*112035*
Volkova S.A., Podgornaya L.V., Kuzicheva E.V.
GVUZ «UDHTU», Dnepropetrovsk
A NON-LINEAR SYSTEM USING A NON-SMOOTH TEMPORAL TRANSFORMATION
Solutions
of differential equations of motion for mechanical systems with periodic
impulsive excitation are represented in a special form which contains a
standard pair of non-smooth periodic functions and possesses the structure of
an algebra without division. This form is also suitable in the case of
excitation with a periodic series of discontinuities of the first kind. All
transformations are illustrated on the Duffing oscillator under a parametric
non-equidistant pulsed forcing with a dipole-like shift of the impulses,
although the technique can be applied to more general cases. An explicit form
of analytical solutions has been obtained for periodic regimes. These solutions
and numerical simulations indicate a principal role of the impulses' shift.
Namely, the system performs periodic, multiperiodic and stochastic-like
dynamical regimes if varying a parameter of the shift. The analytical approach
is based on the limit of linear system under equidistant distribution of the
impulses and asymptotically takes into account the dipole-like shift and
non-linearity.
The
representation is based on a proposition that an arbitrary periodic function x(t) (the period of which is normalized to four)
can be expressed as
(1)
wheret is the saw-tooth piecewise-linear
function of argument t and period equal to four; 
(-1<
<1)
is a parameter characterizing the slope of the "saw". Note that the
period has been normalized to four, not to 2
. A
convenience of this choice is going to be discussed later. The second one
consists of the Dirac functions 
.
A FORMULATION OF THE PROBLEM
In
order to illustrate the transformations consider the Duffiing oscillator under
a parametric pulsed excitation. Define a coordinate of the system x=x(t),which is described by the following
differential equation of motion
(2)
where
is a small parameter; the dot denotes
differentiation with respect to time, t;pand qare constant parameters. The parametric pulsed excitation is expressed
by means of the generalized second derivative of a saw-tooth piecewise-linear
function 
of argument tand period equal to four. 
is a parameter characterizing the slope of
the "saw".
THE NON-SMOOTH TRANSFORMATION OF THE SYSTEM
It
was mentioned above, that system (2) will be transformed first in order to
eliminate the singular Dirac's functions. The case in point is the non-smooth
temporal transformation in a manifold of periodic regimes. To provide this
transformation, a periodic solution with the period T=4is represented due to (1) by the following
expression [1-3].
Note that the derivative 
is a piecewise constant function. Hence, it
can be verified that e2 = 
, where
and 
In the symmetric case (
the
relation reduces to equation 
.
Another relation which will be taken into account is 
(3)
The first formal
derivative is 
=
(4)
Substituting (1) into (2) and taking into account (3)
and (4) gives
(5)
(6)
The
related periodic solutions are Àî = ±Ñ0 = À, (7)
Expression
(7) indicates a branching of curves p=p(q)on a plane of the system parameters, pq, when a non-zerothslope (
) of the saw-tooth function appears. The
branching generate instability regions on the parameters plane and strongly
affects on the system dynamics.
CONCLUSIONS
In
this paper the non-smooth temporal transformation has been applied to construct
a family of periodic solutions of a weakly non-linear system under the
parametric impulsive excitation. The transformation eliminates singular terms
and reduces in the equation of motion to a standard weakly non-linear boundary
value problem. To solve this problem asymptotic expansions were applied. As a
result explicit form analytical solutions in terms of elementary functions have
been obtained for small asymmetry of the distribution of impulses' sequences,
(the dipole-like shift of each two neighboring impulses). The solutions and
numerical simulations show a principal role of the shifts of the impulses'
sequences.
REFERENCES
1.
V.
N. PilipchukTemporal transrmations and visualization diagrams for non smooth
periodic motions // International Journal of Bifurcation and Chaos, Vol.15,
No.6 . – 2005.
2.
V.
N. PilipchukA Non-Linear system using a
non-smooth temporal transformation // Journal of Sound and Vibration. – 1999.
-307-327.
3.
V.
N. PilipchukNon-smooth time decomposition for non linear models
Driven by random impulses // Chaos, Solitons and fractals 14 2004 129-143