// Numbas version: exam_results_page_options
{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Simple Hertz contact without friction.</p>"}, "timing": {"allowPause": true, "timedwarning": {"message": "", "action": "none"}, "timeout": {"message": "", "action": "none"}}, "navigation": {"preventleave": true, "browse": true, "reverse": true, "showresultspage": "oncompletion", "onleave": {"message": "", "action": "none"}, "showfrontpage": true, "allowregen": true}, "showQuestionGroupNames": false, "percentPass": "50", "name": "Contact Mechanics", "feedback": {"showtotalmark": true, "intro": "", "feedbackmessages": [], "advicethreshold": 0, "allowrevealanswer": true, "showanswerstate": true, "showactualmark": true}, "showstudentname": true, "duration": 0, "question_groups": [{"pickingStrategy": "all-ordered", "name": "Group", "pickQuestions": 1, "questions": [{"name": "Contact Mechanics: Twin-disc test", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Contact.png", "/srv/numbas/media/question-resources/Contact.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}], "tags": [], "metadata": {"description": "<p>A gold ball on a steel table - effect of its own weight.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>The size of the contact and the pressure between the contacting surfaces depends on:</p>\n<ol>\n<li>the <strong>materials</strong>, and here we assume them to be linear, elastic and isotropic;</li>\n<li>the <strong>geometry</strong>, and here we assume them to be either flat (infinite radius) or having a constant radius;</li>\n<li>the <strong>applied load</strong>, and here we assume the load is applied normal to the contact, i.e., there is no friction.</li>\n</ol>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/Contact.png\"/></p>\n<p><strong>Materials:</strong> A single material property - the <strong>Elastic Contact Modulus</strong> ($E^*$) - combines the elastic properties of both contacting surfaces into an equivalent stiffness:</p>\n<p>\\[{1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2}\\]</p>\n<p>where $E_1$ and $E_2$ are the Young's elastic moduli of the two materials, and $\\nu_1$ and $\\nu_2$ are the corresponding Poisson's ratios.</p>\n<p><strong>Geometry:</strong> The two simplest forms of contact between curved surfaces are:</p>\n<ul>\n<li>2D: cylindrical surfaces aligned along their length, or a cylinder on a flat surface, creating <strong>a rectangular contact area</strong>;</li>\n<li>3D: spherical surfaces, or a sphere on a flat surface, creating <strong>a circular contact area</strong>.</li>\n</ul>\n<p>In either case, an <strong>equivalent radius</strong> $R$ can be defined. If both surfaces are convex, e.g., two balls touching, then:</p>\n<p>\\[{1 \\over R} = {1 \\over R_1} + {1 \\over R_2}\\]</p>\n<p>where $R_1$ is the radius of Surface 1 and $R_2$ is the radius of Surface 2. If, however, one surface is concave, e.g., a ball in a cup, then:</p>\n<p>\\[{1 \\over R} = {1 \\over R_1} - {1 \\over R_2}\\]</p>\n<p>where $R_1$ is the radius of (convex) Surface 1 and $R_2$ is the radius of (concave) Surface 2. (The 'cup' radius must be larger than the 'ball' radius.)</p>\n<p><strong>Applied Load:</strong> This is a little tricky because the same notation, $P$, is used to mean different things:</p>\n<ul>\n<li>In 2D, $P$ is the applied load per unit length of the cylindrical contact. [Units: N/m]</li>\n<li>In 3D, $P$ is simply the applied load. [Units: N]</li>\n</ul>\n<p><strong>2D Contact:</strong> The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:</p>\n<p>\\[p_0 = \\left({P E^* \\over \\pi R}\\right)^{1 \\over 2}\\]</p>\n<p>\\[a = \\left({4 P R \\over \\pi E^*}\\right)^{1 \\over 2}\\]</p>\n<p><strong>3D Contact:</strong> The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:</p>\n<p>\\[p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3}\\]</p>\n<p>\\[a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3}\\]</p>", "advice": "<ul>\n<li>The equivalent radius, $R$, is given by:</li>\n</ul>\n<p>${1 \\over R} = {1 \\over R_1} + {1 \\over R_2} =&nbsp;{1 \\over \\var{diawheel} \\div 2} + {1 \\over \\var{diarail} \\div 2}$ [units: mm$^{-1}$]</p>\n<p>which can be rearranged to give $R = \\var{siground(R*1000,3)}$mm.</p>\n<ul>\n<li>The Elastic Contact Modulus, $E^*$, is given by:</li>\n</ul>\n<p>${1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2} ={1 - 0.3^2 \\over&nbsp;209 \\times 10^9} + {1 - 0.3^2 \\over 209 \\times 10^9}$</p>\n<p>which can be rearranged to give $E^* = \\var{siground(ECM,3)}$GPa.</p>\n<ul>\n<li>This is line (2D) contact (between parallel cylinders), so $P$ is the load per unit length, i.e., $\\var{load}$kN $\\div \\var{width}$mm, and the peak contact pressure, $p_0$, is given by:</li>\n</ul>\n<p>$p_0 = \\left({P E^* \\over \\pi R}\\right)^{1 \\over 2} =\\left({ \\left( \\var{load} \\times 10^3 \\div \\var{width} \\times 10^{-3} \\right) \\times \\var{siground(ECM,3)} \\times 10^9 \\over \\pi \\times \\var{siground(R*1000,3)} \\times 10^{-3}} \\right)^{1 \\over 2} = \\var{siground(p0,3)}$MPa.</p>\n<ul>\n<li>The semi-contact width, $a$, is likewise given by:</li>\n</ul>\n<p>$a = \\left({4 P R \\over \\pi E^*}\\right)^{1 \\over 2} = \\left({4 \\times\\left( \\var{load} \\times 10^3 \\div \\var{width} \\times 10^{-3} \\right) \\times&nbsp;\\var{siground(R*1000,3)} \\times 10^{-3} \\over \\pi \\times \\var{siground(ECM,3)} \\times 10^9}\\right)^{1 \\over 2} = \\var{siground(scw,3)}$mm.</p>\n<ul>\n<li>The area of the contact patch is rectangular, and is thus the width of the contact (along the cylindrical axis) times twice the semi-contact width, $a$, from above:</li>\n</ul>\n<p>area = $2 \\times \\var{siground(scw,3)}$mm $\\times \\var{width}$mm $= \\var{siground(area,3)}$mm$^2$.</p>", "rulesets": {}, "variables": {"scw": {"name": "scw", "group": "Ungrouped variables", "definition": "(4 * P * R / (pi * ECM * 10^9))^(1/2) * 1000", "description": "<p>Semi-contact width. [Units: mm]</p>", "templateType": "anything"}, "p0": {"name": "p0", "group": "Ungrouped variables", "definition": "(P * ECM*10^9 / (pi * R))^(1/2) / 10^6", "description": "<p>Peak contact pressure. [Units: MPa]</p>", "templateType": "anything"}, "load": {"name": "load", "group": "Ungrouped variables", "definition": "random(4..7#0.5)", "description": "<p>Applied load. [Units: kN]</p>", "templateType": "anything"}, "diarail": {"name": "diarail", "group": "Ungrouped variables", "definition": "random(40..47#0.1)", "description": "<p>Rail disc diameter. [Units: mm]</p>", "templateType": "anything"}, "ECM": {"name": "ECM", "group": "Ungrouped variables", "definition": "1/((1-0.3^2)/209+(1-0.3^2)/209)", "description": "<p>Elastic Contact Modulus. [Units: GPa]</p>", "templateType": "anything"}, "width": {"name": "width", "group": "Ungrouped variables", "definition": "random(5..10)", "description": "<p>Track width. [Units: mm]</p>", "templateType": "anything"}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "2*scw*width", "description": "<p>Contact area. [Units: mm$^2$]</p>", "templateType": "anything"}, "P": {"name": "P", "group": "Ungrouped variables", "definition": "(load*1000)/(width/1000)", "description": "<p>Load per unit length. [Units: N/m]</p>", "templateType": "anything"}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "1/(2000/diawheel+2000/diarail)", "description": "<p>Equivalent radius. [Units: m]</p>", "templateType": "anything"}, "diawheel": {"name": "diawheel", "group": "Ungrouped variables", "definition": "random(40..47#0.1)", "description": "<p>Wheel disc diameter. [Units: mm]</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["diawheel", "diarail", "R", "ECM", "width", "load", "P", "p0", "scw", "area"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Reference values:</p>\n<ul>\n<li>Steel: Density, $\\rho = 8\\ 050$ kg/m$^3$; Young's modulus, $E = 209$ GPa; Poisson's ratio, $\\nu = 0.3$.</li>\n</ul>\n<p>In a twin-disc test, a 'wheel' disc (machined from a steel train wheel) with diameter $\\var{diawheel}$ mm runs against a 'rail' disc (machined from a steel railway rail) with diameter $\\var{diarail}$ mm.</p>\n<p>The disc surfaces are cylindrical with the axes aligned. The width of the surfaces is $\\var{width}$ mm.</p>\n<p>The applied load is $\\var{load}$ kN.</p>\n<p>Calculate:</p>\n<ul>\n<li>the equivalent radius: [[0]] [Units: mm]</li>\n<li>the Elastic Contact Modulus: $E^* =$ [[1]] [Units: GPa]</li>\n<li>the peak contact pressure: $p_0 =$ [[2]] [Units: MPa]</li>\n<li>the semi-contact width: $a =$ [[3]] [Units: mm]</li>\n<li>the area of the contact patch: [[4]] [Units: mm$^2$]</li>\n</ul>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "R*1000", "maxValue": "R*1000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ECM", "maxValue": "ECM", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p0", "maxValue": "p0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "scw", "maxValue": "scw", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "area", "maxValue": "area", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Contact Mechanics: Ball on table", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Contact.png", "/srv/numbas/media/question-resources/Contact.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}], "tags": [], "metadata": {"description": "<p>A gold ball on a steel table - effect of its own weight.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>The size of the contact and the pressure between the contacting surfaces depends on:</p>\n<ol>\n<li>the <strong>materials</strong>, and here we assume them to be linear, elastic and isotropic;</li>\n<li>the <strong>geometry</strong>, and here we assume them to be either flat (infinite radius) or having a constant radius;</li>\n<li>the <strong>applied load</strong>, and here we assume the load is applied normal to the contact, i.e., there is no friction.</li>\n</ol>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/Contact.png\"/></p>\n<p><strong>Materials:</strong> A single material property - the <strong>Elastic Contact Modulus</strong> ($E^*$) - combines the elastic properties of both contacting surfaces into an equivalent stiffness:</p>\n<p>\\[{1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2}\\]</p>\n<p>where $E_1$ and $E_2$ are the Young's elastic moduli of the two materials, and $\\nu_1$ and $\\nu_2$ are the corresponding Poisson's ratios.</p>\n<p><strong>Geometry:</strong> The two simplest forms of contact between curved surfaces are:</p>\n<ul>\n<li>2D: cylindrical surfaces aligned along their length, or a cylinder on a flat surface, creating <strong>a rectangular contact area</strong>;</li>\n<li>3D: spherical surfaces, or a sphere on a flat surface, creating <strong>a circular contact area</strong>.</li>\n</ul>\n<p>In either case, an <strong>equivalent radius</strong> $R$ can be defined. If both surfaces are convex, e.g., two balls touching, then:</p>\n<p>\\[{1 \\over R} = {1 \\over R_1} + {1 \\over R_2}\\]</p>\n<p>where $R_1$ is the radius of Surface 1 and $R_2$ is the radius of Surface 2. If, however, one surface is concave, e.g., a ball in a cup, then:</p>\n<p>\\[{1 \\over R} = {1 \\over R_1} - {1 \\over R_2}\\]</p>\n<p>where $R_1$ is the radius of (convex) Surface 1 and $R_2$ is the radius of (concave) Surface 2. (The 'cup' radius must be larger than the 'ball' radius.)</p>\n<p><strong>Applied Load:</strong> This is a little tricky because the same notation, $P$, is used to mean different things:</p>\n<ul>\n<li>In 2D, $P$ is the applied load per unit length of the cylindrical contact. [Units: N/m]</li>\n<li>In 3D, $P$ is simply the applied load. [Units: N]</li>\n</ul>\n<p><strong>2D Contact:</strong> The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:</p>\n<p>\\[p_0 = \\left({P E^* \\over \\pi R}\\right)^{1 \\over 2}\\]</p>\n<p>\\[a = \\left({4 P R \\over \\pi E^*}\\right)^{1 \\over 2}\\]</p>\n<p><strong>3D Contact:</strong> The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:</p>\n<p>\\[p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3}\\]</p>\n<p>\\[a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3}\\]</p>", "advice": "<ul>\n<li>The weight of the&nbsp;sphere is the volume times the density times the acceleration from gravity, i.e.:</li>\n</ul>\n<p>weight $= \\rho g \\times {4 \\over 3} \\pi r^3 = 19320 \\times 9.81 \\times {4 \\over 3} \\times \\pi \\times \\left( {\\var{diameter} \\times 10^{-2} \\over 2} \\right)^3 = \\var{siground(weight,3)}$N.</p>\n<ul>\n<li>The Elastic Contact Modulus, $E^*$, is given by:</li>\n</ul>\n<p>${1 \\over E^*} = {1 - 0.4^2 \\over 79 \\times 10^9} + {1 - 0.3^2 \\over 209 \\times 10^9}$</p>\n<p>which can be rearranged to give $E^* = \\var{siground(ECM,3)}$GPa.</p>\n<ul>\n<li>This is spherical (3D) contact, with the equivalent radius equal to the radius of the sphere. The peak contact pressure, $p_0$, is therefore given by:</li>\n</ul>\n<p>$p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3} = \\left({6 \\times&nbsp;\\var{siground(weight,3)} \\times \\left( \\var{siground(ECM,3)} \\times 10^9 \\right)^2 \\over \\pi^3 \\left( \\var{diameter} \\times 10^{-2} \\div 2 \\right)^2}\\right)^{1 \\over 3} = \\var{siground(p0,3)}$MPa.</p>\n<ul>\n<li>Similarly, the semi-contact width, $a$, is given by:</li>\n</ul>\n<p>$a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3} = \\left({3 \\times \\var{siground(weight,3)} \\times \\var{diameter} \\times 10^{-2} \\div 2 \\over 4 \\times \\var{siground(ECM,3)} \\times 10^9} \\right)^{1 \\over 3} = \\var{siground(scw,3)}$mm.</p>\n<ul>\n<li>The area of the contact patch is circular with radius equal to $a$, i.e.:</li>\n</ul>\n<p>area $= \\pi a^2 = \\pi \\left( \\var{siground(scw,3)} \\times 10^{-3} \\right)^2 = \\var{siground(area,3)}$mm$^2$.</p>", "rulesets": {}, "variables": {"weight": {"name": "weight", "group": "Ungrouped variables", "definition": "volume*19320*9.81", "description": "<p>Weight of gold ball. [Units: N]</p>", "templateType": "anything"}, "ECM": {"name": "ECM", "group": "Ungrouped variables", "definition": "1/((1-0.4^2)/79+(1-0.3^2)/209)", "description": "<p>Elastic Contact Modulus. [Units: GPa]</p>", "templateType": "anything"}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "pi*scw^2", "description": "<p>Contact area. [Units: mm$^2$]</p>", "templateType": "anything"}, "p0": {"name": "p0", "group": "Ungrouped variables", "definition": "(6 * weight * (ECM*10^9)^2 / (pi^3 * R^2))^(1/3) / 10^6", "description": "<p>Peak contact pressure. [Units: MPa]</p>", "templateType": "anything"}, "scw": {"name": "scw", "group": "Ungrouped variables", "definition": "(3 * weight * R / (4 * ECM * 10^9))^(1/3) * 1000", "description": "<p>Semi-contact width. [Units: mm]</p>", "templateType": "anything"}, "diameter": {"name": "diameter", "group": "Ungrouped variables", "definition": "random(6..40)", "description": "<p>Diameter of gold ball. [Units: cm]</p>", "templateType": "anything"}, "volume": {"name": "volume", "group": "Ungrouped variables", "definition": "pi*R^3*4/3", "description": "<p>Volume of gold ball. [Units: m3]</p>", "templateType": "anything"}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "diameter/200", "description": "<p>Radius of gold ball. [Units: m]</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["diameter", "R", "volume", "weight", "ECM", "p0", "scw", "area"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Reference values:</p>\n<ul>\n<li>Gold: Density, $\\rho = 19\\ 320$ kg/m$^3$; Young's modulus, $E = 79$ GPa; Poisson's ratio, $\\nu = 0.4$.</li>\n<li>Steel: Density, $\\rho = 8\\ 050$ kg/m$^3$; Young's modulus, $E = 209$ GPa; Poisson's ratio, $\\nu = 0.3$.</li>\n<li>Gravity: $g = 9.81$ m/s$^2$.</li>\n</ul>\n<p>A gold sphere of diameter $\\var{diameter}$ cm sits on a flat steel table.</p>\n<p>Calculate:</p>\n<ul>\n<li>the weight of the ball: [[0]] [Units: N]</li>\n<li>the Elastic Contact Modulus: $E^* =$ [[1]] [Units: GPa]</li>\n<li>the peak contact pressure: $p_0 =$ [[2]] [Units: MPa]</li>\n<li>the semi-contact width: $a =$ [[3]] [Units: mm]</li>\n<li>the area of the contact patch: [[4]] [Units: mm$^2$]</li>\n</ul>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "weight", "maxValue": "weight", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ECM", "maxValue": "ECM", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p0", "maxValue": "p0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "scw", "maxValue": "scw", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "area", "maxValue": "area", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Contact Mechanics: Ball joint", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Contact.png", "/srv/numbas/media/question-resources/Contact.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}], "tags": [], "metadata": {"description": "<p>Hertz contact calculation for&nbsp;ball joint.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>The size of the contact and the pressure between the contacting surfaces depends on:</p>\n<ol>\n<li>the <strong>materials</strong>, and here we assume them to be linear, elastic and isotropic;</li>\n<li>the <strong>geometry</strong>, and here we assume them to be either flat (infinite radius) or having a constant radius;</li>\n<li>the <strong>applied load</strong>, and here we assume the load is applied normal to the contact, i.e., there is no friction.</li>\n</ol>\n<p style=\"text-align: center;\"><img src=\"resources/question-resources/Contact.png\"/></p>\n<p><strong>Materials:</strong> A single material property - the <strong>Elastic Contact Modulus</strong> ($E^*$) - combines the elastic properties of both contacting surfaces into an equivalent stiffness:</p>\n<p>\\[{1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2}\\]</p>\n<p>where $E_1$ and $E_2$ are the Young's elastic moduli of the two materials, and $\\nu_1$ and $\\nu_2$ are the corresponding Poisson's ratios.</p>\n<p><strong>Geometry:</strong> The two simplest forms of contact between curved surfaces are:</p>\n<ul>\n<li>2D: cylindrical surfaces aligned along their length, or a cylinder on a flat surface, creating <strong>a rectangular contact area</strong>;</li>\n<li>3D: spherical surfaces, or a sphere on a flat surface, creating <strong>a circular contact area</strong>.</li>\n</ul>\n<p>In either case, an <strong>equivalent radius</strong> $R$ can be defined. If both surfaces are convex, e.g., two balls touching, then:</p>\n<p>\\[{1 \\over R} = {1 \\over R_1} + {1 \\over R_2}\\]</p>\n<p>where $R_1$ is the radius of Surface 1 and $R_2$ is the radius of Surface 2. If, however, one surface is concave, e.g., a ball in a cup, then:</p>\n<p>\\[{1 \\over R} = {1 \\over R_1} - {1 \\over R_2}\\]</p>\n<p>where $R_1$ is the radius of (convex) Surface 1 and $R_2$ is the radius of (concave) Surface 2. (The 'cup' radius must be larger than the 'ball' radius.)</p>\n<p><strong>Applied Load:</strong> This is a little tricky because the same notation, $P$, is used to mean different things:</p>\n<ul>\n<li>In 2D, $P$ is the applied load per unit length of the cylindrical contact. [Units: N/m]</li>\n<li>In 3D, $P$ is simply the applied load. [Units: N]</li>\n</ul>\n<p><strong>2D Contact:</strong> The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:</p>\n<p>\\[p_0 = \\left({P E^* \\over \\pi R}\\right)^{1 \\over 2}\\]</p>\n<p>\\[a = \\left({4 P R \\over \\pi E^*}\\right)^{1 \\over 2}\\]</p>\n<p><strong>3D Contact:</strong> The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:</p>\n<p>\\[p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3}\\]</p>\n<p>\\[a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3}\\]</p>", "advice": "<p>Reference values:</p>\n<ul>\n<li>HDPE: Young's modulus, $E = 1$ GPa; Poisson's ratio, $\\nu = 0.45$.</li>\n<li>Steel: Young's modulus, $E = 209$ GPa; Poisson's ratio, $\\nu = 0.3$.</li>\n</ul>\n<p>A ball joint has a steel ball of diameter $\\var{diaball}$ mm in a HDPE cup of diameter $\\var{diacup}$ mm. The maximum applied load is $\\var{load}$ N.</p>\n<p>Calculate:</p>\n<ul>\n<li>Because the cup is concave not convex, the radius is effectively negative. The equivalent radius is therefore given by:</li>\n</ul>\n<p>${1 \\over R} =&nbsp;{1 \\over R_1} - {1 \\over R_2} = {1 \\over \\var{diaball} \\div 2} - {1 \\over \\var{diacup} \\div 2}$ [units: mm$^{-1}$]</p>\n<p>which can be rearranged to give $R = \\var{siground(R*1000,3)}$mm.</p>\n<ul>\n<li>The Elastic Contact Modulus, $E^*$, is given by:</li>\n</ul>\n<p>${1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2} = {1 - 0.45^2 \\over 1 \\times 10^9} + {1 - 0.3^2 \\over 209 \\times 10^9}$</p>\n<p>which can be rearranged to give $E^* = \\var{siground(ECM,3)}$GPa.</p>\n<ul>\n<li>This is spherical (3D) contact, with the equivalent radius equal to the radius of the sphere. The peak contact pressure, $p_0$, is therefore given by:</li>\n</ul>\n<p>$p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3} = \\left({6 \\times&nbsp;\\var{load} \\times \\left( \\var{siground(ECM,3)} \\times 10^9 \\right)^2 \\over \\pi^3 \\left( \\var{siground(R*1000,3)} \\times 10^{-3} \\right)^2}\\right)^{1 \\over 3} = \\var{siground(p0,3)}$MPa.</p>\n<ul>\n<li>Similarly, the semi-contact width, $a$, is given by:</li>\n</ul>\n<p>$a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3} = \\left({3 \\times \\var{load} \\times \\var{siground(R*1000,3)} \\times 10^{-3} \\over 4 \\times \\var{siground(ECM,3)} \\times 10^9} \\right)^{1 \\over 3} = \\var{siground(scw,3)}$mm.</p>\n<ul>\n<li>The area of the contact patch is circular with radius equal to $a$, i.e.:</li>\n</ul>\n<p>area $= \\pi a^2 = \\pi \\left( \\var{siground(scw,3)} \\times 10^{-3} \\right)^2 = \\var{siground(area,3)}$mm$^2$.</p>", "rulesets": {}, "variables": {"load": {"name": "load", "group": "Ungrouped variables", "definition": "random(500..1500)", "description": "<p>Applied load. [Units: N]</p>", "templateType": "anything"}, "ECM": {"name": "ECM", "group": "Ungrouped variables", "definition": "1/((1-0.45^2)/1+(1-0.3^2)/209)", "description": "<p>Elastic Contact Modulus. [Units: GPa]</p>", "templateType": "anything"}, "p0": {"name": "p0", "group": "Ungrouped variables", "definition": "(6 * load * (ECM*10^9)^2 / (pi^3 * R^2))^(1/3) / 10^6", "description": "<p>Peak contact pressure. [Units: MPa]</p>", "templateType": "anything"}, "diaball": {"name": "diaball", "group": "Ungrouped variables", "definition": "random(20..30)", "description": "<p>Diameter of ball. [Units: mm]</p>", "templateType": "anything"}, "diacup": {"name": "diacup", "group": "Ungrouped variables", "definition": "diaball+random(1..4)", "description": "<p>Diameter of cup. [Units: mm]</p>", "templateType": "anything"}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "pi*scw^2", "description": "<p>Contact area. [Units: mm$^2$]</p>", "templateType": "anything"}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "1/(2000/diaball-2000/diacup)", "description": "<p>Equivalent radius. [Units: m]</p>", "templateType": "anything"}, "scw": {"name": "scw", "group": "Ungrouped variables", "definition": "(3 * load * R / (4 * ECM * 10^9))^(1/3) * 1000", "description": "<p>Semi-contact width. [Units: mm]</p>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["diaball", "diacup", "R", "ECM", "load", "p0", "scw", "area"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Reference values:</p>\n<ul>\n<li>HDPE: Young's modulus, $E = 1$ GPa; Poisson's ratio, $\\nu = 0.45$.</li>\n<li>Steel: Young's modulus, $E = 209$ GPa; Poisson's ratio, $\\nu = 0.3$.</li>\n</ul>\n<p>A ball joint has a steel ball of diameter $\\var{diaball}$ mm in a HDPE cup of diameter $\\var{diacup}$ mm. The maximum applied load is $\\var{load}$ N.</p>\n<p>Calculate:</p>\n<ul>\n<li>the equivalent radius: $R =$ [[0]] [Units: mm]</li>\n<li>the Elastic Contact Modulus: $E^* =$ [[1]] [Units: GPa]</li>\n<li>the peak contact pressure: $p_0 =$ [[2]] [Units: MPa]</li>\n<li>the semi-contact width: $a =$ [[3]] [Units: mm]</li>\n<li>the area of the contact patch: [[4]] [Units: mm$^2$]</li>\n</ul>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "R*1000", "maxValue": "R*1000", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ECM", "maxValue": "ECM", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "p0", "maxValue": "p0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "scw", "maxValue": "scw", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "area", "maxValue": "area", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "sigfig", "precision": "3", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "type": "exam", "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}], "extensions": [], "custom_part_types": [], "resources": [["question-resources/Contact.png", "/srv/numbas/media/question-resources/Contact.png"]]}