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{"name": "Twin Disc: Creepage and Wear", "extensions": [], "custom_part_types": [], "resources": [["question-resources/TwinDisc.png", "/srv/numbas/media/question-resources/TwinDisc.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Twin Disc: Creepage and Wear", "tags": [], "metadata": {"description": "<p>Estimating the wear rate of a disc from a twin-disc experiment.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Engineering surfaces in sliding contact with inevitably get damaged over time. The gradual removal of material from a surface is called wear, and one of the ways it can be tested and measured is through twin-disc tests. Two discs - usually of different materials - are run against each other under load. Disc mass is measured before and after testing, and the change in diameter - and thus depth of material lost - per cycle can thus be estimated. Creepage and load are usually kept constant during the test, where creepage ($\\gamma$) is the difference in surface velocity divided by average surface velocity:</p>\n<p>$\\gamma = {v_2 - v_1 \\over 0.5 (v_1 +&nbsp; v_2)}$</p>\n<p><img style=\"width: 20%;\" src=\"resources/question-resources/TwinDisc.png\"/></p>", "advice": "<p>A twin-disc test is performed to assess the wear characteristics of a new alloy under rolling-sliding contact. Disc 1 is the new alloy, and has radius {R1}mm, initial mass {mass1}g and density {rho1}kg/m${}^3$. Disc 2 has radius {R2}mm. Both discs have running width {width}mm.&nbsp;The test lasts for {cycles} cycles - one cycle is one full rotation of Disc 1. The speed of Disc 1 is fixed as {omega1} r.p.m. (revolutions per minute) and a creepage of {creepage}% is maintained throughout.</p>\n<ul>\n<li>The surface speed of Disc 1 is $v_1 = \\omega_1 R_1 = \\var{omega1} \\times {2 \\pi \\over 60} \\times \\var{R1} = \\var{omega1*2*pi*R1/60} \\approx \\var{v1_ans}$&nbsp;mm/s.</li>\n<li>The surface speed of Disc 2 is&nbsp;$v_2 = v_1 \\times {2 + \\gamma \\over 2 - \\gamma} = v_1 \\times {\\var{2+creepage/100} \\over \\var{2 - creepage/100}} = \\var{v1*(2+creepage/100)/(2-creepage/100)} \\approx \\var{v2_ans}$&nbsp;mm/s.</li>\n<li>The angular speed of Disc 2 is&nbsp;$\\omega_2 \\times {60 \\over 2 \\pi} = {v_2 \\over R_2} \\times {60 \\over 2 \\pi} = \\var{(v2/R2)*60/(2*pi)} \\approx \\var{omega2_ans}$&nbsp;r.p.m.</li>\n</ul>\n<p>Following the test, the mass of Disc 1 is measured again, and the mass has reduced by {dmass1}g.</p>\n<ul>\n<li>If the total thickness of material removed from the disc surface during the test is $\\delta R_1$, then the volume of material lost is $V = \\text{width } \\times \\pi \\left(R_1^2-(R_1-\\delta R_1)^2\\right)$, with corresponding mass $\\rho_1 V$. Working backwards:</li>\n</ul>\n<p>{dmass1}g $=\\rho_1 V = \\rho_1 \\times \\text{width } \\times \\pi \\left(R_1^2-(R_1-\\delta R_1)^2\\right)$</p>\n<p>which can be rearranged:</p>\n<p>$\\pi \\left(R_1^2-(R_1-\\delta R_1)^2\\right) = {\\var{dmass1} \\times 10^{-3} \\over \\var{rho1} \\times \\var{width} \\times 10^{-3}}$</p>\n<p>and further simplified:</p>\n<p>$\\delta R_1 = R_1 - \\sqrt{R_1^2-{\\var{dmass1} \\over \\var{rho1} \\pi \\times \\var{width}}} = \\var{((R1/1000) - sqrt((R1/1000)^2-(dmass1 / (rho1*pi*width))))*1000}$mm</p>\n<p>The average wear rate of Disc 1 (i.e., depth of material removed per cycle) over the test is&nbsp;$\\delta R_1 / \\var{cycles} = \\var{wear_calc} \\approx \\var{siground(wear_calc,3)}$ nm/cycle.</p>", "rulesets": {}, "extensions": [], "variables": {"R1": {"name": "R1", "group": "Ungrouped variables", "definition": "random(20..24)", "description": "<p>Radius of Disc 1 [mm].</p>", "templateType": "anything"}, "R2": {"name": "R2", "group": "Ungrouped variables", "definition": "random(20..24)+0.5", "description": "<p>Radius of Disc 2 [mm].</p>", "templateType": "anything"}, "Creepage": {"name": "Creepage", "group": "Ungrouped variables", "definition": "-5+2*random(0..5)", "description": "<p>Creepage [%].</p>", "templateType": "anything"}, "omega1": {"name": "omega1", "group": "Ungrouped variables", "definition": "400+random(-10..30)*5", "description": "<p>Angular velocity of Disc 1 [r.p.m.]</p>", "templateType": "anything"}, "omega2": {"name": "omega2", "group": "Ungrouped variables", "definition": "(v2/R2)*60/(2*pi)", "description": "<p>Angular velocity of Disc 2 [r.p.m.]</p>", "templateType": "anything"}, "v1": {"name": "v1", "group": "Ungrouped variables", "definition": "R1*omega1*2*pi/60", "description": "<p>Surface velocity of Disc 1 [mm/s].</p>", "templateType": "anything"}, "v1_ans": {"name": "v1_ans", "group": "Ungrouped variables", "definition": "siground(v1,3)", "description": "<p>v1 - rounded to 3 s.f. [mm/s].</p>", "templateType": "anything"}, "v2": {"name": "v2", "group": "Ungrouped variables", "definition": "v1*(2+Creepage/100)/(2-Creepage/100)", "description": "<p>Surface velocity of Disc 2 [mm/s].</p>", "templateType": "anything"}, "v2_ans": {"name": "v2_ans", "group": "Ungrouped variables", "definition": "siground(v2,3)", "description": "<p>v2 - rounded to 3 s.f. [mm/s].</p>", "templateType": "anything"}, "omega2_ans": {"name": "omega2_ans", "group": "Ungrouped variables", "definition": "siground(omega2,3)", "description": "<p>omega2&nbsp;- rounded to 3 s.f. [r.p.m.].</p>", "templateType": "anything"}, "rho1": {"name": "rho1", "group": "Ungrouped variables", "definition": "random(2800,4500,7850,8750)", "description": "<p>Density of Disc 1 [kg/m3].</p>", "templateType": "anything"}, "width": {"name": "width", "group": "Ungrouped variables", "definition": "random(5..12)", "description": "<p>Running surface width [mm].</p>", "templateType": "anything"}, "mass1": {"name": "mass1", "group": "Ungrouped variables", "definition": "siground(1.2*rho1*width*pi*(R1/1000)^2,2)", "description": "<p>Initial mass of Disc 1 [g].</p>", "templateType": "anything"}, "wear": {"name": "wear", "group": "Ungrouped variables", "definition": "random(5..15)/10", "description": "<p>Depth of material lost per cycle [nm/cycle].</p>", "templateType": "anything"}, "cycles": {"name": "cycles", "group": "Ungrouped variables", "definition": "random(1..5)*100000", "description": "<p>Number of test cycles (for Disc 1).</p>", "templateType": "anything"}, "dR1": {"name": "dR1", "group": "Ungrouped variables", "definition": "wear*cycles/1000000", "description": "<p>Change in Disc 1 radius during test [mm].</p>", "templateType": "anything"}, "dmass1": {"name": "dmass1", "group": "Ungrouped variables", "definition": "siground(rho1*width*pi*((R1/1000)^2-((R1-dR1)/1000)^2),3)", "description": "<p>Change in Disc1 mass [g].</p>", "templateType": "anything"}, "wear_calc": {"name": "wear_calc", "group": "Ungrouped variables", "definition": "(R1-1000*sqrt((R1/1000)^2-dmass1/(rho1*width*pi)))*1000000/cycles", "description": "<p>Calculated wear rate [nm/cycle].</p>", "templateType": "anything"}, "wear_ans_lo": {"name": "wear_ans_lo", "group": "Ungrouped variables", "definition": "siground(wear_calc*0.995,3)", "description": "", "templateType": "anything"}, "wear_ans_hi": {"name": "wear_ans_hi", "group": "Ungrouped variables", "definition": "siground(wear_calc*1.005,3)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["R1", "R2", "Creepage", "omega1", "omega2", "v1", "v1_ans", "v2", "v2_ans", "omega2_ans", "rho1", "width", "mass1", "cycles", "dR1", "dmass1", "wear", "wear_calc", "wear_ans_lo", "wear_ans_hi"], "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>A twin-disc test is performed to assess the wear characteristics of a new alloy under rolling-sliding contact. Disc 1 is the new alloy, and has radius {R1}mm, initial mass {mass1}g and density {rho1}kg/m${}^3$. Disc 2 has radius {R2}mm. Both discs have running width {width}mm. The test lasts for {cycles} cycles - one cycle is one full rotation of Disc 1. The speed of Disc 1 is fixed as {omega1} r.p.m. (revolutions per minute) and a creepage of {creepage}% is maintained throughout.</p>\n<ul>\n<li>The surface speed of Disc 1 is&nbsp;[[0]] mm/s.</li>\n<li>The surface speed of Disc 2 is&nbsp;[[1]] mm/s.</li>\n<li>The angular speed of Disc 2 is&nbsp;[[2]] r.p.m.</li>\n</ul>\n<p>Following the test, the mass of Disc 1 is measured again, and the mass has reduced by {dmass1}g.</p>\n<ul>\n<li>The average wear rate of Disc 1 (i.e., depth of material removed per cycle) over the test is&nbsp;[[3]] nm/cycle.</li>\n</ul>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "v1", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v1_ans", "maxValue": "v1_ans", "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": true, "customName": "v2", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "v2_ans", "maxValue": "v2_ans", "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": true, "customName": "omega2", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "omega2_ans", "maxValue": "omega2_ans", "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": true, "customName": "wearrate", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "wear_ans_lo", "maxValue": "wear_ans_hi", "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", "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}]}]}], "contributors": [{"name": "Francis Franklin", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1887/"}]}