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buckling_spring.html
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<html>
<head>
<title>
BUCKLING_SPRING - The Buckling Spring
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
BUCKLING_SPRING <br> The Buckling Spring
</h1>
<hr>
<p>
<b>BUCKLING_SPRING</b>
is a MATLAB program which
illustrates solutions of the buckling spring equations.
</p>
<p>
We are given three points A, B, and C:
<ul>
<li>
A is at the origin (0,0);
</li>
<li>
B has coordinates (X,Y) with Y nonnegative, and the ray from A to B
makes an angle of THETA with the horizontal axis.
</li>
<li>
C is at the point (2*X,0).
</li>
</ul>
</p>
<p>
Springs:
<ul>
<li>
A spring extends from A to B; it is normally of length 1, and is
currently of length L.
</li>
<li>
A spring extends from B to C; it is normally of length 1, and is
currently of length L.
</li>
<li>
A spring force is also exerted, which tends to draw the two
springs together, proportional to the angle between the two springs.
</li>
</ul>
</p>
<p>
Forces:
<li>
A vertical load MU is applied at point B (downward is positive);
</li>
<li>
A horizontal load LAMBDA is applied at point C (leftware is positive);
</li>
<li>
The spring force is applied perpendicularly to the axes of the two springs.
</li>
</p>
<p>
If we compute F(1), the force along the axis of one spring, and
F(2), the force perpendicular to the axis of one spring, we have that
F(L,THETA,LAMBDA,MU) is given by:
<ul>
<li>
F(1) = - 2 ( 1 - L ) + 2 * LAMBDA * cos ( THETA ) + MU * sin ( THETA )
</li>
<li>
F(2) = 0.5 * THETA - 2 * LAMBDA * L * sin ( THETA ) + MU * L * cos ( THETA )
</li>
</ul>
</p>
<p>
To explore these equations, we can first solve for MU and LAMBDA in terms
of L and THETA:
<ul>
<li>
LAMBDA(L,THETA) = (1-L) * cos(THETA) + 0.25 * THETA * sin(THETA) / L
</li>
<li>
MU(L,THETA) = 2 * (1-L) * sin(THETA) - 0.5 * THETA * cos(THETA) / L
</li>
</ul>
</p>
<p>
We can then study the behavior of solutions by choosing a fixed value of
THETA (say pi/8), and plotting LAMBDA(L,THETA) versus MU(L,THETA) over a range
of values of L, say from 0.25 to 1.75. Recall that L = 1 when the springs
are at their "natural" length. This approach is taken by the function
BUCKLING_SPRING_L.
</p>
<p>
We can also plot the same data, but instead fix a value of L, typically in
the range of 0.25 to 1.75, and the plot LAMBDA(L,THETA) versus MU(L,THETA) over a range
of values of THETA, say from -3pi/8 to +3pi/8. This approach is taken by
the function BUCKLING_SPRING_THETA.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
To draw lines of constant L, with THETA varying along a line:
<blockquote>
<b>buckling_spring_l</b> ( <i>l_num</i>, <i>theta_num</i> )
</blockquote>
where
<ul>
<li>
<i>l_num</i> is the number of L lines to draw;
</li>
<li>
<i>theta_num</i> is the number of THETA values along each L line.
</li>
</ul>
</p>
<p>
To draw lines of constant THETA, with L varying along a line:
<blockquote>
<b>buckling_spring_theta</b> ( <i>l_num</i>, <i>theta_num</i> )
</blockquote>
where
<ul>
<li>
<i>l_num</i> is the number of L values along each THETA line;
</li>
<li>
<i>theta_num</i> is the number of THETA lines to draw.
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>BUCKLING_SPRING</b> is available in
<a href = "../../math_src/buckling_spring/buckling_spring.html">a Mathematica version</a> and
<a href = "../../m_src/buckling_spring/buckling_spring.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/test_con/test_con.html">
TEST_CON</a>,
a FORTRAN program which
sets up sample problems for continuation, including the buckling spring.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Tim Poston, Ian Stewart,<br>
Catastrophe Theory and its Applications,<br>
Dover, 1996,<br>
ISBN13: 978-0486692715,<br>
LC: QA614.58.P66.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "buckling_spring_l.m">buckling_spring_l.m</a>,
chooses equally spaced values of THETA between -3pi/8 and +3pi/8,
and plots curves of LAMBDA(L,THETA) versus MU(L,THETA) over a range
of values of L, from 0.25 to 1.75.
</li>
<li>
<a href = "buckling_spring_l.png">buckling_spring_l.png</a>
a PNG image of the plot, using 101 L values and 101 THETA values.
</li>
</ul>
</p>
<p>
<ul>
<li>
<a href = "buckling_spring_theta.m">buckling_spring_theta.m</a>,
chooses equally spaced values of L between 0.25 and 1.75
and plots curves of LAMBDA(L,THETA) versus MU(L,THETA) over a range
of values of THETA, from -3pi/8 to +3pi/8.
</li>
<li>
<a href = "buckling_spring_theta.png">buckling_spring_theta.png</a>
a PNG image of the plot, using 101 L values and 101 THETA values.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../m_src.html">
the MATLAB directory</a>.
</p>
<hr>
<i>
Last revised on 04 August 2009.
</i>
<!-- John Burkardt -->
</body>
</html>