MECHANICS

M. Z. LITVIN-SEDOI

EQUATIONS OF MOTION OF THE BASIC BODY OF A SYSTEM OF RIGID BODIES OF VARIABLE COMPOSITION

(Presented by Academician I. I. Artobolevskii on 6 IV 1961)

A rigid body of variable composition is a body of variable composition \((^1)\) in which the change of composition can occur only through the separation of material particles from the instantaneous outer surface of this body; moreover, the aggregate of particles that at each given instant of time are located in the region bounded by the outer surface of the body is a rigid body. By the instantaneous outer surface of a body of variable composition is meant the outer surface of the rigid body that at the given instant coincides with the body of variable composition.

Consider a system \(G\) of a finite number of rigid bodies of variable composition (shown schematically in Fig. 1), formed by attaching to the basic body \(G_0\) \(s\) kinematic chains in such a way that in each chain every body \(G_\nu^{(\sigma)}\) \((\nu = 1, 2, \ldots, n_\sigma;\ \sigma = 1, 2, \ldots, s)\) has up to 6 degrees of freedom inclusive with respect to the preceding body \(G_{\nu-1}^{(\sigma)}\); by the body \(G_0^{(\sigma)}\) the body \(G_0\) is meant. It is assumed that each of the bodies of the system \(G\) contains a part that remains unchanged during the entire time \(0 \le t \le T\) \((0 < T \le \infty)\) of consideration of this system, and that the constraints in the system \(G\) are imposed on the aforementioned unchanged parts of the bodies \(G_0, G_\nu^{(\sigma)}\). Some systems of type \(G\) were considered in works \((^{2-5})\); a general theory of a system of bodies of arbitrary form was presented in the report by A. I. Lur’e “Problems in the Theory of Relative Motion” at the First Congress on Mechanics (January 1960) \((^6)\). Here the equations of motion of the body \(G_0\) are given for prescribed laws of motion of all bodies \(G_\nu^{(\sigma)}\) relative to the bodies \(G_{\nu-1}^{(\sigma)}\) and for prescribed external and reactive forces.

Fig. 1

Introduce the notation: \(N_\lambda^{(\sigma)} x_\lambda^{(\sigma)} y_\lambda^{(\sigma)} z_\lambda^{(\sigma)}\) \((\lambda = 0, 1, 2, \ldots, n_\sigma)\) is a system \(\binom{\sigma}{\lambda}\) of coordinates (rectangular, right-handed) connected with the unchanged part of the body \(G_\lambda^{(\sigma)}\); \(c_\lambda^{(\sigma)}\) is the row matrix of the coordinates of the center of mass of the body \(G_\lambda^{(\sigma)}\) in the system \(\binom{\sigma}{\lambda}\); \(m_\lambda^{(\sigma)}\) is the mass of the body \(G_\lambda^{(\sigma)}\); \(I_\lambda^{(\sigma)}\) is the matrix of the moments of inertia (denoted in the usual way \((^7)\)) of the body \(G_\lambda^{(\sigma)}\) with respect to the axes of the system \(\binom{\sigma}{\lambda}\):

\[ I_\lambda^{(\sigma)} = \begin{Vmatrix} A_\lambda^{(\sigma)} & -F_\lambda^{(\sigma)} & -E_\lambda^{(\sigma)}\\ -F_\lambda^{(\sigma)} & B_\lambda^{(\sigma)} & -D_\lambda^{(\sigma)}\\ -E_\lambda^{(\sigma)} & -D_\lambda^{(\sigma)} & C_\lambda^{(\sigma)} \end{Vmatrix}, \qquad k_\nu^{(\sigma)} = \frac{1}{2}\left(A_\nu^{(\sigma)} + B_\nu^{(\sigma)} + C_\nu^{(\sigma)}\right), \]

\(m\) is the mass of the system \(G\); \(l_{pq}^{(\sigma)}=\|l_{ij}^{(pq,\sigma)}\|\) \((i,j=1,2,3;\; p,q=0,1,2,\ldots,n_\sigma;\; l_{12}^{(pq,\sigma)}=\cos(x_p^{(\sigma)},y_q^{(\sigma)}))\) is the matrix of direction cosines between the axes of the systems \(\binom{\sigma}{p}\) and \(\binom{\sigma}{q}\); \(h_{\nu-1}^{(\sigma)}\) is the row matrix of the coordinates of the origin \(N_\nu^{(\sigma)}\) in the system \(\binom{\sigma}{\nu-1}\); \(\omega_{pq}^{(\sigma)}\) is the row matrix of the projections of the instantaneous angular velocity of the system \(\binom{\sigma}{p}\) relative to the system \(\binom{\sigma}{q}\), taken on the axes of the system \(\binom{\sigma}{p}\); \(V_\lambda^{(\sigma)}\) is the row matrix of the projections of the principal vector of forces, external with respect to the system \(G\) and applied to the body \(G_\lambda^{(\sigma)}\), as well as of the reactive forces applied to this body, taken on the axes of the system \(\binom{\sigma}{\lambda}\); \(M_\lambda^{(\sigma)}\) is the row matrix of the projections of the principal moment of the aforementioned forces with respect to the origin \(N_\lambda^{(\sigma)}\), taken on the axes of the system \(\binom{\sigma}{\lambda}\); \(\sim\) is the operator that transforms the row matrix \(a=\|a_1,a_2,a_3\|\) into the matrix \(\tilde a\) (8)

\[ \tilde a= \begin{Vmatrix} 0 & -a_3 & a_2\\ a_3 & 0 & -a_1\\ -a_2 & a_1 & 0 \end{Vmatrix}; \]

\(E\) is the unit matrix \((3\times3)\); a prime denotes matrix transposition. The masses \(m_\lambda^{(\sigma)}\) and the moments of inertia \(A_\lambda^{(\sigma)},\ldots,F_\lambda^{(\sigma)}\) are, generally speaking, nonincreasing functions of time; the elements of the matrices \(c_\lambda^{(\sigma)}, h_{\nu-1}^{(\sigma)}\), and \(l_{\nu(\nu-1)}^{(\sigma)}\) are functions of time. Denote by \(v\) the row matrix of the projections of the absolute velocity of the origin \(N_0\) on the axes of the system \(N_0x_0y_0z_0\), associated with the invariable part of the principal body; by \(\omega\), the row matrix of the projections of the absolute angular velocity of the system \(x_0y_0z_0\) on the same axes, and

\[ S_\nu^{(\sigma)}=\sum_{\varkappa=1}^{\nu}h_{\varkappa-1}^{(\sigma)}l_{(\varkappa-1)0}^{(\sigma)}. \]

The motion of the origin \(N_0\) is described by the equations

\[ \frac{dv}{dt}=v\tilde\omega+\frac{1}{m}\sum_{\sigma=1}^{s}\sum_{\nu=1}^{n_\sigma} \{V_0-m_0c_0+V_\nu^{(\sigma)}l_{\nu0}^{(\sigma)} +m_\nu^{(\sigma)}[2(c_\nu^{(\sigma)}\dot l_{\nu0}^{(\sigma)} +\dot S_\nu^{(\sigma)})\tilde\omega \]
\[ -(S_\nu^{(\sigma)}+c_\nu^{(\sigma)}l_{\nu0}^{(\sigma)})(\tilde\omega^2-\dot{\tilde\omega}) -(c_\nu^{(\sigma)}\ddot l_{\nu0}^{(\sigma)}+\ddot S_\nu^{(\sigma)})]\}, \tag{1} \]

and the motion of the system \(x_0y_0z_0\) about the origin \(N_0\) is described by the equations

\[ \frac{d\omega}{dt}\sum_{\sigma=1}^{s}\sum_{\nu=1}^{n_\sigma} \{I_0+l_{\nu0}^{(\sigma)\prime}I_\nu^{(\sigma)}l_{\nu0}^{(\sigma)} -m_\nu^{(\sigma)}[(\tilde S_\nu^{(\sigma)})^2+U_\nu^{(\sigma)} +U_\nu^{(\sigma)\prime}]\} = \]
\[ =M_0+\omega I_0\tilde\omega' +(\dot v-v\tilde\omega) \left[m_0\tilde c_0+\sum_{\sigma=1}^{s}\sum_{\nu=1}^{n_\sigma} m_\nu^{(\sigma)}(\tilde S_\nu^{(\sigma)} +l_{\nu0}^{(\sigma)\prime}\tilde c_\nu^{(\sigma)}l_{\nu0}^{(\sigma)})\right] + \]
\[ +\sum_{\sigma=1}^{s}\sum_{\nu=1}^{n_\sigma} \{m_\nu^{(\sigma)}(\ddot S_\nu^{(\sigma)}-2\dot S_\nu^{(\sigma)}\tilde\omega +S_\nu^{(\sigma)}\tilde\omega^2) (\tilde S_\nu^{(\sigma)} +l_{\nu0}^{(\sigma)\prime}\tilde c_\nu^{(\sigma)}l_{\nu0}^{(\sigma)}) + \]
\[ +m_\nu^{(\sigma)}c_\nu^{(\sigma)} [(\tilde\omega_{\nu0}^{(\sigma)})^2-\dot{\tilde\omega}_{\nu0}^{(\sigma)} +2\tilde\omega_{\nu0}^{(\sigma)}l_{\nu0}^{(\sigma)}\tilde\omega l_{\nu0}^{(\sigma)\prime} +l_{\nu0}^{(\sigma)}\tilde\omega^2l_{\nu0}^{(\sigma)\prime}] l_{\nu0}^{(\sigma)}\tilde S_\nu^{(\sigma)} + \]
\[ +[\omega_{\nu0}I_\nu^{(\sigma)}\tilde\omega_{\nu0}^{(\sigma)} -\dot\omega_{\nu0}^{(\sigma)}I_\nu^{(\sigma)} +2\omega l_{\nu0}^{(\sigma)\prime}(I_\nu^{(\sigma)}-k_\nu^{(\sigma)}E) \tilde\omega_{\nu0}^{(\sigma)}]l_{\nu0}^{(\sigma)} \]
\[ -\omega l_{\nu0}^{(\sigma)\prime}I_\nu^{(\sigma)}l_{\nu0}^{(\sigma)}\tilde\omega -M_\nu^{(\sigma)}l_{\nu0}^{(\sigma)} -V_\nu^{(\sigma)}l_{\nu0}^{(\sigma)} \sum_{\varkappa=1}^{\nu}l_{(\varkappa-1)0}^{(\sigma)\prime} \tilde h_{\varkappa-1}^{(\sigma)} l_{(\varkappa-1)0}^{(\sigma)}\}, \tag{2} \]

where

\[ U_\nu^{(\sigma)}=\tilde S_\nu^{(\sigma)}l_{\nu0}^{(\sigma)\prime}\tilde c_\nu^{(\sigma)}l_{\nu0}^{(\sigma)}. \]

In this case,

\[ \dot l_{x0}^{(\sigma)}=-\widetilde{\omega}_{x0}^{(\sigma)}l_{x0}^{(\sigma)},\qquad \ddot l_{x0}^{(\sigma)}=\left[(\widetilde{\omega}_{x0}^{(\sigma)})^2-\dot{\widetilde{\omega}}_{x0}^{(\sigma)}\right]l_{x0}^{(\sigma)}, \]

\[ l_{x0}^{(\sigma)}=\prod_{\alpha=1}^{x}l_{(x-\alpha+1)(x-\alpha)}^{(\sigma)},\qquad \omega_{x0}^{(\sigma)}=\sum_{\beta=0}^{x-1}\omega_{(x-\beta)(x-\beta-1)}^{(\sigma)}l_{(x-\beta)0}^{(\sigma)}. \]

The derivative \(\dot v\) also enters the right-hand side of equation (2), the derivative \(\dot\omega\) enters the right-hand side of equation (1), and the matrix of moments of inertia of the system \(G\) enters the left-hand side of equation (2) as a multiplier of the derivative \(\dot\omega\). However, in computations with the aid of computing devices this circumstance is not restrictive. In principle, the system of equations (1) and (2) is solvable with respect to the derivatives \(\dot v\) and \(\dot\omega\), but the form of the right-hand sides thereby becomes unnecessarily complicated.

Equations (1) and (2) may also describe the relative motion of any one of the bodies \(G_\nu^{(\sigma)}\), if the absolute motion of the body \(G_0\) and the relative motions of all the remaining bodies of the system \(G\) are prescribed. Equations (1) and (2) are also applicable to a system \(G^*\) of more complex structure, differing from the system \(G\) by the presence of parallel chains of bodies (still open), branching from all or some of the bodies \(G_\nu^{(\sigma)}\). The system \(G^*\) is reduced to the system \(G\) by introducing additional chains, branching from the body \(G_0\) and regarded as massless on the segments from the body \(G_0\) to the corresponding parallel branch.

Moscow State University
named after M. V. Lomonosov

Received
6 IV 1961

REFERENCES

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