Formulation and solutions of fractional continuously variable order mass spring damper systems controlled by viscoelastic and viscous-viscoelastic dampers

The article presents the formulation and a new approach to find analytic solutions for fractional continuously variable order dynamic models viz. Fractional continuously variable order mass-spring damper systems. Here, we use the viscoelastic and viscous-viscoelastic dampers for describing the damping nature of the oscillating systems, where the order of fractional derivative varies continuously. Here, we handle the continuous changing nature of fractional order derivative for dynamic systems, which has not been studied yet. By successive iteration method, here we find the solution of fractional continuously variable order mass-spring damper systems, and then give a close form solution. We then present and discuss the solutions obtained in the cases with continuously variable order of damping for this oscillator with graphical plots.


Introduction
In past few years, the fractional order physical models [1,2] have seen much attention by researchers due to dynamic behaviour and the viscoelastic behaviour of material [3]. Thus the fractional order model is remarkably used for describing the frequency distribution of the structural damping systems [4][5][6][7]. Several authors have modelled the dynamic system based on fractional calculus. Rossikhin and Shitikova [8] have done analysis on viscoelastic single-mass system by considering the damped vibration. Enelund and Josefson [9] used finite element method for analysis of fractionally damped viscoelastic material. The exact solution of fractional order of 1/2 was obtained by Elshehawey et al. [10]. The Green function approach for finding solution of dynamic system was studied by Agrawal [11], which followed by the Mittag-leffler function proposed by Miller [12]. By using fractional Green function and Laplace transform, Hong et al. [13] has obtained the solution of single-degree freedom mass-spring system of order 1 0 < < α . The analytical solution of fractional systems massspring and spring-damper system formed by using Mittag-Leffler function was analysed by Gomez-Aguilar et al [14]. The fractional Maxwell model for viscous-damper model and its analytical solution was proposed by Makris and Constantinous [15], Choudhury et al. [16].
Other methods like Fourier transform [17][18][19] and Laplace transform [19][20][21][22] have been proposed by researchers to find the solution of fractional damper systems. Recently Saha Ray et al. [23] used the Adomian decomposition method to determine the analytical solution of dynamic system of order onehalf and proclaim that the acquired solutions coincided with the solutions obtained through eigenvector expansion method given by Suarez and Shokooh [24]. Naber [25] used Caputo approach to study linear damping system. The generalization of linear oscillator to form the fractional oscillator has been studied by Stanislavsky [26]. The variable order structure is described in Laplace domain by Das [7]. Here in this article we extend the concept of continuously variable order structure of differential equation and get time domain solution.
The objectives of this article are first the mathematical formulation of fractional continuously variable order spring-mass damping systems and then analysing approximate analytical solution of fractional continuous-variable order models,in which damping are controlled by viscoelastic and viscousviscoelastic dampers. Due to dynamic varying nature of fractional order derivative of damper material, it is very difficult to obtain the analytic solutions of the system. The solutions for fractional continuously variable dynamic models have been newly studied in this article. The linear damping natures of the systems have been taken here for modelling the problems. The changing property of the guide, on which the motion takes place, results in oscillation of the systems, which are modelled here by fractional continuously variable-order . The obtained results have been plotted for showing the nature of oscillation, with continuously variable damping order. q The contents of the paper are organised as follows. Some mathematical aspects of fractional calculus including Riemann-Liouville approach and Mittag-Leffler function have been presented in Section 2. Section 3 presents the algorithm of successive iteration method. The dynamic fractional continuouslyvariable order mass-spring damping systems have been formulated in Section 4. The successful implementations of proposed successive iteration method for finding the analytical solutions of fractional dynamic systems have been discussed in Section 5. The numerical simulations for the results as obtained have been studied in Section 6. Section 7 concludes the paper.

2.
Mathematical aspects of fractional calculus

Definition (Riemann-Liouville):
There are several definitions of fractional derivative [1,2] that have been proposed in past. Here we review the most frequently used definitions of fractional integral viz. Riemann-Liouville integral, which is defined as follows: α is defined as and the fractional derivative viz.Riemann-Liouville derivative of order where is positive integer, with m m m < ≤ − α 1 .

Definition (Mittag-Leffler function):
The two-parameter generalized Mittag-Leffler function [1,2] defined by means of series expansion is as follows:

Basic principle of proposed successive iteration method
For defining the brief outline of proposed method, let us consider the differential equation in the following form g Ru Lu = + (3.1) where and L R are invertible linear operator and remaining of the linear part respectively. The detailing of this method is described in Das [7], and Saha Ray et al [23], and the symbolic part described here-in this section will be clear in the solutions in subsequent sections. The general solution of the eq. (3.1) can be written as where the complete solution of is . By using the property of invertible linear operator, we can write the equivalent expression of eq. (3.2) as following form: For initial value problem, we define the inverse linear operator for that is n-fold derivative operator, and its inverse that is 1 − L w be n-fold integration operation from 0 to t. If we take where the integration constants A and B are determined from the given condition.
So the general solution of eq.

The problem formulation for mass-spring damper system
Damping is defined as restraining of vibratory or oscillatory motion; that means it reduces, restricts and prevents the oscillation of an oscillatory system. When the damping force is viscoelastic, it has both viscous and elastic characteristic to prevent or damp the oscillation of the system. When the system attains a pure viscous friction at high speed and viscoelastic friction at low-speed the damping force is called viscous-viscoelastic. Similarly, when the system attains a pure viscoelastic friction at high speed and viscous friction at low-speed the damping force is called viscoelastic-viscous. The damping force is expressed in the form of fractional derivative of position [7,[27][28][29][30], with damping constant . Here in this article order of fractional derivative is taken as , which varies continuously. c q 4.1.

Free oscillation with viscoelastic damping
Firstly, we consider the free oscillation of fractional continuously system with single-degree freedom. Here the mas " -variable order mass-spring damper s " displaced from its equilibrium position and then it vibrates freely without any external force where " " denotes the acceleration w The total force on the body is given as F (4.1.5) which is equal to We can model the above described problem as differential equation (FDE) with viscoelastic oscillato v f damping coefficient c and given as: where q is a continuously variable fractional order viscoelastic oscillator. B e Newton's secon law, due to os ody with mass " m " experiences a total force Net F which is given as hile the mass oscillates.
a continuous-variable order linear fractional r, which is described as where is a continuously variable fraction is continuously variable order be defined as q fractional order al order viscoelastic damper, let us assume that the q ( ) 0.5 0.5 tanh q v = + (4.18) here v is velocity possessed by the system i.e.
. This continuously variable order is shown in figure-4. In this case, the continuous-variable damper order that is q is Therefore with the oscillation process the fractional orde viscoelastic damping changes continuously with position, time from value half to almost unit that also changes the damping order of frac r of y, and tional differential equation (4.1.6). This example we will solve subsequently.
For free oscillation case there is no external force, we take 0 0 = F . So the governing eq. (4.1.6) changes to The eq. (4.1.9) can be made continuously-variable order initial value problem by assigning suitable ed for initial conditions. In this case, the continuous-variable-order Initial value problem is well pos the initial conditions 0 ) 0 ( and = x

Forced oscillation with viscous-viscoelastic or viscoelastic-viscous damping.
Consider a fractional continuously mass-spring damper system with a mass " m " cillating smoothly and repeatedl out its equilibrium position and vibrating freely with external rce 0 F on a variable viscous-viscoelastic or viscoelastic-viscous path of length L . Therefore the generalized d ( ) , will be having a conti usly variable order q wh on position ( ) h is x t that is also depending on where at the present instant with-in l length L , the system is positioned. Say we formulate the viscous-viscoelastic damping by , and viscoelastic-viscous damping by For the case of viscous-viscoelastic damping, at the beginning position, that is ( ) x t = ±L the system starts with a pure viscous friction with order of derivative as 1 q = , and here the velocity is low; whereas at high speed that is at = the damping is viscoelastic friction with order of its derivative For the case of viscoelastic-viscous damping at very low speeds i.e. at position ( ) x t = ± ( ) 0 x t = L , the order of derivative is zero, that is there is no friction, whereas at the high speed at , the order of the damping is half. In this article we will solve the case with viscousviscoelastic damping.

Fig 2: A mass-spring oscillator sliding on a continuous order guide when external force is applied
The following figure gives the difference in the two cases

Fig 3: viscous-viscoelastic and viscoelastic-viscous oscillators
Suppose body has small impulsive force which is applied externally which is denote as and given as The problem is well defined for ( ) 1 x t L < and no information about the viscoelastic force is known beyond the limit of the guide. The eq. (4.2.3) can be made continuous-variable order initial value problem by assigning suitable initial conditions. The continuous-order initial value problem is well posed for the initial conditions and These are rest condition at the start with an impulse force ) (t δ .

Application of proposed successive iteration method for solution of fractional continuously-variable order mass-spring damper system
The present section includes the analytic solutions (obtained via successive iteration method) for fractional continuously-variable order spring-mass damper systems for free oscillation with viscoelastic damping and force oscillation with viscous-viscoelastic damping. The successive iteration method has been implemented here for finding the analytical solutions for proposed systems.

Implementation of successive iteration method for free oscillation of mass-spring viscoelastic damping system
Consider eq. (4.1.9) with initial conditions, given as which tend to the equilibrium states of the proposed dynamic system at the beginning process. The equation (4.1.9) can be written in the following form here is the continuously-variable order of a viscous-viscoelastic oscillator, which is defined as in (4.1.8) that we re-write again as following with q t n t = Δ .
Here the changes with the small change with time say q t Δ with velocity and N n ∈ in (4.1.8), depicted in figure-4. So for each we have the new solution, which continuously changes throughout the oscillation period. In the successive iteration method that we will use subsequently, we use symbol

Fig 4: Figure representing continuously variable viscoelastic oscillator
By successive iterative method eq. (5.1.2) can be written as Here the inverse linear operator is taken as , that is 2-fold definite integration operation from to t .The eq. (5.1.4) can be rewritten as By using initial conditions (5.1.1), we can calculate initially for first iteration as q We use the Euler's formula for fractional integration of a power-function that is So by successive iteration method, we have the following iterations   By generalizing the solution by successive iteration method, we have the following, where q is time dependent too, call that n t The value of is variable and that in this case depends on velocity or q ( ) x t ′ , and also with t given by expression (4.1.8). With rearrangement in above series solution we can write the above series as following compact expression as following The choice of time step of 0.01 is for convenience. Ideally it should be as small as possible. A smaller value of time step that 0.01 gives a very large time to obtain the the solution in computer; and a larger value of time step gives inaccurate results. The idea is to simulate the results for a continuously variable order; and we found the 0.01 time-step to be convenient for our 600 steps iteration, which are plotted in the graphs.

Application of successive iteration method for forced oscillation of spring-mass viscous-viscoelastic damping system
Let us consider eq.
Here the inverse linear operator is taken as .The eq. (5.2.4) can be rewritten as By using initial conditions (5.2.1) and by taking unit length of regime i.e. 1 = L , we can calculate initially for first iteration as q ( ) First iteration: Therefore equation (5.2.5) can be written as So by successive iteration method, we have So by successive iteration method, we have m t m The value of q is variable and that in this case depends on position ( ) x t given by expression (5.2.3).
With rearrangement in above series solution we can write the above series as following compact expression as following In the second case the forcing function starts the oscillations. Both are cases of free running damped oscillator, with continuously varying order of damping, that is q throughout the oscillation period.
In the present section, the displacement-time graphs have been presented for fractional continuously variable order mass-spring damper systems for free oscillation with viscoelastic damping and forced oscillation with viscous-viscoelastic damping. Thus the two fractional differential equations with continuous variable order are following; with the initial conditions stated as above sections.
Here n ω is called the natural or angular frequency and η is called the damping ratio of the system.

Numerical simulation of fractional continuously-variable order mass-spring damper system for free oscillation with viscoelastic damping
For case-one, the body oscillates without implementation of any external forces with the continuous change of the fractional continuously variable order viscoelastic damper q , with    When 05 . 0 = η or near to or equal zero, the system oscillates at its natural frequency n ω and the system is called 'un-damped'. That means the oscillation will continue almost forever, like simple harmonic motion. ii. When 5 . 0 = η , the system oscillates at higher than the natural frequency and the system is called 'under-damped', like classical damped integer order oscillator. In this situation the oscillation gradually tends to zero. However relation of damped natural frequency say d ω to n ω in the case of continuously variable order damping case is to be developed; like we have in for integer order systems i.e. given by iii. When 1 = η , the system oscillates quickly and it converges to zero as quickly as possible and the system is similar as called 'critical damped', like in integer order damped oscillators. In this situation the oscillation returns to equilibrium in the shortest period of time. iv. Whenη = π , the system oscillates a littleas compared to critical damping and it converges to zero slowly and the system is similar as called 'over damped', like in integer order damped oscillators. We point out here, the definitions regarding natural frequency, damped-frequency, under-damped oscillation, critically-damped oscillation, and over-damped oscillations in the continuously variable fractional order damped oscillator, that we have developed needs to be re-defined with respect to the variable fractional order of the system. But here we have drawn similarity with the classical integer order damped oscillator system. q

Conclusion
In this article, we modelled the fractional continuously-variable order mass-spring damper systems for free oscillation with viscoelastic damping and forced oscillation with viscous-viscoelastic damping. The approach is new in the sense of changing of behaviour of guide continuously with the small change of time with respect to both viscoelastic and viscous-viscoelastic oscillator of order . We also find the analytical solutions of the fractional continuously-variable order mass-spring damper systems by the successive iteration method. The graphical plots also have been presented for different values of damping ratio. From the graph we have given the conclusion for the nature of damping viz. undamped, underdamped, critical damping and over damping of the system, similar to those for integer order classical damped oscillator system, however, though these parameters in context of continuously variable order damping oscillator systems need to be developed. From this new method developed in this article we conclude that the proposed method is highly effective for finding the solution for the fractional dynamic model, where the fractional order of damping changes continuously. This development has immense potential in study of various physical dynamic systems.