Abstract
Full Text
MECHANICS
G. L. GRODZOVSKII, Yu. N. IVANOV, and V. V. TOKAREV
ON THE MOTION OF A BODY OF VARIABLE MASS WITH CONSTANT EXPENDITURE OF POWER IN A GRAVITATIONAL FIELD
(Presented by Academician L. I. Sedov, 1 VIII 1960)
In the present work the general case is investigated of optimization of the reactive motion of a body of variable mass in the gravitational field of two centers, with a constant expenditure of power \(N = \mathrm{const}\), and the corresponding variational problem is considered.
- For a prescribed trajectory of motion, the acceleration due to reactive thrust is a given quantity
\[ a = a(t) = - \frac{V}{m}\frac{dm}{dt}, \]
where \(V\) is the exhaust velocity of the working body, and the useful reactive power is written in the form
\[ N = - \frac{dm}{dt}\frac{V^2}{2}; \]
hence we obtain
\[ \frac{a^2}{2N} = - \frac{1}{m^2}\frac{dm}{dt}. \tag{1} \]
Integrating equation (1), we arrive at the following law for the change of the body’s weight with time \(t\):
\[ G = G_0 \left/\left(1 + \frac{G_0}{2Ng}\int_0^T a^2\,dt\right)\right., \tag{2} \]
where \(g\) is the terrestrial acceleration of gravity, \(G_0\) is the initial weight.
Let us denote the specific weight of the power source by \(\alpha = G_N/N\). The relative total initial weight of the reserve of expelled mass \(G_M\) and of the power source \(G_N\) will be equal to
\[ \overline{G} = \frac{G_M + G_N}{G_0} = \alpha \frac{N}{G_0} + 1 - 1 \left/\left(1 + \frac{G_0}{2Ng}\int_0^T a^2\,dt\right)\right. . \tag{3} \]
For a prescribed function \(a(t)\), the quantity \(\overline{G}\) has a minimum
\[ \overline{G}_{\min} = 2\sqrt{\Phi} - \Phi \quad \text{when} \quad (G_N/G_0)_{\mathrm{opt}} = (\alpha N/G_0)_{\mathrm{opt}} = \sqrt{\Phi} - \Phi, \]
where
\[ \Phi = \frac{\alpha}{2g}\int_0^T a^2\,dt. \]
In the case of a multistage reduction of power with a corresponding reduction of weight, the maximum relative useful weight is expressed by the formula
\[ \overline{G}_{\mathrm{p.\ max}} = (1 + \Phi_1 - 2\sqrt{\Phi_1}) \prod_{i=2}^{n} \left(\frac{1 - \Phi_i}{1 + \Phi_i}\right)^2, \tag{4} \]
where
\[ \sum_{i=1}^{n}\Phi_i = \Phi \]
is prescribed. The optimal relation between the \(\Phi_i\) can be obtained by differentiating (4).
The variation of \(\overline{G}_{\min}\) and \(G_N/G_0\) as functions of \(\Phi\) is given in Fig. 1.
- It follows from Sec. 1 that the minimum of \(\overline{G}\) requires a minimum of the integral.
\[
\int_0^T a^2\,dt.
\]
As an application, let us consider the simplest plane motion along a gently sloping spiral in a central gravitational field for small values of \(a\).
We write the equation of motion in polar coordinates \(r,\psi\) in the form
\[
a_r=\dot v_r-\frac{v_\psi^2}{r}+\frac{g_0R_0^2}{r^2},
\]
\[
a_\psi=\frac{1}{r}\frac{d}{dt}(rv_\psi),\qquad
v_r=\dot r,\qquad v_\psi=r\dot\psi, \tag{5}
\]
where \(g_0\) is the acceleration of gravity at the radius \(R_0\).
For realizing spiral motion, put \(a_\psi=k(t)g_0,\ a_r=v_r^{*}\); then
\[
v_\psi=R_0\sqrt{g_0/r},\qquad
k(t)=-R_0^2\frac{d}{dt}\left(\frac{1}{rv_\psi}\right).
\]
Fig. 1
After integration, assuming that at \(t=0\), \(r=R_0,\ v_\psi=v_0=\sqrt{g_0R_0}\), we obtain the following law of spiral motion:
\[
\frac{r}{R_0}=\left[1-\frac{\displaystyle\int_0^t k(t)\,dt}{R_0/v_0}\right]^2. \tag{6}
\]
Let us note that in this motion the transition from a given radius \(R_0^{*}\) to a given radius \(R_1\) in a given time is determined by the integral
\[
\int_0^T k(t)\,dt.
\]
To ensure, in this case, the minimum of \(\bar G\), a minimum of the integral
\[
\int_0^T k^2(t)\,dt
\]
is required. According to the Cauchy–Bunyakovsky inequality,
\[
T\int_0^T k^2(t)\,dt\geq\left[\int_0^T k(t)\,dt\right]^2,
\]
therefore, in the spiral motion under consideration, the minimum \(\bar G\) is attained when \(k(t)=\mathrm{const}\).
Fig. 2
- Let us consider the problem of the optimal transfer of a body of variable mass in time \(T\) between two given points (with given velocities) with minimum \(\bar G\). This leads to the variational problem of finding the extremals of the integral
\[ I=\int_0^T a^2(t)\,dt \]
in the class of continuous functions \(a(t)\).
Let us consider plane motion in the gravitational field of two gravitating centers, one of which is at rest, while the other rotates about it with angular velocity \(\omega\) along a circle of radius \(r_0\). To study the character of the motion in the region of predominant influence of one of the centers, it is expedient to choose a coordinate system associated with this center (see Fig. 2). The equations of motion in the chosen coordinate systems can
* It can be shown that in this case \(a_r\) is of order \(ka_\psi\), i.e., for small \(k\) and for total energy of motion not close to zero, the radial component is negligible.
write in a unified form:
\[ \ddot r_i-r_i(\dot\psi_i+\omega)^2=a_{r_i}-\frac{k_i}{r_i^2}-\mathfrak R_i; \tag{7} \]
\[ r_i\ddot\psi_i+2\dot r_i(\dot\psi_i+\omega)=a_{\psi_i}-\Psi_i,\qquad i=1,2, \tag{8} \]
where
\[ \mathfrak R_1=\frac{k_2}{r_0^2}\cos\psi_1-\frac{k_2}{r_2^2}\cos(\psi_1-\psi_2),\quad \mathfrak R_2=\frac{k_1}{r_1^2}\cos(\psi_1-\psi_2), \]
\[ \Psi_1=\frac{k_2}{r_0^2}\sin\psi_1+\frac{k_2}{r_2^2}\sin(\psi_1-\psi_2),\quad \Psi_2=\frac{k_1}{r_1^2}\sin(\psi_1-\psi_2), \]
\(k_1\) and \(k_2\) are the gravitational constants.
It is required to find trajectories \(r(t)\) and \(\psi(t)\) that minimize the integral
\[ I=\int_0^T\left\{\left[\ddot r_i-r_i(\dot\psi_i+\omega)^2+\frac{k_i}{r_i^2}-\mathfrak R_i\right]^2+ \left[r_i\ddot\psi_i+2\dot r_i(\dot\psi_i+\omega)\Psi_i\right]^2\right\}\,dt. \tag{9} \]
We shall represent the Euler equations of this variational problem in the form
\[ \dot a_{r_i}=\frac{1}{v_{r_i}}\left[\frac{a_{r_i}^2+a_{\psi_i}^2}{2} +a_{r_i}\left(\frac{v_{\psi_i}^2}{r_i}-\frac{k_i}{r_i^2}\right) -\lambda_i-v_i\frac{v_{\psi_i}}{r_i}\right], \tag{10} \]
\[ \dot a_{\psi_i}=\frac{1}{r_i}\left(a_{\psi_i}v_{r_i}-2a_{r_i}v_{\psi_i}+v_i\right); \tag{11} \]
\[ \dot v_i=a_{\psi_i}\Psi'_{\psi_i}+a_{r_i}\mathfrak R'_{\psi_i}; \tag{12} \]
\[ \dot\lambda_i=-v_{r_i}\left(a_{\psi_i}\Psi'_{r_i}+a_{r_i}\mathfrak R'_{r_i} +\frac{\Psi_i}{r_i}a_{\psi_i}\right)-v_i\frac{\Psi_i}{r_i} -\dot a_{r_i}\mathfrak R_i-\frac{v_{\psi_i}}{r_i}\left(v_i-2a_{r_i}\Psi_i\right); \]
\[ v_{r_i}=\dot r_i,\qquad v_{\psi_i}=r_i(\dot\psi_i+\omega). \tag{14} \]
The origin of the functions \(v\) and \(\lambda\) will be clarified by considering motion in a central field (\(k_2=0;\ \omega=0\)). In this case \(\mathfrak R_1=0\) and \(\Psi_1=0\), i.e. \(\dot v=0\) and \(\dot\lambda=0;\ v\equiv\mathrm{const}\) and \(\lambda\equiv\mathrm{const}\) are integrals of the system. Let us simplify the equations of motion (7), replacing differentiation with respect to time by differentiation with respect to radius:
\[ v'_r v_r-\left(v_\psi^2/r-k/r^2\right)=a_r,\qquad v'_\psi v_r+v_r v_\psi/r=a_\psi, \]
and consider the following variational problem: to find the trajectory providing
\[ \min \int_{r_1}^{r_2} a^2\,\frac{dr}{v_r} \]
under the additional isoperimetric conditions: a prescribed time of displacement from \(r_1\) to \(r_2\)
\[ \left(T=\int_{r_1}^{r_2}\frac{dr}{v_r}\right) \]
and a prescribed polar angle of displacement
\[ \Delta\psi=\int_{r_1}^{r_2}\frac{v_\psi}{r}\,\frac{dr}{v_r}, \]
i.e., we find the extremals of the integral
\[ \int_{r_1}^{r_2}\left(\frac{a_r^2+a_\psi^2}{v_r}+\varkappa_1\frac{1}{v_r} +\varkappa_2\frac{v_\psi}{r v_r}\right)\,dr. \tag{15} \]
After carrying out the differentiation and returning to the variable \(t\) (\(dt=dr/v_r\)), we obtain expressions for \(\dot a_r\) and \(\dot a_\psi\) coinciding with (10) and (11) for \(\varkappa=2\lambda\) and \(\varkappa_2=2v^*\).
* In work (1), devoted to the optimal programming of the thrust vector during motion in a central field with \(N=\mathrm{const}\), the constant \(v\) is omitted, which narrows the class of possible solutions.
Thus, in a central field the functions \(\lambda\) and \(\nu\) have the meaning of constants under the conditions of isoperimetry.
- Thus, the solution of the problem posed in Sec. 3 reduces to the joint integration of two systems of equations (7) and (10)—(14).
As an example, let us consider motion in a force-free field: \(k_1 = 0\), \(k_2 = 0\). In this case equations (7), (10)—(14) in a rectangular coordinate system will be written in the form
\[ \frac{d^2 \bar r}{dt^2}=\bar a; \tag{16} \]
\[ \frac{d^4 \bar r}{dt^4}=0, \tag{17} \]
whence
\[ \bar a=\bar b_1 t+\bar b_2,\qquad \bar r=\frac{1}{6}\bar b_1 t^3+\frac{1}{2}\bar b_2 t^2+\bar b_3 t+\bar b_4. \tag{18} \]
The vectors \(\bar b_1\), \(\bar b_2\), \(\bar b_3\), and \(\bar b_4\) are determined from the boundary conditions according to the prescribed coordinates and velocities at two points and the time of transfer \(T\) between them. Let us note that in this case, along the optimal trajectory, the acceleration \(\bar a\) varies linearly with time (see (18)), while in planar motion the ratio of the acceleration components is a fractional-linear function of time, as in the work\({}^{2}\), where the case of constant reactive thrust is investigated.
- Let us investigate the singularities of the system of equations obtained. For simplicity we consider motion in a central field. As is seen from equation (10), when the radial velocity vanishes, \(v_r=0\), \(\dot a_r\to\infty\), unless the numerator of the right-hand side of this equation also vanishes at these points. Eliminating \(a_r\) from equations (7), we arrive at an equation linear with respect to \(\tfrac12 \dot v_r^2\); its solution is written in the form
\[ \frac{1}{2}\dot v_r^{\,2} = c v_r + v_r\int \frac{\dot f v_r-\tfrac12 f^2+h}{v_r^2}\,dv_r, \tag{19} \]
where \(f=v_\psi^2/r-k/r^2\), \(h=\tfrac12 a_\psi^2-\lambda-\nu v_\psi/r\), and \(c\) is the constant of integration.
The integrand in (19) can be expanded at the point \(v_r=0\) in powers of \(v_r\); in doing so the derivatives \(d^k/dv_r^k\) will be finite in the case of finiteness of \(\dot v_r\). Integrating this expansion and letting \(v_r\) tend to zero, we find, in the case of finite \(c\),
\[ \left(\frac{1}{2}\dot v_r^{\,2}\right)_{v_r=0} = \left(\frac{1}{2}f^2-h\right)_{v_r=0}, \]
or, replacing \(f\) and \(h\) by their values and substituting \(\dot v_r=a_r+f\), we obtain
\[ \frac{1}{2}a_r^2+a_r f+\frac{1}{2}f^2 = \frac{1}{2}f^2-\frac{1}{2}a_\psi^2+\lambda+\nu\dot\psi. \tag{20} \]
It is evident that the condition of finiteness of \(\tfrac12\dot v_r^{\,2}\), and hence also of \(a_r\), consists in the vanishing of the numerator of the expression \(\dot a_r\) at the point \(v_r=0\), and the point itself is a singular point of the “node” type.
As a result of differentiating \(\tfrac12\dot v_r^{\,2}\) with respect to \(v_r\), we obtain
\[ \ddot v_r = c+\dot f+ \int \frac{1}{v_r} \left[ \dot f-\frac{d}{dv_r}\left(\frac{1}{2}f^2-h\right) \right]dv_r. \tag{21} \]
As \(v_r\to0\), the last integral tends to zero (this can be shown by expanding the integral in a neighborhood of the point \(v_r=0\) and letting \(v_r\to0\)). Hence the value of the constant \(c\) is determined—this is the value of \(a_r\) at \(v_r=0\).
The investigation carried out makes it possible to use methods of numerical integration for solving the problem in the general case.
Central Aerohydrodynamic Institute
named after N. E. Zhukovsky
Received
24 VII 1960
CITED LITERATURE
- J. H. Irving, E. K. Blum, Vistas in Astronautics, 2, Second Annual Astronautics Symposium, 1959.
- D. E. Okhotsimsky, T. M. Eneev, Uspekhi Fizicheskikh Nauk, 58, issue 1a (1957).