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<title>pylib.numerical.ode_model module — pylib 2019.12.21 documentation</title>
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<div class="section" id="module-pylib.numerical.ode_model">
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<span id="pylib-numerical-ode-model-module"></span><h1>pylib.numerical.ode_model module<a class="headerlink" href="#module-pylib.numerical.ode_model" title="Permalink to this headline">¶</a></h1>
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<p>Mathmatical models governed by ordinary differential equations.</p>
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<p>Describes initial value problems as systems of first order ordinary
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differential equations.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Date</dt>
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<dd class="field-odd"><p>2020-01-08</p>
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</dd>
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</dl>
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<span class="target" id="module-ode_model"></span><dl class="function">
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<dt id="pylib.numerical.ode_model.disk">
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<code class="sig-name descname">disk</code><span class="sig-paren">(</span><em class="sig-param">d</em>, <em class="sig-param">e</em>, <em class="sig-param">T</em>, <em class="sig-param">method=''</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/pylib/numerical/ode_model.html#disk"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#pylib.numerical.ode_model.disk" title="Permalink to this definition">¶</a></dt>
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<dd><p>Rotation of an eccentric disk.</p>
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<dl class="field-list simple">
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<dt class="field-odd">Parameters</dt>
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<dd class="field-odd"><ul class="simple">
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<li><p><strong>d</strong> (<em>float</em>) – diameter</p></li>
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<li><p><strong>e</strong> (<em>float</em>) – eccentricity</p></li>
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<li><p><strong>T</strong> (<em>float</em>) – torque</p></li>
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<li><p><strong>method</strong> – the method to use, default = ‘’.</p></li>
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</ul>
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</dd>
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<dt class="field-even">Returns</dt>
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<dd class="field-even"><p><p>disk function. This function is independent of the time.</p>
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<ul class="simple">
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<li><p>For method = ‘’: f(x, t=0) -> (xp1, xp2, xp3, xp4, xp5, xp6).
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x is (x, y, phi, x’, y’, phi’) and the return values are (x’,
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y’, phi’, x’‘, y’‘, phi’‘)</p></li>
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<li><p>For method = ‘nm’: f(xn, xpn, xppn, t=0) -> (N, dN, dNp, dNpp).
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xn are the values of the function (x, y, phi), xpn are first
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derivative values of the function (x’, y’, phi’) and xppn are
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the second derivative values of the function (x’‘, y’‘, phi’‘).
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The return values are (N, dN, dNp, dNpp)</p></li>
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<li><p>For method = ‘nmmdk’: f(xn, xpn, xppn, t=0) ->
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(rm, rmx, rmxpp, rd, rdx, rdxp, rk, rkx, f).
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xn are the values of the function (x, y, phi), xpn are first
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derivative values of the function (x’, y’, phi’) and xppn are
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the second derivative values of the function (x’‘, y’‘, phi’‘).
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The return values are (rm, rmx, rmxpp, rd, rdx, rdxp, rk, rkx,
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f)</p></li>
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</ul>
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</p>
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</dd>
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<dt class="field-odd">Return type</dt>
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<dd class="field-odd"><p>function</p>
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</dd>
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</dl>
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<p>Model</p>
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<blockquote>
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<div><div class="math notranslate nohighlight">
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\[\begin{split}\begin{vmatrix}
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\ddot{x} + \cos(\varphi)\ddot{\varphi} + 2d \,\dot{x} - \sin(\varphi) \,\dot{\varphi}^2 + 2d\cos(\varphi)\, \dot{\varphi} + x &=&
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0 \\
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\ddot{y} - \sin(\varphi)\ddot{\varphi} + 2d \,\dot{y} - \cos(\varphi) \,\dot{\varphi}^2 + 2d\sin(\varphi)\, \dot{\varphi} + y &=&
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0 \\
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\ddot{\varphi} + e\,y\sin(\varphi) - e\,x\cos(\varphi) &=& t
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\end{vmatrix}
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\\
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\begin{vmatrix}
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\ddot{x} + \cos(\varphi)\ddot{\varphi} &=&
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-2d \,\dot{x} + \sin(\varphi) \,\dot{\varphi}^2 -2d\cos(\varphi)\, \dot{\varphi} - x \\
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\ddot{y} - \sin(\varphi)\ddot{\varphi} &=&
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-2d \,\dot{y} + \cos(\varphi) \,\dot{\varphi}^2 -2d\sin(\varphi)\, \dot{\varphi} - y \\
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\ddot{\varphi} &=& t - e\,y\sin(\varphi) + e\,x\cos(\varphi)
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\end{vmatrix}\end{split}\]</div>
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<div class="math notranslate nohighlight">
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\[\begin{split}\mathbf{M}(\mathbf{x}) \cdot \mathbf{\ddot{x}} &=
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\mathbf{f}(\mathbf{x}, \mathbf{\dot{x}})
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\\
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\begin{bmatrix}
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1 & 0 & \cos \varphi \\
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0 & 1 & -\sin \varphi \\
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0 & 0 & 1
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\end{bmatrix} \cdot
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\begin{bmatrix}
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\ddot{x} \\ \ddot{y} \\ \ddot{\varphi}
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\end{bmatrix} &= \begin{bmatrix}
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-2d \,\dot{x} + \sin(\varphi) \,\dot{\varphi}^2 -2d\cos(\varphi)\, \dot{\varphi} - x \\
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-2d \,\dot{y} + \cos(\varphi) \,\dot{\varphi}^2 -2d\sin(\varphi)\, \dot{\varphi} - y \\
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t - e\,y\sin(\varphi) + e\,x\cos(\varphi)
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\end{bmatrix}\end{split}\]</div>
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</div></blockquote>
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<p>returns</p>
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<blockquote>
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<div><div class="math notranslate nohighlight">
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\[\begin{split}x_1 &= x &\quad x_4 &= \dot{x}_1 = \dot{x} &\quad \dot{x}_4 &= \ddot{x} \\
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x_2 &= y &\quad x_5 &= \dot{x}_2 = \dot{y} &\quad \dot{x}_5 &= \ddot{y} \\
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x_3 &= \varphi &\quad x_6 &= \dot{x}_3 = \dot{\varphi} &\quad \dot{x}_6 &= \ddot{\varphi} \\\end{split}\]</div>
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<div class="math notranslate nohighlight">
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\[\begin{split}\dot{q} &= f(x) \\
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\begin{bmatrix}
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\dot{x}_1 \\
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\dot{x}_2 \\
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\dot{x}_3 \\
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\dot{x}_4 \\
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\dot{x}_5 \\
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\dot{x}_6
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\end{bmatrix} &= \begin{bmatrix}
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x_4 \\
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x_5 \\
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x_6 \\
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\begin{bmatrix}
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1 & 0 & \cos x_3 \\
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0 & 1 & -\sin x_3 \\
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0 & 0 & 1
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\end{bmatrix}^{-1} \cdot \begin{bmatrix}
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-2d \,x_4 + \sin(x_3) \,x_6^2 -2d\cos(x_3)\, x_6 - x_1 \\
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-2d \,x_5 + \cos(x_3) \,x_6^2 -2d\sin(x_3)\, x_6 - x_2 \\
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t - e\,x_2\sin(x_3) + e\,x_1\cos(x_3)
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\end{bmatrix}
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\end{bmatrix}\end{split}\]</div>
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</div></blockquote>
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<p>Three explicit differential equations of order 2 reducted to a
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system of 3x2 first-order differential equations.</p>
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<div class="admonition seealso">
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<p class="admonition-title">See also</p>
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<p><a class="reference internal" href="pylib.numerical.ode.html#pylib.numerical.ode.newmark_newtonraphson" title="pylib.numerical.ode.newmark_newtonraphson"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pylib.numerical.ode.newmark_newtonraphson()</span></code></a> and
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<a class="reference internal" href="pylib.numerical.ode.html#pylib.numerical.ode.newmark_newtonraphson_mdk" title="pylib.numerical.ode.newmark_newtonraphson_mdk"><code class="xref py py-meth docutils literal notranslate"><span class="pre">pylib.numerical.ode.newmark_newtonraphson_mdk()</span></code></a></p>
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