merge solutions to master - phugoid 1 and 2

.gitignore

@@ -1,2 +1 @@
-
-lessons/phugoid/.ipynb_checkpoints/01_01_Pendulum-checkpoint.ipynb
+lessons/phugoid/.ipynb_checkpoints/*checkpoint.ipynb

lessons/.gitignore

@@ -0,0 +1 @@
+.ipynb_checkpoints

lessons/01_phugoid/phugoid.py

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+# Script to plot the flight path of the phugoid using Lanchester's model.
+# It uses the sign convention and formulae provided by Lanchester (1909).
+
+import numpy
+import matplotlib.pyplot as plt
+
+def radius_of_curvature(y, yt, C):
+    """Returns the radius of curvature of the flight path at any point.
+    
+    Parameters
+    ---------
+    y : float
+        current depth below the reference horizontal line.
+    yt : float
+        initial depth below the reference horizontal line.
+    C : float
+        constant of integration.
+
+    Returns
+    -------
+    radius : float
+        radius of curvature.
+    """
+    #return 2./(C*(1./y)**1.5 - 2./3./yt)
+    return -1*yt / (1./3 - C/2.*(yt/y)**1.5)
+
+def rotate(x, y, xCenter, yCenter, angle):
+    """Returns the new position of the point.
+
+    Parameters
+    ---------
+    x : float
+        previous x-position of the point
+    y : float
+        previous y-position of the point.
+    xCenter : float
+        x-location of the center of rotation.
+    yCenter : float
+        y-location of the center of rotation.
+    angle : float
+        angle of rotation
+
+    Returns
+    -------
+    xCenter_new : float
+        new x-location of the center of rotation.
+    yCenter_new : float
+        new y-location of the center of rotation.
+    """
+    dx = x - xCenter
+    dy = y - yCenter
+    xNew = dx*numpy.cos(angle) - dy*numpy.sin(angle)
+    yNew = dx*numpy.sin(angle) + dy*numpy.cos(angle)
+    return xCenter + xNew, yCenter + yNew
+
+def plot_flight_path(yt, y0, theta0):
+    """Plots the flight path.
+
+    Parameters
+    ---------
+    yt : float
+        trim height of the glider.
+    y0 : float
+        initial height of the glider.
+    theta0 : float
+        initial orientation of the glider.
+
+    Returns
+    -------
+    None : None
+    """
+    # arrays to store the coordinates of the flight path
+    N = 1000
+    y = numpy.zeros(N)
+    x = numpy.zeros(N)
+
+    # set initial conditions
+    y[0] = y0
+    x[0] = 0.
+    theta = theta0
+
+    # calculate the constant C
+    #C = numpy.sqrt(y[0])*(numpy.cos(theta) - y[0]/yt/3.)
+    C = (numpy.cos(theta) - 1./3*y[0]/yt)*(y[0]/yt)**.5
+
+    # incremental distance along the flight path
+    ds = 1 
+
+    #obtain the curve coordinates
+    for i in range(1,N):
+        normal = numpy.array([numpy.cos(theta+numpy.pi/2.), numpy.sin(theta+numpy.pi/2.)])
+        R = radius_of_curvature(y[i-1], yt, C)
+        center = numpy.array([x[i-1]+normal[0]*R, y[i-1]+normal[1]*R])
+        dtheta = ds/R
+        x[i], y[i] = rotate(x[i-1], y[i-1], center[0], center[1], dtheta)
+        theta = theta + dtheta
+
+    # generate a plot
+    plt.figure(figsize=(10,6))
+    plt.plot(x, -y, color = 'k', ls='-', lw=2.0, label="$z_t=\ %.1f,\\,z_1=\ %.1f,\\,\\theta_1=\ %.2f$" % (yt, y[0], theta0))
+    plt.axis('equal')
+    plt.title("Flight path for $C$ = %.3f" % C, fontsize=18)
+    plt.xlabel("$x$", fontsize=18)
+    plt.ylabel("$z$", fontsize=18)
+    plt.legend()
+    plt.show()
+
+# End of File

styles/numericalmoocstyle.css

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+<style>
+
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+    font-family: "Computer Modern";
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