On nonlinear dynamic of a non-ideal Duffing system with fractional damping

Fractional damping is appearing in different contexts in any systems with memory and hysteresis. Such damping is defined by a fractional derivative term, in contrary to classical viscous damping which takes into account the first order derivative. In this work, we characterize the nonlinear dynamics of a non-ideal Duffing system, with fractional damping using nonlinear dynamical tools. The non-ideal excitation originates from a DC electric motor with limited power supply driving an unbalanced rotating mass. The response of the system is investigated with the voltage as a control parameter. Numerical simulations show the occurrence of regular and non-regular motions, which are investigated via bifurcation diagrams and phase plane portraits.


Introduction
The study of problems that involve the coupling of several systems was explored widely in the last years in function of the change of constructive characteristics of the machines and structures.In this way, some phenomena are observed in dynamical systems composed by supporting structures and rotating machines, where the unbalancing of the rotating parts is the main cause of vibrations.In the study of these systems, for a more realistic formulation one has to consider the action of an energy source with limited power (non-ideal), that is, to consider the influence of the oscillatory system on the driving force and vice versa.Recently a number of non-ideal vibrating systems has been studied, for some examples, see [1][2][3], besides others.The above mentioned results consider a nonlinear stiffness, but in all of them the damping element of the structure exerts a linear viscous damping force.However, many materials show a behaviour that differs from a linear viscous damper.Different models for viscoelastic behaviour applying the concept of fractional derivatives were discussed [4].Fractional calculus have been applied in several fields like as physics, chemistry, finances, and bioengineering [5][6].In Ref. [7], it is shown that the fractional order duffing excited by a ideal system can produce a chaotic attractor.The idea in this work is study the nonlinear dynamic of a non-ideal Duffing system with fractional damping.The figure 1 shows the non-ideal fractional oscillator.The vibrating system under consideration consists of a main structure of mass 1 m , constrained by a linearly viscous element with damping coefficient μ and a pseudoelastic SMD.The structure is also connected with a non-ideal DC motor, with driving rotor of moment of inertia M J and an unbalanced mass 0 m with eccentricity r.
The electrical scheme of the DC motor is shown in Fig. 1 The non-dimensional fractional derivative is defined as: . In this way is possible to rewrite the Eq. 1 in the dimensionless form:

governing equations
The corresponding equations that govern the angular motion φ of the DC motor are typically written in the form [8]. The equations of motion are given by Here the non-dimensional quantities are The damping device exerts a force d F given by and is thus proportional to the νth time derivative of the relative displacement.For is a nondimensional coefficient for the damping force, and is possible to note that when ν = 1 the coefficient ( ) that is a classical nondimensional coefficient for the damping force.derivative of the relative displacement.For ν = 1 the force is a linear viscous damping force and for ν = 0 the force is a linear restoring force.It is assumed that 0 ≤ ν ≤ 1 and in the interval 0 < ν < 1 the device contributes with both stiffness and damping.c is the coefficient and has the dimension mass•time ν−2.

Numerical Results
The numerical simulation of the vibrating system was carried out for parameters of the DC motor and mechanical parameters.
Numerical dimensionless In order to characterize quantitatively the attractors involved in this study, we computed the 0-1 test proposed by [9].In are vectors.Finally, the median is taken of K -values (Eq.7) corresponding to 100 different values.
Figure 2 shows the bifurcation diagram for the non-ideal oscillator in terms of the control parameter 1 U .For low values of it we have limit cycle (periodic attractors, see Fig. 3(a) and K = 0).This attractors suffer computed the 0-1 test proposed by [9].In the last years, several researches have been applied this test to characterize the attractor of the dynamical system mainly in systems where exist difficulties to calculate the Lyapunov exponents.A complete overview about theoretical aspects and applications of the 0-1 test can be found in [10].
This test yields a binary type of result: 0 for regular (periodic/quasi periodic) and 1 for chaotic signals.where c is a constant, (0, ) c π ∈ .If the system dynamics is chaotic, the motion in the ( ) , p q plane would be unbounded, while it would be bounded for regular (periodic/quasiperiodic) motion.The mean square displacement is expressed as [10] square displacement is expressed as [10]  M n c p j n c p j c N q j n c q j c For regular dynamics, it is quite likely that ( , ) M n c will be a bounded function of n but not for chaotic dynamics.From this one can define the asymptotic growth rate of the mean square displacement using covariance and variance measures:

Figure 1 :
Figure 1: (a) Non-ideal fractional oscillator and (b) the electrical schematic representation of the DC motor Fig. 3(a) and K = 0).This attractors suffer an abrupt change at 1 U ≈ 3.7, where a bifurcation occurs, and after the bifurcation, there appears chaotic motion (see Fig. 3(b) and K = 0.99)

Figure 2 :
Figure 2: Bifurcation diagram showing displacement as a function of 1 U and a linear

Figure 4 :
Figure 4: Bifurcation diagram showing displacement as a function of v and 1 = 4 U