3240force.fodt

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   <text:p text:style-name="Text_20_body">But</text:p>
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         <mrow>
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            <mi>,</mi>
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        <annotation encoding="StarMath 5.0">r_d
=
r_d
left(
r,
z
right)
</annotation>
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       <semantics>
        <mrow>
         <mrow>
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           <mi>d</mi>
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          <mo stretchy="false">=</mo>
          <mrow>
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         </mrow>
         <mrow>
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          <mrow>
           <mrow>
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             <mi>d</mi>
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          </mrow>
          <mo fence="true" form="postfix" stretchy="true">)</mo>
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        </mrow>
        <annotation encoding="StarMath 5.0">z_d
=
z 
- 
s
left(
r
-
r_d
right)
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Text_20_body">Assuming smooth cones</text:p>
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       <semantics>
        <mrow>
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          <msub>
           <mi>F</mi>
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          </msub>
          <msub>
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         <mo stretchy="false">=</mo>
         <mi>s</mi>
        </mrow>
        <annotation encoding="StarMath 5.0">F_r
over
F_z
=
s</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="Text_20_body"><draw:frame draw:style-name="fr1" draw:name="Object10" text:anchor-type="as-char" svg:width="0.7429in" svg:height="0.9154in" draw:z-index="7"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mfrac>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mfrac>
             <mrow>
              <mo stretchy="false">∂</mo>
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             </mrow>
             <mrow>
              <mo stretchy="false">∂</mo>
              <mi>r</mi>
             </mrow>
            </mfrac>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mfrac>
             <mrow>
              <mo stretchy="false">∂</mo>
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             </mrow>
             <mrow>
              <mo stretchy="false">∂</mo>
              <mi>z</mi>
             </mrow>
            </mfrac>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
         </mfrac>
         <mo stretchy="false">=</mo>
         <mi>s</mi>
        </mrow>
        <annotation encoding="StarMath 5.0">{
left(
{partial E}
over
{partial r}
right)
}
over
{
left(
{partial E}
over
{partial z}
right)
}
=
s</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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=
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r_d
{partial r_d}
over
{partial r}
+
k_z
left(
z_d - l
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{partial z_d}
over
{partial r}</annotation>
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over
{partial r}
=
k_r
r_d
{partial r_d}
over
{partial r}
-
k_z
left(
z 
- 
s
left(
r
-
r_d
right)
- l
right)
s
left(
1
-
{partial r_d}
over
{partial r}
right)</annotation>
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   <text:p text:style-name="Text_20_body">How to solve for r_d?</text:p>
   <text:p text:style-name="Text_20_body">Since lateral stiffness of the LMU is ten times stiffer than the vertical stiffness, assume r_d is zero.</text:p>
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{partial E}
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{partial y}
=
k_z
left(
z 
- 
s
sqrt
{
x^2
+
y^2
}
- l
right)
{sy}
over
{sqrt
{
x^2
+
y^2
+
%epsilon
}}
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{partial E}
over
{partial z}
=
-
k_z
left(
z 
- 
s
sqrt
{
x^2
+
y^2
}
- l
right)</annotation>
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   <text:p text:style-name="Text_20_body">More general potential</text:p>
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