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   <text:h text:style-name="P6" text:outline-level="1">Small Vibrations</text:h>
   <text:p text:style-name="P1">Solve for natural frequencies and mode shapes.</text:p>
   <text:p text:style-name="P2">Find static solution</text:p>
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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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    </draw:frame>= static solution</text:p>
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   <text:p text:style-name="P1">To find transfer function, let</text:p>
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   <text:p text:style-name="Heading_20_2">Forced vibration about static equilibrium</text:p>
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</annotation>
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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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partial
{bold nitalic F}
left(
t,
{bold nitalic y}_s,
{bold nitalic 0}
right)
}
over
{partial {bold nitalic y}}
%DELTA
{bold nitalic y}
+
{
partial
{bold nitalic F}
left(
t,
{bold nitalic y}_s,
{bold nitalic 0}
right)
}
over
{partial {bold nitalic dot y}}
%DELTA
{bold nitalic dot y}
+
{
partial
{bold nitalic F}
left(
t,
{bold nitalic y}_s,
{bold nitalic 0}
right)
}
over
{partial f}
f
=
{bold nitalic 0}

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   <text:p text:style-name="P1">Is the matrix invertable?</text:p>
   <text:p text:style-name="P1"><text:soft-page-break/></text:p>
   <text:p text:style-name="Heading_20_2"/>
   <text:p text:style-name="P4">Simple pendulum example</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>
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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>
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   <text:p text:style-name="Standard">The DAE in implicit form:</text:p>
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{dot z}_1 - z_3 =0
newline
&quot;&quot;
{dot z}_2 - z_4 =0
newline
&quot;&quot;
{dot z}_3 + z_1 z_5 = 0
newline
&quot;&quot;
{dot z}_4 + z_2 z_5 + g  = 0
newline
&quot;&quot;
z^2_1 + z^2_2 - L^2 = 0

</annotation>
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   <text:p text:style-name="Standard">Find static equilibrium solution</text:p>
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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 + g  = 0
newline
&quot;&quot;
z^2_1 + z^2_2 - L^2 = 0

</annotation>
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left lline
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   <text:p text:style-name="Heading_20_2"><text:soft-page-break/>Find the mode</text:p>
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newline
&quot;&quot;
z_2 = z_4 over {i sqrt {g over L}} = 0
newline
&quot;&quot;
z_3 = 1 (arbitrary) 
newline
&quot;&quot;
z_4 = 0 
newline
&quot;&quot;
z_5 = 0 

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