Full Text
Analysis of a Certain Nonlinear System by the Method of Separation of Motions
E. I. Gerashchenko, A. F. Kleimenov
- In [1] a method was proposed for forcing a sliding mode by organizing sliding modes of high orders. Acceleration of sliding gives the control system the properties of essentially nonlinear systems, increases the “robustness” of the regulator with respect to the parameters of the plant, and improves the quality of regulation. However, in implementing a forced sliding mode the structure of the regulator is inevitably complicated. This leads to considerable difficulties in the mathematical analysis of the system. In article [1] the stability of the system regulator—linear plant was investigated by the method of separation of motions into fast and slow components (this method was proposed by Academician A. A. Andronov and his students and has been successfully applied in the theory of discontinuous oscillations [2]). However, the controlled plant is, generally speaking, nonlinear. Below we show the application of the method of separation of motions to the investigation of the system considered in [1], with a set of nonlinearities of the controlled plant characteristic for practice (saturation, backlash, and a dead zone).
We consider the following problem. A system is given
\[ \frac{dx_1}{dt}=x_2,\quad \frac{dx_2}{dt}=\Phi(x_3,\dot{x}_3), \tag{1} \]
\[ \frac{dx_3}{dt}=-ax_3-bx_2-cx_1-K\Psi(|x_1|\operatorname{sign}\sigma_1), \]
\[ \sigma_1=x_3+Ax_2+B|x_1|\operatorname{sign}\sigma_2,\quad \sigma_2=Cx_1+x_2, \]
where \(x=(x_1,x_2,x_3)\) is the controlled vector quantity; \(\Phi(x_3,\dot{x}_3)\) is a piecewise-linear (possibly multivalued) function describing nonlinearities of the saturation type with respect to the coordinate \(x_3\) and backlash; \(\Psi(|x_1|\operatorname{sign}\sigma_1)\) is likewise a piecewise-linear function describing the dead zone of the switching device or regulator; \(C\) is a positive constant.
It is necessary to estimate the influence of the functions \(\Phi\) and \(\Psi\) on the stability of the zero solution of system (1), and also to determine the parameters of self-oscillations, if they arise.
- Let us consider the influence of saturation on the coordinate \(x_3\), assuming that saturation is the only nonlinearity in the controlled plant. In this case
\[ \Phi(x_3,\dot{x}_3)= \begin{cases} -r_0, & \text{for } x_3\le -r_0,\\ x_3, & \text{for } |x_3|<r_0,\\ r_0, & \text{for } x_3\ge r_0; \end{cases} \]
\[ \Psi(|x_1|\operatorname{sign}\sigma_1)=|x_1|\operatorname{sign}\sigma_1, \]
where the positive quantity \(r_0\) indicates the level of limitation of the coordinate \(x_3\).
We shall assume that sufficient conditions for the occurrence of a sliding regime of the second order are fulfilled. According to [1], these conditions have the form: \(K>K_0>0,\ B>B_0>0\), where \(K_0,\ B_0\) are sufficiently large numbers. From system (1) it is seen that, for sufficiently large \(K\), the change of the coordinate \(x_3\) occurs with greater speed than the change of the coordinates \(x_1\) and \(x_2\). Therefore it is expedient to make the following change of coordinates and time:
\[ x_1=\nu y_1,\quad x_2=\nu y_2,\quad x_3=y_3,\quad t=\nu\tau\quad (K=\nu^{-2}). \tag{2} \]
For sufficiently large values of \(K\), the quantity \(\nu\) becomes a small parameter.
After the substitution (2), system (1) takes the form:
\[ \frac{dy_1}{d\tau}=\nu y_2,\quad \frac{dy_2}{d\tau}=\Phi\left(y_3,\frac{dy_3}{d\tau}\right), \tag{3} \]
\[ \frac{dy_3}{d\tau}=-\nu ay_3-\nu^2by_2-\nu^2cy_1-|y_1|\operatorname{sign}\sigma_1, \]
\[ \sigma_1=y_3+\nu Ay_2+b_0|y_1|\operatorname{sign}\sigma_2,\quad \sigma_2=y_2+Cy_1. \]
(We have put \(B=b_0\nu^{-1}\) and shall assume \(b_0>0\).)
For sufficiently small \(\nu\), according to [1], the motions in system (3) can be separated into fast (in the plane \(y_1=\mathrm{const}\)) and slow (along the \(y_1\)-axis) components.
The fast motions are described by the system obtained from (3) for \(\nu=0\):
\[ y_1(\tau)=y_{10}=\mathrm{const},\quad \frac{dy_2}{d\tau}=\Phi\left(y_3,\frac{dy_3}{d\tau}\right),\quad \frac{dy_3}{d\tau}=-|y_1|\operatorname{sign}\sigma_1. \tag{4} \]
We note that system (4) describes the motions of system (3) with accuracy up to \(O(\nu)\) as \(\nu\to 0\).
Let, for definiteness, \(y_{10}<0\). Then the representative point, moving according to system (4), arrives at the surface \(\sigma_1=0\), which in the plane \(y_1=y_{10}\) is represented by two half-lines
\[ \sigma_1= \begin{cases} \sigma_1^{+}=y_3+b_0|y_{10}|, & \text{for } \sigma_2>0,\\ \sigma_1^{-}=y_3-b_0|y_{10}|, & \text{for } \sigma_2\le 0. \end{cases} \]
The trajectories of system (4) in the region of discontinuity of the surface \(\sigma_1=0\) are represented by a closed cycle composed of segments of straight lines \(r_0y_3\pm y_{10}y_2=\mathrm{const}\) at points of the region \(|y_3|>r_0\) and parts of parabolas \(y_3^2\pm 2y_{10}y_2=\mathrm{const}\) at points of the region \(|y_3|\le r_0\). Moreover, this cycle is symmetric with respect to the \(y_3\)-axis and the line \(\sigma_2=0\) in the plane \(y_1=y_{10}\). Taking into account that \(C>0\), one can compute that the mean value of the coordinate \(y_2(\tau)\) over the period of revolution of the representative point along the cycle \((T)\) is positive, i.e.
\[ \frac{1}{T}\int_{0}^{T} y_2(\tau)\,d\tau>0. \]
As a consequence, the coordinate \(y_1(\tau)\), by virtue of the equation \(\dfrac{dy_1}{d\tau}=v y_2\), increases on the average, over one revolution along the closed cycle, by a certain amount \(\Delta y_1\), depending on \(y_{10}\). In particular, in the domain \(|y_3|\leq r_0\) we have
\[ \Delta y_1 = v\int_{\tau}^{\tau+T} y_2(\tau)\,d\tau = -4vCb_0y_{10}. \]
Thus, the representative point, moving according to system (3) (or (1)), asymptotically approaches the origin. The presence of the constraint on the coordinate \(x_3\) does not affect the stability of the zero solution of system (1). The effect of the constraint is manifested only in a slowing down of the transient process in the domain \(|x_3|>r_0\).
- Let us consider the combined influence of backlash and of the constraint in the coordinate \(x_3\). In this case the functions \(\Phi\) and \(\Psi\) have the following form:
\[ \Phi\left(x_3,\frac{dx_3}{dt}\right) = \begin{cases} \Phi_1\left(x_3,\dfrac{dx_3}{dt}\right), & \text{for } \left|\Phi_1\left(x_3,\dfrac{dx_3}{dt}\right)\right|\leq r_0,\\[1.2em] r_0, & \text{for } \Phi_1\left(x_3,\dfrac{dx_3}{dt}\right)>r_0,\\[1.2em] -r_0, & \text{for } \Phi_1\left(x_3,\dfrac{dx_3}{dt}\right)<-r_0, \end{cases} \]
where \(r_0\) is, as before, the constraint on the coordinate \(x_3\), and the function \(\Phi_1\left(x_3,\dfrac{dx_3}{dt}\right)\) describes the backlash (or dry friction):
\[ \Phi_1\left(x_3,\frac{dx_3}{dt}\right) = \begin{cases} x_3+\Delta, & \text{for } \dfrac{dx_3}{dt}<0,\\[1.2em] x_3-\Delta, & \text{for } \dfrac{dx_3}{dt}>0, \end{cases} \]
for
\[ \frac{dx_3}{dt}=0 \qquad \left| \Phi_1\left(x_3,\frac{dx_3}{dt}\right)-x_3 \right| <\Delta. \]
The positive number \(2\Delta\) denotes the magnitude of the backlash. Usually \(\Delta\ll r_0\), \(\Psi(x)\equiv x\).
Still assuming the quantities \(K\) and \(B\) to be sufficiently large, we pass to new coordinates and time, using the substitution (2). System (1) is transformed to the form (3) (only the definition of the function \(\Phi\left(x_3,\dfrac{dx_3}{dt}\right)\) changes). Separating the motions into fast and slow ones, we again obtain the equations of fast motion in the form (4). Let \(|y_{10}|>r_0/b_0\). The trajectory, by virtue of the equations of fast motion for a point lying on the line of discontinuity of the right-hand side of system (4), is a closed cycle composed of arcs of the parabolas
\[ (y_3\pm \Delta)^2 \pm 2y_{10}y_2=\mathrm{const} \]
for \(|y_3|<r_0-\Delta\), and of straight-line segments
\[ r_0y_3\pm y_{10}y_2=\mathrm{const} \]
for \(|y_3| > r_0 + \Delta\). For \(|r_0 - y_3| < \Delta\) either the backlash is selected,* if
\[ y_3 \frac{d y_3}{d \tau} < 0 \]
(in this case the point moves along a line segment); or, if
\[ y_3 \frac{d y_3}{d \tau} > 0, \]
the motion takes place along a parabola. If \(T\) denotes the period of revolution of the representative point along the closed cycle, then, since \(C > 0\), we have
\[ y_1 \cdot \int_{\tau}^{\tau+T} y_2(\tau)\,d\tau < 0. \]
Consequently, the quantity \(|y_1(\tau)|\), changing by virtue of the equations of the slow motions, decreases monotonically on the average.
Fig. 1
Together with it, the quantities \(|y_2(\tau)|\) and \(|y_3(\tau)|\) also decrease on the average. In the case
\[ \frac{\Delta}{b_0} < |y_{10}| < \frac{r_0}{b_0} \]
the qualitative picture of the motion remains the same.
A qualitatively new character of the motion arises when \(y_1(\tau)\) becomes equal in modulus to \(\Delta/b_0\). For definiteness, put
\[ y_1(\tau) = -\frac{\Delta}{b_0}. \]
Let the representative point be at point 1 with the backlash selected (Fig. 1, a, b). From point 1, by virtue of the equations of the fast motions,
\[ y_1 = y_{10} = -\frac{\Delta}{b_0}, \qquad \frac{d y_2}{d \tau} = y_3 + \Delta, \qquad \frac{d y_3}{d \tau} = -\frac{\Delta}{b_0}, \]
the representative point along a parabola reaches the half-line
\[ \sigma_1^{+} = y_3 + b_0 |y_{10}| = 0 \]
at point 2, at which
\[ y_2 = \frac{C\Delta}{b_0} + 2\Delta b_0, \qquad y_3 = -\Delta. \]
Since the sliding conditions are considered to be satisfied, the representative point will not leave the half-line \(\sigma_1^{+}=0\). Moreover, at point 2 we have
\[ \frac{d y_2}{d \tau}=0, \]
*) If the value of the function \(\Phi\!\left(x_3,\dfrac{dx_3}{dt}\right)\) is represented by a point on the backlash characteristic (Fig. 1, b), then selection of the backlash corresponds to the motion of the point parallel to the \(x_3\) axis.
therefore the trajectory, by virtue of the equations of fast motions, passing through point 1, ends at the “stationary” point 2. Taking into account that at point 2 \(y_2 > 0\), it is easy to see that, by virtue of the equations of slow motions, the coordinate \(y_1(\tau)\) increases and changes sign. In this case the quantity \(y_3(\tau)\), related to the quantity \(y_1(\tau)\) by the relation \(\sigma_1^+ = y_3 + b_0 |y_1| = 0\), first increases to zero, and when \(y_1(\tau)\) changes sign, it again decreases. As long as \(\dfrac{dy_3}{d\tau} > 0\), the backlash is taken up from the value \(-\Delta\) to zero; when \(\dfrac{dy_3}{d\tau} < 0\), the backlash is taken up in the reverse direction from zero to \(-\Delta\) (Fig. 1, b). The backlash will be completely taken up at the moment when \(y_1(\tau)\) becomes equal to \(\dfrac{\Delta}{b_0}\). Further motion of the point will occur by virtue of the system
\[ \frac{dy_1}{d\tau} = \nu y_2,\quad \frac{dy_2}{d\tau} = y_3 + \Delta,\quad y_3 = -b_0 |y_1| \]
along the surface \(\sigma_1^+ = 0\) up to the surface of discontinuity \(\sigma_2 = 0\). In this case the coordinate \(y_1(\tau)\) will still increase. The trajectory of the representative point in the plane \((y_1, y_2)\) will be the ellipse
\[ \nu y_2^2 + b_0\left(y_1 - \frac{\Delta}{b_0}\right)^2 = \mathrm{const}. \tag{5} \]
At the moment when \(y_2(\tau)\) vanishes (point 4), the quantity \(y_1(\tau)\) reaches its maximum value
\[ y_{1\max} = \frac{\Delta}{b_0} \left[ 1 + C\left(1 + \frac{2b_0^2}{C}\right)\sqrt{\frac{\nu}{b_0}} \right]. \tag{6} \]
At this same moment the quantity \(\dfrac{dy_3}{d\tau}\) will change sign and the backlash will begin to be taken up, as a result of which the motion will occur by virtue of the system
\[ \frac{dy_1}{d\tau} = \nu y_2,\quad \frac{dy_2}{d\tau} = -C\Delta\left(1+\frac{2b_0^2}{C}\right)\sqrt{\frac{\nu}{b_0}}. \]
The trajectory of the representative point in projection onto the plane \((y_1, y_2)\) will be the parabola
\[ \nu y_2^2 + C\Delta\left(1+\frac{2b_0^2}{C}\right) \sqrt{\frac{\nu}{b_0}}\,y_1 = \mathrm{const}, \tag{7} \]
along which the point reaches the line of discontinuity \(\sigma_2 = y_2 + C y_1 = 0\). We note that, in the motion along the ellipse (5), the absolute value of the rate of change of the coordinate \(y_2(\tau)\) is smaller than in the motion along the parabola (7). Therefore the parabola (7) will pass to the right of the ellipse (5) (Fig. 2), and the representative point, moving along the sliding surface, will reach the plane of discontinuity \(\sigma_2 = 0\) at a value \(y_1(\tau) > \dfrac{\Delta}{b_0}\).
Having left the plane of discontinuity (in Figs. 1, 2, point 5), the representative point will move by virtue of the equations of fast motions toward the plane-
stiffness \(\sigma_1=0\). Having made a number of “fast” revolutions about the line \(\sigma_2=\dfrac{d\sigma_2}{d\tau}=0\), the representative point will arrive at the discontinuity line \(\sigma_1=\sigma_2=0\) at the value \(y_1(\tau)=\dfrac{\Delta}{b_0}\) (point 7 in Fig. 1, a). In this case the backlash will remain unselected and the motion will continue by virtue of the system
\[ \frac{dy_1}{d\tau}=v y_2,\qquad \frac{dy_2}{d\tau}=0,\qquad y_3=b_0|y_1|. \]
Further, the motion proceeds analogously to the transition described above from point 2 to point 7, until the representative point again reaches position 1. The quantity \(|y_1|\) then attains its greatest value
\[ |y_1|=\frac{\Delta}{b_0}\left(1+C\sqrt{\frac{v}{b_0}}\right). \tag{8} \]
Comparing formulas (6) and (8), one may conclude that the maximum deviation of the quantity \(y_1\), in absolute value, for the initial position of the representative point on the discontinuity plane \(\sigma_2=0,\ |y_{10}|=\dfrac{\Delta}{b_0}\), and with selected backlash, is greater than for the same initial position and unselected backlash. The representative point will return to position 1, generally speaking, with partially selected backlash; therefore the maximum deviation of the modulus of \(y_1(\tau)\) from the equilibrium position during the subsequent cycle will lie between the quantities determined by formulas (6) and (8). Consequently, for the amplitude of the self-oscillations that arise one may write the following inequalities:
\[ \frac{\Delta}{b_0}\left(1+C\sqrt{\frac{v}{b_0}}\right) \leq A_{y_1}\leq \frac{\Delta}{b_0}\left[1+C\left(1+\frac{2b_0^2}{C}\right)\sqrt{\frac{v}{b_0}}\right]. \tag{9} \]
Fig. 2
Approximately integrating the equations of motion of the representative point from position 1 to positions 2, 3, etc., we estimate the value of the period of self-oscillation in the time \(\tau\):
\[ \frac{4}{v(C+2b_0^2)}+O\left(\frac{1}{\sqrt{v}}\right) \leq T_\tau \leq \frac{4}{vC}+O\left(\frac{1}{\sqrt{v}}\right), \tag{10} \]
or in the time \(t\):
\[ \frac{4}{C+2b_0^2}+O\left(\frac{1}{\sqrt{B}}\right) \leq T_t \leq \frac{4}{C}+O\left(\frac{1}{\sqrt{B}}\right). \tag{11} \]
Thus, the presence of backlash causes the appearance of self-oscillations near the equilibrium position with amplitude determined by the inequalities
\[ \frac{\Delta}{B}\left(1+C\sqrt{\frac{1}{B}}\right) \leq A_{x_1}\leq \frac{\Delta}{B}\left[1+C\left(1+\frac{2b_0^2}{C}\right)\sqrt{\frac{1}{B}}\right], \]
and with period determined by formulas (10) and (11).
4. We investigate the influence of the dead zone of the switching device, assuming that backlash and limitation are absent. In this case
\[ \Phi\left(x_3,\,-\frac{dx_3}{dt}\right)=x_3, \]
\[ \Psi(z)= \begin{cases} z-\delta, & \text{for } z\geq \delta,\\ 0, & \text{for } |z|<\delta,\\ z+\delta, & \text{for } z\leq -\delta. \end{cases} \]
The positive quantity \(2\delta\) denotes the size of the dead zone.
In system (1) we make a change of coordinates and time
\[ y_1=x_1,\quad y_2=x_2,\quad y_3=\mu x_3,\quad t=\mu\tau\quad (K=\mu^{-2}). \tag{12} \]
In the new variables system (1) takes the form:
\[ \frac{dy_1}{d\tau}=\mu y_2,\quad \frac{dy_2}{d\tau}=y_3, \tag{13} \]
\[ \frac{dy_3}{d\tau}=-(a\mu y_3+b\mu^2y_2+c\mu^2y_1)-\Psi(|y_1|\operatorname{sign}\sigma_1), \]
\[ \sigma_1=y_3+A\mu y_2+b_0|y_1|\operatorname{sign}\sigma_2, \]
\[ \sigma_2=y_2+Cy_1\quad (B=b_0\mu^{-1}). \]
If the representative point is in the region \(|y_1|\gg\delta\), then the qualitative picture of the motion will be the same as in the case \(\Psi(z)\equiv z\). The presence of the dead zone will be reflected only in a decrease of the absolute value of the rate of change of the coordinate \(y_3\) by the amount \(\delta\).
At the moment when the quantity \(y_1(\tau)\) becomes comparable with the quantity \(\delta\), the character of the motion changes. Indeed, for \(|y_1(\tau)|\leq\delta\) the control process becomes uncontrollable (the correcting action \(|y_1|\operatorname{sign}\sigma_1\) does not act on the controlled object). As a consequence, self-oscillations arise in the system.
Let at the initial moment the representative point be on the discontinuity line of the surface \(\sigma_1=0\) at the point \((-\delta, C\delta, b_0\delta)\). Since \(y_{2\Phi}>0\), the quantity \(y_1(\tau)\) will increase. The motion will occur according to the system
\[ \frac{dy_1}{d\tau}=\mu y_2,\quad \frac{dy_2}{d\tau}=y_3,\quad \frac{dy_3}{d\tau}=-a\mu y_3-\mu^2(by_2+cy_1). \tag{14} \]
Integrating system (14) approximately, we find the particular solution corresponding to the chosen initial conditions:
\[ y_1(\tau)=-\delta+\mu\left(C\delta\tau+b_0\delta\frac{\tau^2}{2!}\right)+O(\mu^2), \]
\[ y_2(\tau)=C\delta+b_0\delta\left(\tau-\frac{\mu a\tau^2}{2}\right)+O(\mu^2), \tag{15} \]
\[ y_3(\tau)=b_0\delta-\mu ab_0\delta\tau+O(\mu^2). \]
Carrying out simple calculations, we find that after the time
\[ \tau_1=\frac{2}{\sqrt{\mu b_0}}+O(\mu^0) \]
the representative point will reach the plane \(y_1=\delta\) at the point
\[ M_1=\left(\delta;\;2\delta\sqrt{\frac{b_0}{\mu}}+O(\mu^0);\; b_0\delta+O(\sqrt{\mu})\right). \]
Let us note that at the point \(M_1\) the rates of change of the coordinates \(y_1(\tau)\), \(y_2(\tau)\), \(y_3(\tau)\) are quantities of orders \(O(\sqrt{\mu})\), \(O(\mu^0)\), \(O(\mu)\), respectively; that is, the coordinate \(y_2(\tau)\) changes most rapidly, then the coordinate \(y_1(\tau)\), and the “slowest” coordinate is \(y_3(\tau)\). From the point \(M_1\) the representative point continues its motion under the linear system
\[ \frac{dy_1}{d\tau}=\mu y_2,\quad \frac{dy_2}{d\tau}=y_3,\quad \frac{dy_3}{d\tau}=-\mu a y_3+O(\mu^2)-y_1+\delta \tag{16} \]
until it reaches the surface \(\sigma_1^+=0\). Approximately integrating system (16) and taking the point \(M_1\) as the initial point, we find the time of transition of the representative point from the point \(M_1\) to the point \(M_2\) on the surface \(\sigma_1^+=0\):
\[ \tau_2=\sqrt{2}\,\sqrt[4]{b_0\mu^{-1}}+O(\mu^0)\quad \text{as } \mu\to 0. \]
The point \(M_2\) will have the coordinates:
\[ \left(-\delta+2\sqrt{2}\,\delta\, b_0^{3/4}\mu^{1/4}+O(\sqrt{\mu}),\right. \]
\[ \left.2\delta b_0^{1/2}\mu^{-1/2}-2\sqrt{2}\,\delta b_0^{5/4}\mu^{-1/4}+O(\mu^0),\; -b_0\delta+O(\sqrt{\mu})\right). \]
It is easy to see that the transition from the point \(M_1\) to the point \(M_2\) will occur along a trajectory of fast motion. Let us note that, after reaching the surface \(\sigma_1^+=0\), the quantity \(\bigl[|y_1|-\delta\bigr]\) has order \(O(\mu^{1/4})\), while the coordinate \(y_3(\tau)\) has order \(O(\mu^0)\). Hence the term \(\Psi\bigl(|y_1|\operatorname{sign}\sigma_1\bigr)\) becomes dominant in the third equation of system (13). As a result, the surface \(\sigma_1=0\) becomes a sliding surface, and the subsequent motion will take place in the sliding mode. In this case the coordinate \(y_1(\tau)\) will increase and attain its maximum value at a certain point \(M_3\), where the coordinate \(y_2(\tau)\) vanishes. Solving the equations of the sliding mode
\[ \frac{dy_1}{d\tau}=\mu y_2,\quad \frac{dy_2}{d\tau}=-b_0y_1,\quad y_3(\tau)=-b_0y_1(\tau) \]
with the initial conditions at the point \(M_2\), we determine \(\tau_3\), the time of transition of the representative point from position \(M_2\) to position \(M_3\), and the magnitude of the maximum deviation of the coordinate \(y_1\):
\[ \tau_3=\frac{\operatorname{arc\,tg}2}{\sqrt{\mu b_0}}=\frac{1.1\ldots}{\sqrt{\mu b_0}}, \]
\[ y_{1\max}=\delta\left(\cos\sqrt{\mu b_0}\,\tau_3 +2\sin\sqrt{\mu b_0}\,\tau_3\right)+O(\mu^{1/4}) =\sqrt{5}\,\delta+O(\mu^{1/4}). \]
Continuing to move along the sliding surface, the representative point will pass from the point \(M_3\) to the point \(M_4\), lying on the plane of discontinuity of the surface \(\sigma_1=0\). It is not difficult to calculate that the point \(M_4\) will have the coordinates
\[ \left(\sqrt{5}\,\delta+O(\mu^{1/4});\; -C\sqrt{5}\,\delta+O(\mu^{1/4});\; -b_0\sqrt{5}\,\delta+O(\mu^{1/4})\right), \]
and the time of transition from the point \(M_2\) to the point \(M_4\) is
\[ \tau_4=\tau_3+O(\sqrt{\mu}). \]
In the region of discontinuity of the surface \(\sigma_1=0\), the representative point will perform rapid rotations about the line \(\sigma_2=\dfrac{d\sigma_2}{d\tau}=0\) and “slowly” move toward the origin; moreover, the angular velocity will decrease as the quantity \(y_1(\tau)\) decreases. The time for the coordinate \(y_1(\tau)\) to decrease from the value \(\sqrt{5}\,\delta\) to the value \(\delta\) will be approximately equal to
\[ \tau_5=\frac{2(\sqrt{5}-1)}{C\mu(\sqrt{5}+1)} =\frac{0.764}{\mu C}. \]
Thereafter the motion will occur, by virtue of system (14), analogously to the motion from the point \(\widetilde M_1\). Consequently, the trajectory of the representative point near the equilibrium position will be a limit cycle. The approximate values of the self-oscillation parameters are determined by the formulas:
\[ T_\tau=2\left[\frac{\tau_1}{2}+\tau_2+\tau_4+\tau_5+\frac{\tau_1}{2}\right] =\frac{1.528}{C\mu}+O\left(\frac{1}{\sqrt{\mu}}\right); \tag{17} \]
\[ T_t=\frac{1.528}{C}+O(\sqrt{\mu});\qquad y_{1\max}=x_{1\max}=\sqrt{5}\,\delta+O(\mu^{1/4}). \]
Obviously, the estimates (17) will be the more accurate, the smaller the value of \(\mu\).
- In conclusion, we note the following. Above, in the investigation of various nonlinearities, two changes of coordinates, (2) and (12), were used. In this, the choice of one or the other substitution was determined by the desire to separate the motions into fast and slow components in such a way that the simplified system would preserve the specific character of the complete system due to the particular nonlinearity. In other words, with a properly chosen substitution, the nonlinearities under investigation, upon the formal substitution \(\nu,\mu=0\), must not lose their physical meaning. In our case, when the nonlinearities have the form \(F(x_k,\dot x_k)\), this condition was fulfilled by choosing such a substitution for which \(x_k=y_k\).
References
-
Barbashin E. A., Gerashchenko E. I. Differential Equations, 1, No. 1, 25–32, 1965.
-
Andronov A. A., Vitt A. A., Khaikin S. E. Theory of Oscillations. Fizmatgiz, 1959.
Received by the editors
April 16, 1965
Sverdlovsk Branch of the Mathematical Institute
named after V. A. Steklov