Abstract
Full Text
MECHANICS
P. V. KHARLAMOV
A KINEMATIC INTERPRETATION OF ONE SOLUTION OF THE PROBLEM OF THE MOTION OF A BODY HAVING A FIXED POINT
(Presented by Academician P. Ya. Kochina, 29 IV 1964)
Dividing the components of the angular velocity by the constant $\nu$ and introducing dimensionless quantities
\[ v_i=\frac{\gamma_i}{\Gamma}\quad (i=1,2,3),\qquad \mu=\frac{\lambda}{B\nu},\qquad R^2=1+\frac{2H}{B\nu^2}+\frac{\Gamma^2}{B^2\nu^4},\qquad a=\frac{\Gamma}{B\nu^2}>0, \tag{1} \]
we write the solution indicated in the article \((^1)\) in the form
\[ p=\cos\alpha,\qquad r=\sin\alpha; \tag{2} \]
\[ \frac{dq}{d\tau}=-\sqrt{f(q)}\qquad \left(\tau=\frac{\nu}{2}t\right); \tag{3} \]
\[ v_1=\frac{1}{2a}\left[(R^2-1-a^2+q^2)\cos\alpha-\sqrt{f(q)}\sin\alpha\right],\quad v_2=\frac{1}{a}(\mu-q), \tag{4} \]
\[ v_3=\frac{1}{2a}\left[(R^2-1-a^2+q^2)\sin\alpha+\sqrt{f(q)}\cos\alpha\right]. \]
Here
\[ f(q)=4a^2-4(\mu-q)^2-(R^2-1-a^2+q^2)^2. \tag{5} \]
The moving hodograph of the angular velocity of the body is given by equations (2), (3). It is a segment of a straight line parallel to the second coordinate axis. It intersects the straight line joining the center of gravity of the body with the fixed point $O$. For real $q$ the function $f(q)$ is nonnegative (see (3)); therefore the end points of the moving hodograph are determined by the roots of the equation $f(q)=0$, among which, in view of $f(q_0)\geq 0$, $f(\pm\infty)<0$, there are at least two real ones.
Directing the $\zeta$-axis of the fixed coordinate system $O\xi\eta\zeta$ along the unit vector $v_i$, we write the equations of the fixed hodograph \((^2)\):
\[ \omega_\zeta=\frac{1}{2a}(R^2-1-a^2+2\mu q-q^2),\qquad \omega_\xi^2+\omega_\eta^2+\omega_\zeta^2=1+q^2; \tag{6} \]
\[ \frac{d\alpha}{dq}= \frac{2a\sqrt{f(q)}}{4a^2(1+q^2)-(R^2-a^2-1+2\mu q-q^2)^2}. \tag{7} \]
From (6) we conclude that the fixed hodograph lies on the surface of revolution
\[ \left[\omega_\xi^2+\omega_\eta^2+(\omega_\zeta+a)^2-R^2-2\mu^2\right]^2 = 4\mu^2(R^2-1-a^2+\mu^2-2a\omega_\zeta). \tag{8} \]
We shall confine ourselves here to the analysis of the case $\mu=0$ (which includes also the Bobylev \((^3)\)—Steklov \((^4)\) solution). In this case the surface (8) is a sphere, whose center, in consequence of (1), lies above the point of support of the body (Fig. 1). We introduce the new variable
\[ u=\cos\beta=\frac{R^2+a^2-1-q^2}{2aR}. \tag{9} \]
Fig. 1
Fig. 3
Fig. 2
and denote
\[ u_1=\frac{R-1}{a},\qquad u_2=\frac{R+1}{a}>0,\qquad u'=\frac{R^2+a^2-1}{2aR}=\frac{u_1u_2+1}{u_1+u_2}; \tag{10} \]
then from (7), (3)
\[ \frac{d\alpha}{du} = -\frac{1}{1-u^2} \sqrt{\frac{(u_2-u)(u-u_1)}{(u_1+u_2)(u'-u)}}\,; \tag{11} \]
\[ \frac{du}{dt} = \sqrt{8aR\,(u'-u)(u_2-u)(u-u_1)}. \tag{12} \]
The geodesic curvature of the fixed hodograph
\[ \chi= \frac{u_1+u_2-2u}{2R\sqrt{(u_2-u)(u-u_1)}} \]
vanishes at
\[ u_*=\frac{u_1+u_2}{2}=\frac{R}{a}. \]
In this position the plane \(ONM\) is tangent to the sphere (Fig. 1). From (11), (9), (10)
\[ (u_2-u)(u-u_1)(u'-u)\geq 0,\qquad -1\leq u\leq 1, \]
\[ u_2>|u_1|,\qquad |u_1|<1,\qquad u'>u_1. \]
Three cases are possible.
\(1^\circ.\) \(u_2 > 1\). In this case \(-1 < u_1 \leqslant u \leqslant u' < 1 < u_2\). The line \(\alpha=\alpha(u)\) is tangent to the parallel \(u=u'\), while on the parallel \(u=u_1\) it is tangent to a meridian. The successive positions of the moving hodograph on the fixed one are indicated in Fig. 2a for the case \(u_* > u'\) and in Fig. 2b for the case \(u_* < u'\).
\(2^\circ.\) \(u_2 < 1\). In this case \(-1 < u_1 \leqslant u \leqslant u_2 < 1 < u'\). The line \(\alpha=\alpha(u)\) is nowhere tangent to the parallels of the sphere, but approaches the parallels \(u=u_1,\ u=u_2\), touching the meridians. On the parallel \(u=u_*\) the geodesic curvature changes sign (Fig. 2b).
\(3^\circ.\) \(u_2=1\). From (10) \(a=R+1,\ u'=1\), and equations (11), (12)
\[ \frac{d\alpha}{du} = \frac{1}{1-u^2} \sqrt{\frac{u-u_1}{1+u_1}}, \qquad \frac{du}{d\tau} = 2(1-u)\sqrt{2R(R+1)(u-u_1)} \]
show that \(\alpha=\alpha(u),\ u=u(\tau)\) are elementary functions, with \(u \to 1\) as \(\tau \to \infty\). The line \(\alpha=\alpha(u)\), winding without bound around the pole of the sphere, has finite length (Fig. 3).
Institute of Hydrodynamics
Siberian Branch of the Academy of Sciences of the USSR
Received
29 IV 1964
REFERENCES
- P. V. Kharlamov, Prikl. matem. i mekh., 28, 1 (1964).
- P. V. Kharlamov, Prikl. matem. i mekh., 28, 3 (1964).
- D. N. Bobylev, Tr. Otd. fiz. nauk Obshch. liubit. estestv., 8, 2 (1896).
- V. A. Steklov, Tr. Otd. fiz. nauk Obshch. liubit. estestv., 8, 2 (1896).