Update documentation

_sources/intro.ipynb

@@ -45,7 +45,7 @@
     "  + \\int_{S_0} \\rho \\mathbf{\\big[p \\times (\\boldsymbol{\\omega} \\times p)\\big](v_r \\cdot {n})} \\, dS\n",
     "```\n",
     "\n",
-    "The remainder of this paper concerns the torque-free motions of axisymmetric variable mass systems so $\\mathbf{M^* = 0}$. Note that though this non-whirling flow assumption is not realistic, it has been shown that the magnitude of the transverse angular rates are unaffected by such a flow assumption[2]. As stability information in this paper is derived from the magnitude of the angular rates, the non-whirling flow assumption is sufficient.\n",
+    "The remainder of this paper concerns the torque-free motions of axisymmetric variable mass systems so $\\mathbf{M^* = 0}$. Note that though this non-whirling flow assumption is not realistic, it has been shown that the magnitude of the transverse angular rates are unaffected by such a flow assumption {cite}`ekewang3`. As stability information in this paper is derived from the magnitude of the angular rates, the non-whirling flow assumption is sufficient.\n",
     "\n",
     "Referring back to Figure {numref}`fig-general-variable-mass-system`, $\\mathbf{b}_1$, $\\mathbf{b}_2$, $\\mathbf{b}_3$ are a dextral set of unit vectors affixed in B and parallel to its principal directions. The instantaneous central inertia dyadic of this axisymmetric system is:\n",
     "\n",

intro.html

@@ -490,7 +490,7 @@ <h1>Equations of Motion and the Angular Velocity of an Axisymmetric Systems<a cl
   + \boldsymbol{\omega} \times (\mathbf{I^*} \cdot \boldsymbol{\omega}) 
   + \frac{^B d \mathbf{I}^*}{dt} \cdot \boldsymbol{\omega} \\
   + \int_{S_0} \rho \mathbf{\big[p \times (\boldsymbol{\omega} \times p)\big](v_r \cdot {n})} \, dS\end{split}\]</div>
-<p>The remainder of this paper concerns the torque-free motions of axisymmetric variable mass systems so <span class="math notranslate nohighlight">\(\mathbf{M^* = 0}\)</span>. Note that though this non-whirling flow assumption is not realistic, it has been shown that the magnitude of the transverse angular rates are unaffected by such a flow assumption[2]. As stability information in this paper is derived from the magnitude of the angular rates, the non-whirling flow assumption is sufficient.</p>
+<p>The remainder of this paper concerns the torque-free motions of axisymmetric variable mass systems so <span class="math notranslate nohighlight">\(\mathbf{M^* = 0}\)</span>. Note that though this non-whirling flow assumption is not realistic, it has been shown that the magnitude of the transverse angular rates are unaffected by such a flow assumption <span id="id2">[<a class="reference internal" href="refs.html#id13" title="S.-M. Wang and F.O. Eke. Rotational dynamics of axisymmetric variable mass systems. J. Appl. Mech., 62(4):970-974, 1995.">Wang and Eke, 1995</a>]</span>. As stability information in this paper is derived from the magnitude of the angular rates, the non-whirling flow assumption is sufficient.</p>
 <p>Referring back to Figure <a class="reference internal" href="#fig-general-variable-mass-system"><span class="std std-numref">Fig. 1</span></a>, <span class="math notranslate nohighlight">\(\mathbf{b}_1\)</span>, <span class="math notranslate nohighlight">\(\mathbf{b}_2\)</span>, <span class="math notranslate nohighlight">\(\mathbf{b}_3\)</span> are a dextral set of unit vectors affixed in B and parallel to its principal directions. The instantaneous central inertia dyadic of this axisymmetric system is:</p>
 <div class="math notranslate nohighlight" id="equation-eq3">
 <span class="eqno">(3)<a class="headerlink" href="#equation-eq3" title="Permalink to this equation">#</a></span>\[\mathbf{I}^* \triangleq I \, \mathbf{b}_1 \mathbf{b}_1 + I \, \mathbf{b}_2 \mathbf{b}_2 + J \, \mathbf{b}_3 \mathbf{b}_3\]</div>
@@ -539,7 +539,7 @@ <h2>Spin rate<a class="headerlink" href="#spin-rate" title="Permalink to this he
 <p>Examining Equation <a class="reference internal" href="#equation-eq17">(17)</a>, the spin rate may decay, stay constant, grow, or fluctuate. Another observation that can be added to this is that the spin rate retains the polarity of its initial condition; if the initial condition is positive, then the spin rate is always non-negative, and vice versa. Further analysis in this chapter assumes a positive value for the initial spin rate. Additionally, if <span class="math notranslate nohighlight">\(k_3^2\)</span> is assumed to be a constant in Equation <a class="reference internal" href="#equation-eq17">(17)</a>, then</p>
 <div class="math notranslate nohighlight" id="equation-eq18">
 <span class="eqno">(18)<a class="headerlink" href="#equation-eq18" title="Permalink to this equation">#</a></span>\[\omega_3 = \omega_{30} \bigg( \frac{m(0)}{m}\bigg) ^{1 - \frac{R^2}{2k_3^2}}.\]</div>
-<p>Equation <a class="reference internal" href="#equation-eq18">(18)</a> asserts that the spin rate cannot fluctuate for a system with constant axial radius of gyration; it can only grow or decay monotonically, or stay constant. It can then also be inferred that a time-varying axial radius of gyration can lead to fluctuations in the spin rate. Thus, the radius of gyration has a crucial effect on the spin rate. These comments on the spin rate are in agreement with the work of Snyder and Warner <span id="id2">[<a class="reference internal" href="refs.html#id32" title="V.W. Snyder and G.G. Warner. A re-evaluation of jet damping. J. Spacecr. Rockets, 5:364-366, 1968.">Snyder and Warner, 1968</a>]</span>, and Wang and Eke <span id="id3">[<a class="reference internal" href="refs.html#id13" title="S.-M. Wang and F.O. Eke. Rotational dynamics of axisymmetric variable mass systems. J. Appl. Mech., 62(4):970-974, 1995.">Wang and Eke, 1995</a>]</span>.</p>
+<p>Equation <a class="reference internal" href="#equation-eq18">(18)</a> asserts that the spin rate cannot fluctuate for a system with constant axial radius of gyration; it can only grow or decay monotonically, or stay constant. It can then also be inferred that a time-varying axial radius of gyration can lead to fluctuations in the spin rate. Thus, the radius of gyration has a crucial effect on the spin rate. These comments on the spin rate are in agreement with the work of Snyder and Warner <span id="id3">[<a class="reference internal" href="refs.html#id32" title="V.W. Snyder and G.G. Warner. A re-evaluation of jet damping. J. Spacecr. Rockets, 5:364-366, 1968.">Snyder and Warner, 1968</a>]</span>, and Wang and Eke <span id="id4">[<a class="reference internal" href="refs.html#id13" title="S.-M. Wang and F.O. Eke. Rotational dynamics of axisymmetric variable mass systems. J. Appl. Mech., 62(4):970-974, 1995.">Wang and Eke, 1995</a>]</span>.</p>
 </section>
 <section id="transverse-rate">
 <h2>Transverse rate<a class="headerlink" href="#transverse-rate" title="Permalink to this heading">#</a></h2>