Ìàòåìàòèêà / 4. Ïðèêëàäíàÿ ìàòåìàòèêà
Àipanov Azamat Sh.
Süleyman Demirel
University, Faculty of Economics,
Almaty, Kazakhstan
GLOBAL
ASYMPTOTIC STABILITY OF OSCILLATORY-ROTARY MOTION OF A MATHEMATICAL PENDULUM
1. Introduction. The problem
covered is a stability of the mathematical pendulum moving under the force of
gravity, viscous friction and external force with constant moment. It is
assumed that initial deviation and velocity of the pendulum may be very large,
so that it is capable of exhibiting not only oscillatory, but also rotational
motion. When the damping coefficient is sufficiently large
the pendulum's motion becomes stable due to the dissipation of energy by the
friction force. The pendulum eventually comes to a damped oscillations mode and
converges to the equilibrium point. When
is small there is a
possibility that the pendulum will start to rotate without stopping if the
external force moment acting on it prevails. Thus there is a threshold value
such that when the
damping coefficient
, the motion of the pendulum tends to its equilibrium
position with any initial conditions.
Several attempts to determine the value of made by
F. Tricomi, L. Amerio, G. Seifert, W. Hayes,
C. Böhm and other scientists in 30th – 50th years of the XX century
were not successful. Only lower and upper estimates in the forms of
and
were found. It was not
until 1954 when Japanese mathematician M. Urabe from the University of
Hiroshima proposed a numerical algorithm [1, 2] for calculating
. However, it turned out in some cases that the algorithm
doesn't converge. As for the analytical formula for finding
, the problem has not been solved yet.
The modification of Urabe's method proposed by the
author [3] has an increased rate of convergence. The results obtained here can
be used for the analysis of stability of systems with angular coordinates
(phase systems) where oscillation and rotation processes take place. These are,
for example, rotor systems in mechanics, synchronous generators in electric
power industry, phase-locked loop (PLL) systems in radio technology and others.
2. The problem
statement. The mathematical pendulum represents a rod of length , with a mass
on its end. There is
the force of gravity
and an external force
with constant moment
acting on the
pendulum. It moves in a viscous ambient where the force of friction is equal
and is directed
opposite to the velocity vector (
is a friction coefficient which depends on ambient
viscosity,
is a velocity of the
pendulum). It is assumed that the inequality
is satisfied.
The motion of the mathematical pendulum considered
here is described by the system of differential equations [4]
(1)
which can be reduced to the form
(2)
where ;
is an angle of the
pendulum's deviation from the vertical;
;
is an angular velocity; a parameter
is a damping coefficient;
is a
-periodic function;
.
On the set the function
turns to zero at two points:
at
and at
. We designate
and point out that the following
relationships are true:
,
;
,
;
,
.
The stationary set of the system (1) on
the plane
consists of the stable
equilibrium points (
,
) and the unstable equilibrium points (
,
), where
.
Let's denote the solution of the system (1) with the
initial conditions ,
as the functions
,
, where
.
D e f i n i t i o n . The system (1) is called globally asymptotically stable, if its
solution (,
) for any initial conditions
,
goes with
to some (stable or unstable)
equilibrium state from the set
.
We consider on the plane two special phase
trajectories:
which starts from the
unstable equilibrium point (
,
) and
which tends to the unstable
equilibrium point (
,
). The images of the phase trajectories
,
corresponding to the cases a)
, b)
and c)
are presented on the Figure 1.
T h e o r e m 1 .
(E.A. Barbashin and V.A. Tabuyeva [4, p. 70]). There exists a unique value of such that the following relationships hold
true
(Figure 1):
a) when
,
(3)
b) when
, (4)
c) when
. (5)
Thereby, as follows from Theorem 1, the two phase trajectories and
merge together:
,
, when
(Figure 1 (b)). The
criterion of the global asymptotic stability of the system (1) has a form of an
inequality
, setting the problem of determining the value of
.
3. Modification of
the Urabe's algorithm. The Urabe's original algorithm [1, 2] is based
on assumption of analyticity of the function on the segment
. But as the Figure 1 (c) suggests, when
the phase trajectory
cannot be represented as power
series on the segment
. This is the main disadvantage of the Urabe's algorithm. The
modification of this method is rather based on Theorem 1 by E.A. Barbashin
and V.A. Tabuyeva and the expansions of the functions
,
into Taylor series on the segments
,
respectively.
Step 1. Expanding the
function on the segment
into the Taylor power series
we get
(6)
where . Repeat the same procedure with
on the segment
(7)
where . Whereas
and
is a
-periodic function the following equalities hold true:
,
, moreover
,
.
Step 2. The sought-for
value of lies on the segment
, where
can be taken as any of the lower
estimations
and
can be taken as any of the
upper estimations
. Let us take the first approximation equal to
.
Step 3. We seek the
function which satisfies the equation
(8)
on the segment in the form of power series
(9)
where ,
Step 4. Plugging the
series (6) and (9) into the formula (8) we get the equation
which can be written as . Equating the coefficients
,
to zero we get the system of
equations
From this we find the values of :
Here is a positive root of the
quadratic equation
because
,
.
Step 5. Plugging the found
coefficients into the formula (9) we
calculate the value
.
Step 6. We are searching
for the function that satisfies the equation
(10)
on the segment in the form of power series
(11)
where ,
Step 7. Plugging the series
(7) and (11) into the formula (10) we get the equation
which can be reduced to . Equating the coefficients
,
to zero, we get the system
of equations
From this we find the values of :
Here is a negative root of the
quadratic equation
since
,
.
Step 8. Plugging the found
coefficients into the formula (11) we
calculate the value of
.
Step 9 (dichotomy
method). Let's consider the following three cases:
9a) If , then from (3) follows that $
, i.e. the value of
lies in the segment
. We take the next approximation equal to
.
9b) If , then from (4) follows that
, i.e. we found
– the sought-for value of
the damping coefficient. The algorithm stops.
9c) If , then from (5) follows that
, i.e. the value of
lies in the segment
. We take the next approximation equal to
.
During the computer implementation of this algorithm
the fulfillment of the condition serves as a criterion of
stoppage of the algorithm on the step 9b, where
is a sufficiently small
number which specifies the accuracy of calculations. If this inequality
doesn't hold true, on the step 9a or 9c we determine the segment
which contains
and the length of this
segment is going to be half the length of the initial segment
.
Next we repeat the steps 3-9 taking the value of instead of
; determine the segment
which contains
and select the middle point
of this segment as a next approximation
and so forth. Thus, on the
th iteration the length of
is going to be
times smaller than the
length of
, which means that it is possible to reach very high accuracy in a
reasonable amount of iterations. Furthermore, one should consider as many
summands in the expansions (6), (7), (9) and (11) as needed to achieve the
given accuracy of calculations.
4. Numerical
example. Let us look at the numerical example illustrating the efficiency of the
proposed modification of the Urabe's algorithm. We consider the equations of
the mathematical pendulum motion (1) with following values of parameters
,
,
. (12)
Using the formulas for lower and upper estimates of we can determine the
interval
in which
lies. Notably, there are
following lower estimations [4] in the form of inequality
:
(a) F. Tricomi estimate:
; (13)
(b) W. Hayes estimate:
;
(14)
(c) C. Böhm estimate:
.
(15)
There are also the upper estimates [4] in the form of :
(d) G. Seifert estimate:
;
(16)
(e) F. Tricomi estimate:
; (17)
(f) L. Amerio estimate:
. (18)
For the example considered here the following estimates were gained from
the formulas (13)-(18):
(a) ; (b)
; (c)
;
(d) ; (e)
; (f)
.
Thus, we have the estimates for the value of
. In the examined example
, so the conditions
are also satisfied.
Meanwhile it is impossible to say anything about the mutual position of
and
(what of the next
inequalities holds true:
or
?). So we are not able to say whether the system is stable or unstable
based on the estimates (13)-(18).
Computer calculations using the modified algorithm of
M. Urabe showed that . From this we can see that the criterion
is satisfied, which means
that the system (1) with the given values of parameters (12) is globally
asymptotically stable.
The calculations based on the original algorithm
carried out for the different values ,
,
, where
is the number of
terms in the Taylor series expansion, don't converge. On the other hand, the
iteration process in the modified algorithm rapidly converges to the value
, the condition
, where
, which signifies the end of the process, is fulfilled on the 24th
iteration for
.
5. Conclusion. The paper contains
the results related to the stability analysis of the mathematical pendulum: the
global asymptotic stability of the system under the condition , the theorem by E.A. Barbashin and V.A. Tabuyeva in existence and
uniqueness of the threshold value
, the upper and lower estimates of
by F. Tricomi, L. Amerio,
G. Seifert, W. Hayes, C. Böhm, and the modification of the
Urabe's method for numerical calculation
. The Urabe's original algorithm has a following drawback. When
the function
has a positive value at the
point
, while it is undefined at the point
when
. That is why the modified algorithm uses the expansion of the function
on the segment
and the expansion of the
function
on the segment
so that there is no need to calculate
. The fact that the functions are expanded into Taylor series on the
segments of smaller length also leads to the increase in accuracy of the
algorithm.
The values of on the segment
and
on the segment
(as the solutions of the
differential equations (8) and (11) respectively) can be also calculated
numerically using Runge – Kutta method.
References:
1. M. Urabe. Infinitesimal deformation of the periodic
solution of the second kind and its application to the equation of a pendulum
// J. Sci. Hiroshima Univ., Ser. A, Vol. 18, 1954. – P. 183-219.
2. M. Urabe. The least upper bound of a damping
coefficient ensuring the existence of a periodic motion of a pendulum under
constant torque // J. Sci. Hiroshima Univ., Ser. A, Vol. 18, 1955. – P. 379-389.
3. A.Sh. Aipanov. Modification of the Urabe's method //
Theses of Int. Sci. Students Conf. "VII Kolmogorov Readings", Sect.
Math., Moscow, 2007. – P. 13-14 (in Russian).
4. E.A. Barbashin, V.A. Tabuyeva. Dynamic systems with
cylindrical phase space. – Moscow: Nauka, 1969 (in Russian).