move notes out of year subfolders
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itags: [ mmme2044 ]
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d="m 548.98,146.16 -0.66,-3.97 -4.34,-10.47 -4,-10.1 -3.34,-11.55 9.56,-10.47 2.78,10.11 3.01,7.58 2.67,5.41 2.67,3.25 4.67,4.33 6,4.33 6.01,1.8 h 6.67 z"
|
||||
style="fill:#254061;fill-opacity:1;fill-rule:evenodd;stroke:none"
|
||||
id="path194" />
|
||||
<path
|
||||
d="m 548.98,146.16 -0.66,-3.97 -4.34,-10.47 -4,-10.1 -3.34,-11.55 9.56,-10.47 2.78,10.11 3.01,7.58 2.67,5.41 2.67,3.25 4.67,4.33 6,4.33 6.01,1.8 h 6.67 z"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
||||
id="path196" />
|
||||
<path
|
||||
d="m 529.67,148.64 h 15.58 l 4.33,-2.88 32.04,-10.22 h 22.58 l 6.01,-6.61 v -9.38 l 30.37,-21.51 h 6.12 l 21.58,17.79 v 29.93 h -39.83 l -6.89,11.29 -59.85,0.97 -12.46,1.44 -0.67,6.13 -2,4.32 -3.34,2.53 -4.34,1.08 -9.34,-16.95 z"
|
||||
style="fill:#4f81bd;fill-opacity:1;fill-rule:evenodd;stroke:none"
|
||||
id="path198" />
|
||||
<path
|
||||
d="m 529.67,148.64 h 15.58 l 4.33,-2.88 32.04,-10.22 h 22.58 l 6.01,-6.61 v -9.38 l 30.37,-21.51 h 6.12 l 21.58,17.79 v 29.93 h -39.83 l -6.89,11.29 -59.85,0.97 -12.46,1.44 -0.67,6.13 -2,4.32 -3.34,2.53 -4.34,1.08 -9.34,-16.95 z"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
||||
id="path200" />
|
||||
<path
|
||||
d="m 658.68,144.3 c 0,6.99 -5.21,12.66 -11.64,12.66 -6.43,0 -11.64,-5.67 -11.64,-12.66 0,-6.99 5.21,-12.66 11.64,-12.66 6.43,0 11.64,5.67 11.64,12.66 z"
|
||||
style="fill:#c0504d;fill-opacity:1;fill-rule:evenodd;stroke:none"
|
||||
id="path202" />
|
||||
<path
|
||||
d="m 658.68,144.3 c 0,6.99 -5.21,12.66 -11.64,12.66 -6.43,0 -11.64,-5.67 -11.64,-12.66 0,-6.99 5.21,-12.66 11.64,-12.66 6.43,0 11.64,5.67 11.64,12.66 z"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
||||
id="path204" />
|
||||
<path
|
||||
d="m 655.32,144.48 c 0,4.9 -3.68,8.88 -8.22,8.88 -4.54,0 -8.22,-3.98 -8.22,-8.88 0,-4.9 3.68,-8.88 8.22,-8.88 4.54,0 8.22,3.98 8.22,8.88 z"
|
||||
style="fill:#4f81bd;fill-opacity:1;fill-rule:evenodd;stroke:none"
|
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id="path206" />
|
||||
<path
|
||||
d="m 655.32,144.48 c 0,4.9 -3.68,8.88 -8.22,8.88 -4.54,0 -8.22,-3.98 -8.22,-8.88 0,-4.9 3.68,-8.88 8.22,-8.88 4.54,0 8.22,3.98 8.22,8.88 z"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
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id="path208" />
|
||||
<path
|
||||
d="M 689.04,240 V 98.04"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
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id="path210" />
|
||||
<path
|
||||
d="M 668.28,115.92 V 98.16"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
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id="path212" />
|
||||
<path
|
||||
d="M 610.44,120 V 98.28"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
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id="path214" />
|
||||
<path
|
||||
d="M 529.68,148.68 V 98.04"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
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id="path216" />
|
||||
<path
|
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d="M 706.44,98.04 H 512.28"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
||||
id="path218" />
|
||||
<g
|
||||
id="g220">
|
||||
<g
|
||||
id="g222"
|
||||
clip-path="url(#clipPath226)">
|
||||
<path
|
||||
d="m 700.95,24.983 c 6.5,14.144 13,28.287 13.91,40.39 0.91,12.102 -4.32,30.99 -8.23,32.452 -3.92,1.461 -15.17,-8.637 -15.48,-23.429 -0.3,-14.791 11.07,-52.025 13.96,-65.7022"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.75;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
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id="path228" />
|
||||
</g>
|
||||
</g>
|
||||
<path
|
||||
d="m 513.18,98.04 c -6.09,-5.768 -12.3,-11.656 -12.19,-26.196 0.11,-14.54 9.64,-50.59 12.97,-60.924"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
||||
id="path230" />
|
||||
<path
|
||||
d="M 684.27,255.48 H 539.66 l -6.23,-2.88 -2.67,-3.85 -1.56,-4.56 0.67,-87.95 9.34,0.24 v 78.82 l 0.89,4.8 2.23,2.88 4.45,0.49 124.58,0.36 1,5.4 2.56,1.44 4.12,-0.72 2.33,-2.52 0.67,-9.25 h 7.12 v 6.73 3.36 l -0.44,3.85 z"
|
||||
style="fill:#ff0000;fill-opacity:1;fill-rule:evenodd;stroke:none"
|
||||
id="path232" />
|
||||
<path
|
||||
d="M 684.27,255.48 H 539.66 l -6.23,-2.88 -2.67,-3.85 -1.56,-4.56 0.67,-87.95 9.34,0.24 v 78.82 l 0.89,4.8 2.23,2.88 4.45,0.49 124.58,0.36 1,5.4 2.56,1.44 4.12,-0.72 2.33,-2.52 0.67,-9.25 h 7.12 v 6.73 3.36 l -0.44,3.85 z"
|
||||
style="fill:none;stroke:#000000;stroke-width:0.72;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
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id="path234" />
|
||||
<path
|
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d="m 550.68,242.88 120.28,0.24 5.67,-0.72 3.44,-1.32 1.56,-3.25 0.33,-7.68 -0.33,-96.23 h -5 v 96.23 l -0.22,3.6 -1.78,2.88 -2.67,0.84 -4.33,0.24 -116.95,-0.24 z"
|
||||
style="fill:#595959;fill-opacity:1;fill-rule:evenodd;stroke:none"
|
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id="path236" />
|
||||
<path
|
||||
d="m 550.68,242.88 120.28,0.24 5.67,-0.72 3.44,-1.32 1.56,-3.25 0.33,-7.68 -0.33,-96.23 h -5 v 96.23 l -0.22,3.6 -1.78,2.88 -2.67,0.84 -4.33,0.24 -116.95,-0.24 z"
|
||||
style="fill:none;stroke:#000000;stroke-width:1.2;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:none;stroke-opacity:1"
|
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id="path238" />
|
||||
<path
|
||||
d="m 653.68,298.72 -35.16,-37.31 1.48,-1.4 35.16,37.31 z m -34.42,-38.01 -0.92,6.47 -3.28,-10.92 10.71,3.92 z"
|
||||
style="fill:#c00000;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="path240" />
|
||||
<g
|
||||
id="g242">
|
||||
<g
|
||||
id="g244"
|
||||
clip-path="url(#clipPath248)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,718.61,218.64)"
|
||||
style="font-variant:normal;font-weight:normal;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
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id="text252"><tspan
|
||||
x="0 13.26648 23.787479 30.500879 35.049961 45.611038 56.132038 65.731201 70.340401"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan250">Optional </tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<g
|
||||
id="g254">
|
||||
<g
|
||||
id="g256"
|
||||
clip-path="url(#clipPath260)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,718.61,194.64)"
|
||||
style="font-variant:normal;font-weight:normal;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
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id="text264"><tspan
|
||||
x="0 4.6092 15.1302 25.651199 35.631119 42.62508 47.154121 55.55088 65.15004 72.98568"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan262">inner case</tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<g
|
||||
id="g266">
|
||||
<g
|
||||
id="g268"
|
||||
clip-path="url(#clipPath272)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,661.66,309.5)"
|
||||
style="font-variant:normal;font-weight:normal;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="text276"><tspan
|
||||
x="0 13.3266 23.847601 30.3606 40.340519 47.33448 51.863522 60.240238 69.839401 77.675041 87.654961 92.183998 98.256119 114.26808 123.50652 132.58464 137.11368 147.63467 157.61459"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan274">Outer case (may be </tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<g
|
||||
id="g278">
|
||||
<g
|
||||
id="g280"
|
||||
clip-path="url(#clipPath284)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,661.66,285.5)"
|
||||
style="font-variant:normal;font-weight:normal;font-size:20.064px;font-family:Calibri;-inkscape-font-specification:Calibri;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="text288"><tspan
|
||||
x="0 7.002336 17.535936 28.069536 38.603134 48.595009 55.597343 59.99136 68.398178 78.971901 88.321732 94.802399 104.79427 115.32787"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan286">rubber coated)</tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<g
|
||||
id="g290">
|
||||
<g
|
||||
id="g292"
|
||||
clip-path="url(#clipPath296)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,704.93,113.95)"
|
||||
style="font-variant:normal;font-weight:normal;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="text300"><tspan
|
||||
x="0 12.64524 22.2444 29.158199 35.631119 45.611038 52.605 57.214199 65.049843 75.570839 82.564796 87.093842 97.614838"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan298">Garter spring</tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<g
|
||||
id="g302">
|
||||
<g
|
||||
id="g304"
|
||||
clip-path="url(#clipPath308)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,734.81,56.304)"
|
||||
style="font-variant:normal;font-weight:normal;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="text312"><tspan
|
||||
x="0 10.36068 17.274481 21.88368 37.795441 47.3946 54.448681"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan310">Primary</tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<g
|
||||
id="g314">
|
||||
<g
|
||||
id="g316"
|
||||
clip-path="url(#clipPath320)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,734.81,32.304)"
|
||||
style="font-variant:normal;font-weight:normal;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="text324"><tspan
|
||||
x="0 7.83564 17.815559 27.414721 31.923719 36.532921 47.053921 56.49276 61.061878 65.671082 70.180077"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan322">sealing lip</tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<g
|
||||
id="g326">
|
||||
<g
|
||||
id="g328"
|
||||
clip-path="url(#clipPath332)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,468.96,72.168)"
|
||||
style="font-variant:normal;font-weight:normal;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="text336"><tspan
|
||||
x="0 13.26648 23.787479 30.500879 35.049961 45.611038 56.132038 65.731201 70.340401 74.769241 82.604881 92.584801 100.92144 111.48252 122.00352 132.58464 142.18381 149.23788"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan334">Optional secondary</tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<g
|
||||
id="g338">
|
||||
<g
|
||||
id="g340"
|
||||
clip-path="url(#clipPath344)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,468.96,48.168)"
|
||||
style="font-variant:normal;font-weight:normal;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="text348"><tspan
|
||||
x="0 7.83564 17.815559 27.414721 31.923719 36.532921 47.053921 56.49276 61.061878 65.671082 70.180077"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan346">sealing lip</tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<path
|
||||
d="m 712.56,208.15 -29.55,-9.11 0.6,-1.95 29.55,9.12 z m -29.25,-10.09 2.39,6.08 -8.24,-7.88 11.25,-1.87 z"
|
||||
style="fill:#c00000;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="path350" />
|
||||
<path
|
||||
d="m 698.96,122.75 -41.69,20.9 0.92,1.82 41.69,-20.9 z m -41.23,21.81 1.36,-6.39 -6.83,9.13 11.4,-0.01 z"
|
||||
style="fill:#c00000;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="path352" />
|
||||
<path
|
||||
d="m 727.08,60.929 -80.33,36.017 0.83,1.861 80.34,-36.016 z m -79.92,36.947 1.64,-6.322 -7.22,8.826 11.39,0.48 z"
|
||||
style="fill:#c00000;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="path354" />
|
||||
<path
|
||||
d="m 520.29,72.48 24.84,39.81 -1.74,1.08 -24.84,-39.81 z m 23.97,40.35 2.17,-6.16 1.07,11.35 -9.73,-5.95 z"
|
||||
style="fill:#c00000;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="path356" />
|
||||
<g
|
||||
id="g670">
|
||||
<path
|
||||
d="m 496.92,42.96 225,-3.6"
|
||||
style="fill:none;stroke:#000000;stroke-width:1.2;stroke-linecap:butt;stroke-linejoin:round;stroke-miterlimit:10;stroke-dasharray:9.6, 3.6, 1.2, 3.6;stroke-dashoffset:0;stroke-opacity:1"
|
||||
id="path672" />
|
||||
</g>
|
||||
<g
|
||||
id="g674">
|
||||
<g
|
||||
id="g676"
|
||||
clip-path="url(#clipPath680)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,470.4,286.99)"
|
||||
style="font-variant:normal;font-weight:bold;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri-Bold;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="text684"><tspan
|
||||
x="0 11.523 18.476879 34.769402 45.550919 53.627041 64.388519 75.210121 85.290237 92.404442 97.334282 105.711"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan682">Atmospheric </tspan></text>
|
||||
</g>
|
||||
</g>
|
||||
<g
|
||||
id="g686">
|
||||
<g
|
||||
id="g688"
|
||||
clip-path="url(#clipPath692)">
|
||||
<text
|
||||
xml:space="preserve"
|
||||
transform="matrix(1,0,0,-1,470.4,262.99)"
|
||||
style="font-variant:normal;font-weight:bold;font-size:20.04px;font-family:Calibri;-inkscape-font-specification:Calibri-Bold;writing-mode:lr-tb;fill:#003399;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="text696"><tspan
|
||||
x="0 7.9959602 12.9258 23.7474"
|
||||
y="0"
|
||||
sodipodi:role="line"
|
||||
id="tspan694">side</tspan></text>
|
||||
</g>
|
||||
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|
||||
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|
||||
d="m 492.21,259.18 30.58,-51.03 -1.75,-1.04 -30.58,51.03 z m 29.7,-51.55 2.28,6.12 0.87,-11.37 -9.62,6.13 z"
|
||||
style="fill:#c00000;fill-opacity:1;fill-rule:nonzero;stroke:none"
|
||||
id="path698" />
|
||||
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|
||||
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|
||||
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|
||||
---
|
||||
author: Akbar Rahman
|
||||
date: \today
|
||||
title: MMME2044 // Linkages
|
||||
tags: [ mmme2044, linkages, uni ]
|
||||
uuid: ae955a7e-bc65-4055-a79d-38b57c5a00a5
|
||||
---
|
||||
|
||||
# Freedom and Constraints
|
||||
|
||||
A body that is free in space can move in three, independent perpendicular directions and rotation
|
||||
in three ways about those directions.
|
||||
It is said to have 6 degrees of freedom (DoF).
|
||||
|
||||
The number of degrees of freedom are the number of components of motion that are required in order
|
||||
to generate the motion.
|
||||
|
||||
If a joint is constrained to move along a line, it has 1 degree of freedom.
|
||||
If a joint is constrained to a plane with no rotation then it has two degrees of freedom.
|
||||
|
||||
## Gruebler's Equation of Degrees of Freedom
|
||||
|
||||
$$M = 3L - 2J - 3G$$
|
||||
|
||||
where $M$ is degrees of freedom, $L$ is number of links, $J$ is number of joints, and $G$ is number
|
||||
of grounded links.
|
||||
|
||||
|
||||
|
||||
- if DoF > 0, it is a mechanism with relative motions
|
||||
- if DoF = 0, it is a structure without motion
|
||||
- if DoF < 0, it is a preloaded (or overly constrained) structure without motion but possibly
|
||||
additional stress
|
||||
|
||||
|
||||
|
||||
## Grashof Condition
|
||||
|
||||
Four bar linkage is the simplest possible pin-jointed mechanism.
|
||||
|
||||
The Grashof condition is a simple relationship that predicts the linkage behaviour based only on
|
||||
the lengths of the links.
|
||||
A linkage is a Grashof linkage if at least one link is capable of making a full revolution and
|
||||
the following equation holds true:
|
||||
|
||||
$$S + L < P + Q$$
|
||||
|
||||
where
|
||||
|
||||
- $S$ is the length of shortest link
|
||||
- $L$ is the length of longest link
|
||||
- $P$ is the length of one remaining link
|
||||
- $Q$ is the length of other remaining link
|
||||
|
||||
### Triple Rockers
|
||||
|
||||
If $S + L > P + Q$ then it is not a Grashof linkage and is known as a triple rocker.
|
||||
|
||||
|
||||
|
||||
### Double Cranks / Crank Rockers
|
||||
|
||||
$$S + L = P + Q$$
|
||||
|
||||
|
||||
|
||||
## Limit Conditions
|
||||
|
||||
Toggle is an important test to check if the linkage can reach all specified positions without
|
||||
encountering a limit or toggle position, also called a station configuration.
|
||||
|
||||
Toggle positions may be determined by collinearity of two links
|
||||
|
||||
|
||||
|
||||
## Transmission Angle
|
||||
|
||||
Transmission angle, $\mu$, is the angle between the output link and the coupler.
|
||||
|
||||
It is a useful test to check the quality of the force transmission in a linkage.
|
||||
|
||||
Ideally $\mu = 90$ deg but at minimum $\mu \ge 40$ deg.
|
||||
@@ -1,157 +0,0 @@
|
||||
---
|
||||
author: Akbar Rahman
|
||||
date: \today
|
||||
title: MMME2044 // Seals
|
||||
tags: [ uni, mmme2044, seals ]
|
||||
uuid: ff74e8ad-090d-47eb-b9cc-41ccdefb9a63
|
||||
---
|
||||
|
||||
# Function
|
||||
|
||||
- used to prevent or limit leakage of fluids or particulates
|
||||
- keep dirt out
|
||||
- allow motion
|
||||
- allow disassembly
|
||||
|
||||
# Classification of Seals
|
||||
|
||||
|
||||
|
||||
## O-Rings
|
||||
|
||||
- good for static and low speed applications
|
||||
- requires interference fit
|
||||
- application is codified---use a standard where possible (BS 4518:1982, BS ISO 3601-1~5)
|
||||
- sized by internal diameter (ID) and section diameter (A)
|
||||
- works by having the rubber o-ring squeezed against the two surfaces
|
||||
|
||||
- pre-compression ensures sealing when there is no pressure
|
||||
|
||||
### Challenger Space Shuttle Disaster
|
||||
|
||||
- Space Shuttle Challenger broke apart 73 seconds into flight
|
||||
- at the low temperatures, the o-rings lost their resilience
|
||||
- the booster cases distorted during launch
|
||||
- insufficient groove width in o-ring design
|
||||
|
||||
|
||||
|
||||
### Common Forms of Failure
|
||||
|
||||
- Abrasion---sealing face is worn away. Can be reduced with smoother surfaces, harder materials,
|
||||
better lubricant, and reduced speed
|
||||
- Compression set---the sealing face is permanently distorted. Can be solved with more temperature
|
||||
resistant o-rings
|
||||
- Installation damage---can be avoided with chamfers and blending radii
|
||||
- Extrusion---the o-ring is forced into the sealing clearence. can be avoided with:
|
||||
|
||||
- smaller clearances < 0.13 mm;
|
||||
- harder material
|
||||
- different cross section
|
||||
- backup rings---come in a range of shapes and sizes (BS 5106:1988)
|
||||
|
||||
- Spiral damage---o-ring is twisted and leaks. can be avoided with:
|
||||
|
||||
- lower reciprocating speed
|
||||
- harder material
|
||||
- different cross sections (e.g. quadrant seals, which have a x-shaped section)
|
||||
|
||||
# Groove Dimensions
|
||||
|
||||
- up to 100 bar for static loading
|
||||
- up to 14 bar & 3.8 m/s for dynamic loading
|
||||
|
||||
> i don't know when this is the case but it just says this in the
|
||||
> [lecture slides](./lecture_slides/Lecture 6 Seals_v1.0.pdf)
|
||||
> (and i haven't watched the lecture yet)
|
||||
|
||||
- $F$---radial depth
|
||||
- $E$---groove width
|
||||
- $G$---total clearance
|
||||
- $C$---lead-in chamfer
|
||||
- $R$---radius
|
||||
|
||||
## Groove in Cylinder
|
||||
|
||||
|
||||
|
||||
## Groove in Piston
|
||||
|
||||
|
||||
|
||||
## Example of a Dimensions Table
|
||||
|
||||
|
||||
|
||||
# O-Ring Properties
|
||||
|
||||
Material | Hardness (Shore A) | Temperature Range ($^{\circ}$C) | Uses
|
||||
-------- | ------------------ | ------------------------------- | ---------------
|
||||
Nitrile rubber | 70 | -30 to 110 | general purpose, fuel, oils, water
|
||||
Silicone | 70 | -60 to 200 | medical, body fluid resistant, low temp, alcohol, oxygen
|
||||
Viton (fluorocarbon) | 85 | -20 to 200 | high temperature, extreme chemicals
|
||||
Polyurethane | 95 | -30 to 110 | high temps, oil, gas, hydraulics, good wear
|
||||
|
||||
Shore A context:
|
||||
|
||||
Item | Hardness (Shore A)
|
||||
------------- | ------------------
|
||||
Rubber band | 20
|
||||
Pencil rubber | 40
|
||||
Tire tread | 70
|
||||
Shoe heel | 80
|
||||
|
||||
# Gaskets
|
||||
|
||||
- a material or combination of materials clamped between two separable members to form a joint
|
||||
|
||||
## Gasket Anatomy
|
||||
|
||||
1. Base material---a compliant element such as cork, cellulose, or a fibre
|
||||
2. Binders---temperature and chemical resistant material such as rubber, elastomers, and resins
|
||||
3. Fillers---fill voids in base material. usually rubbers, elastomers, or resins
|
||||
4. Reinforcements---prevents distortion. usually a metal or fabric core
|
||||
|
||||
|
||||
|
||||
# Seals in Rotating Shafts
|
||||
|
||||
- low pressure and low temperature---o rings, radial lip seals, axial lip seals
|
||||
- high pressure---high performance lip seals, mechanical seals
|
||||
- high temperature---labrynth seals
|
||||
|
||||
## O-Rings
|
||||
|
||||
- o-rings can be used in dynamic applications
|
||||
- friction is high but sealing is good within operating range (14 bar, 3.8 m/s)
|
||||
- surface roughness must not exceed 0.4 mm Ra, should be around 0.1 mm Ra
|
||||
- spiral and abrasion are main failure modes, if installed correctly
|
||||
|
||||
# Radial Lip Seal Anatomy
|
||||
|
||||
- low pressure---0.3-0.6 bar, up to 7 bar with backing plate
|
||||
- moderate speed---up to 16 m/s
|
||||
- high temperature
|
||||
- normal environmental conditions
|
||||
- slight leakage permitted
|
||||
- normally made of nitrile rubber compounds
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
## Types of Lip Seal
|
||||
|
||||
- BS ISO 6149-1:2007 defines 4 basic types of rotary shaft lip seal:
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
- $D_1$---nominal diameter of shaft
|
||||
- $D_2$---nominal diameter of housing
|
||||
- $b$---nominal seal width
|
||||
- $A$---air side
|
||||
- $B$---fluid side
|
||||
|
||||
|
||||
@@ -1,235 +0,0 @@
|
||||
---
|
||||
author: Akbar Rahman
|
||||
date: \today
|
||||
title: MMME2044 // Shaft Design
|
||||
tags: [ mmme2044, shafts, uni ]
|
||||
uuid: 8e0928a6-c20c-4f80-9691-beb2defa4022
|
||||
---
|
||||
|
||||
# Shaft Design Considerations
|
||||
|
||||
- Function and loading
|
||||
- Size and connection to components
|
||||
- Material selection and treatments
|
||||
- Deflection and rigidity
|
||||
- Stress and strength
|
||||
- Critical speed
|
||||
- Manufacturing constraints
|
||||
|
||||
# Shaft-Hub Connections
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
# Shaft-Shaft Connections
|
||||
|
||||
Shaft-shaft connections can use either rigid or flexible couplings.
|
||||
|
||||
|
||||
|
||||
# Location of Bearing on Shaft
|
||||
|
||||
|
||||
|
||||
# Shaft Loading
|
||||
|
||||
- Axial stresses
|
||||
|
||||
- cause by self-weight in vertical shafts
|
||||
- due to axial restraint at bearings and associated axial load
|
||||
|
||||
- bending stress
|
||||
|
||||
- due to bending moment in belt drivers, gear forces, mounted component weights
|
||||
- dynamic forces which can load to fatigue and resonance
|
||||
|
||||
- shear stresses
|
||||
|
||||
- due to torque load/direct shear
|
||||
|
||||
# Shaft Diameter (ASME Design Code)
|
||||
|
||||
$$d = \left[
|
||||
\frac{32n_s}{\pi}
|
||||
\sqrt{
|
||||
\left(\frac{M}{\sigma_e}\right)^2
|
||||
+ \frac34 \left(\frac{T}{\sigma_y}\right)^2
|
||||
}
|
||||
\right]^{\frac13}
|
||||
$$
|
||||
|
||||
where:
|
||||
|
||||
- $n_s$ is [safety factor](#safety-factor)
|
||||
- $M$ is max bending moment (Nm)
|
||||
- $T$ is max torque (Nm)
|
||||
- $\sigma_e$ is endurance limit stress (Pa)
|
||||
- $\sigma_y$ is yield strength of shaft (Pa)
|
||||
|
||||
|
||||
<details>
|
||||
|
||||
<summary>
|
||||
|
||||
#### ASME Design Code Derivation
|
||||
|
||||
</summary>
|
||||
|
||||
- Bending moment creates alternating tensile/compressive stresses ($\sigma_a$):
|
||||
|
||||
\begin{align}
|
||||
\sigma &= \frac{My}{I} \\
|
||||
I &= \frac{\pi d^4}{64}
|
||||
\end{align}
|
||||
|
||||
Combine to get:
|
||||
|
||||
$$\sigma_a = \frac{32M}{\pi d^3}$$
|
||||
|
||||
- Torque normally generates constant shear stress ($\tau_m$):
|
||||
|
||||
\begin{align}
|
||||
\tau &= \frac{TR}{J} \\
|
||||
J &= \frac{\pi d^4}{32}
|
||||
\end{align}
|
||||
|
||||
Combine to get:
|
||||
|
||||
$$\rightarrow \tau_m = \frac{16T}{\pi d^3}$$
|
||||
|
||||
- von Mises stress in plane stress condition:
|
||||
|
||||
$$\sigma_\text{von Mises} = (\sigma^2 + 3\tau^2)^{\frac12}$$
|
||||
|
||||
- ASME Fatigue Failure Criterion
|
||||
|
||||
$$\left(\frac{n_s\sigma_a}{\sigma_e}\right)^2 + \left(\frac{n_s\sigma_m}{\sigma_Y}\right)^2 = 1$$
|
||||
|
||||
Finally substitute $\sigma_a$ and $\tau_m$ to make $d$ the subject.
|
||||
|
||||
</details>
|
||||
|
||||
## Endurance Limit Stress, $\sigma_e$
|
||||
|
||||
$\sigma_e$ is the level of stress with which fatigue failure wouldn't occur in cycling or
|
||||
alternating load conditions:
|
||||
|
||||
|
||||
|
||||
- $\sigma$ at failure decreases with the number of cycles up until a certain point ($\sigma_e$ after
|
||||
around $10^6$ cycles)
|
||||
- if you never exceed this point then the material will last for "infinite" cycles
|
||||
- most steels have this fatigue behaviour so they are often used for shafts
|
||||
|
||||
Time for a silly equation:
|
||||
|
||||
$$\sigma_e = k_ak_bk_ck_dk_ek_fk_g\sigma_e'$$
|
||||
|
||||
where:
|
||||
|
||||
- $k_a$ --- surface factor
|
||||
- $k_b$ --- size factor
|
||||
- $k_c$ --- reliability factor
|
||||
- $k_d$ --- temperature factor
|
||||
- $k_e$ --- duty cycle factor
|
||||
- $k_f$ --- fatigue stress concentration factor
|
||||
- $k_g$ --- miscellaneous effects factor
|
||||
- $\sigma_e'$ --- endurance limit of test specimen
|
||||
|
||||
this is a joke of an equation
|
||||
|
||||
## Reducing Stress Concentrations
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
## Critical Speed of Shaft (Natural Frequency)
|
||||
|
||||
- operational speed of shat should be half the critical speed
|
||||
- Centre of Mass should be on the Centre of Rotation
|
||||
- in practice this is not the case
|
||||
- this imbalance causes a deflection (due to centrifugal force, $mr\omega^2$
|
||||
- the critical speed (or natural frequency) is the speed at which the shaft is unstable
|
||||
|
||||
when this is the case it may cause damage to the shaft, bearings, and other destructive
|
||||
vibrations
|
||||
|
||||
## Critical Speed Equation
|
||||
|
||||
$$\omega_c = \sqrt\frac{g}{\delta_\text{st}}$$
|
||||
|
||||
where $g$ is acceleration due to gravity and $\delta_\text{st}$ is
|
||||
[static deflection](#shaft-deflection) of the shaft.
|
||||
|
||||
## Rayleigh Ritz Equation
|
||||
|
||||
When you have multiple masses the Rayleigh-Ritz equation may be more suitable:
|
||||
|
||||
$$\omega_c = \sqrt{g\frac{\Sigma w_i\delta_i}{\Sigma w_i\delta_i^2}} $$
|
||||
|
||||
where $w_i$ is the weight of node $i$ and $\delta_i$ is the static deflection at node $i$.
|
||||
|
||||
# Shaft Deflection
|
||||
|
||||
- Shaft deflection is required to determine the critical speed.
|
||||
- Macaulay's method for the deflection in beam bending:
|
||||
|
||||
$$\frac{\mathrm{d}^2y}{\mathrm{d}x^2} = \frac{M}{EI}$$
|
||||
$$y = \int^x_0\int^x_0\left(\frac{M}{EI}\right) \mathrm{d}x + C_1x + C_2$$
|
||||
|
||||
## Shaft Deflection Equations
|
||||
|
||||
|
||||
|
||||
- Maximum deflection:
|
||||
|
||||
$$\delta_\text{max} = \frac{PL^3}{3EI}$$
|
||||
|
||||
- Deflection at any point $x$
|
||||
|
||||
$$\delta = \frac{Px^3}{6EI}(3L-x)$$
|
||||
|
||||
|
||||
|
||||
- Maximum deflection:
|
||||
|
||||
$$\delta_\text{max} = \frac{PL^3}{48EI}$$
|
||||
|
||||
- Deflection at any point:
|
||||
|
||||
$$\delta = \frac{Px}{12EI} \left(\frac{3L^2}{4} - x^2 \right)$$
|
||||
|
||||
|
||||
|
||||
|
||||
- Maximum deflection
|
||||
|
||||
$$\delta_\text{max} = \frac{Pb^2L}{3EI}$$
|
||||
|
||||
- Deflection at any point
|
||||
|
||||
- For $0 \le x \le a$:
|
||||
|
||||
$$\delta = \frac{Pbx}{6aEI}(x^2-a^2)$$
|
||||
|
||||
- For $0 \le z \le b$:
|
||||
|
||||
$$\delta = \frac{Pbx}{6aEI}\left(z^3 - b(2L+b) + 2b^2L\right)$$
|
||||
|
||||
# Safety Factor
|
||||
|
||||
Safety Factor (also known as reserve factor) is a simple way to accommodate for uncertainties in
|
||||
design.
|
||||
|
||||
Simply multiply the maximum stresses and loads you expect by the safety factor, $n_s$, and assume
|
||||
that as your maximum stress and load.
|
||||
|
||||
$n_s$ | Operational conditions and use of materials
|
||||
----- | -------------------------------------------
|
||||
1.25-1.50 | Reliable materials under controlled conditions, known stresses with certainty
|
||||
1.50-2.00 | Well-known materials under reasonably constant environmental condition, known stresses
|
||||
2.00-2.50 | Average materials subjected to known loads and stresses and environment (LSE)
|
||||
2.50-3.00 | Lesser well-known materials under average conditions LSE
|
||||
3.00-3.40 | Untried materials under average conditions of stresses and environment, or well known materials under uncertain LSE
|
||||
@@ -1 +0,0 @@
|
||||
itags: [ mmme2045 ]
|
||||
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@@ -1,18 +0,0 @@
|
||||
---
|
||||
author: Akbar Rahman
|
||||
date: \today
|
||||
title: MMME2045 // Materials in Design
|
||||
tags: [ uni, mmme2045, materials ]
|
||||
uuid: eb10418c-4bfa-4b3f-941b-cbc53e3d5eee
|
||||
---
|
||||
|
||||
# Material Selection
|
||||
|
||||
# Terminology
|
||||
|
||||
- Light alloys --- metals with densities of less than 4.5 gcm$^{-1}$ and typically less than 3
|
||||
characterised by higher strength to weight ratio than typical steel alloys
|
||||
- Cast --- as-cast condition. Cooling rate and grain refiners are key to improve strength and toughness
|
||||
- Wrought --- (rolled, extruded, forged) Properties rely on appropriate heat treatment or working
|
||||
such as precipitation hardening or work hardening
|
||||
|
||||
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|
||||
itags: [ mmme2046 ]
|
||||
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