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Update rigid_body_kinetics
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@mopolz2
mopolz2 committed Sep 13, 2024
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148 changes: 109 additions & 39 deletions src/pages/dyn/rigid_body_kinetics.astro
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Expand Up @@ -18,12 +18,42 @@ import SubSubSubSection from "../../components/SubSubSubSection.astro"
import Warning from "../../components/Warning.astro"
import CalloutCard from "../../components/CalloutCard.astro"
import CalloutContainer from "../../components/CalloutContainer.astro"
import DisplayTable from "../../components/DisplayTable.astro"
---
<Layout title="Rigid Body Kinetics">

<div slot="navtree">
<ul class='list-group list-group-flush py-0'>
<li class='list-group-item py-0'><a class='text-decoration-none subsection' href='#center_of_mass'>Center of mass</a></li>
<ul class='list-group list-group-flush py-0'>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#recap_com'>Recap</a></li>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#basic_shapes_com'>Basic shapes</a></li>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#simple_shapes_com'>Simplified shapes</a></li>
</ul>
<li class='list-group-item py-0'><a class='text-decoration-none subsection' href='#moment_of_inertia'>Moment of inertia</a></li>
<ul class='list-group list-group-flush py-0'>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#parallel_axis_theorem'>Parallel axis theorem</a></li>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#additive_theorem'>Additive theorem</a></li>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#basic_shapes_moi'>Basic shapes</a></li>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#simple_shapes_moi'>Simplified shapes</a></li>
</ul>
</li>
<li class='list-group-item py-0'><a class='text-decoration-none subsection' href='#instant_center_center'>Instantaneous center</a></li>
<ul class='list-group list-group-flush py-0'>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#graphical_rules_ic'>Graphical rules for finding M</a></li>
</ul>
<li class='list-group-item py-0'><a class='text-decoration-none subsection' href='#app_rg_kinetics'>Applications</a></li>
<ul class='list-group list-group-flush py-0'>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#accelerating_braking'>Accelerating and braking</a></li>
<li class='list-group-item py-0'><a class='text-decoration-none subsubsection' href='#banked_turns'>Banked turns</a></li>
</ul>
</ul>
</div>


<Section title="Rigid Body Kinetics"></Section>

<SubSection title="Center of mass (COM)">
<BlueText>Complete in "Center of mass"</BlueText>
<SubSection title="Center of mass (COM)" id="center_of_mass">

<p>
The total mass of a rigid body is as follows:
Expand Down Expand Up @@ -233,12 +263,32 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
</div>
</Example>

<RedText>Add to this section the information shown in Fig fig:COM</RedText>

<Image src="/Dynamics/RigidBodyKineticsFigs/CenterofMass.png" width="8" id="COM"></Image>
</SubSection>

<SubSubSection title="Basic Shapes">
<SubSubSection title="Recap" id="recap_com" >

<DisplayTable id="rvp-tc" class_="mb-3">
<tr>
<th>Type of shape</th>
<th>Operation</th>
</tr>
<tr>
<td>Simple shapes</td>
<td> Symmetry tables </td>
</tr>
<tr>
<td>Combination of simple shapes</td>
<td>Find each c.o.m and then combine</td>
</tr>
<tr>
<td>Complex shapes</td>
<td>Integrate</td>
</tr>
</DisplayTable>

</SubSubSection>

<SubSubSection title="Basic Shapes" id="basic_shapes_com">
<p>
The centers of mass listed below are all computed directly from the integral <a href="#rcm-cm">#rcm-cm</a>. Note
the center of mass provided is the vector from point \(O\) (the reference origin).
Expand Down Expand Up @@ -344,7 +394,7 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
</DisplayEquationCustom>
</SubSubSection>

<SubSubSection title="Simplified shapes">
<SubSubSection title="Simplified shapes" id="simple_shapes_com">
<p>
The centers of mass listed below are all special cases of the basic shapes given in Section <a href="#rcm-bs">#rcm-bs</a>. Other
special cases can be easily obtained by similar methods, or directly computing the integral.
Expand Down Expand Up @@ -457,8 +507,7 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
</DisplayEquationCustom>
</SubSubSection>

<SubSection title="Mass moment of Inertia">
<BlueText>Complete in "Moments of Inertia"</BlueText>
<SubSection title="Moment of Inertia" id="moment_of_inertia">

<p>
The moment of inertia of a body, written <InlineEquation equation="I_{P,\\hat{a}}" />, is
Expand Down Expand Up @@ -674,8 +723,7 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
</CalloutContainer>
</SubSection>

<SubSubSection title="Parallel axis theorem">
<BlueText>Complete in "Moments of Inertia"</BlueText>
<SubSubSection title="Parallel axis theorem" id="parallel_axis_theorem">

<DisplayEquationCustom title="Parallel axis theorem." id="rem-el" background="True" derivation="True">
<DisplayEquation equation="I_{P,\\hat{a}} = I_{C,\\hat{a}} + m \\, d^2"/>
Expand Down Expand Up @@ -836,8 +884,7 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
</p>
</SubSubSection>

<SubSubSection title="Additive theorem">
<BlueText>Complete in "Moments of Inertia"</BlueText>
<SubSubSection title="Additive theorem" id="additive_theorem">

<DisplayEquationCustom title="Adding moments of inertia." id="rem-ea" background="True" derivation="True">
<div class="w-100 d-flex flex-row align-items-center">
Expand Down Expand Up @@ -958,7 +1005,7 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
</Example>
</SubSubSection>

<SubSubSection title="Basic shapes">
<SubSubSection title="Basic shapes" id="basic_shapes_moi">
<p>
The moments of inertia listed below are all computed
directly from the integrals <a href="#rem-ec">#rem-ec</a>.
Expand Down Expand Up @@ -1101,7 +1148,7 @@ import CalloutContainer from "../../components/CalloutContainer.astro"

</SubSubSection>

<SubSubSection title="Simplified shapes">
<SubSubSection title="Simplified shapes" id="simple_shapes_moi">
<p>
The moments of inertia listed below are all special cases of
the basic shapes given in Section <a
Expand Down Expand Up @@ -1266,41 +1313,53 @@ import CalloutContainer from "../../components/CalloutContainer.astro"


</SubSubSection>
<SubSection title="Instantaneous centers">
<RedText>Add this information:</RedText>
<SubSection title="Instantaneous center (M)" id="instant_center">

<Itemize>
<Item><RedText>For a rigid body moving in 2D (rotating and possibly translating)</RedText></Item>
<Item><RedText>Instantaneous Center “M” is the point on or off the rigid body that has zero
velocity at that instant (i.e. no translation at this point)</RedText></Item>
<Item><RedText>Point that the body rotates about (at that instant in time)</RedText></Item>
<Item>For a rigid body moving in 2D (rotating and possibly translating)</Item>
<Item>Instantaneous center “M” is the point on or off the rigid body that has zero velocity at that instant (i.e. no translation at this point)</Item>
<Item>Point that the body rotates about (at that instant in time)</Item>
</Itemize>

<RedText>Add some examples as shown in Figs \ref fig:ICexamples1 , \ref fig:ICexamples2</RedText>

<Image src="/Dynamics/RigidBodyKineticsFigs/ICexamples1.png" width="5"></Image>

<Image src="/Dynamics/RigidBodyKineticsFigs/ICexamples2.png" width="5"></Image>

<RedText>Add: Graphical rules for finding (M) (Instantaneous Center) as shown in Fig \ref fig:ICgraphRules</RedText>

<Image src="/Dynamics/RigidBodyKineticsFigs/ICgraphicalRules.png" width="5"></Image>
</SubSection>

<SubSection title="Applications"></SubSection>
<SubSubSection title="Cargo Ships">
<RedText>This topic is in L22-Notes, slides 3-4, refer to Fig \ref fig:AppCargoShip</RedText>. Application for "Center of mass".
<SubSubSection title=" Graphical rules for finding M" id="graphical_rules_ic">

<p>
(Assuming that figure is drawn to scale, including velocity vectors)
</p>

<ol>
<li> Draw lines perpendicular to velocities </li>
<ul>
<li> If the lines intersect at a single point, that point is M </li>
<li> If the lines are colinear:
<ul>
<li> Draw 2 lines that connect the tips of velocity vectors </li>
<li> If the lines intersect at a single point, that point is M </li>
</ul>
</ul>
</ol>

<Warning title="Pay attention to:" id="icm-wa">
<Itemize>
<Item> Consistent direction of rotation. </Item>
<Item> Consistent speeds <InlineEquation equation=" v = \\omega \\, r" />. </Item>
<Item> Body may not be rotating (pure translation). </Item>
</Itemize>

</Warning>

<Image src="/Dynamics/RigidBodyKineticsFigs/AppCargoShip.png" width="5"></Image>
</SubSubSection>

<SubSubSection title="Tuned Mass Damper">
<RedText>This topic is in L24-Notes, slide 9, include the information in Fig \ref fig:AppCargoShip , and this link to a YouTube video <a href="https://www.youtube.com/watch?v=GzMuF-LMGaM ">https://www.youtube.com/watch?v=GzMuF-LMGaM</a></RedText>. Application for "Rigid body kinetics".
</SubSection>

<Image src="/Dynamics/RigidBodyKineticsFigs/AppCargoShip.png" width="5"></Image>
</SubSubSection>
<SubSection title="Applications" id="app_rg_kinetics" ></SubSection>

<SubSubSection title="Accelerating and braking">
<BlueText>Complete in "Accelerating and braking". Include all the information starting at "2D rigid body model".</BlueText> Refer back to \ref sub:PartKin_acce
<SubSubSection title="Accelerating and braking" id="accelerating_braking">

</SubSubSection>
<SubSubSubSection title="2D rigid body model" >
<p>
Expand Down Expand Up @@ -1374,6 +1433,7 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
more weight during braking.
</p>
</SubSubSubSection>

<SubSubSubSection title="Traction, acceleration, and braking" >
<p>
The maximum force with which the car can push against the
Expand Down Expand Up @@ -1677,7 +1737,7 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
</p>
</SubSubSubSection>

<SubSubSection title="Banked turns">
<SubSubSection title="Banked turns" id="banked_turns">
<BlueText>Complete in "Banked turns". Include all the information starting at "2D rigid body model".</BlueText> Refer back to \ref sub:PartKin_turns

</SubSubSection>
Expand Down Expand Up @@ -1910,6 +1970,16 @@ import CalloutContainer from "../../components/CalloutContainer.astro"
</tr>
</table>
</SubSubSection>

<SubSubSection title="Cargo Ships">
<RedText>This topic is in L22-Notes, slides 3-4.</RedText> Application for "Center of mass".
</SubSubSection>

<SubSubSection title="Tuned Mass Damper">
<RedText>This topic is in L24-Notes, slide 9. YouTube video <a href="https://www.youtube.com/watch?v=GzMuF-LMGaM ">link</a></RedText>Application for "Rigid body kinetics".

</SubSubSection>

</Layout>

<script src="/dyn/rigid_body_kinetics/canvases.js" is:inline></script>

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