Abstract Generated abstract
This paper studies the motion in a Newtonian force field of a rigid body with fluid-filled cavities and a fixed point, using previously established equations and integrals for the system. Under specified relations among the principal inertia coefficients and parameters describing the force field and fluid effects, it derives a special solution that expresses the direction variables in terms of angular velocity components. Substitution into the known integrals reduces the determination of two angular velocity components to algebraic relations depending on the remaining component, while the time dependence of that component is obtained by quadrature. The solution also shows the dependence of the energy integral on the other relations and identifies a ten-parameter family of motions satisfying a stated constraint among the constants.
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MECHANICS
E. I. Kharlamova
A NEW SOLUTION OF THE PROBLEM OF THE MOTION IN A NEWTONIAN FORCE FIELD OF A BODY HAVING CAVITIES FILLED WITH FLUID
(Presented by Academician P. Ya. Kochina on 9 III 1964)
Assuming that the body with cavities filled with fluid has a fixed point, we write the equations of its motion in the notation of work \([1]\)*
\[ A_1\frac{d\omega_1}{dt} = (A_2-A_3)(\omega_2\omega_3-\varepsilon\gamma_2\gamma_3) +\lambda_2\omega_3-\lambda_3\omega_2 +e_2\gamma_3-e_3\gamma_2, \tag{1} \]
\[ \frac{d\gamma_1}{dt}=\omega_3\gamma_2-\omega_2\gamma_3 \qquad (1\ 2\ 3). \]
The known integrals are
\[ \gamma_1^2+\gamma_2^2+\gamma_3^2=\Gamma^2, \tag{2} \]
\[ (A_1\omega_1+\lambda_1)\gamma_1 +(A_2\omega_2+\lambda_2)\gamma_2 +(A_3\omega_3+\lambda_3)\gamma_3=m, \tag{3} \]
\[ A_1\omega_1^2+A_2\omega_2^2+A_3\omega_3^2 +\varepsilon(A_1\gamma_1^2+A_2\gamma_2^2+A_3\gamma_3^2) - \]
\[ {}-2(e_1\gamma_1+e_2\gamma_2+e_3\gamma_3)=2h. \tag{4} \]
Under the conditions
\[ A_1=A_2+A_3^{**},\qquad e_1=0,\qquad \lambda_1=0, \]
\[ (A_2^2\lambda_2^2+A_3^2\lambda_3^2)\varepsilon = A_2^2e_2^2+A_3^2e_3^2 \]
the equations (1) have a solution in which
\[ \gamma_1= \frac{(A_2+A_3)(A_2^2\lambda_2^2+A_3^2\lambda_3^2)} {(A_2-A_3)(A_2^2e_2\lambda_2-A_3^2e_3\lambda_3)} (\omega_1-s), \tag{5} \]
\[ \gamma_2= (A_2^2e_2^2+A_3^2e_3^2)^{-1} \left\{ A_2A_3(e_3\lambda_2+e_2\lambda_3) \left(\omega_3-\frac{\lambda_3}{A_2}\right) +\right. \]
\[ \left. +(A_2^2e_2\lambda_2-A_3^2e_3\lambda_3) \left(\omega_2-\frac{\lambda_2}{A_3}\right) \right\}, \tag{6} \]
\[ \gamma_3= (A_3^2e_3^2+A_2^2e_2^2)^{-1} \left\{ A_3A_2(e_2\lambda_3+e_3\lambda_2) \left(\omega_2-\frac{\lambda_2}{A_3}\right) +\right. \]
\[ \left. +(A_3^2e_3\lambda_3-A_2^2e_2\lambda_2) \left(\omega_3-\frac{\lambda_3}{A_2}\right) \right\}. \]
* The quantities \(\mu_i, R_i\) of work \([1]\) are denoted here by \(e_i, \gamma_i\), respectively, and the notation of the constant integral (2) has been changed.
** In article \([2]\) an example is given showing that this condition can be realized in a body with cavities filled with fluid.
Substituting (5), (6)′ into the integrals (2), (3), we determine \(\omega_2,\omega_3\) as functions of \(\omega_1\):
\[ \left(\omega_2-\frac{\lambda_2}{A_3}\right)^2+ \left(\omega_3-\frac{\lambda_3}{A_2}\right)^2 = \]
\[ = (A_2^2 e_2^2 + A_3^2 e_3^2) \left\{ \frac{\Gamma^2}{A_2^2\lambda_2^2 + A_3^2\lambda_3^2} - \frac{(A_2+A_3)^2(A_2^2\lambda_2^2 + A_3^2\lambda_3^2)} {(A_2-A_3)^2(A_2^2 e_2\lambda_2 - A_3^2 e_3\lambda_3)} (\omega_1-s)^2 \right\}, \]
\[ (A_2^2 e_2\lambda_2 - A_3^2 e_3\lambda_3) \left[ A_2\left(\omega_2-\frac{\lambda_2}{A_3}\right)^2 - A_3\left(\omega_3-\frac{\lambda_3}{A_2}\right)^2 \right] + \tag{7} \]
\[ + A_2A_3(A_2+A_3)(e_3\lambda_2+e_2\lambda_3) \left(\omega_2-\frac{\lambda_2}{A_3}\right) \left(\omega_3-\frac{\lambda_3}{A_2}\right) + \]
\[ + (A_2+A_3)(A_2^2\lambda_2^2 + A_3^2\lambda_3^2) \left[ \left(\omega_2-\frac{\lambda_2}{A_3}\right)\frac{e_2}{A_3} + \left(\omega_3-\frac{\lambda_3}{A_2}\right)\frac{e_3}{A_2} \right] = \]
\[ = (A_2^2 e_2^2 + A_3^2 e_3^2) \left\{ m - \frac{(A_2+A_3)^2(A_2^2\lambda_2^2 + A_3^2\lambda_3^2)} {(A_2-A_3)(A_2^2 e_2\lambda_2 - A_3^2 e_3\lambda_3)} \omega_1(\omega_1-s) \right\}. \]
Thus, from (6), \(\gamma_2,\gamma_3\) are determined as functions of \(\omega_1\), after which the first equation (1) determines, by quadrature, the dependence of \(\omega_1\) on \(t\).
The integral (4) in the indicated solution is dependent; it can be composed from the integrals (5), (6), (7). The constants \(h,m,\Gamma,s\) are connected by the relation
\[ (A_2+A_3)(A_2^2\lambda_2^2 + A_3^2\lambda_3^2)^2 \left[ \frac{A_2\lambda_2^2}{A_3^2} + \frac{A_3\lambda_3^2}{A_2^2} + (A_2+A_3)s^2 - 2h \right] + \]
\[ + 2(A_2-A_3)(A_2^2e_2\lambda_2 - A_3^2e_3\lambda_3) (A_2^2\lambda_2^2 + A_3^2\lambda_3^2)m + \]
\[ + \left[ 4(A_2^2e_2\lambda_2 - A_3^2e_3\lambda_3)^2 + A_2A_3(A_2+A_3)(e_3\lambda_2+e_2\lambda_3)^2 \right]A_2A_3\Gamma^2 = 0. \]
The indicated solution contains 10 independent parameters:
\[ A_2,\ A_3,\ \frac{e_2}{e_3},\ \lambda_2,\ \lambda_3,\ \Gamma,\ s,\ m,\ \omega_1^0,\ \psi_0. \]
Novosibirsk
State University
Received
5 III 1964
REFERENCES CITED
- P. V. Kharlamov, Journal of Applied Mechanics and Technical Physics, No. 4, 17 (1963).
- E. I. Kharlamova, Doklady AN SSSR, 125, No. 5, 996 (1959).