3200force.fodt

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{bold nitalic dot u},
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{bold nitalic ddot u}
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   <text:p text:style-name="Standard">Force is applied to endframe Ie. Magnitude and direction is defined wrt to endframe Ke.</text:p>
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   <text:p text:style-name="Standard">Tension is positive. </text:p>
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t_IeJe
=
t_IeJe
left(
t,
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{bold nitalic s},
{bold nitalic u},
{bold nitalic %lambda},
{bold nitalic v},
{bold nitalic dot s},
{bold nitalic dot u},
{bold nitalic a},
{bold nitalic ddot s},
{bold nitalic ddot u}
right)

newline
&quot;&quot;~~~~
=
t_IeJe
left(
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{bold nitalic %theta}_LeMe,
{bold nitalic u},
{bold nitalic F}_jt,
t,
{bold nitalic v}_LeMeNe,
{bold nitalic %omega}_LeMeNe,
{bold nitalic dot u},
{bold nitalic a}_LeMeNe,
{bold nitalic %alpha}_LeMeNe,
{bold nitalic ddot u}
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   <text:p text:style-name="Standard">Force on endframe Ie is</text:p>
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{partial {bold nitalic q}}
=
{partial t_IeJe}
over
{partial {bold nitalic q}}
{bold nitalic u}_IeJeO
+
t_IeJe
{partial {bold nitalic u}_IeJeO}
over
{partial {bold nitalic q}}
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     </draw:image>
    </draw:frame></text:p>
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       <semantics>
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          </mstyle>
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         </msub>
         <mo stretchy="false">=</mo>
         <mrow>
          <mrow>
           <msub>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">r</mi>
            </mstyle>
            <mi mathvariant="italic">IeJeO</mi>
           </msub>
           <mo stretchy="false">×</mo>
           <msub>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">F</mi>
            </mstyle>
            <mi mathvariant="italic">IeO</mi>
           </msub>
          </mrow>
          <mo stretchy="false">−</mo>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">T</mi>
           </mstyle>
           <mi mathvariant="italic">IeO</mi>
          </msub>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic T}_JeO
=
{bold nitalic r}_IeJeO
times
{bold nitalic F}_IeO
-
{bold nitalic T}_IeO
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="equation">It is possible to define the reaction acting on a floating Je which is always coincident with Ie. Then</text:p>
   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object58" text:anchor-type="as-char" svg:width="0.9016in" svg:height="0.2201in" draw:z-index="55"><draw:object>
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       <semantics>
        <mrow>
         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">T</mi>
          </mstyle>
          <mi mathvariant="italic">JeO</mi>
         </msub>
         <mo stretchy="false">=</mo>
         <mrow>
          <mo stretchy="false">−</mo>
          <msub>
           <mstyle mathvariant="bold">
            <mi mathvariant="normal">T</mi>
           </mstyle>
           <mi mathvariant="italic">IeO</mi>
          </msub>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic T}_JeO
=
-
{bold nitalic T}_IeO
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="equation">See the use of floating Je in the derivation of generalized forces in 40dyna.doc. Using floating Je allows a more concise derivation.</text:p>
   <text:h text:style-name="Heading_20_3" text:outline-level="3">General Action-Reaction,<draw:frame draw:style-name="fr1" draw:name="Object61" text:anchor-type="as-char" svg:width="2.6598in" svg:height="0.2311in" draw:z-index="56"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
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         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">T</mi>
          </mstyle>
          <mi mathvariant="italic">IeO</mi>
         </msub>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
            <mi>t</mi>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">q</mi>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">s</mi>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">u</mi>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">λ</mi>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">v</mi>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mover accent="true">
              <mi mathvariant="normal">s</mi>
              <mo stretchy="false">˙</mo>
             </mover>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mover accent="true">
              <mi mathvariant="normal">u</mi>
              <mo stretchy="false">˙</mo>
             </mover>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mi mathvariant="normal">a</mi>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mover accent="true">
              <mi mathvariant="normal">s</mi>
              <mo stretchy="false">¨</mo>
             </mover>
            </mstyle>
            <mi>,</mi>
            <mstyle mathvariant="bold">
             <mover accent="true">
              <mi mathvariant="normal">u</mi>
              <mo stretchy="false">¨</mo>
             </mover>
            </mstyle>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
         <mi>,</mi>
         <msub>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">T</mi>
          </mstyle>
          <mi mathvariant="italic">JeO</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">{bold nitalic T}_IeO
left(
t,
{bold nitalic q},
{bold nitalic s},
{bold nitalic u},
{bold nitalic %lambda},
{bold nitalic v},
{bold nitalic dot s},
{bold nitalic dot u},
{bold nitalic a},
{bold nitalic ddot s},
{bold nitalic ddot u}
right)

,
{bold nitalic T}_JeO
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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               <mi>,</mi>
               <msub>
                <mstyle mathvariant="bold">
                 <mi mathvariant="normal">v</mi>
                </mstyle>
                <mi mathvariant="italic">LeMeNe</mi>
               </msub>
               <mi>,</mi>
               <msub>
                <mstyle mathvariant="bold">
                 <mi mathvariant="normal">ω</mi>
                </mstyle>
                <mi mathvariant="italic">LeMeNe</mi>
               </msub>
               <mi>,</mi>
               <mstyle mathvariant="bold">
                <mover accent="true">
                 <mi mathvariant="normal">u</mi>
                 <mo stretchy="false">˙</mo>
                </mover>
               </mstyle>
               <mi>,</mi>
               <msub>
                <mstyle mathvariant="bold">
                 <mi mathvariant="normal">a</mi>
                </mstyle>
                <mi mathvariant="italic">LeMeNe</mi>
               </msub>
               <mi>,</mi>
               <msub>
                <mstyle mathvariant="bold">
                 <mi mathvariant="normal">α</mi>
                </mstyle>
                <mi mathvariant="italic">LeMeNe</mi>
               </msub>
               <mi>,</mi>
               <mstyle mathvariant="bold">
                <mover accent="true">
                 <mi mathvariant="normal">u</mi>
                 <mo stretchy="false">¨</mo>
                </mover>
               </mstyle>
              </mrow>
             </mrow>
             <mo fence="true" form="postfix" stretchy="true">)</mo>
            </mrow>
           </mrow>
          </mtd>
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        </mtable>
        <annotation encoding="StarMath 5.0">&quot;&quot;
t_IeJe
=
t_IeJe
left(
t,
{bold nitalic q},
{bold nitalic s},
{bold nitalic u},
{bold nitalic %lambda},
{bold nitalic v},
{bold nitalic dot s},
{bold nitalic dot u},
{bold nitalic a},
{bold nitalic ddot s},
{bold nitalic ddot u}
right)

newline
&quot;&quot;~~~~
=
t_IeJe
left(
{bold nitalic r}_LeMeNe,
{bold nitalic %theta}_LeMe,
{bold nitalic u},
{bold nitalic F}_jt,
t,
{bold nitalic v}_LeMeNe,
{bold nitalic %omega}_LeMeNe,
{bold nitalic dot u},
{bold nitalic a}_LeMeNe,
{bold nitalic %alpha}_LeMeNe,
{bold nitalic ddot u}
right)</annotation>
       </semantics>
      </math>
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=
t_IeJe
{bold nitalic u}_IeJeO
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over
{partial {bold nitalic q}}
=
{partial t_IeJe}
over
{partial {bold nitalic q}}
{bold nitalic u}_IeJeO
+
t_IeJe
{partial {bold nitalic u}_IeJeO}
over
{partial {bold nitalic q}}
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   <text:h text:style-name="Heading_20_4" text:outline-level="4">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object89" text:anchor-type="as-char" svg:width="0.3528in" svg:height="0.2201in" draw:z-index="84"><draw:object>
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   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object31" text:anchor-type="as-char" svg:width="2.2661in" svg:height="0.4382in" draw:z-index="130"><draw:object>
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