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MECHANICS
Academician I. I. ARTОBOLEVSKII, V. S. LOSHCHININ
A CHARACTERISTIC CRITERION FOR AN ASYMPTOTICALLY STABLE LIMIT REGIME OF MOTION OF A MACHINE UNIT
- In a number of works \((^{1-3})\), the question of the nonuniformity of the running of a machine and of the distribution of the inertial forces arising in this process between the initial and permanent motions in the sense of N. E. Zhukovskii \((^4)\) was investigated. The characteristic criterion of the regime of motion of a machine unit introduced in them,
\[ \chi(\varphi)=\frac{2}{\omega}\frac{d\omega}{d\varphi}=2[\ln\omega]_{\varphi}^{\prime}, \tag{1} \]
from the dynamical point of view, for any value of the angle of rotation \(\varphi\) of the driving link, indicates the relative magnitude of the inertial forces of the initial motion in comparison with the inertial forces of the permanent motion.
For reasons that will become clear in the present article, in the dynamical calculation of machine units the problem of investigating the behavior and finding the characteristic criterion \(\chi_0(\varphi)\) of the asymptotically stable limit regime \(T=T_0(\varphi)\) of motion of the machine unit acquires great importance \((^5)\). It is assumed that the equation of motion of the driving link is represented in the form
\[ dT/d\varphi=M(\varphi,T), \tag{2} \]
where:
\(1^\circ\). The reduced moment \(M(\varphi,T)\) of all acting forces is a function defined and continuous in the strip \(0\leq T\leq \tilde T,\ -\infty<\varphi<+\infty\), where \(\tilde T\) is the maximum possible value of the kinetic energy of the motion of the machine unit that can be imparted to it by the acting forces.
\(2^\circ\). \(M(\varphi,0)>0,\ M(\varphi,\tilde T)\leq 0.\)
\(3^\circ\). The steepness of the reduced moment of all acting forces is bounded by certain constants \(-\lambda_2\leq M_T'(\varphi,T)\leq -\lambda_1\) \((0<\lambda_1\leq\lambda_2)\).
The reduced moment of inertia of the masses of all links is assumed to be a function of the angle of rotation \(\varphi\) of the driving link, \(I=I(\varphi)\).
- If \(T=T_0(\varphi)=I(\varphi)\omega_0^2(\varphi)/2\) is an asymptotically stable limit energy regime of motion of the machine unit, then for any value of the angle of rotation \(\varphi\) the equality
\[ \frac{\omega_0^2(\varphi)}{2}\frac{dI}{d\varphi} +I(\varphi)\omega_0(\varphi)\frac{d\omega_0}{d\varphi} =M[\varphi,T_0(\varphi)]. \tag{3} \]
will hold. Hence, following (1), we find the law of variation of the characteristic criterion \(\chi_0(\varphi)\) of the limit energy regime \(T=T_0(\varphi)\) as a function of the angle of rotation \(\varphi\) of the driving link:
\[ \chi_0(\varphi)=\frac{2}{\omega_0}\frac{d\omega_0}{d\varphi} =\frac{M[\varphi,T_0(\varphi)]}{T_0(\varphi)} -\frac{\dot I(\varphi)}{I(\varphi)} =\frac{\dot T_0(\varphi)}{T_0(\varphi)} -\frac{\dot I(\varphi)}{I(\varphi)}. \tag{4} \]
To each of the possible energy regimes \(T=T(\varphi)\) of motion of a machine unit there corresponds uniquely a characteristic criterion
\[ \chi(\varphi)=M[\varphi,T(\varphi)]/T(\varphi)-\dot I(\varphi)/I(\varphi). \tag{5} \]
Consequently, (5) may be regarded as an operator defined in the functional space of all possible energy regimes of motion of the machine unit.
Taking (4) and (5) into account, we find an expression for the difference of the characteristic criteria \(\chi(\varphi)\) and \(\chi_0(\varphi)\) of the regimes \(T=T(\varphi)\) and \(T=T_0(\varphi)\):
\[ \chi(\varphi)-\chi_0(\varphi) = \left[\ln \frac{T(\varphi)}{T_0(\varphi)}\right]'_{\varphi} = \frac{T_0(\varphi)}{T(\varphi)} \left[\frac{T(\varphi)}{T_0(\varphi)}\right]'_{\varphi}. \tag{6} \]
An investigation of the behavior of this difference makes it possible to verify that the following holds here.
Theorem 1. If the reduced moment \(M(\varphi,T)\) of all acting forces satisfies conditions \(1^\circ, 2^\circ, 3^\circ\), then for the characteristic criterion \(\chi(\varphi)\) of any energy regime \(T=T(\varphi)\) different from the asymptotically stable limiting regime \(T=T_0(\varphi)\), the relation \(\chi(\varphi)\ne\chi_0(\varphi)\) holds on the interval on which the regime \(T=T(\varphi)\) is defined.
From the theorem it follows that
Corollary. Two different energy regimes \(T=T_1(\varphi)\) and \(T=T_2(\varphi)\) of the motion of a machine unit correspond to different characteristic criteria \(\chi_1(\varphi)\) and \(\chi_2(\varphi)\): \(\chi_1(\varphi)\ne\chi_2(\varphi)\).
The dynamical meaning of the theorem and of its corollary consists in the fact that, under the conditions considered, for different energy regimes there cannot exist identically coinciding laws of distribution of inertial forces between the initial and steady motions.
3. Among all possible characteristic criteria corresponding to different energy regimes of motion, a special role belongs to the characteristic criterion \(\chi_0(\varphi)\) of the asymptotically stable limiting regime \(T=T_0(\varphi)\) of motion of the machine unit.
Theorem 2. If the reduced moment \(M(\varphi,T)\) of all acting forces satisfies conditions \(1^\circ, 2^\circ, 3^\circ\), then the characteristic criterion \(\chi(\varphi)\) of any energy regime \(T=T(\varphi)\), as the angle of rotation \(\varphi\) of the driving link increases, approaches without bound the characteristic criterion \(\chi_0(\varphi)\) of the asymptotically stable limiting regime \(T=T_0(\varphi)\) of motion of the machine unit:
\[ \lim_{\varphi\to+\infty}\left|\chi(\varphi)-\chi_0(\varphi)\right|=0. \tag{7} \]
Indeed, using relations (4) and (5) and carrying out obvious identical transformations, we find
\[ \chi(\varphi)-\chi_0(\varphi) = \frac{1}{T(\varphi)} \{M[\varphi,T(\varphi)]-M[\varphi,T_0(\varphi)]\} + \frac{M[\varphi,T_0(\varphi)]}{T(\varphi)T_0(\varphi)} [T_0(\varphi)-T(\varphi)]. \tag{8} \]
On the basis of Lagrange’s theorem and condition \(3^\circ\),
\[ \left|M[\varphi,T(\varphi)]-M[\varphi,T_0(\varphi)]\right| = \left|M'_T(\varphi,c)\right|\left|T(\varphi)-T_0(\varphi)\right| \le \lambda_2\left|T_0(\varphi)-T(\varphi)\right|. \tag{9} \]
Moreover, taking into account the definition of the inertial curve \(T=\tau(\varphi)\) of motion of the machine unit (5), we obtain
\[ \left|M[\varphi,T_0(\varphi)]\right| = \left|M'_T(\varphi,c_0)\right| \left|T_0(\varphi)-\tau(\varphi)\right| \le \lambda_2(\tau^*-\tau_*), \tag{10} \]
where
\[ \tau_*=\inf_{|\varphi|<\infty}\tau(\varphi),\qquad \tau^*=\sup_{|\varphi|<\infty}\tau(\varphi). \]
Relations (8), (9), and (10) make it possible to find the estimate
\[ \left|\chi(\varphi)-\chi_0(\varphi)\right| \le \frac{\left|T(\varphi)-T_0(\varphi)\right|}{|T(\varphi)|} \left\{\lambda_2+\frac{\lambda_2(\tau^*-\tau_*)}{\tau_*}\right\} = \left|1-\frac{T_0(\varphi)}{T(\varphi)}\right| \frac{\lambda_2\tau^*}{\tau_*}. \tag{11} \]
Hence, taking into account theorem 5 of paper (5), we obtain the limiting equality (7).
Thus, whichever of the possible energetic regimes \(T = T(\varphi)\) of motion of the machine aggregate is taken, the corresponding characteristic criterion \(\chi(\varphi)\), for all sufficiently large values of the angle of rotation \(\varphi\) of the reduction member, will prove to be arbitrarily close to the characteristic criterion \(\chi_0(\varphi)\) of the asymptotically stable limiting regime \(T = T_0(\varphi)\). In this sense \(\chi_0(\varphi)\) acts as the limiting characteristic criterion.
It follows from this that if, in the dynamic calculation of a machine aggregate, we wish to take comprehensive account of the influence of the inertial forces of the initial motion in comparison with the influence on the machine members of the inertial forces of the permanent motion, then we must naturally investigate the behavior of the characteristic criterion \(\chi_0(\varphi)\), since it is precisely this criterion that expresses the basic tendency in the distribution of inertial forces between the indicated motions.
In this connection, in the dynamic calculation of a machine aggregate there may arise the problem of estimating (from below) those values of the angle of rotation \(\varphi\) of the reduction member for which the characteristic criterion \(\chi(\varphi)\) of the regime \(T = T(\varphi)\) reproduces, with accuracy up to \(\varepsilon'\), the criterion \(\chi_0(\varphi)\) of the asymptotically stable limiting regime \(T = T_0(\varphi)\):
\[ |\chi(\varphi)-\chi_0(\varphi)|<\varepsilon . \tag{12} \]
In solving it, one may proceed from the natural assumption that the initial conditions determining the regime \(T = T(\varphi)\) are chosen so as to satisfy the inequality \(0 \leq T(\varphi_0)=T_0 \leq \tau^*\).
Theorem 3. Under the conditions considered, relation (12) is satisfied at least for all values of the angle of rotation \(\varphi\) of the reduction member satisfying the inequality
\[ \varphi \geq \varphi_0+\frac{1}{\lambda_1}\ln\left(1+\frac{\lambda_2\tau^*}{\tau_* \varepsilon}\right). \tag{13} \]
The method for computing the constants entering into this estimate is indicated in (5).
4. In investigating the behavior of the characteristic criterion \(\chi_0(\varphi)\) for limiting regimes most widespread in practice, we need certain other forms of it, equivalent to (4).
First of all, note that
\[ \chi_0(\varphi)=\left[\ln\frac{T_0(\varphi)}{I(\varphi)}\right]'_{\varphi}, \tag{14} \]
i.e., the characteristic criterion is equal to the rate of change of the logarithm of the normalized kinetic energy \(T_0(\varphi)/I(\varphi)=\omega_0^2(\varphi)/2\) of the motion of the machine aggregate. Using the concept of the inertial curve \(T=\tau(\varphi)\) and Lagrange’s theorem, we obtain
\[ M[\varphi,T_0(\varphi)]=M'_T(\varphi,c_0)[T_0(\varphi)-\tau(\varphi)], \quad c_0 \in (\tau(\varphi),T_0(\varphi)). \tag{15} \]
Therefore, from relations (4) and (15) there follows the possibility of representing the criterion \(\chi_0(\varphi)\) in the form
\[ \chi_0(\varphi)=M'_T(\varphi,c_0)\left[1-\frac{\tau(\varphi)}{T_0(\varphi)}\right]-\frac{\dot I(\varphi)}{I(\varphi)} . \tag{16} \]
Theorem 4. In order that the limiting energetic regime \(T = T_0(\varphi)\) of the motion of a machine aggregate be stationary, it is necessary and sufficient that, in every position of the reduction member,
\[ \chi_0(\varphi)=-\dot I(\varphi)/I(\varphi)=-[\ln I(\varphi)]'_{\varphi}. \tag{17} \]
In this case the computation of the characteristic criterion \(\chi_0(\varphi)\) presents no difficulty. The theorem has a simple dynamic meaning: in the case of a stationary energetic regime, the law of distribution of inertial forces between the initial and permanent motions of the machine...
is completely determined by the distribution of masses and the intensity of its change in any position of the drive link.
Theorem 5. Suppose that the reduced moment \(M(\varphi,T)\) of all acting forces satisfies conditions \(1^\circ, 2^\circ, 3^\circ\) and, moreover, that for the inertial curve \(T=\tau(\varphi)\) there exists a finite limit
\[ \lim_{\varphi\to+\infty}\tau(\varphi)=\tau_0,\qquad 0<\tau_0\leq T. \tag{18} \]
Then the characteristic criterion \(\chi_0(\varphi)\) of the asymptotically stable limiting regime \(T=T_0(\varphi)\) of motion of the machine unit, as \(\varphi\to+\infty\), satisfies the relation
\[ \lim_{\varphi\to+\infty}\left[\chi_0(\varphi)+\dot I(\varphi)/I(\varphi)\right]=0. \tag{19} \]
In the case under consideration, the regime \(T=T_0(\varphi)\) is quasistationary \((^5)\), and
\[ \lim_{\varphi\to+\infty}T_0(\varphi)=\tau_0. \]
Consequently, the characteristic criterion \(\chi_0(\varphi)\) of the quasistationary energy regime of motion \(T=T_0(\varphi)\), for all sufficiently large values of the angle of rotation \(\varphi\), becomes arbitrarily close to the quantity \(-\dot I(\varphi)/I(\varphi)\); therefore
\[ \chi_0(\varphi)\doteq -\dot I(\varphi)/I(\varphi) \quad \text{for the same values of } \varphi . \tag{20} \]
Corollary. In the case of a constant reduced moment of inertia \(I(\varphi)\equiv \mathrm{const}\), the characteristic criterion \(\chi_0(\varphi)\) of the quasistationary limiting regime \(T=T_0(\varphi)\) tends to zero as the angle of rotation \(\varphi\) of the drive link increases:
\[ \lim_{\varphi\to+\infty}\chi_0(\varphi)=0. \tag{21} \]
Theorem 6. If the reduced moment \(M(\varphi,T)\) of all acting forces satisfies conditions \(1^\circ, 2^\circ\), has negative steepness, \(M_T'(\varphi,T)<0\), and, moreover, \(I(\varphi)\) and \(M(\varphi,T)\) are periodic functions with a common period \(\xi\) with respect to the angle of rotation \(\varphi\),
\[ I(\varphi+\xi)=I(\varphi),\qquad M(\varphi+\xi,T)=M(\varphi,T), \tag{22} \]
then the characteristic criterion \(\chi_0(\varphi)\) of the asymptotically stable limiting regime \(T=T_0(\varphi)\) of motion of the machine unit is periodic with the same period \(\xi\):
\[ \chi_0(\varphi+\xi)=\chi_0(\varphi). \tag{23} \]
In this case, the regime \(T=T_0(\varphi)\) is also periodic with period \(\xi\) \((^5)\).
Theorem 7. Suppose that the reduced moment \(M(\varphi,T)\) of all acting forces satisfies conditions \(1^\circ, 2^\circ, 3^\circ\) and, moreover, \(I(\varphi)\) and \(M(\varphi,T)\) are almost-periodic functions with respect to the angle of rotation \(\varphi\), uniformly with respect to \(T\), \(0\leq T\leq \bar T\).
Then the characteristic criterion \(\chi(\varphi)\) of the asymptotically stable limiting regime \(T=T_0(\varphi)\) of motion of the machine unit is also almost-periodic.
One can also verify that, under the conditions considered, the regime \(T=T_0(\varphi)\) itself will be almost-periodic.
State Scientific Research Institute of Machine Science
Received
13 I 1969
CITED LITERATURE
- I. I. Artobolevskii, Izv. AN SSSR, OTN, No. 12 (1952).
- I. I. Artobolevskii, DAN, 87, No. 1 (1952).
- I. I. Artobolevskii, Collection of Works on Agricultural Mechanics, 2 (1954).
- N. E. Zhukovskii, Complete Collected Works, 1, 1937.
- V. S. Lopatin, Tr. Inst. mashinovedeniya, 22, issue 88, Publishing House of the Academy of Sciences of the USSR, 1961.