C\u002Fstrong> — reduced elastic modulus;\u003C\u002Fli>\u003Cli>\u003Cstrong>$E_1$, $E_2$\u003C\u002Fstrong> — Young's moduli of the contacting bodies;\u003C\u002Fli>\u003Cli>\u003Cstrong>$\\nu_1$, $\\nu_2$\u003C\u002Fstrong> — Poisson's ratios of the contacting bodies.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>Equivalently:\u003C\u002Fp>\u003Cp>$$E^*=\\left(\\frac{1-\\nu_1^2}{E_1}+\\frac{1-\\nu_2^2}{E_2}\\right)^{-1}.$$\u003C\u002Fp>\u003C\u002Fdiv>\u003Cdiv class=\"formula-chunk\">\u003Cp>For two convex spherical surfaces along a corresponding principal direction, the reduced radius can be written as:\u003C\u002Fp>\u003Cp>$$\\frac{1}{R^*}=\\frac{1}{R_1}+\\frac{1}{R_2}.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$R^*$\u003C\u002Fstrong> — reduced radius of curvature;\u003C\u002Fli>\u003Cli>\u003Cstrong>$R_1$, $R_2$\u003C\u002Fstrong> — radii of curvature of the contacting surfaces.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>Equivalently, for two convex surfaces:\u003C\u002Fp>\u003Cp>$$R^*=\\frac{R_1R_2}{R_1+R_2}.$$\u003C\u002Fp>\u003Cp>For other combinations of curvature, the sign of an individual radius follows the adopted geometric convention. General three-dimensional contact requires the principal curvatures of both surfaces.\u003C\u002Fp>\u003Cp>The calculator below corresponds to two convex surfaces with positive radii.\u003C\u002Fp>\u003Cdiv data-formula-calculator-config=\"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\">\u003C\u002Fdiv>\u003C\u002Fdiv>\u003Cp>For a cylinder on a plane, $R^*=R$ in the transverse plane. For two convex cylinders, their curvatures add; for a convex-concave pair, curvature signs follow the adopted convention.\u003C\u002Fp>\u003Ch2>Contact width and pressure\u003C\u002Fh2>\u003Cdiv class=\"formula-chunk\">\u003Cp>For two long parallel cylindrical surfaces in the plane Hertz contact model, use the normal load per unit length:\u003C\u002Fp>\u003Cp>$$F'=\\frac{F}{L}.$$\u003C\u002Fp>\u003Cp>The half-width of the contact strip is:\u003C\u002Fp>\u003Cp>$$b=\\sqrt{\\frac{4F'R^*}{\\pi E^*}}.$$\u003C\u002Fp>\u003Cp>The maximum contact pressure is:\u003C\u002Fp>\u003Cp>$$p_0=\\frac{2F'}{\\pi b}.$$\u003C\u002Fp>\u003Cp>The pressure distribution across the strip is:\u003C\u002Fp>\u003Cp>$$p(x)=p_0\\sqrt{1-\\frac{x^2}{b^2}},\\qquad |x|\\le b.$$\u003C\u002Fp>\u003Cul>\u003Cli>\u003Cstrong>$F$\u003C\u002Fstrong> — total normal force;\u003C\u002Fli>\u003Cli>\u003Cstrong>$L$\u003C\u002Fstrong> — effective contact length;\u003C\u002Fli>\u003Cli>\u003Cstrong>$F'$\u003C\u002Fstrong> — normal load per unit length;\u003C\u002Fli>\u003Cli>\u003Cstrong>$b$\u003C\u002Fstrong> — contact-strip half-width;\u003C\u002Fli>\u003Cli>\u003Cstrong>$R^*$\u003C\u002Fstrong> — reduced radius in the contact plane;\u003C\u002Fli>\u003Cli>\u003Cstrong>$E^*$\u003C\u002Fstrong> — reduced elastic modulus;\u003C\u002Fli>\u003Cli>\u003Cstrong>$p_0$\u003C\u002Fstrong> — maximum contact pressure;\u003C\u002Fli>\u003Cli>\u003Cstrong>$p(x)$\u003C\u002Fstrong> — pressure at transverse coordinate $x$.\u003C\u002Fli>\u003C\u002Ful>\u003Cp>End effects along the roller length are not represented by this idealized two-dimensional model.\u003C\u002Fp>\u003C\u002Fdiv>\u003Cp>The full contact-strip width is $2b$. Pressure $p(x)$ is maximum at the centerline, where $p(0)=p_0$, and decreases to zero at $x=\\pm b$.\u003C\u002Fp>\u003Ch2>Example\u003C\u002Fh2>\u003Cp>Let a steel roller of radius $R=20\\ \\text{mm}$ contact a steel plane over $L=50\\ \\text{mm}$ under $F=5000\\ \\text{N}$. For identical steels with $E_1=E_2=210000\\ \\text{MPa}$ and $\\nu_1=\\nu_2=0.30$, $E^*\\approx115385\\ \\text{MPa}$ and $F'=100\\ \\text{N\u002Fmm}$.\u003C\u002Fp>\u003Cp>Then:\u003C\u002Fp>\u003Cp>$$b=\\sqrt{\\frac{4\\cdot100\\cdot20}{\\pi\\cdot115385}}\\approx0.149\\ \\text{mm}.$$\u003C\u002Fp>\u003Cp>The maximum pressure is:\u003C\u002Fp>\u003Cp>$$p_0=\\frac{2\\cdot100}{\\pi\\cdot0.149}\\approx427\\ \\text{MPa}.$$\u003C\u002Fp>\u003Ch2>Edge effects\u003C\u002Fh2>\u003Cp>A real roller has finite length $L$. Misalignment, sharp edges, and nonuniform load distribution can raise pressure near the ends, an effect absent from the ideal two-dimensional model. Crowning or other profile modifications may be used to reduce edge concentration.\u003C\u002Fp>\u003Ch2>Applications\u003C\u002Fh2>\u003Cp>The line-contact model is used for roller bearings, rollers, wheels on rails, and other components where $L$ is much greater than the elastic strip width $2b$.\u003C\u002Fp>\u003Ch2>Procedure\u003C\u002Fh2>\u003Cdiv class=\"algorithm-chunk\">\u003Col>\u003Cli>Identify the initial contact type and local radii of curvature $R_1$ and $R_2$ of the surfaces.\u003C\u002Fli>\u003Cli>Check the assumptions of elastic Hertz theory: small contact area, small deformation, smooth surfaces, and no significant plasticity.\u003C\u002Fli>\u003Cli>Calculate the reduced elastic modulus $E^*$.\u003C\u002Fli>\u003Cli>Determine the reduced curvature or reduced radius $R^*$ for the relevant geometry.\u003C\u002Fli>\u003Cli>Use the normal force $F$, or load per unit length $F'=F\u002FL$ for line contact, to determine the dimensions of the contact area or strip.\u003C\u002Fli>\u003Cli>Calculate the maximum contact pressure $p_0$ and, where needed, the pressure distribution $p(r)$ or $p(x)$.\u003C\u002Fli>\u003Cli>Assess the subsurface stress state and the contact-strength criterion relevant to the component.\u003C\u002Fli>\u003Cli>Check cyclic loading, friction, lubrication, roughness, misalignment, and edge effects when they are significant.\u003C\u002Fli>\u003C\u002Fol>\u003C\u002Fdiv>",[14,18,21,24],{"id":15,"code":16,"type":17,"locale":7},124,"hertz-reduced-elastic-modulus","formula",{"id":19,"code":20,"type":17,"locale":7},125,"hertz-reduced-radius",{"id":22,"code":23,"type":17,"locale":7},128,"hertz-cylinder-line-contact",{"id":25,"code":26,"type":27,"locale":7},129,"hertz-contact-calculation-algorithm","algorithm",[29,33,37],{"id":30,"name":31,"path":32},45,"Strength of Materials","strength-of-materials",{"id":34,"name":35,"path":36},126,"Contact Stresses","strength-of-materials\u002Fcontact-stresses",{"id":5,"name":6,"path":8},[],[],{"en":41,"uk":42},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Fstrength-of-materials\u002Fcontact-stresses\u002Fcontact-of-cylinders-and-rollers","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fopir-materialiv\u002Fkontaktni-napruzhennia\u002Fkontakt-tsylindriv-ta-rolikiv",1787712541131]