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   <text:h text:style-name="P3" text:outline-level="1">Small Vibrations</text:h>
   <text:p text:style-name="P1">Solve for natural frequencies and mode shapes.</text:p>
   <text:h text:style-name="Heading_20_2" text:outline-level="2">Find static solution</text:h>
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   <text:p text:style-name="P1">Note the use of lambda&apos;s instead of mu&apos;s for static analysis.</text:p>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Natural Frequencies and Modes</text:h>
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   <text:p text:style-name="P1">Substitute into linearized equations</text:p>
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                <mrow>
                 <mrow>
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                  <mi>,</mi>
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                  <mi>,</mi>
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              </mrow>
              <mrow>
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">State Input Vibration About Static Equilibrium</text:h>
   <text:h text:style-name="Heading_20_3" text:outline-level="3">Preferred Solution</text:h>
   <text:p text:style-name="Standard">Let a small scalar motion<draw:frame draw:style-name="fr1" draw:name="Object70" text:anchor-type="as-char" svg:width="0.3846in" svg:height="0.1925in" draw:z-index="21"><draw:object>
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    </draw:frame>be applied to a joint in the system at static equilibrium.</text:p>
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d
exp
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i
%omega
t
right)</annotation>
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   <text:p text:style-name="P1">Assuming small variations</text:p>
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%DELTA
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left(
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%DELTA
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+
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left(
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over
{partial d}
d
=
{bold nitalic 0}

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   <text:p text:style-name="P1">Is the matrix invertable? It should be when away from the natural frequencies or poles.</text:p>
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    </draw:frame></text:p>
   <text:p text:style-name="Text_20_body">Remove the dynamic equation associated with<draw:frame draw:style-name="fr1" draw:name="Object64" text:anchor-type="as-char" svg:width="0.3598in" svg:height="0.2134in" draw:z-index="59"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="normal">Δ</mi>
         <msub>
          <mi>y</mi>
          <mi>j</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">%DELTA y_j
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame>since prescribing<draw:frame draw:style-name="fr1" draw:name="Object65" text:anchor-type="as-char" svg:width="0.3598in" svg:height="0.2134in" draw:z-index="60"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="normal">Δ</mi>
         <msub>
          <mi>y</mi>
          <mi>j</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">%DELTA y_j
</annotation>
       </semantics>
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     </draw:object><draw:image>
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     </draw:image>
    </draw:frame>actually introduces a force in the dynamic equation which is not included above.</text:p>
   <text:p text:style-name="Text_20_body">Solve for <draw:frame draw:style-name="fr2" draw:name="Object52" text:anchor-type="as-char" svg:width="1.8929in" svg:height="0.4752in" draw:z-index="61"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mfrac>
          <mrow>
           <msub>
            <mi>Y</mi>
            <mi>i</mi>
           </msub>
           <mrow>
            <mo fence="true" form="prefix" stretchy="true">(</mo>
            <mrow>
             <mi mathvariant="normal">ω</mi>
            </mrow>
            <mo fence="true" form="postfix" stretchy="true">)</mo>
           </mrow>
          </mrow>
          <msub>
           <mi>Y</mi>
           <mi>j</mi>
          </msub>
         </mfrac>
         <mstyle mathvariant="normal">
          <mi mathvariant="normal">;</mi>
         </mstyle>
         <mspace width="2em"/>
         <mrow>
          <mi>i</mi>
          <mo stretchy="false">=</mo>
          <mn>1</mn>
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         <mi>,</mi>
         <mo stretchy="false">⋯</mo>
         <mi>,</mi>
         <mi>n</mi>
         <mstyle mathvariant="normal">
          <mi mathvariant="normal">;</mi>
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         <mspace width="2em"/>
         <mrow>
          <mi>i</mi>
          <mo stretchy="false">≠</mo>
          <mi>j</mi>
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        </mrow>
        <annotation encoding="StarMath 5.0">{
Y_i
left( 
%omega
right)
}
over
Y_j
{nitalic ;}
~
i=1, dotsaxis,n
{nitalic ;}
~ 
i neq j</annotation>
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       AAAfAIoAAQA7AAAAAwA1AAAACQBUaG9ybmRhbGUAAAAAAACnAQAAAAAAAAAABQAAAAAAAgD/
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     </draw:image>
    </draw:frame>which is the transfer function from state <draw:frame draw:style-name="fr1" draw:name="Object53" text:anchor-type="as-char" svg:width="0.352in" svg:height="0.2134in" draw:z-index="62"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="normal">Δ</mi>
         <msub>
          <mi>y</mi>
          <mi>j</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">%DELTA y_j
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame>to state <draw:frame draw:style-name="fr1" draw:name="Object54" text:anchor-type="as-char" svg:width="0.3429in" svg:height="0.2134in" draw:z-index="63"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="normal">Δ</mi>
         <msub>
          <mi>y</mi>
          <mi>i</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">%DELTA y_i
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AACWAAEAAgAAAAkAjAABAAAAAAA=
      </office:binary-data>
     </draw:image>
    </draw:frame>.</text:p>
   <text:h text:style-name="Heading_20_3" text:outline-level="3">Alternative solution 2</text:h>
   <text:p text:style-name="Text_20_body">Apply an action only force<draw:frame draw:style-name="fr1" draw:name="Object66" text:anchor-type="as-char" svg:width="0.3717in" svg:height="0.2016in" draw:z-index="64"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi>f</mi>
         <mrow>
          <mo fence="true" form="prefix" stretchy="false">(</mo>
          <mrow>
           <mi>t</mi>
          </mrow>
          <mo fence="true" form="postfix" stretchy="false">)</mo>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">f(t)
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAfgDAADo
       AQAAJAAAAJYAAQACAAAACQCLAAEAAgAAAP//gQABABAAAAAAAAAAAAAAAPcDAADnAQAAlQAB
       AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxl
       AAAAAAAApwEAAAAAAAAAAAUAAAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcA
       AQAFAAAA/////wCGAAEABAAAAAAAAAByAAIAFwAAALkAAAByAQAAAQBmdwAAAAAA//8BAGYA
       jAABAAAAAACLAAEAAgAAAB8AigABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACnAQAA
       //8AAAAABQAAAAAAAAD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////
       AIYAAQAEAAAAAAAAAHIAAgAaAAAAwgEAAHIBAAABAAAAKACQAAAAAAD//wEAKACMAAEAAAAA
       AIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxlAAAAAAAApwEAAAAAAAAAAAUA
       AAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAA
       AAAAAAByAAIAFwAAAGECAAByAQAAAQB0dwAAAAAA//8BAHQAjAABAAAAAACLAAEAAgAAAB8A
       igABADwAAAADADYAAAAKAE9wZW5TeW1ib2wAAAAAAACnAQAA//8AAAAABQAAAAAAAAD/AwAA
       AAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/////AIYAAQAEAAAAAAAAAHIAAgAa
       AAAA/wIAAHIBAAABAAAAKQCQAAAAAAD//wEAKQCMAAEAAAAAAJUAAQAEAAAAAAAAAJYAAQAC
       AAAACQCMAAEAAAAAAA==
      </office:binary-data>
     </draw:image>
    </draw:frame>to the component affecting<draw:frame draw:style-name="fr1" draw:name="Object67" text:anchor-type="as-char" svg:width="0.3598in" svg:height="0.2134in" draw:z-index="65"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mi mathvariant="normal">Δ</mi>
         <msub>
          <mi>y</mi>
          <mi>j</mi>
         </msub>
        </mrow>
        <annotation encoding="StarMath 5.0">%DELTA y_j
</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AQAAAAAAAAAABQAAAAAAAgD/AwAAAAAAAAAAAP8DAAAAAACIAAEAAgAAAAEAhwABAAUAAAD/
       ////AIYAAQAEAAAAAAAAAHIAAgAXAAAA9wEAAHIBAAABAHm6AAAAAAD//wEAeQCMAAEAAAAA
       AIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxlAAAAAAAA/QAAAAAAAAAAAAUA
       AAAAAAIA/wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAA
       AAAAAAByAAIAFwAAAP8CAADcAQAAAQBqSAAAAAAA//8BAGoAjAABAAAAAACVAAEABAAAAAAA
       AACWAAEAAgAAAAkAjAABAAAAAAA=
      </office:binary-data>
     </draw:image>
    </draw:frame>. Derive<draw:frame draw:style-name="fr2" draw:name="Object68" text:anchor-type="as-char" svg:width="0.5165in" svg:height="0.4339in" draw:z-index="66"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mfrac>
         <mrow>
          <mstyle mathvariant="bold">
           <mi mathvariant="normal">Y</mi>
          </mstyle>
          <mrow>
           <mo fence="true" form="prefix" stretchy="true">(</mo>
           <mrow>
            <mi mathvariant="normal">ω</mi>
           </mrow>
           <mo fence="true" form="postfix" stretchy="true">)</mo>
          </mrow>
         </mrow>
         <mi>f</mi>
        </mfrac>
        <annotation encoding="StarMath 5.0">{
{bold nitalic Y}
left( %omega right)
} over f</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AAQAAAAAAAAAlgABAAIAAAAJAIsAAQACAAAAHwCKAAEAOwAAAAMANQAAAAkAVGhvcm5kYWxl
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       AIYAAQAEAAAAAAAAAHIAAgAaAAAAEQIAAHIBAAABAAAAKAAzAAAAAAD//wEAKACMAAEAAAAA
       AIsAAQACAAAAHwCKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAAAKcBAAD//wAAAAAF
       AAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQA
       AAAAAAAAcgACABoAAAB7AgAAjQEAAAEAAADJAyQBAAAAAP//AQDJA4wAAQAAAAAAiwABAAIA
       AAAfAIoAAQA8AAAAAwA2AAAACgBPcGVuU3ltYm9sAABqAAAA4AAAAP//AAAAAAUAAAAAAAAA
       /wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAABy
       AAIAGgAAALkDAAByAQAAAQAAACkAMwAAAAAA//8BACkAjAABAAAAAACLAAEAAgAAAB8AhQAB
       AAUAAAAAAAAAAYQAAQAFAAAAAAAAAACKAAEAPAAAAAMANgAAAAoAT3BlblN5bWJvbAAAAAAA
       AKcBAAD//wAAAAAFAAAAAAAAAP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAA
       AP////8AhgABAAQAAAAAAAAAZwABABAAAACEAAAA9wEAABIEAAALAgAAjAABAAAAAACLAAEA
       AgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAKcBAAAAAAAAAAAFAAAAAAAC
       AP8DAAAAAAAAAAAA/wMAAAAAAIgAAQACAAAAAQCHAAEABQAAAP////8AhgABAAQAAAAAAAAA
       cgACABcAAAD3AQAAngMAAAEAZncAAAAAAP//AQBmAIwAAQAAAAAAlQABAAQAAAAAAAAAlgAB
       AAIAAAAJAIwAAQAAAAAA
      </office:binary-data>
     </draw:image>
    </draw:frame>from the method below. Then</text:p>
   <text:p text:style-name="Text_20_body"><draw:frame draw:style-name="fr2" draw:name="Object69" text:anchor-type="as-char" svg:width="2.6984in" svg:height="0.4528in" draw:z-index="67"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mrow>
         <mrow>
          <mfrac>
           <mrow>
            <msub>
             <mi>Y</mi>
             <mi>i</mi>
            </msub>
            <mrow>
             <mo fence="true" form="prefix" stretchy="true">(</mo>
             <mrow>
              <mi mathvariant="normal">ω</mi>
             </mrow>
             <mo fence="true" form="postfix" stretchy="true">)</mo>
            </mrow>
           </mrow>
           <msub>
            <mi>Y</mi>
            <mi>j</mi>
           </msub>
          </mfrac>
          <mo stretchy="false">=</mo>
          <mfrac>
           <mrow>
            <msub>
             <mi>Y</mi>
             <mi>i</mi>
            </msub>
            <mrow>
             <mo fence="true" form="prefix" stretchy="true">(</mo>
             <mrow>
              <mi mathvariant="normal">ω</mi>
             </mrow>
             <mo fence="true" form="postfix" stretchy="true">)</mo>
            </mrow>
           </mrow>
           <mi>f</mi>
          </mfrac>
         </mrow>
         <mfrac>
          <mi>f</mi>
          <mrow>
           <msub>
            <mi>Y</mi>
            <mi>j</mi>
           </msub>
           <mrow>
            <mo fence="true" form="prefix" stretchy="true">(</mo>
            <mrow>
             <mi mathvariant="normal">ω</mi>
            </mrow>
            <mo fence="true" form="postfix" stretchy="true">)</mo>
           </mrow>
          </mrow>
         </mfrac>
         <mstyle mathvariant="normal">
          <mi mathvariant="normal">;</mi>
         </mstyle>
         <mspace width="2em"/>
         <mrow>
          <mi>i</mi>
          <mo stretchy="false">=</mo>
          <mn>1</mn>
         </mrow>
         <mi>,</mi>
         <mo stretchy="false">⋯</mo>
         <mi>,</mi>
         <mi>n</mi>
         <mstyle mathvariant="normal">
          <mi mathvariant="normal">;</mi>
         </mstyle>
         <mspace width="2em"/>
         <mrow>
          <mi>i</mi>
          <mo stretchy="false">≠</mo>
          <mi>j</mi>
         </mrow>
        </mrow>
        <annotation encoding="StarMath 5.0">{
Y_i
left( 
%omega
right)
}
over
Y_j
=
{
Y_i
left( %omega right)
} over f
f over
{
Y_j
left( %omega right)
}
{nitalic ;}
~
i=1, dotsaxis,n
{nitalic ;}
~ 
i neq j</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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       AAAAAACLAAEAAgAAAB8AigABADsAAAADADUAAAAJAFRob3JuZGFsZQAAAAAAAKcBAAAAAAAA
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       /wMAAAAAAAAAAAD/AwAAAAAAiAABAAIAAAABAIcAAQAFAAAA/////wCGAAEABAAAAAAAAABy
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   <text:h text:style-name="Heading_20_2" text:outline-level="2">Forced vibration about static equilibrium</text:h>
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    </draw:frame>be applied on the system at static equilibrium.</text:p>
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f
exp
left(
i
%omega
t
right)</annotation>
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   <text:p text:style-name="P1">Assuming small variations</text:p>
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left(
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over
{partial {bold nitalic y}}
%DELTA
{bold nitalic y}
+
{
partial
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left(
t,
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right)
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over
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%DELTA
{bold nitalic dot y}
+
{
partial
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left(
t,
{bold nitalic y}_s,
{bold nitalic 0}
right)
}
over
{partial f}
f
=
{bold nitalic 0}

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     </draw:image>
    </draw:frame></text:p>
   <text:p text:style-name="P1">Is the matrix invertable? It should be when away from the natural frequencies or poles.</text:p>
   <text:p text:style-name="P1"/>
   <text:h text:style-name="Heading_20_2" text:outline-level="2"/>
   <text:h text:style-name="P5" text:outline-level="2">Simple pendulum example</text:h>
   <text:p text:style-name="Text_20_body">A point mass,<draw:frame draw:style-name="fr2" draw:name="Object57" text:anchor-type="as-char" svg:width="0.2071in" svg:height="0.1835in" draw:z-index="68"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mi>m</mi>
        <annotation encoding="StarMath 5.0">m</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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     </draw:image>
    </draw:frame>, suspended distance,<draw:frame draw:style-name="fr2" draw:name="Object58" text:anchor-type="as-char" svg:width="0.178in" svg:height="0.1835in" draw:z-index="69"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mi>L</mi>
        <annotation encoding="StarMath 5.0">L</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAd4BAADk
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       jAABAAAAAACVAAEABAAAAAAAAACWAAEAAgAAAAkAjAABAAAAAAA=
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     </draw:image>
    </draw:frame>, from a pivot.</text:p>
   <text:p text:style-name="Text_20_body">Gravity,<draw:frame draw:style-name="fr2" draw:name="Object59" text:anchor-type="as-char" svg:width="0.1717in" svg:height="0.1835in" draw:z-index="70"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mi>g</mi>
        <annotation encoding="StarMath 5.0">g</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAdsBAADk
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      </office:binary-data>
     </draw:image>
    </draw:frame>, is down. <draw:frame draw:style-name="fr2" draw:name="Object60" text:anchor-type="as-char" svg:width="0.1602in" svg:height="0.1835in" draw:z-index="71"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mi>x</mi>
        <annotation encoding="StarMath 5.0">x</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAbgBAADk
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     </draw:image>
    </draw:frame>is to the right. <draw:frame draw:style-name="fr2" draw:name="Object61" text:anchor-type="as-char" svg:width="0.1602in" svg:height="0.1835in" draw:z-index="72"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mi>y</mi>
        <annotation encoding="StarMath 5.0">y</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
      <office:binary-data>VkNMTVRGAQAxAAAAAAAAAAEAGwAAAAAAAAAAAAAAAAABAAAAAQAAAAEAAAABAAAAAc4BAADk
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      </office:binary-data>
     </draw:image>
    </draw:frame>is up.</text:p>
   <text:p text:style-name="Standard">DAE of simple pendulum</text:p>
   <text:p text:style-name="Standard">Define state variables:</text:p>
   <text:p text:style-name="MTDisplayEquation"><draw:frame draw:style-name="fr1" draw:name="Object20" text:anchor-type="as-char" svg:width="0.4862in" svg:height="1.102in" draw:z-index="25"><draw:object>
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       <semantics>
        <mtable>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>z</mi>
              <mn>1</mn>
             </msub>
             <mo stretchy="false">=</mo>
             <mi>x</mi>
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           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>z</mi>
              <mn>2</mn>
             </msub>
             <mo stretchy="false">=</mo>
             <mi>y</mi>
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           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>z</mi>
              <mn>3</mn>
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             <mo stretchy="false">=</mo>
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              <mi>x</mi>
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          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>z</mi>
              <mn>4</mn>
             </msub>
             <mo stretchy="false">=</mo>
             <mover accent="true">
              <mi>y</mi>
              <mo stretchy="false">˙</mo>
             </mover>
            </mrow>
           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <msub>
              <mi>z</mi>
              <mn>5</mn>
             </msub>
             <mo stretchy="false">=</mo>
             <mi mathvariant="normal">λ</mi>
            </mrow>
           </mrow>
          </mtd>
         </mtr>
        </mtable>
        <annotation encoding="StarMath 5.0">&quot;&quot;
z_1 = x 
newline
&quot;&quot;
z_2 = y 
newline
&quot;&quot;
z_3 = dot x 
newline
&quot;&quot;
z_4 = dot y 
newline
&quot;&quot;
z_5 = %lambda 

</annotation>
       </semantics>
      </math>
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   <text:p text:style-name="Standard">The DAE in implicit form:</text:p>
   <text:p text:style-name="MTDisplayEquation"><draw:frame draw:style-name="fr1" draw:name="Object22" text:anchor-type="as-char" svg:width="1.4071in" svg:height="1.1256in" draw:z-index="26"><draw:object>
      <math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
       <semantics>
        <mtable>
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          <mtd>
           <mrow>
            <mtext/>
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             <mrow>
              <msub>
               <mover accent="true">
                <mi>z</mi>
                <mo stretchy="false">˙</mo>
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               <mn>1</mn>
              </msub>
              <mo stretchy="false">−</mo>
              <msub>
               <mi>z</mi>
               <mn>3</mn>
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             <mo stretchy="false">=</mo>
             <mn>0</mn>
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         <mtr>
          <mtd>
           <mrow>
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             <mrow>
              <msub>
               <mover accent="true">
                <mi>z</mi>
                <mo stretchy="false">˙</mo>
               </mover>
               <mn>2</mn>
              </msub>
              <mo stretchy="false">−</mo>
              <msub>
               <mi>z</mi>
               <mn>4</mn>
              </msub>
             </mrow>
             <mo stretchy="false">=</mo>
             <mn>0</mn>
            </mrow>
           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mi>m</mi>
            <mrow>
             <msub>
              <mover accent="true">
               <mi>z</mi>
               <mo stretchy="false">˙</mo>
              </mover>
              <mn>3</mn>
             </msub>
             <mo stretchy="false">+</mo>
             <msub>
              <mi>z</mi>
              <mn>1</mn>
             </msub>
            </mrow>
            <mrow>
             <msub>
              <mi>z</mi>
              <mn>5</mn>
             </msub>
             <mo stretchy="false">=</mo>
             <mn>0</mn>
            </mrow>
           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mi>m</mi>
            <mrow>
             <msub>
              <mover accent="true">
               <mi>z</mi>
               <mo stretchy="false">˙</mo>
              </mover>
              <mn>4</mn>
             </msub>
             <mo stretchy="false">+</mo>
             <msub>
              <mi>z</mi>
              <mn>2</mn>
             </msub>
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            <mrow>
             <mrow>
              <msub>
               <mi>z</mi>
               <mn>5</mn>
              </msub>
              <mo stretchy="false">+</mo>
              <mi mathvariant="italic">mg</mi>
             </mrow>
             <mo stretchy="false">=</mo>
             <mn>0</mn>
            </mrow>
           </mrow>
          </mtd>
         </mtr>
         <mtr>
          <mtd>
           <mrow>
            <mtext/>
            <mrow>
             <mrow>
              <msubsup>
               <mi>z</mi>
               <mn>1</mn>
               <mn>2</mn>
              </msubsup>
              <mo stretchy="false">+</mo>
              <msubsup>
               <mi>z</mi>
               <mn>2</mn>
               <mn>2</mn>
              </msubsup>
              <mo stretchy="false">−</mo>
              <msup>
               <mi>L</mi>
               <mn>2</mn>
              </msup>
             </mrow>
             <mo stretchy="false">=</mo>
             <mn>0</mn>
            </mrow>
           </mrow>
          </mtd>
         </mtr>
        </mtable>
        <annotation encoding="StarMath 5.0">&quot;&quot;
{dot z}_1 - z_3 =0
newline
&quot;&quot;
{dot z}_2 - z_4 =0
newline
&quot;&quot;
m{dot z}_3 + z_1 z_5 = 0
newline
&quot;&quot;
m{dot z}_4 + z_2 z_5 + mg  = 0
newline
&quot;&quot;
z^2_1 + z^2_2 - L^2 = 0

</annotation>
       </semantics>
      </math>
     </draw:object><draw:image>
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   <text:p text:style-name="Standard">Find static equilibrium solution</text:p>
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            <mrow>
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             <mrow>
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        <annotation encoding="StarMath 5.0">&quot;&quot;
z_3 =0
newline
&quot;&quot;
z_4 =0
newline
&quot;&quot;
z_1 z_5 = 0
newline
&quot;&quot;
z_2 z_5 + mg  = 0
newline
&quot;&quot;
z^2_1 + z^2_2 - L^2 = 0

</annotation>
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   <text:h text:style-name="Heading_20_3" text:outline-level="3">For small vibration about static equilibrium</text:h>
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   <text:h text:style-name="Heading_20_3" text:outline-level="3"><text:soft-page-break/>Find the mode</text:h>
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