diff --git a/_sources/intro.ipynb b/_sources/intro.ipynb index 2a5332a..de7f47a 100644 --- a/_sources/intro.ipynb +++ b/_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", diff --git a/intro.html b/intro.html index 9592037..9e1da93 100644 --- a/intro.html +++ b/intro.html @@ -490,7 +490,7 @@

Equations of Motion and the Angular Velocity of an Axisymmetric Systems -

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.

+

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 [Wang and Eke, 1995]. As stability information in this paper is derived from the magnitude of the angular rates, the non-whirling flow assumption is sufficient.

Referring back to Figure Fig. 1, \(\mathbf{b}_1\), \(\mathbf{b}_2\), \(\mathbf{b}_3\) 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:

(3)#\[\mathbf{I}^* \triangleq I \, \mathbf{b}_1 \mathbf{b}_1 + I \, \mathbf{b}_2 \mathbf{b}_2 + J \, \mathbf{b}_3 \mathbf{b}_3\]
@@ -539,7 +539,7 @@

Spin rate(17), 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 \(k_3^2\) is assumed to be a constant in Equation (17), then

(18)#\[\omega_3 = \omega_{30} \bigg( \frac{m(0)}{m}\bigg) ^{1 - \frac{R^2}{2k_3^2}}.\]
-

Equation (18) 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 [Snyder and Warner, 1968], and Wang and Eke [Wang and Eke, 1995].

+

Equation (18) 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 [Snyder and Warner, 1968], and Wang and Eke [Wang and Eke, 1995].

Transverse rate#

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