better torque fromulation for the turning steamboat

examples-gallery/plot_Mississippi_steamboat.py

@@ -165,6 +165,23 @@
 
 Tdrag = -cS*aS * u*AS.z.cross(AS.y)
 forces.append((AS, Tdrag))
+# %%
+# If :math:`u \neq 0`, the boat will rotate and a drag torque will act on it.
+# Let's look at the situation from the center of the boat to its bow. At a
+# distance :math:`r` from the center, The speed of a point a r is :math:`u \cdot r`.
+# The area is :math:`dr`, hence the force is :math:`-c_S (u \cdot r)^2 dr`. The
+# leverage at this point is :math:`r`, hence the torque is
+# :math:`-c_S (u \cdot r)^2 \cdot r dr`.
+# Hence total torque is:
+# :math:`-c_S u^2 \int_{0}^{a_S/2} r^3 \, dr` =
+# :math:`\frac{1}{4} c_S u^2 \frac{a_S}{16}`
+#
+# The same is from the center to the stern, hence the total torque is
+# :math:`\frac{1}{32} c_S u^2 a_S^4`.
+# Same again across the front, with :math:`b_S` instead of :math:`a_S`, hence
+# the total torque is :math:`\frac{1}{32} c_S u^2 b_S^4`.
+tB = -cS*u**2*(aS**4 + bS**4)/32 * sm.tanh(20*u) * N.z
+forces.append((AS, tB))
 
 # %%
 # Set up the rigid bodies