4500collis.fodt

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    </draw:frame>when there is joint friction.</text:p>
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+
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{partial {bold nitalic dot q}_J}
{bold nitalic dot q}_J
=
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over
{partial {bold nitalic q}_I}
{bold nitalic dot q}_I
+
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over
{partial {bold nitalic q}_J}
{bold nitalic dot q}_J
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=
f_impz
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matrix
{
%mu
{v_IeJeIe1}
over
{sqrt {v^2_IeJeIe1 + v^2_IeJeIe2}+%epsilon}
##
%mu
{v_IeJeIe2}
over
{sqrt {v^2_IeJeIe1 + v^2_IeJeIe2}+%epsilon}
##
-1
}
right rbrace
=
f_scalar
{bold nitalic f}_imp
left(
{bold nitalic dot q}
right)</annotation>
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   <text:p text:style-name="Standard">Refer to the solution for impulsive force using:</text:p>
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        <mrow>
         <msub>
          <mi>f</mi>
          <mi mathvariant="italic">scalar</mi>
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         <mo stretchy="false">=</mo>
         <msub>
          <mi>f</mi>
          <mi mathvariant="italic">impz</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">f_scalar
=
f_impz
</annotation>
       </semantics>
      </math>
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            <mtable>
             <mtr>
              <mtd>
               <mrow>
                <mi>μ</mi>
                <mfrac>
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                    </msubsup>
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                </mfrac>
               </mrow>
              </mtd>
             </mtr>
             <mtr>
              <mtd>
               <mrow>
                <mi>μ</mi>
                <mfrac>
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              </mtd>
             </mtr>
             <mtr>
              <mtd>
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        <annotation encoding="StarMath 5.0">{bold nitalic f}_imp
left(
{bold nitalic dot q}
right)
=
left lbrace
matrix
{
%mu
{v_IeJeIe1}
over
{sqrt {v^2_IeJeIe1 + v^2_IeJeIe2}+%epsilon}
##
%mu
{v_IeJeIe2}
over
{sqrt {v^2_IeJeIe1 + v^2_IeJeIe2}+%epsilon}
##
-1
}
right rbrace
</annotation>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="equation">During integration, look for<draw:frame draw:style-name="fr1" draw:name="Object8" text:anchor-type="as-char" svg:width="0.5083in" svg:height="0.2138in" draw:z-index="12"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <msub>
         <mi>P</mi>
         <mrow>
          <mi mathvariant="italic">scalar</mi>
          <mn>0</mn>
         </mrow>
        </msub>
        <annotation encoding="StarMath 5.0">P_scalar0
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAQwFAAAf
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      </office:binary-data>
     </draw:image>
    </draw:frame>when the normal impact velocity is zero, i.e.</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object15" text:anchor-type="as-char" svg:width="1.4173in" svg:height="0.2354in" draw:z-index="8"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="italic">vIeJeIe</mi>
         <mn>3</mn>
         <mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <msub>
             <mi>P</mi>
             <mrow>
              <mi mathvariant="italic">scalar</mi>
              <mn>0</mn>
             </mrow>
            </msub>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mo stretchy="false">=</mo>
          <mn>0</mn>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">vIeJeIe3
left(
P_scalar0
right)
=
0
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/
       ////AIYAAQAEAAAAAAAAAHIAAgAaAAAAeQsAAI0BAAABAAAAPQBOAQAAAAD//wEAPQCMAAEA
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       BAAAAAAAAAByAAIAFwAAANEMAACNAQAAAQAw1AAAAAAA//8BADAAjAABAAAAAACVAAEABAAA
       AAAAAACWAAEAAgAAAAkAjAABAAAAAAA=
      </office:binary-data>
     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">Stop integration when</text:p>
   <text:p text:style-name="Standard"><draw:frame draw:style-name="fr1" draw:name="Object16" text:anchor-type="as-char" svg:width="1.4457in" svg:height="0.2236in" draw:z-index="9"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
           <mi>P</mi>
           <mi mathvariant="italic">scalar</mi>
          </msub>
          <mo stretchy="false">=</mo>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mrow>
             <mn>1</mn>
             <mo stretchy="false">+</mo>
             <mi>e</mi>
            </mrow>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
         </mrow>
         <msub>
          <mi>P</mi>
          <mrow>
           <mi mathvariant="italic">scalar</mi>
           <mn>0</mn>
          </mrow>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">P_scalar
=
left(
1 + e
right)
P_scalar0</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AQBlugAAAAAA//8BAGUAjAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAFN0YXJT
       eW1ib2wAANgAAAD0AQAA//8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAA
       AAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAaAAAAMwkAAKcBAAABAAAAn+BdAAAA
       AAD//wEAn+CMAAEAAAAAAIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxlAAAA
       AAAApwEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAF
       AAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAANIJAACNAQAAAQBQAgEAAAAA//8BAFAAjAAB
       AAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAP0AAAAAAAAA
       AAAFAAAAAAACAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgAB
       AAQAAAAAAAAAcgACACkAAAD1CgAA9wEAAAcAc2NhbGFyMPoCAAAAAP//BwBzAGMAYQBsAGEA
       cgAwAIwAAQAAAAAAlQABAAQAAAAAAAAAlgABAAIAAAAJAIwAAQAAAAAA
      </office:binary-data>
     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Standard">where e is the coefficient of restitution.</text:p>
   <text:p text:style-name="Standard">Compute the after collision states and continue dynamic integration.</text:p>
   <text:p text:style-name="Standard"/>
  </office:text>
 </office:body>
</office:document>