2040kineij.fodt

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 <office:body>
  <office:text>
   <text:sequence-decls>
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    <text:sequence-decl text:display-outline-level="0" text:name="Table"/>
    <text:sequence-decl text:display-outline-level="0" text:name="Text"/>
    <text:sequence-decl text:display-outline-level="0" text:name="Drawing"/>
    <text:sequence-decl text:display-outline-level="0" text:name="Figure"/>
   </text:sequence-decls>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">Relative Orbit Angle,<draw:frame draw:style-name="fr1" draw:name="Object169" text:anchor-type="as-char" svg:width="1.1244in" svg:height="0.2417in" draw:z-index="0"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <msub>
          <mi>ϕ</mi>
          <mi mathvariant="italic">IeJe</mi>
         </msub>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
           <mrow>
            <mi>t</mi>
            <mi>,</mi>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">q</mi>
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             <mi>I</mi>
            </msub>
            <mi>,</mi>
            <msub>
             <mstyle mathvariant="bold">
              <mi mathvariant="normal">q</mi>
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             <mi>J</mi>
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            <mi>,</mi>
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         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">%phi_IeJe
left(
t,
{bold nitalic q}_I,
{bold nitalic q}_J,
right)</annotation>
       </semantics>
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   <text:p text:style-name="Standard">Endframe I cannot be in line with endframe J z-axis. </text:p>
   <text:p text:style-name="Standard">Endframe J cannot be in line with endframe I z-axis.</text:p>
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         <mi>I</mi>
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    </draw:frame><text:s/>and <draw:frame draw:style-name="fr1" draw:name="Object60" text:anchor-type="as-char" svg:width="0.2161in" svg:height="0.2165in" draw:z-index="42"><draw:object>
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    </draw:frame><text:s/>need not be parallel.</text:p>
   <text:p text:style-name="Text_20_body">Can be used for all kinds of gear pairs or pulleys. </text:p>
   <text:p text:style-name="Text_20_body">Define</text:p>
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      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <msub>
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           <mi mathvariant="italic">IeJe</mi>
          </msub>
          <mo stretchy="false">=</mo>
          <msup>
           <mi>tan</mi>
           <mrow>
            <mo stretchy="false">−</mo>
            <mn>1</mn>
           </mrow>
          </msup>
         </mrow>
         <mrow>
          <mo fence="true" form="prefix" stretchy="true">(</mo>
          <mrow>
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            </msub>
            <msub>
             <mi>x</mi>
             <mi mathvariant="italic">IeJeIe</mi>
            </msub>
           </mfrac>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">%phi_IeJe =tan^-1 left(y_IeJeIe over x_IeJeIe right) </annotation>
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     </draw:image>
    </draw:frame></text:p>
   <text:h text:style-name="Heading_20_3" text:outline-level="3">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object171" text:anchor-type="as-char" svg:width="0.3681in" svg:height="0.2201in" draw:z-index="25"><draw:object>
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       <semantics>
        <msub>
         <mi>ϕ</mi>
         <mi mathvariant="italic">IeJe</mi>
        </msub>
        <annotation encoding="StarMath 5.0">%phi_IeJe
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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=
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left(
{partial y}
over
{partial {bold nitalic %alpha}}
{1}
over
{x}
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{y}
over
{x^2}
{partial x}
over
{partial {bold nitalic %alpha}}
right)
newline
&quot;&quot;~~~~~~{}
=
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left(
{partial y}
over
{partial {bold nitalic %alpha}}
x
-
y
{partial x}
over
{partial {bold nitalic %alpha}}
right)</annotation>
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newline
&quot;&quot;~~~~~~~~~
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{partial^2 %phi_IeJe}
over
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left lbrace 
matrix{
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over
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{partial y}
over
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{partial y}
over
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{partial x}
over
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{partial x}
over
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+2 y^2
{partial x}
over
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{partial y}
over
{partial {bold nitalic %beta}}   ## 
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{partial^2 y}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
x
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over
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over
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{partial y}
over
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left(
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x
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y
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left(
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+
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            </msup>
            <msub>
             <mi>ϕ</mi>
             <mi mathvariant="italic">IeJe</mi>
            </msub>
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            <mstyle mathvariant="bold">
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             <mrow>
              <mrow>
               <msup>
                <mi>x</mi>
                <mn>2</mn>
               </msup>
               <mo stretchy="false">+</mo>
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                <mn>2</mn>
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        <annotation encoding="StarMath 5.0">&quot;&quot;
{partial^2 %phi_IeJe}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
=
{1} over {left( x^2+y^2 right)^2}
left lbrace 
matrix{
2 xy
left(
{partial x}
over
{partial {bold nitalic %alpha}}
{partial x}
over
{partial {bold nitalic %beta}} 
-
{partial y}
over
{partial {bold nitalic %alpha}}
{partial y}
over
{partial {bold nitalic %beta}} right)  ## 
+left( x^2+y^2 right)
left(
{partial^2 y}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
x
-
y
{partial^2 x}
over
{partial {bold nitalic %alpha} partial {bold nitalic %beta}}
right)
  ## 
+left( y^2-x^2 right)
left(
{partial x}
over
{partial {bold nitalic %alpha}}
{partial y}
over
{partial {bold nitalic %beta}}
+
{partial y}
over
{partial {bold nitalic %alpha}}
{partial x}
over
{partial {bold nitalic %beta}} 
right)
} 
right rbrace
</annotation>
       </semantics>
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newline
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=
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Orbit Angular Velocity,<draw:frame draw:style-name="fr1" draw:name="Object1" text:anchor-type="as-char" svg:width="1.5402in" svg:height="0.2457in" draw:z-index="9"><draw:object>
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       <semantics>
        <mrow>
         <msub>
          <mover accent="true">
           <mi>ϕ</mi>
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         <mrow>
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           <mrow>
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left(
t,
{bold nitalic q}_I,
{bold nitalic q}_J,
{bold nitalic v}_I,
{bold nitalic v}_J
right)</annotation>
       </semantics>
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   <text:p text:style-name="equation"><draw:frame draw:style-name="fr1" draw:name="Object13" text:anchor-type="as-char" svg:width="3.1665in" svg:height="0.7008in" draw:z-index="35"><draw:object>
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       <semantics>
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         <mrow>
          <msub>
           <mover accent="true">
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          <mo stretchy="false">=</mo>
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             </mover>
            </mrow>
           </mrow>
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          </mrow>
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         </mrow>
         <mrow>
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          <mrow>
           <mrow>
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             <mi>y</mi>
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            <mrow>
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            </mover>
           </mrow>
          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{dot %phi}_IeJe
=
{1} over {1+left(y over x right)^2}
left(
{dot y}

{1}
over
{x}
-
{y}
over
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        <annotation encoding="StarMath 5.0">&quot;&quot;
{partial {dot %phi}_IeJe}
over
{partial {bold nitalic %alpha}}
=
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left(
2{x}
{partial x}
over
{partial {bold nitalic %alpha}}
+
2{y}
{partial y}
over
{partial {bold nitalic %alpha}}
right)

left(
{dot y}
x
-
y
{dot x}
right)

newline
&quot;&quot;~~~~~~~~~
+
{1} over {left( x^2+y^2 right)}
left(
{partial dot y}
over
{partial {bold nitalic %alpha}}
x
+
{dot y}
{partial x}
over
{partial {bold nitalic %alpha}}
-
{partial y}
over
{partial {bold nitalic %alpha}}
{dot x}
-
y
{partial dot x}
over
{partial {bold nitalic %alpha}}
right)
</annotation>
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+
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left(
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y
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newline
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x
+
{dot y}
{dot x}
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{dot y}
{dot x}
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y
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newline
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x
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y
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left(
left( x^2+y^2 right)
left(
{ddot y}
x
-
y
{ddot x}
right)
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{dot x}
+
2{y}
{dot y}
right)

left(
{dot y}
x
-
y
{dot x}
right)
right)
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              <mo stretchy="false">¨</mo>
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              <mn>3</mn>
             </msup>
            </mrow>
            <mrow>
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              <mi>x</mi>
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             </mover>
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             <mn>2</mn>
            </mrow>
            <msup>
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            </msup>
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            <mrow>
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            <mi>x</mi>
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            <mrow>
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        <annotation encoding="StarMath 5.0">{{ddot %phi}_IeJe}
=
{1} over {left( x^2+y^2 right)^2}
left(
x^3 {ddot y} - x^2 y {ddot x} + x y^2{ddot y} - y^3 {ddot x}
- 2{x}^2{dot x}{dot y} + 2 x y {dot x}^2 - 2 x y {dot y}^2 + 2 y^2 {dot x}{dot y}
right)
</annotation>
       </semantics>
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=
{1} over {left( x^2+y^2 right)^2}
left(
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right)
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">Partial Derivative of<draw:frame draw:style-name="fr1" draw:name="Object19" text:anchor-type="as-char" svg:width="0.3681in" svg:height="0.2201in" draw:z-index="49"><draw:object>
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