On Optimal Rigid Body Rotation with Application of Internal Forces

Abstract

The article describes the result obtained for the problem of a rigid body’s maximum rotation in a given time interval by moving a movable internal mass. The mass movement is achieved by applying limited force. Previously, similar problems were considered in which the displacements of internal mass were assumed to be kinematic with restrictions on the point’s speed. The obtained result is described by analytical, easily verifiable formulas. The optimal trajectory of the moving mass is a spiral that coils around the center of mass of a rigid body with a frequency increasing to infinity. The obtained numerical results relate to the design of other optimal trajectories that cannot be analyzed analytically.

APPENDIX

Proof of Statement 3. Let us rewrite system (4.25) in the form of the following system of integral equations:

$$\begin{gathered} {{z}_{1}}(\tau ) = {{e}^{{2\tau }}}\left[ {{{z}_{1}}(0) - 4\int\limits_0^\tau {{{e}^{{ - 2s}}}{{z}_{2}}(s)ds} } \right], \\ \frac{{\left| {{{z}_{2}}(\tau )} \right|}}{{\sqrt {1 + z_{2}^{2}(\tau )} }} = \frac{{\left| {{{z}_{2}}(0)} \right|}}{{\sqrt {1 + z_{2}^{2}(0)} }}\exp \left[ {\int\limits_0^\tau {\frac{{z_{1}^{2}(s) - 1}}{{z_{1}^{2}(s) + 1}}ds} } \right]. \\ \end{gathered} $$

(A1)

We are interested in solutions of the system (A1), which continue over the entire infinite interval $0 \leqslant \tau < + \infty$. We will name such solutions acceptable. From the second equation of system (A1), it follows that any acceptable solution ${{z}_{2}}(\tau )$ (at ${{z}_{2}}(0) \ne 0$) does not vanish on any finite time interval $\tau$. Then, from the first equation of system (A1) we obtain that at ${{z}_{1}}(0){{z}_{2}}(0) < 0$, the solution ${{z}_{1}}(\tau )$ tends to infinity at $\tau \to + \infty$ no slower than ${{e}^{{2\tau }}}$. In this case, the second equation of system (A1) becomes inconsistent at $\tau \to + \infty$, since the left side of this equation is limited for all $\tau$ (less than 1), while the right side tends to $+ \infty$ at $\tau \to + \infty$, as ${{e}^{\tau }}$. Thus, the solutions of the system do not continue for an infinite time interval at ${{z}_{1}}(0){{z}_{2}}(0) < 0$; i.e., there are no acceptable solutions.

Let’s consider the case ${{z}_{1}}(0){{z}_{2}}(0) > 0$. Due to symmetry, it is sufficient to consider the case ${{z}_{1}}(0) > 0$, ${{z}_{2}}(0) > 0$. Therefore, we have ${{z}_{2}}(\tau ) > 0$, $\tau \in [0, + \infty )$ and the right side of the first equation of system (A1) remains positive all the time (otherwise, there is such ${{\tau }_{1}}$ that ${{z}_{1}}({{\tau }_{1}}){{z}_{2}}({{\tau }_{1}}) < 0$, and, due to the above, there are no acceptable solutions. Thus, in this case, for an acceptable solution, we need

$$0 < 4\int\limits_0^{ + \infty } {{{e}^{{ - 2s}}}} {{z}_{2}}(s)ds \leqslant {{z}_{1}}(0).$$

Let us show that, for an acceptable solution, the following exact equality must be satisfied:

$$4\int\limits_0^{ + \infty } {{{e}^{{ - 2s}}}} {{z}_{2}}(s)ds = {{z}_{1}}(0).$$

(A2)

Indeed, otherwise, due to the monotonicity of the integral $\int_0^\tau {{{e}^{{ - 2s}}}} {{z}_{2}}(s)ds$ on $\tau$ and the first equation of system (A1), violation of equality (A2) would lead to an infinite increase at $\tau \to + \infty$ (as ${{e}^{{2\tau }}}$) of function ${{z}_{1}}(\tau )$. This, as noted above, would be contradictory for the second equation of system (A1) (in the case of an acceptable solution).

Using equality (A2), the first equation of system (A1) can be rewritten in the following form:

$${{z}_{1}}(\tau ) = 4{{e}^{{2\tau }}}\int\limits_\tau ^{ + \infty } {{{e}^{{ - 2s}}}} {{z}_{2}}(s)ds.$$

(A3)

Let us assume that the limit of function $\left| {{{z}_{1}}(s)} \right|$ at $s \to + \infty$ exists and is finite: $\mathop {\lim }\limits_{s \to + \infty } \left| {{{z}_{1}}(s)} \right| = z_{*}^{{}} < + \infty$. Let us consider three cases:

(1) $z_{*}^{{}} > 1$ . Then, the integral $\int_0^\tau {\frac{{z_{1}^{2}(s) - 1}}{{z_{1}^{2}(s) + 1}}ds}$ is positive and diverges at $\tau \to + \infty$, the second equation of system (P1) is contradictory, and there are no acceptable solutions.

(2) $z_{*}^{{}} < 1$. Then, the integral $\int_0^\tau {\frac{{z_{1}^{2}(s) - 1}}{{z_{1}^{2}(s) + 1}}ds}$ is negative and diverges at $\tau \to + \infty$ and we obtain $\mathop {\lim }\limits_{\tau \to + \infty } {{z}_{2}}(\tau ) = 0$ from the second equation of the system (A1). From here, using Eq. (A3), we obtain that $\mathop {\lim }\limits_{s \to + \infty } \left| {{{z}_{1}}(s)} \right| = {{z}_{ * }} = 0$. Considering the system of linear approximation for the original system (4.25) in the vicinity of the singular point $\left\{ {{{z}_{1}} = 0,\;{{z}_{2}} = 0} \right\}$, we obtain

$$\frac{{d{{z}_{1}}}}{{d\tau }} = 2{{z}_{1}} - 4{{z}_{2}},\quad \frac{{d{{z}_{2}}}}{{d\tau }} = {{z}_{2}};\quad \tau \in [0, + \infty ).$$

The eigenvalues of this system are ${{\lambda }_{1}} = 2$ and ${{\lambda }_{2}} = 1$. Therefore, the solutions cannot possibly enter this point, which indicates the inconsistency of the obtained limiting values for the original nonlinear system (4.25). Thus, there are no acceptable solutions in this case either.

(3) ${{z}_{ * }} = 1$. Then, if the integral $\int_0^\tau {\frac{{z_{1}^{2}(s) - 1}}{{z_{1}^{2}(s) + 1}}ds}$ diverges at $\tau \to + \infty$, it must be negative (otherwise, the second equation of system (A1) is inconsistent. In addition, in this case, the second equation of system (A1) leads to the limiting relation $\mathop {\lim }\limits_{\tau \to + \infty } {{z}_{2}}(\tau ) = 0$. According to relation (A3), this limiting relation gives limiting relation $\mathop {\lim }\limits_{\tau \to + \infty } {{z}_{1}}(\tau ) = z_{*}^{{}} = 0$. The resulting contradiction indicates the absence of acceptable solutions in this case.

If the integral $\int_0^\tau {\frac{{z_{1}^{2}(s) - 1}}{{z_{1}^{2}(s) + 1}}ds}$ converges at $\tau \to + \infty$, then, according to the second equation of system (A1), there is a limit $\mathop {\lim }\limits_{\tau \to + \infty } {{z}_{2}}(\tau ) = {{z}_{{2 \text{*} }}}$. Passing in equality (A3) to the limit at $\tau \to + \infty$, we obtain the relation $1 = 2{{z}_{2\text{*}}} \to {{z}_{2\text{*}}} = 1{\text{/}}2$. The resulting limit is the first singular point from (4.26). The linearized equations of the original system (4.25) in the vicinity of this singular point have the form $z_{1}^{'} = 2{{z}_{1}} - 4{{z}_{2}}$, $z_{2}^{'} = \frac{5}{8}{{z}_{1}}$. The eigenvalues of the resulting system are ${{\lambda }_{{1,2}}} = 1 \pm i\sqrt {\frac{3}{2}}$. Thus, in this case also, the solutions cannot possibly enter the point under consideration. This indicates the absence of acceptable solutions in the case under consideration.

The last remaining case, when $\mathop {\lim }\limits_{s \to + \infty } \left| {{{z}_{1}}(s)} \right| = z_{*}^{{}} = + \infty$, also leads to the absence of acceptable solutions, since then, at $\tau \to + \infty$, the second equation of system (A1) is inconsistent. Statement 3 has been proven.