// Numbas version: finer_feedback_settings {"name": "Contact Mechanics", "metadata": {"description": "

Simple Hertz contact without friction.

", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": "50", "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Contact Mechanics", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Twin-disc test w/ units", "extensions": ["quantities"], "custom_part_types": [{"source": {"pk": 19, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/19/edit"}, "name": "Engineering Accuracy with units", "short_name": "engineering-answer", "description": "

A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "siground(settings['correctAnswer'],4)", "hint": {"static": true, "value": ""}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nswitch( \n right and good_units and right_sign, add_credit(1.0,'Correct.'),\n right and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\n right and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but wrong or missing units.'),\n close and good_units, add_credit(settings['C1'],'Close.'),\n close and not good_units, add_credit(settings['C3'],'Answer is close, but wrong or missing units.'),\n incorrect('Wrong answer.')\n)\n\ninterpreted_answer:\nqty(student_scalar, student_units)\n\n\n\ncorrect_quantity:\nsettings[\"correctAnswer\"]\n\n\n\ncorrect_units:\nunits(correct_quantity)\n\n\nallowed_notation_styles:\n[\"plain\",\"en\"]\n\nmatch_student_number:\nmatchnumber(studentAnswer,allowed_notation_styles)\n\nstudent_scalar:\nmatch_student_number[1]\n\nstudent_units:\nreplace_regex('ohms','ohm',\n replace_regex('\u00b0', ' deg',\n replace_regex('-', ' ' ,\n studentAnswer[len(match_student_number[0])..len(studentAnswer)])),\"i\")\n\ngood_units:\ntry(\ncompatible(quantity(1, student_units),correct_units),\nmsg,\nfeedback(msg);false)\n\n\nstudent_quantity:\nswitch(not good_units, \n student_scalar * correct_units, \n not right_sign,\n -quantity(student_scalar, student_units),\n quantity(student_scalar,student_units)\n)\n \n\n\npercent_error:\ntry(\nscalar(abs((correct_quantity - student_quantity)/correct_quantity))*100 \n,msg,\nif(student_quantity=correct_quantity,0,100))\n \n\nright:\npercent_error <= settings['right']\n\n\nclose:\nright_sign and percent_error <= settings['close']\n\nright_sign:\nsign(student_scalar) = sign(correct_quantity)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "switch( \n right and good_units and right_sign, add_credit(1.0,'Correct.'),\n right and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\n right and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but wrong or missing units.'),\n close and good_units, add_credit(settings['C1'],'Close.'),\n close and not good_units, add_credit(settings['C3'],'Answer is close, but wrong or missing units.'),\n incorrect('Wrong answer.')\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "qty(student_scalar, student_units)\n\n"}, {"name": "correct_quantity", "description": "", "definition": "settings[\"correctAnswer\"]\n\n"}, {"name": "correct_units", "description": "", "definition": "units(correct_quantity)\n"}, {"name": "allowed_notation_styles", "description": "", "definition": "[\"plain\",\"en\"]"}, {"name": "match_student_number", "description": "", "definition": "matchnumber(studentAnswer,allowed_notation_styles)"}, {"name": "student_scalar", "description": "", "definition": "match_student_number[1]"}, {"name": "student_units", "description": "

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

", "definition": "replace_regex('ohms','ohm',\n replace_regex('\u00b0', ' deg',\n replace_regex('-', ' ' ,\n studentAnswer[len(match_student_number[0])..len(studentAnswer)])),\"i\")"}, {"name": "good_units", "description": "", "definition": "try(\ncompatible(quantity(1, student_units),correct_units),\nmsg,\nfeedback(msg);false)\n"}, {"name": "student_quantity", "description": "

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

", "definition": "switch(not good_units, \n student_scalar * correct_units, \n not right_sign,\n -quantity(student_scalar, student_units),\n quantity(student_scalar,student_units)\n)\n \n"}, {"name": "percent_error", "description": "", "definition": "try(\nscalar(abs((correct_quantity - student_quantity)/correct_quantity))*100 \n,msg,\nif(student_quantity=correct_quantity,0,100))\n "}, {"name": "right", "description": "", "definition": "percent_error <= settings['right']\n"}, {"name": "close", "description": "

Only marked close if the student actually has the right sign.

", "definition": "right_sign and percent_error <= settings['close']"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity) "}], "settings": [{"name": "correctAnswer", "label": "Correct Quantity.", "help_url": "", "hint": "The correct answer given as a JME quantity.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "right", "label": "% Accuracy for right.", "help_url": "", "hint": "Question will be considered correct if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "0.2", "evaluate": true}, {"name": "close", "label": "% Accuracy for close.", "help_url": "", "hint": "Question will be considered close if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "1.0", "evaluate": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units. if correct answer is 100 N and close is ±1%,
99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
If the correct answer is 100 N, both 100 and -100 N are accepted.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value but forgotten units.
This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": ["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": "

A gold ball on a steel table - effect of its own weight.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The size of the contact and the pressure between the contacting surfaces depends on:

the materials, and here we assume them to be linear, elastic and isotropic;
the geometry, and here we assume them to be either flat (infinite radius) or having a constant radius;
the applied load, and here we assume the load is applied normal to the contact, i.e., there is no friction.

Materials: A single material property - the Elastic Contact Modulus ($E^*$) - combines the elastic properties of both contacting surfaces into an equivalent stiffness:

\\[{1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2}\\]

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.

Geometry: The two simplest forms of contact between curved surfaces are:

2D: cylindrical surfaces aligned along their length, or a cylinder on a flat surface, creating a rectangular contact area;
3D: spherical surfaces, or a sphere on a flat surface, creating a circular contact area.

In either case, an equivalent radius $R$ can be defined. If both surfaces are convex, e.g., two balls touching, then:

\\[{1 \\over R} = {1 \\over R_1} + {1 \\over R_2}\\]

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:

\\[{1 \\over R} = {1 \\over R_1} - {1 \\over R_2}\\]

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.)

Applied Load: This is a little tricky because the same notation, $P$, is used to mean different things:

In 2D, $P$ is the applied load per unit length of the cylindrical contact. [Units: N/m]
In 3D, $P$ is simply the applied load. [Units: N]

2D Contact: The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:

\\[p_0 = \\left({P E^* \\over \\pi R}\\right)^{1 \\over 2}\\]

\\[a = \\left({4 P R \\over \\pi E^*}\\right)^{1 \\over 2}\\]

3D Contact: The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:

\\[p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3}\\]

\\[a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3}\\]

", "advice": "

The equivalent radius, $R$, is given by:

${1 \\over R} = {1 \\over R_1} + {1 \\over R_2} = {1 \\over \\var{diawheel} \\div 2} + {1 \\over \\var{diarail} \\div 2}$ [units: mm$^{-1}$]

which can be rearranged to give $R = \\var{siground(R*1000,3)}$mm.

The Elastic Contact Modulus, $E^*$, is given by:

${1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2} ={1 - 0.3^2 \\over 209 \\times 10^9} + {1 - 0.3^2 \\over 209 \\times 10^9}$

which can be rearranged to give $E^* = \\var{siground(ECM,3)}$GPa.

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:

$p_0 = \\left({P E^* \\over \\pi R}\\right)^{1 \\over 2} =\\left({ \\left( \\var{load} \\times 10^3 \\div \\var{width} \\times 10^{-3} \\right) \\times \\var{siground(ECM,3)} \\times 10^9 \\over \\pi \\times \\var{siground(R*1000,3)} \\times 10^{-3}} \\right)^{1 \\over 2} = \\var{siground(p0,3)}$MPa.

The semi-contact width, $a$, is likewise given by:

$a = \\left({4 P R \\over \\pi E^*}\\right)^{1 \\over 2} = \\left({4 \\times\\left( \\var{load} \\times 10^3 \\div \\var{width} \\times 10^{-3} \\right) \\times \\var{siground(R*1000,3)} \\times 10^{-3} \\over \\pi \\times \\var{siground(ECM,3)} \\times 10^9}\\right)^{1 \\over 2} = \\var{siground(scw,3)}$mm.

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:

area = $2 \\times \\var{siground(scw,3)}$mm $\\times \\var{width}$mm $= \\var{siground(area,3)}$mm$^2$.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"scw": {"name": "scw", "group": "Ungrouped variables", "definition": "(4 * P * R / (pi * ECM * 10^9))^(1/2) * 1000", "description": "

Semi-contact width. [Units: mm]

", "templateType": "anything", "can_override": false}, "p0": {"name": "p0", "group": "Ungrouped variables", "definition": "(P * ECM*10^9 / (pi * R))^(1/2) / 10^6", "description": "

Peak contact pressure. [Units: MPa]

", "templateType": "anything", "can_override": false}, "load": {"name": "load", "group": "Ungrouped variables", "definition": "random(4..7#0.5)", "description": "

Applied load. [Units: kN]

", "templateType": "anything", "can_override": false}, "diarail": {"name": "diarail", "group": "Ungrouped variables", "definition": "random(40..47#0.1)", "description": "

Rail disc diameter. [Units: mm]

", "templateType": "anything", "can_override": false}, "ECM": {"name": "ECM", "group": "Ungrouped variables", "definition": "1/((1-0.3^2)/209+(1-0.3^2)/209)", "description": "

Elastic Contact Modulus. [Units: GPa]

", "templateType": "anything", "can_override": false}, "width": {"name": "width", "group": "Ungrouped variables", "definition": "random(5..10)", "description": "

Track width. [Units: mm]

", "templateType": "anything", "can_override": false}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "2*scw*width", "description": "

Contact area. [Units: mm$^2$]

", "templateType": "anything", "can_override": false}, "P": {"name": "P", "group": "Ungrouped variables", "definition": "(load*1000)/(width/1000)", "description": "

Load per unit length. [Units: N/m]

", "templateType": "anything", "can_override": false}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "1/(2000/diawheel+2000/diarail)", "description": "

Equivalent radius. [Units: m]

", "templateType": "anything", "can_override": false}, "diawheel": {"name": "diawheel", "group": "Ungrouped variables", "definition": "random(40..47#0.1)", "description": "

Wheel disc diameter. [Units: mm]

", "templateType": "anything", "can_override": false}}, "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": true, "customName": "Contact Mechanics", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Reference values:

Steel: Density, $\\rho = 8\\ 050$ kg/m$^3$; Young's modulus, $E = 209$ GPa; Poisson's ratio, $\\nu = 0.3$.

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.

The disc surfaces are cylindrical with the axes aligned. The width of the surfaces is $\\var{width}$ mm.

The applied load is $\\var{load}$ kN.

Calculate:

the equivalent radius: [[0]]
the Elastic Contact Modulus: $E^* =$ [[1]]
the peak contact pressure: $p_0 =$ [[2]]
the semi-contact width: $a =$ [[3]]
the area of the contact patch: [[4]]

Important: Remember to state units (if any) with your answer(s).

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "Eq Radius", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(R,\"m\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "ECM", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(ECM*1000000000,\"Pa\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "peak pressure", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(p0*1000000,\"Pa\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "scw", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(scw/1000,\"m\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "area", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(area/1000000,\"m^2\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ball on table w/ units", "extensions": ["quantities"], "custom_part_types": [{"source": {"pk": 19, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/19/edit"}, "name": "Engineering Accuracy with units", "short_name": "engineering-answer", "description": "

A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

A gold ball on a steel table - effect of its own weight.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The size of the contact and the pressure between the contacting surfaces depends on:

the materials, and here we assume them to be linear, elastic and isotropic;
the geometry, and here we assume them to be either flat (infinite radius) or having a constant radius;
the applied load, and here we assume the load is applied normal to the contact, i.e., there is no friction.

Materials: A single material property - the Elastic Contact Modulus ($E^*$) - combines the elastic properties of both contacting surfaces into an equivalent stiffness:

\\[{1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2}\\]

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.

Geometry: The two simplest forms of contact between curved surfaces are:

2D: cylindrical surfaces aligned along their length, or a cylinder on a flat surface, creating a rectangular contact area;
3D: spherical surfaces, or a sphere on a flat surface, creating a circular contact area.

In either case, an equivalent radius $R$ can be defined. If both surfaces are convex, e.g., two balls touching, then:

\\[{1 \\over R} = {1 \\over R_1} + {1 \\over R_2}\\]

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:

\\[{1 \\over R} = {1 \\over R_1} - {1 \\over R_2}\\]

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.)

Applied Load: This is a little tricky because the same notation, $P$, is used to mean different things:

In 2D, $P$ is the applied load per unit length of the cylindrical contact. [Units: N/m]
In 3D, $P$ is simply the applied load. [Units: N]

2D Contact: The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:

\\[p_0 = \\left({P E^* \\over \\pi R}\\right)^{1 \\over 2}\\]

\\[a = \\left({4 P R \\over \\pi E^*}\\right)^{1 \\over 2}\\]

3D Contact: The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:

\\[p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3}\\]

\\[a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3}\\]

", "advice": "

The weight of the sphere is the volume times the density times the acceleration from gravity, i.e.:

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.

The Elastic Contact Modulus, $E^*$, is given by:

${1 \\over E^*} = {1 - 0.4^2 \\over 79 \\times 10^9} + {1 - 0.3^2 \\over 209 \\times 10^9}$

which can be rearranged to give $E^* = \\var{siground(ECM,3)}$GPa.

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:

$p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3} = \\left({6 \\times \\var{siground(weight,3)} \\times \\left( \\var{siground(ECM,3)} \\times 10^9 \\right)^2 \\over \\pi^3 \\left( \\var{diameter} \\times 10^{-2} \\div 2 \\right)^2}\\right)^{1 \\over 3} = \\var{siground(p0,3)}$MPa.

Similarly, the semi-contact width, $a$, is given by:

$a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3} = \\left({3 \\times \\var{siground(weight,3)} \\times \\var{diameter} \\times 10^{-2} \\div 2 \\over 4 \\times \\var{siground(ECM,3)} \\times 10^9} \\right)^{1 \\over 3} = \\var{siground(scw,3)}$mm.

The area of the contact patch is circular with radius equal to $a$, i.e.:

area $= \\pi a^2 = \\pi \\left( \\var{siground(scw,3)} \\times 10^{-3} \\right)^2 = \\var{siground(area,3)}$mm$^2$.

", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"weight": {"name": "weight", "group": "Ungrouped variables", "definition": "volume*19320*9.81", "description": "

Weight of gold ball. [Units: N]

", "templateType": "anything", "can_override": false}, "ECM": {"name": "ECM", "group": "Ungrouped variables", "definition": "1/((1-0.4^2)/79+(1-0.3^2)/209)", "description": "

Elastic Contact Modulus. [Units: GPa]

", "templateType": "anything", "can_override": false}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "pi*scw^2", "description": "

Contact area. [Units: mm$^2$]

", "templateType": "anything", "can_override": false}, "p0": {"name": "p0", "group": "Ungrouped variables", "definition": "(6 * weight * (ECM*10^9)^2 / (pi^3 * R^2))^(1/3) / 10^6", "description": "

Peak contact pressure. [Units: MPa]

", "templateType": "anything", "can_override": false}, "scw": {"name": "scw", "group": "Ungrouped variables", "definition": "(3 * weight * R / (4 * ECM * 10^9))^(1/3) * 1000", "description": "

Semi-contact width. [Units: mm]

", "templateType": "anything", "can_override": false}, "diameter": {"name": "diameter", "group": "Ungrouped variables", "definition": "random(6..40)", "description": "

Diameter of gold ball. [Units: cm]

", "templateType": "anything", "can_override": false}, "volume": {"name": "volume", "group": "Ungrouped variables", "definition": "pi*R^3*4/3", "description": "

Volume of gold ball. [Units: m3]

", "templateType": "anything", "can_override": false}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "diameter/200", "description": "

Radius of gold ball. [Units: m]

", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["diameter", "R", "volume", "weight", "ECM", "p0", "scw", "area"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Contact Mechanics", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Reference values:

Gold: Density, $\\rho = 19\\ 320$ kg/m$^3$; Young's modulus, $E = 79$ GPa; Poisson's ratio, $\\nu = 0.4$.
Steel: Density, $\\rho = 8\\ 050$ kg/m$^3$; Young's modulus, $E = 209$ GPa; Poisson's ratio, $\\nu = 0.3$.
Gravity: $g = 9.81$ m/s$^2$.

A gold sphere of diameter $\\var{diameter}$ cm sits on a flat steel table.

Calculate:

the weight of the ball: [[0]]
the Elastic Contact Modulus: $E^* =$ [[1]]
the peak contact pressure: $p_0 =$ [[2]]
the semi-contact width: $a =$ [[3]]
the area of the contact patch: [[4]]

Important: Remember to state units (if any) with your answer(s).

", "gaps": [{"type": "engineering-answer", "useCustomName": true, "customName": "weight", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(weight,\"N\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "ECM", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(ECM*1000000000,\"Pa\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "peak pressure", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(p0*1000000,\"Pa\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "scw", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(scw/1000,\"m\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}, {"type": "engineering-answer", "useCustomName": true, "customName": "area", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correctAnswer": "quantity(area/1000000,\"m^2\")", "right": "0.2", "close": "1.0", "C1": "75", "C2": "50", "C3": "25"}}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Ball joint w/ units", "extensions": ["quantities"], "custom_part_types": [{"source": {"pk": 19, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/19/edit"}, "name": "Engineering Accuracy with units", "short_name": "engineering-answer", "description": "

A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

Hertz contact calculation for ball joint.

", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "

The size of the contact and the pressure between the contacting surfaces depends on:

the materials, and here we assume them to be linear, elastic and isotropic;
the geometry, and here we assume them to be either flat (infinite radius) or having a constant radius;
the applied load, and here we assume the load is applied normal to the contact, i.e., there is no friction.

Materials: A single material property - the Elastic Contact Modulus ($E^*$) - combines the elastic properties of both contacting surfaces into an equivalent stiffness:

\\[{1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2}\\]

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.

Geometry: The two simplest forms of contact between curved surfaces are:

2D: cylindrical surfaces aligned along their length, or a cylinder on a flat surface, creating a rectangular contact area;
3D: spherical surfaces, or a sphere on a flat surface, creating a circular contact area.

In either case, an equivalent radius $R$ can be defined. If both surfaces are convex, e.g., two balls touching, then:

\\[{1 \\over R} = {1 \\over R_1} + {1 \\over R_2}\\]

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:

\\[{1 \\over R} = {1 \\over R_1} - {1 \\over R_2}\\]

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.)

Applied Load: This is a little tricky because the same notation, $P$, is used to mean different things:

In 2D, $P$ is the applied load per unit length of the cylindrical contact. [Units: N/m]
In 3D, $P$ is simply the applied load. [Units: N]

2D Contact: The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:

\\[p_0 = \\left({P E^* \\over \\pi R}\\right)^{1 \\over 2}\\]

\\[a = \\left({4 P R \\over \\pi E^*}\\right)^{1 \\over 2}\\]

3D Contact: The peak contact pressure, $p_0$, and semi-contact width, $a$, are given by:

\\[p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3}\\]

\\[a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3}\\]

", "advice": "

Reference values:

HDPE: Young's modulus, $E = 1$ GPa; Poisson's ratio, $\\nu = 0.45$.
Steel: Young's modulus, $E = 209$ GPa; Poisson's ratio, $\\nu = 0.3$.

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.

Calculate:

Because the cup is concave not convex, the radius is effectively negative. The equivalent radius is therefore given by:

${1 \\over R} = {1 \\over R_1} - {1 \\over R_2} = {1 \\over \\var{diaball} \\div 2} - {1 \\over \\var{diacup} \\div 2}$ [units: mm$^{-1}$]

which can be rearranged to give $R = \\var{siground(R*1000,3)}$mm.

The Elastic Contact Modulus, $E^*$, is given by:

${1 \\over E^*} = {1 - \\nu_1^2 \\over E_1} + {1 - \\nu_2^2 \\over E_2} = {1 - 0.45^2 \\over 1 \\times 10^9} + {1 - 0.3^2 \\over 209 \\times 10^9}$

which can be rearranged to give $E^* = \\var{siground(ECM,3)}$GPa.

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:

$p_0 = \\left({6 P {E^*}^2 \\over \\pi^3 R^2}\\right)^{1 \\over 3} = \\left({6 \\times \\var{load} \\times \\left( \\var{siground(ECM,3)} \\times 10^9 \\right)^2 \\over \\pi^3 \\left( \\var{siground(R*1000,3)} \\times 10^{-3} \\right)^2}\\right)^{1 \\over 3} = \\var{siground(p0,3)}$MPa.

Similarly, the semi-contact width, $a$, is given by:

$a = \\left({3 P R \\over 4 E^*}\\right)^{1 \\over 3} = \\left({3 \\times \\var{load} \\times \\var{siground(R*1000,3)} \\times 10^{-3} \\over 4 \\times \\var{siground(ECM,3)} \\times 10^9} \\right)^{1 \\over 3} = \\var{siground(scw,3)}$mm.

The area of the contact patch is circular with radius equal to $a$, i.e.:

area $= \\pi a^2 = \\pi \\left( \\var{siground(scw,3)} \\times 10^{-3} \\right)^2 = \\var{siground(area,3)}$mm$^2$.

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Applied load. [Units: N]

", "templateType": "anything", "can_override": false}, "ECM": {"name": "ECM", "group": "Ungrouped variables", "definition": "1/((1-0.45^2)/1+(1-0.3^2)/209)", "description": "

Elastic Contact Modulus. [Units: GPa]

", "templateType": "anything", "can_override": false}, "p0": {"name": "p0", "group": "Ungrouped variables", "definition": "(6 * load * (ECM*10^9)^2 / (pi^3 * R^2))^(1/3) / 10^6", "description": "

Peak contact pressure. [Units: MPa]

", "templateType": "anything", "can_override": false}, "diaball": {"name": "diaball", "group": "Ungrouped variables", "definition": "random(20..30)", "description": "

Diameter of ball. [Units: mm]

", "templateType": "anything", "can_override": false}, "diacup": {"name": "diacup", "group": "Ungrouped variables", "definition": "diaball+random(1..4)", "description": "

Diameter of cup. [Units: mm]

", "templateType": "anything", "can_override": false}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "pi*scw^2", "description": "

Contact area. [Units: mm$^2$]

", "templateType": "anything", "can_override": false}, "R": {"name": "R", "group": "Ungrouped variables", "definition": "1/(2000/diaball-2000/diacup)", "description": "

Equivalent radius. [Units: m]

", "templateType": "anything", "can_override": false}, "scw": {"name": "scw", "group": "Ungrouped variables", "definition": "(3 * load * R / (4 * ECM * 10^9))^(1/3) * 1000", "description": "

Semi-contact width. [Units: mm]

", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["diaball", "diacup", "R", "ECM", "load", "p0", "scw", "area"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Contact Mechanics", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Reference values:

HDPE: Young's modulus, $E = 1$ GPa; Poisson's ratio, $\\nu = 0.45$.
Steel: Young's modulus, $E = 209$ GPa; Poisson's ratio, $\\nu = 0.3$.

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.

Calculate:

the equivalent radius: $R =$ [[0]]
the Elastic Contact Modulus: $E^* =$ [[1]]
the peak contact pressure: $p_0 =$ [[2]]
the semi-contact width: $a =$ [[3]]
the area of the contact patch: [[4]]

Important: Remember to state units (if any) with your answer(s).

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A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.

", "definition": "right_sign and percent_error <= settings['close']"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity) "}], "settings": [{"name": "correctAnswer", "label": "Correct Quantity.", "help_url": "", "hint": "The correct answer given as a JME quantity.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "right", "label": "% Accuracy for right.", "help_url": "", "hint": "Question will be considered correct if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "0.2", "evaluate": true}, {"name": "close", "label": "% Accuracy for close.", "help_url": "", "hint": "Question will be considered close if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "1.0", "evaluate": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units. if correct answer is 100 N and close is ±1%,
99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
If the correct answer is 100 N, both 100 and -100 N are accepted.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value but forgotten units.
This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [["question-resources/Contact.png", "/srv/numbas/media/question-resources/Contact.png"]]}