MECHANICS

Academician I. I. ARTOBOLEVSKII, V. S. LOSHCHININ

AN ITERATIVE PROCESS FOR COMPUTING THE CHARACTERISTIC CRITERION OF A PERIODIC LIMITING REGIME OF MOTION OF A MACHINE UNIT

1. The question of the actual determination of the characteristic criterion \(\chi_\xi(\varphi)\) of the periodic limiting regime \(T = T_\xi(\varphi)\) of motion of a machine unit is closely connected with the dynamic calculation of the machine and therefore has great theoretical and applied significance \((^{1-3})\).

In the general case, in order to find the characteristic criterion \(\chi_\xi(\varphi)\), and to take into account the influence on the machine links of the inertial forces of the initial motion in the sense of N. E. Zhukovskii \((^4)\), it is necessary to know the periodic limiting regime \(T = T_\xi(\varphi)\) of motion of the machine unit. But this latter problem is solvable in quadratures only in rare cases, and therefore the criterion \(\chi_\xi(\varphi)\), generally speaking, is not computed in closed form.

In the present note a uniformly convergent iterative process is constructed, allowing one to compute the characteristic criterion \(\chi_\xi(\varphi)\) with any degree of accuracy; a convenient relation is given that makes it possible at each step of the iterative process to estimate the error with which the approximation \(\chi_k(\varphi)\) reproduces the criterion \(\chi_\xi(\varphi)\).

It is assumed that the equation of motion of the machine unit has been reduced to the form

\[ dT/d\varphi = M(\varphi,T), \tag{1} \]

where:

\(1^\circ.\) The reduced moment \(M(\varphi,T)\) of all acting forces is a function defined and continuous in the strip

\[ 0 \le T \le \hat T,\qquad -\infty < \varphi < +\infty, \tag{2} \]

where \(\hat T\) is the maximum possible value of the kinetic energy of the motion of the machine unit that the acting forces can impart to it.

\(2^\circ.\) \(M(\varphi,0)>0,\quad M(\varphi,\hat T)<0.\)

\(3^\circ.\) The slope of the reduced moment of all acting forces is continuous and negative in the strip (2), \(M_T'(\varphi,T)<0\).

\(4^\circ.\) The reduced moment \(M(\varphi,T)\) has period \(\xi\) with respect to the angle of rotation \(\varphi\): \(M(\varphi+\xi,T)=M(\varphi,T)\).

The reduced moment of inertia of the masses of all links is regarded as a function of the angle of rotation of the reduction link, \(I=I(\varphi)\).

Under the conditions considered, the slope of the moment of all acting forces is bounded below and above by certain negative constants

\[ -\lambda_2 \le M_T'(\varphi,T) \le -\lambda_1 \qquad (0<\lambda_1\le \lambda_2). \tag{3} \]

Introduce the notation

\[ \tau_*=\inf_{0\le \varphi<\xi}\tau(\varphi),\qquad \tau^*=\sup_{0\le \varphi<\xi}\tau(\varphi), \]

where \(T=\tau(\varphi)\) is the inertial curve of the motion of the machine unit \((^5)\).

Starting from an arbitrarily chosen \(\xi\)-periodic function \(T_1(\varphi)\), defined, continuous, and satisfying in the interval \(-\infty<\varphi<+\infty\) the inequality

\[ \tau_* \le T_1(\varphi) \le \tau^*, \tag{4} \]

construct a functional sequence \(T_k(\varphi)\) \((k=1,2,\ldots)\), defined by the recurrence law

\[ T_{k+1}(\varphi)=\frac{e^{-\lambda_2\varphi}}{e^{\lambda_2\xi}-1} \int_{\varphi}^{\varphi+\xi} e^{\lambda_2 t}\{M[t,T_k(t)]+\lambda_2T_k(t)\}\,dt . \tag{5} \]

The latter converges uniformly on the entire number line to the periodic limiting regime

\[ T_k(\varphi)\to T_\xi(\varphi),\qquad -\infty<\varphi<+\infty, \]

as \(k\to\infty\) in (5).

Theorem 1. If the reduced moment \(M(\varphi,T)\) of all acting forces satisfies conditions \(1^0\)—\(4^0\), then the functional sequence

\[ \chi_k(\varphi)=\frac{M[\varphi,T_k(\varphi)]}{T_k(\varphi)}-\frac{\dot I(\varphi)}{I(\varphi)},\qquad -\infty<\varphi<+\infty,\quad k=1,2,\ldots, \tag{6} \]

converges uniformly on the entire number line to the characteristic criterion \(\chi_\xi(\varphi)\) of the periodic limiting regime \(T=T_\xi(\varphi)\) of motion of the machine aggregate:

\[ \chi_k(\varphi)\to \chi_\xi(\varphi),\qquad -\infty<\varphi<+\infty, \]

as \(k\to\infty\).

Indeed, carrying out the obvious identical transformations and using Lagrange’s theorem, we find

\[ \chi_k(\varphi)-\chi_\xi(\varphi)= \frac{1}{T_k(\varphi)} \left\{M_T'(\varphi,c_k)- \frac{M[\varphi,T_\xi(\varphi)]}{T_\xi(\varphi)}\right\} [T_\xi(\varphi)-T_k(\varphi)]. \tag{7} \]

where \(c_k\in (T_k(\varphi),T_\xi(\varphi))\).

It is easy to show that, for all \(k\), the inequalities

\[ \tau_* \leq T_k(\varphi)\leq \tau^*,\qquad -\infty<\varphi<+\infty,\quad k=1,2,\ldots \tag{8} \]

will hold.

On the basis of Lemmas 1 and 2 of [5], replacing in the latter \(T\) by \(\tau^*-\tau_*\), we have

\[ |M_T'(\varphi,c_k)|\leq \lambda_2,\qquad |M[\varphi,T_\xi(\varphi)]|\leq \lambda_2(\tau^*-\tau_*). \tag{9} \]

Therefore, from (7), (8), and (9) we obtain the estimate

\[ |\chi_k(\varphi)-\chi_\xi(\varphi)| \leq \frac{1}{\tau_*} \left\{\lambda_2+\frac{\lambda_2(\tau^*-\tau_*)}{\tau_*}\right\} |T_k(\varphi)-T_\xi(\varphi)| = \frac{\lambda_2\tau^*}{\tau_*^2}|T_k(\varphi)-T_\xi(\varphi)|. \tag{10} \]

By virtue of the uniform convergence of the sequence (5), for every \(\varepsilon>0\), with \(\varepsilon_1=\varepsilon\tau_*^2/\lambda_2\tau^*\), there is a number \(K\), depending only on \(\varepsilon_1\) (and, consequently, on \(\varepsilon\)), such that the inequality

\[ |T_k(\varphi)-T_\xi(\varphi)|<\varepsilon_1 \]

is satisfied for all numbers \(k>K(\varepsilon)\), and moreover at once on the entire number line.

Taking estimate (10) into account, we see that

\[ |\chi_k(\varphi)-\chi_\xi(\varphi)|<\varepsilon \]

for all \(k>K(\varepsilon)\) and \(\varphi\in(-\infty,+\infty)\).

Thus, the constructed uniformly convergent iterative process (6) makes it possible to compute the characteristic criterion \(\chi_\xi(\varphi)\) of the periodic limiting regime \(T=T_\xi(\varphi)\) of motion of the machine aggregate with any desired degree of accuracy.

Theorem 2. Under the conditions considered, for the error \(r_k\) with which the approximation \(\chi_k(\varphi)\) reproduces the characteristic criterion \(\chi_\xi(\varphi)\) of the periodic limiting regime \(T=T_\xi(\varphi)\), at each step

of the iterative process (6) the estimate is valid

\[ r_k=\sup_{0\leqslant \varphi<\xi}\left|\chi_k(\varphi)-\chi_\xi(\varphi)\right| \leqslant \frac{\lambda_2\tau_*}{\tau_*^2} \left(\frac{\lambda_2}{\lambda_1}-1\right)\rho_k,\qquad k=2,3,\ldots, \tag{11} \]

where

\[ \rho_k=\sup_{0\leqslant \varphi<\xi}\left|T_k(\varphi)-T_{k-1}(\varphi)\right|. \tag{12} \]

1. Table 1 gives the results of tabulating the successive approximations \(\chi_k(\varphi)\) to the characteristic criterion \(\chi_{2\pi}(\varphi)\) of a periodic—

Table 1

limit regime \(T=T_{2\pi}(\varphi)\) of a rotor whose motion is described by the equation

\[ \frac{dT}{d\varphi}=M_b(\varphi)-k\left(\sqrt{\frac{2T}{I}}\right)^n. \]

The case not integrable in quadratures is taken, when \(n=4\), with numerical data \(M_b(\varphi)=2+\sin\varphi\) kgf, \(I=1\ \text{kg}\cdot\text{m}^2\), \(k=0{,}01\ \text{kgf}\cdot\text{sec}^4\), corresponding to a slow-speed rotor.

The successive approximations

\[ \chi_k(\varphi)=\frac{M[\varphi,T_k(\varphi)]}{T_k(\varphi)} = \frac{2+\sin\varphi-0{,}04\,T_k^2(\varphi)}{T_k(\varphi)}, \qquad k=1,2,\ldots,6, \]

were computed at the points

\[ \varphi_i=-\pi+\frac{\pi}{8}i,\quad i=0,1,2,\ldots,16, \]

for \(T_1(\varphi)\equiv 7\) J.

Using estimate (11), it is easy to verify that

\[ r_6=\sup_{|\varphi|<\pi}\left|\chi_6(\varphi)-\chi_{2\pi}(\varphi)\right|<3\cdot 10^{-5}. \]

Consequently, the 6th approximation \(\chi_6(\varphi)\) reproduces the characteristic criterion \(\chi_{2\pi}(\varphi)\) with at least 3 significant digits.

It is obvious that \(|\chi_{2\pi}(\varphi)| < 0.126\); therefore, in any position of the rotor the inertial forces of the initial motion will amount to less than \(6.3\%\) of the inertial forces of the permanent motion.

Fig. 1

Figure 1 shows the graphs of the approximations \(T_1(\varphi)\), \(T_2(\varphi)\), \(T_6(\varphi)\) to the periodic limiting regime \(T = T_{2\pi}(\varphi)\) of the rotor motion, and the graphs of the corresponding approximations \(\chi_1(\varphi)\), \(\chi_2(\varphi)\), \(\chi_6(\varphi)\) to the characteristic criterion \(\chi_{2\pi}(\varphi)\).

State Scientific Research Institute of Machine Science

Received
13 I 1969

REFERENCES

1. I. I. Artobolevskii, Izv. AN SSSR, OTN, No. 12 (1952).
2. I. I. Artobolevskii, DAN, 87, No. 1 (1952).
3. I. I. Artobolevskii, Collection of Works on Agricultural Mechanics, 2, 1954.
4. N. E. Zhukovskii, Complete Collected Works, 1, 1937.
5. V. S. Loshchin, Proceedings of the Institute of Machine Science, Seminar on the Theory of Machines and Mechanisms, 23, p. 91, Publishing House of the Academy of Sciences of the USSR, 1961.