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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).