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Abstract

This article presents the formulation and a new approach to find analytic solutions for fractional continuously variable order dynamic models, namely, 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 recursive method, here we find the solution of fractional continuously variable order mass–spring–damper systems and then obtain closed-form solutions. We then present and discuss the solutions obtained in the cases with continuously variable order of damping for oscillator through graphical plots.

Keywords

Fractional continuously variable order viscoelastic oscillator viscous–viscoelastic oscillator Riemann–Liouville fractional integral Riemann–Liouville fractional derivative Mittag-Leffler function successive recursive method

Introduction

In the past few years, the fractional order physical models^1–4 have seen much attention by researchers due to dynamic behaviour and the viscoelastic behaviour of material.⁵ Thus, the fractional order model is remarkably used for describing the frequency distribution of the structural damping systems.^6–8 The motion of an N-degree-of-freedom system in viscoelastic material with respect to fractional damping has been studied by Ingman and Suzdalnitsky.⁹ In this study, Ingman and Suzdalnitsky have transferred the system into Volterra integral equation and then applied iteration method for obtaining the numerical solution. Several authors have modelled the dynamic system based on fractional calculus. Rossikhin and Shitikova¹⁰ have done analysis on viscoelastic single-mass system by considering the damped vibration. Enelund and Josefson¹¹ 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.¹² The Green function approach for finding solution of dynamic system was studied by Agrawal,¹³ which was followed by the Mittag-Leffler function proposed by Miller.¹⁴ By using fractional Green function and Laplace transform, Hong et al.¹⁵ have obtained the solution of single-degree-of-freedom mass–spring system of order $0 < α < 1$ . The analytical solution of fractional systems mass–spring and spring–damper system formed by using Mittag-Leffler function was analysed by Gomez-Aguilar et al.¹⁶ The fractional Maxwell model for viscous-damper model and its analytical solution was proposed by Makris and Constantinou¹⁷ and Choudhury et al.¹⁸

Other methods such as Fourier transform^19–21 and Laplace transform^21–24 have been proposed by researchers to find the solution of fractional damper systems. Recently, Saha Ray and Bera²⁵ used the Adomian decomposition method to determine the analytical solution of dynamic system of order one-half and proclaim that the acquired solutions coincided with the solutions obtained through eigenvector expansion method given by Suarez and Shokooh.²⁶ In this study, Adomian decomposition method have been used by Saha Ray et al. for obtaining a solution for dynamic analysis of a single-degree-of-freedom spring–mass–damper system whose damping is described by a fractional derivative of order 1/2. Naber²⁷ used Caputo approach to study linear damping system. The generalisation of linear oscillator to form the fractional oscillator has been studied by Stanislavsky.²⁸ The variable order structure is described in Laplace domain by Das.⁸ 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 continuously variable order models, in which damping are controlled by viscoelastic and viscous–viscoelastic 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 q. The obtained results have been plotted for showing the nature of oscillation, with continuously variable damping order.

The contents of this article are organised as follows. Some mathematical aspects of fractional calculus including Riemann–Liouville approach and Mittag-Leffler function have been presented in section ‘Mathematical aspects of fractional calculus’. Section ‘Basic principle of proposed successive recursive method’ presents the description of successive recursive method. The dynamic fractional continuously variable order mass–spring–damper systems have been formulated in section ‘The problem formulation for mass–spring–damper system’. The successful implementations of proposed successive recursive method for finding the analytical solutions of fractional dynamic systems have been discussed in section ‘Application of proposed successive recursive method for solution of fractional continuously variable order mass–spring–damper system’. The numerical simulations for the results as obtained have been studied in section ‘Numerical simulations and discussion’. Section ‘Conclusion’ concludes this article.

Mathematical aspects of fractional calculus

Definition (Riemann–Liouville)

There are several definitions of fractional derivative^1–4 that have been proposed in the past. Here, we review the most frequently used definitions of fractional integral, namely, Riemann–Liouville integral, which is defined as follows

J_{t}^{α} f (t) = \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} f (τ) d τ, t > 0, α \in ℜ^{+}

(1)

and the fractional derivative, namely, Riemann–Liouville derivative of order $α$ , is defined as

D_{t}^{α} f (t) = \frac{1}{Γ (m - α)} \frac{d^{m}}{d t^{m}} \int_{0}^{t} {(t - τ)}^{m - α - 1} f (τ) d τ, t > 0, α \in ℜ^{+}

(2)

where m is the positive integer, with $m - 1 \leq α < m$ .

Definition (Mittag-Leffler function)

The two-parameter generalised Mittag-Leffler function^1,2 defined by means of series expansion is as follows

E_{α, β} (z) = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (α k + β)} for α > 0 and β > 0, where z \in C

(3)

Basic principle of proposed successive recursive method

For describing the brief outline of proposed method, let us consider the differential equation in the following form

Lu + Ru = g

(4)

where L and R are the invertible linear operator and remaining of the linear part, respectively. The symbolic part described here in this section will be clear in the solutions in subsequent sections.

The general solution of equation (4) can be written as

u = \sum_{n = 0}^{\infty} u_{n}

(5)

where the complete solution of $Lu = g$ is $u_{0}$ . By using the property of invertible linear operator, we can write the equivalent expression of equation (4) as in the following form

L^{- 1} Lu = L^{- 1} g - L^{- 1} Ru

(6)

For initial value problem, we define the inverse linear operator $L^{- 1}$ for $L = D_{t}^{n}$ that is n-fold derivative operator, and its inverse that is $L^{- 1}$ will be n-fold integration operation from 0 to t. If we take $L = D_{t}^{2}$ , then we have $L^{- 1} Lu = u - u (0) - tu' (0)$ . So, equation (6) reduces to

u = u (0) + tu' (0) + L^{- 1} g - L^{- 1} Ru

(7)

and for boundary value problem, we have

u = A + Bt + L^{- 1} g - L^{- 1} Ru

(8)

where the integration constants A and B are determined from the given conditions. Let

u_{0} = u (0) + tu' (0) + L^{- 1} g

So, the general solution of equation (4) becomes

u = u_{0} - L^{- 1} R \sum_{n = 0}^{\infty} u_{n}

(9)

where

u_{0} = ϕ + L^{- 1} g, ϕ = u (0) + tu' (0)

(10)

and $ϕ$ is the solution of $Lu = 0$ , so that $L ϕ = 0$ .

Thus, we obtain the following recursive formula from equation (9)

u_{n + 1} = - L^{- 1} R u_{n}, n \geq 0

(11)

So, by using equation (11), we can find the value of $u_{1}, u_{2}, \dots$ .

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 characteristics 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,^9,29–33 with damping constant c. In this article, order of fractional derivative is taken as q, which varies continuously.

In this section, we analyse two suitable cases for linear damping of fractional continuously variable order mass–spring–damper system with single-degree-of-freedom:

Case 1. Free oscillation with viscoelastic damping;

Case 2. Forced oscillation with viscous–viscoelastic or viscoelastic–viscous damping.

Free oscillation with viscoelastic damping

First, we consider the free oscillation of fractional continuously variable order mass–spring–damper system with single-degree-of-freedom. Here, the mass m displaced from its equilibrium position and then it vibrates freely without any external force $F_{0}$ (Figure 1).

Figure 1.

A mass–spring oscillator under viscoelastic damping when no external force is applied.

For the displaced mass from equilibrium, the system experiences restoring force $F_{s}$ due to spring constant k, opposing its displacement which is given as

F_{s} = - kx (t)

(12)

and a damping force $F_{d}$ , which is viscoelastic that is described by a fractional continuously variable order derivative due to viscoelastic of damping coefficient c and given as

F_{d} = - {cD}_{t}^{q} x (t)

(13)

where q is a continuously variable fractional order viscoelastic oscillator. By Newton’s second law, due to oscillation, the free body with mass m experiences a total force $F_{Net}$ which is given as

F_{Net} = ma

(14)

where a denotes the acceleration while the mass oscillates

F_{Net} = {mD}_{t}^{2} x (t)

(15)

where $a = D_{t}^{2} x (t)$ .

The total force on the body is given as

F_{Net} = F_{s} + F_{d} + F_{0}

(16)

which is equal to

{mD}_{t}^{2} x (t) = - kx (t) - {cD}_{t}^{q} x (t) + F_{0}

(17)

We can model the above-described problem as a continuously variable order linear fractional differential equation (FDE) with viscoelastic oscillator, which is described as

{mD}_{t}^{2} x (t) + {cD}_{t}^{q} x (t) + kx (t) = F_{0}

(18)

where q is a continuously variable fractional order viscoelastic damper, and let us assume that the fractional order q is a continuously variable order, which is defined as

q = 0.5 + 0.5 \tanh (| v |)

(19)

where v is the velocity possessed by the system, that is, $v (t) = dx (t) / dt = x' (t)$ and $0 < v \leq 1$ . This continuously variable order is shown in Figure 4. In this case, the continuously variable damper order q is maintained to be a function of velocity. For demonstration of a continuously variable fractional order damping, we have assumed the function of q varying with velocity in the form (19). Therefore, with this definition of q in equation (19) at very low speeds, the order q tends to 0.5 and the equation of oscillator is with half-order damping, that is

{mD}_{t}^{2} x (t) + {cD}_{t}^{0.5} x (t) + kx (t) = F_{0}

At high speeds $v = 1$ , the fractional order is $q = 0.8807$ , and the equation of oscillator is tending towards classical integer-order damped oscillator that is described as follows

\begin{array}{l} m D_{t}^{2} x (t) + c D_{t}^{0.8807} x (t) + k x (t) \\ = F_{0}, m D_{t}^{2} x (t) + c D_{t}^{1} x (t) + k x (t) = F_{0} \end{array}

Therefore, with the oscillation process, the fractional order of viscoelastic damping changes continuously with position and time from value half to almost unity and that also changes the damping order of FDE (17). This example will be solved subsequently.

For free oscillation case, there is no external force, we take $F_{0} = 0$ . So, the governing equation (17) changes to

{mD}_{t}^{2} x (t) + {cD}_{t}^{q} x (t) + kx (t) = 0

(20)

Equation (20) can be made continuously variable order initial value problem by assigning suitable initial conditions. In this case, the continuously variable order initial value problem is well posed for the initial conditions $x (0) = 0$ and ${D_{t}^{1} x (t) |}_{t = 0} \equiv D_{t}^{1} x (0) = x' (0) = 1$ .

Forced oscillation with viscous–viscoelastic or viscoelastic–viscous damping

Consider a fractional continuously variable order mass–spring–damper system with a mass m oscillating smoothly and repeatedly about its equilibrium position and vibrating freely with external force $F_{0}$ on a variable viscous–viscoelastic or viscoelastic–viscous path of length L. Therefore, the generalised damping force, that is, $F_{d} = - {cD}_{t}^{q} x (t)$ , will be having a continuously variable order q which depends on position $x (t)$ that also depends on where at the present instant within travel length L, the system is positioned. Say we formulate the viscous–viscoelastic damping by $q = 1 / 2 [1 + (x (t) / L)^{2}]$ and viscoelastic–viscous damping by $q = 1 / 2 [1 - (x (t) / L)^{2}]$ , given in equation (21). Here, $x (t)$ denotes the displacement that is defined for a guide $| x | \leq L$ . The guide is the length where the continuously variable damping force exists, and the motion of the mass is within the guide’s limit. The guide length is thus $2 L$ , from $x = - L$ to $x = + L$ . With q as a viscoelastic–viscous or viscous–viscoelastic type of operator and defined as follows

q = {\begin{matrix} \frac{1 + {(\frac{x (t)}{L})}^{2}}{2}, if viscous - viscoelastic oscillator \\ \frac{1 - {(\frac{x (t)}{L})}^{2}}{2}, if viscoelastic - viscous oscillator \end{matrix}

(21)

For the case of viscous–viscoelastic damping, at the beginning position, that is $x (t) = \pm L$ , the system starts with a pure viscous friction with order of derivative as $q = 1$ , and here $v (\pm L) = x' (\pm L) = 0$ the velocity is low, whereas at high speed that is at $x (t) = 0$ , the damping is viscoelastic friction with order of its derivative $q = 1 / 2$ . For the case of viscoelastic–viscous damping at very low speeds, that is, at position $x (t) = \pm L$ , the order of derivative is zero, that is there is no friction, whereas at the high speed at $x (t) = 0$ , the order of the damping is half. In this article, we will solve the case with viscous–viscoelastic damping (Figure 2).

Figure 2.

A mass–spring oscillator sliding on a continuous-order guide when external force is applied.

Figure 3 gives the difference in the two cases.

Figure 3.

Viscous–viscoelastic and viscoelastic–viscous oscillators.

Suppose the body has been acted upon small impulsive force $F_{0} (t)$ , which is applied externally and is given by

F_{0} (t) = δ (t)

(22)

where $δ (t)$ is defined as $δ (t) = lim_{ε \to 0} 1 / 2 ε$ for $- ε \leq t \leq ε$ , else $δ (t) = 0$ , which is known as Dirac delta function.

So, the equation of motion is

{mD}_{t}^{2} x (t) + {cD}_{t}^{q} x (t) + kx (t) = δ (t)

(23)

The problem is well defined for $| x (t) / L | < 1$ and no information about the viscoelastic force is known beyond the limit of the guide. Equation (23) can be made continuously variable order initial value problem by assigning suitable initial conditions. The continuous-order initial value problem is well posed for the initial conditions $x (0) = 0$ and ${D_{t}^{1} x (t) |}_{t = 0} = 0$ . These are equilibrium state at the start with an impulse force $δ (t)$ .

Application of proposed successive recursive method for solution of fractional continuously variable order mass–spring–damper system

This section includes the analytic solutions (obtained via successive recursive 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 recursive method has been implemented here for finding the analytical solutions for the proposed systems.

Implementation of successive recursive method for free oscillation of mass–spring viscoelastic damping system

Consider equation (20) with initial conditions, given as

x (0) = 0 and {D_{t}^{1} x (t) |}_{t = 0} = x' (0) = 1

(24)

which tends to the equilibrium states of the proposed dynamic system at the beginning process.

Equation (20) can be written in the following form

D_{t}^{2} x (t) + \frac{c}{m} D_{t}^{q} x (t) + \frac{k}{m} x (t) = 0

(25)

where q is the continuously variable order of a viscous–viscoelastic oscillator, which is defined as in equation (19) that we rewrite again as following with discretised $t = n Δ t$

q = {0.5 + 0.5 \tanh (| v |) |}_{t = n Δ t}

(26)

Here, the q changes with the small change in time say $Δ t$ and velocity, where $n \in N$ in equation (26). This has been depicted in Figure 4. So, for each q, we have the new solution, which continuously changes throughout the oscillation period. In the successive recursive method that we will use subsequently, we use $D_{t} x (0)$ , which implies initial velocity that is ${D_{t} x (t) |}_{t = 0} \equiv {D_{t}^{1} x (t) |}_{t = 0} \equiv {dx / dt |}_{t = 0} \equiv x' (0)$ .

Figure 4.

Figure representing continuously variable viscoelastic oscillator.

By successive recursive method, equation (25) can be written as

x (t) = x (0) + t D_{t} x (0) - \frac{k}{m} L^{- 1} x (t) - \frac{c}{m} L^{- 1} D_{t}^{q} x (t)

(27)

Here, the inverse linear operator is taken as $L^{- 1} = D_{t}^{- 2}$ , that is twofold definite integration operation from $0$ to t. Equation (27) can be rewritten as

\begin{matrix} x (t) & = x (0) + t D_{t} x (0) - \frac{k}{m} D_{t}^{- 2} x (t) - \frac{c}{m} D_{t}^{- 2} (D_{t}^{q} x (t)) \\ = x (0) + t D_{t} x (0) - \frac{k}{m} D_{t}^{- 2} x (t) - \frac{c}{m} D_{t}^{- 2 + q} x (t) \end{matrix}

(28)

By using initial conditions (24), we can calculate initially for first recursion q as

\begin{matrix} q & = {0.5 + 0.5 \tanh (| v |) |}_{t = 0} \\ = 0.5 + 0.5 \tanh (1) = 0.8807 \end{matrix}

(29)

First recursion.

Therefore, equation (28) can be written as

x (t) = x (0) + t D_{t} x (0) - \frac{k}{m} D_{t}^{- 2} x (t) - \frac{c}{m} D_{t}^{- 1.1192} x (t)

(30)

So, by successive recursive method, we have the successive terms as follows

\begin{matrix} x_{0} (t) & = x (0) + t D_{t} x (0) = t \\ x_{1} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{0} (t) - \frac{c}{m} D_{t}^{- 1.1192} x_{0} (t) \\ = - \frac{k}{m} \frac{t^{3}}{Γ (4)} - \frac{c}{m} \frac{t^{2.1192}}{Γ (3.1192)} \\ x_{2} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{1} (t) - \frac{c}{m} D_{t}^{- 1.1192} x_{1} (t) \\ = \frac{k^{2}}{m^{2}} \frac{t^{5}}{Γ (6)} + \frac{2 kc}{m^{2}} \frac{t^{4.1192}}{Γ (5.1192)} + \frac{c^{2}}{m^{2}} \frac{t^{3.2384}}{Γ (4.2384)} \\ x_{3} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{2} (t) - \frac{c}{m} D_{t}^{- 1.1192} x_{2} (t) \\ = - \frac{k^{3}}{m^{3}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{3}} \frac{t^{6.1192}}{Γ (7.1192)} \\ - \frac{3 k c^{2}}{m^{3}} \frac{t^{5.3284}}{Γ (6.2384)} - \frac{c^{3}}{m^{3}} \frac{t^{4.3576}}{Γ (5.3576)} \end{matrix}

and so on. So, the solution of equation (20) for $q = 0.880797$ is given as

\begin{matrix} x (t) & = x_{0} (t) + x_{1} (t) + x_{2} (t) + x_{3} (t) \\ = t - \frac{k}{m} \frac{t^{3}}{Γ (4)} - \frac{c}{m} \frac{t^{2.1192}}{Γ (3.1192)} + \frac{k^{2}}{m^{2}} \frac{t^{5}}{Γ (6)} \\ + \frac{2 kc}{m^{2}} \frac{t^{4.1192}}{Γ (5.1192)} + \frac{c^{2}}{m^{2}} \frac{t^{3.2384}}{Γ (4.2384)} \\ - \frac{k^{3}}{m^{3}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{3}} \frac{t^{6.1192}}{Γ (7.1192)} - \frac{3 k c^{2}}{m^{3}} \frac{t^{5.2384}}{Γ (6.2384)} \\ - \frac{c^{3}}{m^{3}} \frac{t^{4.3576}}{Γ (5.3576)} \end{matrix}

(31)

Second recursion.

By differentiation, we obtain from equation (31) $v (t) = x' (t)$ , that is, velocity. Substitution of the values $k / m = ω_{n}^{2}$ , $c / m = 2 η ω_{n}^{3 / 2}$ , $ω_{n} = 2$ and $η = 0.5$ in equation (31) is done to make the equation of motion in similar lines that of a classical damped oscillator equation.^{12,16,22,25,32} We have also solved for other values of $η$ as we report in graphical plots in subsequent section. With this we have by substituting $Δ t = 0.01$ , that is, for the next time step $n = 1$ , $t = 0.001$ , and we obtain $x (0.01) = 0.0099267$ and $x' (0.01) = v (0.01) = 0.984444$ .

From expression (19) of variable order of viscoelastic element, we obtain the next value of the variable order that is

\begin{matrix} q & = {0.5 + 0.5 \tanh (| v |) |}_{t = 0.01} \\ = 0.5 + 0.5 \tanh (0.984444) = 0.877492 \end{matrix}

Here, we note that initially at $t = 0$ , we started with order of viscoelastic element with fractional order $q = 0.8807$ , and in next recursion, for $t = 0.01$ with change in time, we have the new value of fractional order of viscoelastic damper that is $q = 0.877492$ . So, equation (28) becomes with changed fractional order of damping as described below

x (t) = x (0) + t D_{t} x (0) - \frac{k}{m} D_{t}^{- 2} x (t) - \frac{c}{m} D_{t}^{- 1.122508} x (t)

(32)

So, by successive recursive method, we have

\begin{matrix} x_{0} (t) & = x (0) + t D_{t} x (0) = t \\ x_{1} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{0} (t) - \frac{c}{m} D_{t}^{- 1.122508} x_{0} (t) \\ = - \frac{k}{m} \frac{t^{3}}{Γ (4)} - \frac{c}{m} \frac{t^{2.122508}}{Γ (3.122508)} \\ x_{2} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{1} (t) - \frac{c}{m} D_{t}^{- 1.122508} x_{1} (t) \\ = \frac{k^{2}}{m^{2}} \frac{t^{5}}{Γ (6)} + \frac{2 kc}{m^{2}} \frac{t^{4.122508}}{Γ (5.122508)} + \frac{c^{2}}{m^{2}} \frac{t^{3.45016}}{Γ (4.45016)} \\ x_{3} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{2} (t) - \frac{c}{m} D_{t}^{- 1.122508} x_{2} (t) \\ = - \frac{k^{3}}{m^{3}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{3}} \frac{t^{6.122508}}{Γ (7.122508)} \\ - \frac{3 k c^{2}}{m^{3}} \frac{t^{5.45016}}{Γ (6.45016)} - \frac{c^{3}}{m^{3}} \frac{t^{4.67524}}{Γ (5.67524)} \end{matrix}

and so on. So, the solution of equation (20) for $q = 0.877492$ is given as

\begin{matrix} x (t) & = x_{0} (t) + x_{1} (t) + x_{2} (t) + x_{3} (t) \\ = t - \frac{k}{m} \frac{t^{3}}{Γ (4)} - \frac{c}{m} \frac{t^{2.122508}}{Γ (3.122508)} + \frac{k^{2}}{m^{2}} \frac{t^{5}}{Γ (6)} \\ + \frac{2 kc}{m^{2}} \frac{t^{4.122508}}{Γ (5.122508)} + \frac{c^{2}}{m^{2}} \frac{t^{3.45016}}{Γ (4.45016)} \\ - \frac{k^{3}}{m^{3}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{3}} \frac{t^{6.122508}}{Γ (7.122508)} \\ - \frac{3 k c^{2}}{m^{3}} \frac{t^{5.45016}}{Γ (6.45016)} - \frac{c^{3}}{m^{3}} \frac{t^{4.67524}}{Γ (5.67524)} \end{matrix}

(33)

Similarly by taking $ω_{n}$ and using equation (26), the fractional order q for next all recursion can be calculated. Here, $Δ t = 0.01$ , that is, the time step is 0.01 s and $n = 2, 3, 4, \dots$ .

By generalising the solution by successive recursive method, we have the following, where q is time dependent and can be written as $q_{n Δ t}$

\begin{matrix} x_{0} (t) & = x (0) + t D_{t} x (0) = t \\ x_{1} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{0} (t) - \frac{c}{m} D_{t}^{- 2 + q_{n Δ t}} x_{0} (t) \\ = - \frac{k}{m} \frac{t^{3}}{Γ (4)} - \frac{c}{m} \frac{t^{3 - q_{n Δ t}}}{Γ (4 - q_{n Δ t})} \\ x_{2} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{1} (t) - \frac{c}{m} D_{t}^{- 2 + q_{n Δ t}} x_{1} (t) \\ = \frac{k^{2}}{m^{2}} \frac{t^{5}}{Γ (6)} + \frac{2 kc}{m^{2}} \frac{t^{5 - q_{n Δ t}}}{Γ (6 - q_{n Δ t})} + \frac{c^{2}}{m^{2}} \frac{t^{5 - 2 q_{n Δ t}}}{Γ (6 - 2 q_{n Δ t})} \\ x_{3} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{2} (t) - \frac{c}{m} D_{t}^{- 2 + q_{n Δ t}} x_{2} (t) \\ = - \frac{k^{3}}{m^{3}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{3}} \frac{t^{7 - q_{n Δ t}}}{Γ (8 - q_{n Δ t})} \\ - \frac{3 k c^{2}}{m^{3}} \frac{t^{7 - 2 q_{n Δ t}}}{Γ (8 - 2 q_{n Δ t})} - \frac{c^{3}}{m^{3}} \frac{t^{7 - 3 q_{n Δ t}}}{Γ (8 - 3 q_{n Δ t})} \end{matrix}

and so on.

Thus, the cumulative solution is therefore

\begin{matrix} x (t) & = x_{0} (t) + x_{1} (t) + x_{2} (t) + x_{3} (t) + \dots \\ = t - \frac{k}{m} \frac{t^{3}}{Γ (4)} - \frac{c}{m} \frac{t^{3 - q_{n Δ t}}}{Γ (4 - q_{n Δ t})} + \frac{k^{2}}{m^{2}} \frac{t^{5}}{Γ (6)} \\ + \frac{2 kc}{m^{2}} \frac{t^{5 - q_{n Δ t}}}{Γ (6 - q_{n Δ t})} + \frac{c^{2}}{m^{2}} \frac{t^{5 - 2 q_{n Δ t}}}{Γ (6 - 2 q_{n Δ t})} \\ - \frac{k^{3}}{m^{3}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{3}} \frac{t^{7 - q_{n Δ t}}}{Γ (8 - q_{n Δ t})} - \frac{3 k c^{2}}{m^{3}} \frac{t^{7 - 2 q_{n Δ t}}}{Γ (8 - 2 q_{n Δ t})} \\ - \frac{c^{3}}{m^{3}} \frac{t^{7 - 3 q_{n Δ t}}}{Γ (8 - 3 q_{n Δ t})} + \dots \end{matrix}

The value of q is variable and that in this case depends on velocity or $x' (t)$ , and also with t given by expression (19). With rearrangement in above series solution, we can write the above series in a compact form as following

\begin{matrix} x (t) & = \sum_{r = 0}^{\infty} \frac{{(- \frac{k}{m})}^{r}}{r!} t^{2 r + 1} \sum_{j = 0}^{\infty} \frac{{(- \frac{c}{m})}^{j} (j + r)! t^{(2 - q_{n Δ t}) j}}{j! Γ ((2 - q_{n Δ t}) j + 2 r + 2)} \\ = \sum_{r = 0}^{\infty} \frac{{(- \frac{k}{m})}^{r}}{r!} t^{2 r + 1} E_{2 - q_{n Δ t}, \frac{r}{2} + 2}^{(r)} ((- \frac{c}{m}) t^{(2 - q_{n Δ t})}) \\ = \sum_{r = 0}^{\infty} \frac{{(- ω_{n}^{2})}^{r}}{r!} t^{2 r + 1} E_{2 - q_{n Δ t}, \frac{r}{2} + 2}^{(r)} ((- 2 η ω_{n}^{3 / 2}) t^{(2 - q_{n Δ t})}) \end{matrix}

(34)

where $E_{λ, μ}$ defines the Mittag-Leffler function in two parameters given by

E_{λ, μ} (y) = \sum_{j = 0}^{\infty} \frac{y^{j}}{Γ (λ j + μ)} for λ > 0 and μ > 0

and the rth derivative of the two-parameter Mittag-Leffler function is defined as

E_{λ, μ}^{(r)} (y) = \frac{d^{r}}{d y^{r}} E_{λ, μ} = \sum_{j = 0}^{\infty} \frac{(j + r)! y^{j}}{j! Γ (λ j + λ r + μ)}, (r = 0, 1, 2, \dots)

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 0.01 gives a very large time to obtain 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 recursion, which are plotted in the graphs.

Application of successive recursive method for forced oscillation of spring–mass viscous–viscoelastic damping system

Let us consider equation (23) with the homogenous initial conditions

x (0) = 0 and {D_{t}^{1} x (t) |}_{t = 0} = 0

(35)

which is at the equilibrium states of the dynamic system at the beginning process. Equation (23) can be written in the following form

D_{t}^{2} x (t) + \frac{c}{m} D_{t}^{q} x (t) + \frac{k}{m} x (t) = \frac{δ (t)}{m}

(36)

here q is the fractional continuously variable order viscous–viscoelastic oscillator, which is defined as

q = \frac{1 + {{(\frac{x (t)}{L})}^{2} |}_{t = n Δ t}}{2}

(37)

Here, the q changes with the small change in time say $Δ t$ and $n \in N$ in equation (37). So, for each $t = n Δ t$ , we have the new FDE and its solution, which continuously changes throughout the period of oscillation. By successive recursive method, equation (36) can be written as

\begin{array}{l} x (t) = x (0) + t D_{t} x (0) + L^{- 1} \frac{δ (t)}{m} \\ - \frac{k}{m} L^{- 1} x (t) - \frac{c}{m} L^{- 1} D_{t}^{q} x (t) \end{array}

(38)

Here, the inverse linear operator is taken as $L^{- 1} = D_{t}^{- 2}$ . Equation (38) can be rewritten as

\begin{matrix} x (t) & = x (0) + t D_{t} x (0) + D_{t}^{- 2} \frac{δ (t)}{m} - \frac{k}{m} D_{t}^{- 2} x (t) \\ - \frac{c}{m} D_{t}^{- 2} (D_{t}^{q} x (t)) \\ = x (0) + t D_{t} x (0) + D_{t}^{- 2} \frac{δ (t)}{m} - \frac{k}{m} D_{t}^{- 2} x (t) \\ - \frac{c}{m} D_{t}^{- 2 + q} x (t) \end{matrix}

(39)

By using initial conditions (35) and by taking unit length of regime, that is, $L = 1$ , we can calculate initially for first recursion q as

q = \frac{1 + {(\frac{x (0)}{L})}^{2}}{2} = \frac{1}{2}

(40)

First recursion.

Therefore, equation (39) can be written as

\begin{array}{l} x (t) = x (0) + t D_{t} x (0) + D_{t}^{- 2} \frac{δ (t)}{m} \\ - \frac{k}{m} D_{t}^{- 2} x (t) - \frac{c}{m} D_{t}^{- 1.5} x (t) \end{array}

(41)

So, by successive recursive method, we have

\begin{matrix} x_{0} (t) & = x (0) + t D_{t} x (0) + D_{t}^{- 2} \frac{δ (t)}{m} = \frac{t}{m} \\ x_{1} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{0} (t) - \frac{c}{m} D_{t}^{- 1.5} x_{0} (t) \\ = - \frac{k}{m^{2}} \frac{t^{3}}{Γ (4)} - \frac{c}{m^{2}} \frac{t^{2.5}}{Γ (3.5)} \\ x_{2} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{1} (t) - \frac{c}{m} D_{t}^{- 1.5} x_{1} (t) \\ = \frac{k^{2}}{m^{3}} \frac{t^{5}}{Γ (6)} + \frac{2 kc}{m^{3}} \frac{t^{4.5}}{Γ (5.5)} + \frac{c^{2}}{m^{3}} \frac{t^{4}}{Γ (5)} \\ x_{3} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{2} (t) - \frac{c}{m} D_{t}^{- 1.5} x_{2} (t) \\ = - \frac{k^{3}}{m^{4}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{4}} \frac{t^{6.5}}{Γ (7.5)} - \frac{3 k c^{2}}{m^{4}} \frac{t^{6}}{Γ (7)} \\ - \frac{c^{3}}{m^{4}} \frac{t^{5.5}}{Γ (6.5)} \end{matrix}

and so on. So, the solution of equation (23) for ${q (x (t)) |}_{t = 0} = 1 / 2$ is given as

\begin{matrix} x (t) & = x_{0} (t) + x_{1} (t) + x_{2} (t) + x_{3} (t) \\ = \frac{t}{m} - \frac{k}{m^{2}} \frac{t^{3}}{Γ (4)} - \frac{c}{m^{2}} \frac{t^{2.5}}{Γ (3.5)} \\ + \frac{k^{2}}{m^{3}} \frac{t^{5}}{Γ (6)} + \frac{2 kc}{m^{3}} \frac{t^{4.5}}{Γ (5.5)} + \frac{c^{2}}{m^{3}} \frac{t^{4}}{Γ (5)} \\ - \frac{k^{3}}{m^{4}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{4}} \frac{t^{6.5}}{Γ (7.5)} - \frac{3 k c^{2}}{m^{4}} \frac{t^{6}}{Γ (7)} - \frac{c^{3}}{m^{4}} \frac{t^{5.5}}{Γ (6.5)} \end{matrix}

(42)

Second recursion.

By substituting $Δ t = 0.01$ , that is for $n = 1$ , and considering $m = 1$ , $k / m = ω_{n}^{2}$ , $c / m = 2 η ω_{n}^{3 / 2}$ , $ω_{n} = 2$ and $η = 0.5$ in equation (42), we have $x (0.01) = 0.0100078$ . We have also solved for other values of $η$ as we report in graphical plots in subsequent section. By taking unit length, that is, $L = 1$ and $x (0.01) = 0.0100078$ , we find q for second recursion and calculated as follows

q = \frac{1 + {(\frac{x (0.01)}{L})}^{2}}{2} = 0.50005

Equation (39) can be written as

\begin{array}{l} x (t) = x (0) + t D_{t} x (0) + D_{t}^{- 2} \frac{δ (t)}{m} \\ - \frac{k}{m} D_{t}^{- 2} x (t) - \frac{c}{m} D_{t}^{- 1.49995} x (t) \end{array}

(43)

So, by successive recursive method, we have

\begin{matrix} x_{0} (t) & = x (0) + t D_{t} x (0) + D_{t}^{- 2} \frac{δ (t)}{m} = \frac{t}{m} \\ x_{1} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{0} (t) - \frac{c}{m} D_{t}^{- 1.49995} x_{0} (t) \\ = - \frac{k}{m^{2}} \frac{t^{3}}{Γ (4)} - \frac{c}{m^{2}} \frac{t^{2.49995}}{Γ (3.49995)} \\ x_{2} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{1} (t) - \frac{c}{m} D_{t}^{- 1.49995} x_{1} (t) \\ = \frac{k^{2}}{m^{3}} \frac{t^{5}}{Γ (6)} + \frac{2 kc}{m^{3}} \frac{t^{4.49995}}{Γ (5.49995)} + \frac{c^{2}}{m^{3}} \frac{t^{3.9999}}{Γ (4.9999)} \\ x_{3} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{2} (t) - \frac{c}{m} D_{t}^{- 1.49995} x_{2} (t) \\ = - \frac{k^{3}}{m^{4}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{4}} \frac{t^{6.49995}}{Γ (7.49995)} - \frac{3 k c^{2}}{m^{4}} \frac{t^{5.9999}}{Γ (6.9999)} \\ - \frac{c^{3}}{m^{4}} \frac{t^{5.49985}}{Γ (6.49985)} \end{matrix}

and so on. So, the solution of equation (23) for $q = 0.50005$ is given as

\begin{matrix} x (t) & = x_{0} (t) + x_{1} (t) + x_{2} (t) + x_{3} (t) \\ = \frac{t}{m} - \frac{k}{m^{2}} \frac{t^{3}}{Γ (4)} - \frac{c}{m^{2}} \frac{t^{2.49995}}{Γ (3.49995)} + \frac{k^{2}}{m^{3}} \frac{t^{5}}{Γ (6)} \\ + \frac{2 kc}{m^{3}} \frac{t^{4.49995}}{Γ (5.49995)} + \frac{c^{2}}{m^{3}} \frac{t^{3.9999}}{Γ (4.9999)} \\ - \frac{k^{3}}{m^{4}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{4}} \frac{t^{6.49995}}{Γ (7.49995)} - \frac{3 k c^{2}}{m^{4}} \frac{t^{5.9999}}{Γ (6.9999)} \\ - \frac{c^{3}}{m^{4}} \frac{t^{5.49985}}{Γ (6.49985)} \end{matrix}

(44)

Similarly, by taking $t = n Δ t$ and using equation (37), the fractional order q for next all recursions can be calculated. Here, $Δ t = 0.01$ , that is, the time step is 0.01 s and $n = 2, 3, 4, \dots$ .

By generalising the solution by successive recursive method (as carried out in the previous section), we have the following

\begin{matrix} x_{0} (t) & = x (0) + t D_{t} x (0) + D_{t}^{- 2} \frac{δ (t)}{m} = \frac{t}{m} \\ x_{1} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{0} (t) - \frac{c}{m} D_{t}^{- 2 + q_{n Δ t}} x_{0} (t) \\ = - \frac{k}{m^{2}} \frac{t^{3}}{Γ (4)} - \frac{c}{m^{2}} \frac{t^{3 - q_{n Δ t}}}{Γ (4 - q_{n Δ t})} \\ x_{2} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{1} (t) - \frac{c}{m} D_{t}^{- 2 + q_{n Δ t}} x_{1} (t) \\ = \frac{k^{2}}{m^{3}} \frac{t^{5}}{Γ (6)} + \frac{2 kc}{m^{3}} \frac{t^{5 - q_{n Δ t}}}{Γ (6 - q_{n Δ t})} + \frac{c^{2}}{m^{3}} \frac{t^{5 - 2 q_{n Δ t}}}{Γ (6 - 2 q_{n Δ t})} \\ x_{3} (t) & = - \frac{k}{m} D_{t}^{- 2} x_{2} (t) - \frac{c}{m} D_{t}^{- 2 + q_{n Δ t}} x_{2} (t) \\ = - \frac{k^{3}}{m^{4}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{4}} \frac{t^{7 - q_{n Δ t}}}{Γ (8 - q_{n Δ t})} \\ - \frac{3 k c^{2}}{m^{4}} \frac{t^{7 - 2 q_{n Δ t}}}{Γ (8 - 2 q_{n Δ t})} - \frac{c^{3}}{m^{4}} \frac{t^{7 - 3 q_{n Δ t}}}{Γ (8 - 3 q_{n Δ t})} \end{matrix}

and so on.

Thus, the solution is given as

\begin{matrix} x (t) & = x_{0} (t) + x_{1} (t) + x_{2} (t) + x_{3} (t) + \dots \\ = \frac{t}{m} - \frac{k}{m^{2}} \frac{t^{3}}{Γ (4)} - \frac{c}{m^{2}} \frac{t^{3 - q_{n Δ t}}}{Γ (4 - q_{n Δ t})} \\ + \frac{k^{2}}{m^{3}} \frac{t^{5}}{Γ (6)} + \frac{2 kc}{m^{3}} \frac{t^{5 - q_{n Δ t}}}{Γ (6 - q_{n Δ t})} + \frac{c^{2}}{m^{3}} \frac{t^{5 - 2 q_{n Δ t}}}{Γ (6 - 2 q_{n Δ t})} \\ - \frac{k^{3}}{m^{4}} \frac{t^{7}}{Γ (8)} - \frac{3 k^{2} c}{m^{4}} \frac{t^{7 - q_{n Δ t}}}{Γ (8 - q_{n Δ t})} - \frac{3 k c^{2}}{m^{4}} \frac{t^{7 - 2 q_{n Δ t}}}{Γ (8 - 2 q_{n Δ t})} \\ - \frac{c^{3}}{m^{4}} \frac{t^{7 - 3 q_{n Δ t}}}{Γ (8 - 3 q_{n Δ t})} + \dots \end{matrix}

The value of q is variable and that in this case depends on position $x (t)$ given by expression (37). With rearrangement in above series solution, we can write the above series in the following compact form

\begin{matrix} x (t) & = \frac{1}{m} \sum_{r = 0}^{\infty} \frac{{(- \frac{k}{m})}^{r}}{r!} t^{2 r + 1} \sum_{j = 0}^{\infty} \frac{{(- \frac{c}{m})}^{j} (j + r)! t^{(2 - q_{n Δ t}) j}}{j! Γ ((2 - q_{n Δ t}) j + 2 r + 2)} \\ = \frac{1}{m} \sum_{r = 0}^{\infty} \frac{{(- \frac{k}{m})}^{r}}{r!} t^{2 r + 1} E_{2 - q_{n Δ t}, \frac{r}{2} + 2}^{(r)} ((- \frac{c}{m}) t^{(2 - q_{n Δ t})}) \\ = \frac{1}{m} \sum_{r = 0}^{\infty} \frac{{(- ω_{n}^{2})}^{r}}{r!} t^{2 r + 1} E_{2 - q_{n Δ t}, \frac{r}{2} + 2}^{(r)} ((- 2 η ω_{n}^{3 / 2}) t^{(2 - q_{n Δ t})}) \end{matrix}

(45)

where $E_{λ, μ}$ defines the Mittag-Leffler function in two parameters, and the $r -$ derivative of two-parameter Mittag-Leffler function is defined as

E_{λ, μ}^{(r)} (y) = \frac{d^{r}}{d y^{r}} E_{λ, μ} = \sum_{j = 0}^{\infty} \frac{(j + r)! y^{j}}{j! Γ (λ j + λ r + μ)}, (r = 0, 1, 2, \dots)

Numerical simulations and discussion

The solution to the oscillator problem with continuously variable damping order q, defined in first case as viscoelastic damping via expression (26) and in second case as viscous–viscoelastic damping via expression (37) governed by FDEs $D_{t}^{2} x (t) + c / {mD}_{t}^{q} x (t) + k / mx (t) = 0$ and $D_{t}^{2} x (t) + c / {mD}_{t}^{q} x (t) + k / mx (t) = δ (t)$ , respectively, gives same type of solution in compact form expressed in equations (34) and (45), respectively. In the first case though we are not having external forcing function, the initial condition at $t = 0$ is $x (0) = 0$ and ${D_{t}^{1} x (t) |}_{t = 0} = x' (0) = 1$ , that is, the system has initial velocity of unity magnitude. In the second case, we have Dirac delta function as forcing function that is applied at $t = 0$ with system initially at rest, that is, at $t = 0$ , we have $x (0) = 0$ and ${D_{t} x (t) |}_{t = 0} = x' (0) = 0$ . 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 this 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 FDEs with continuously variable order q are as follows, with the initial conditions stated in the above sections

\begin{array}{l} D_{t}^{2} x (t) + 2 η ω_{n}^{3 / 2} D_{t}^{q} x (t) + ω_{n}^{2} x (t) = 0 \\ D_{t}^{2} x (t) + 2 η ω_{n}^{3 / 2} D_{t}^{q} x (t) + ω_{n}^{2} x (t) = δ (t) \end{array}

Here, $ω_{n}$ 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 1, the body oscillates without implementation of any external forces with the continuous change of the fractional continuously variable order viscoelastic damper q, with $q (x' (t)) = 0.5 + 0.5 \tanh (| x' (t) |) |$ . Here, we have taken $0 \leq t \leq 6$ and 600 recursions for plotting the time response of the fractional continuous-order damping system with displacement $x (t)$ and velocity $v (t)$ . Also, in the solution (34), we have taken mass $m = 1$ unit, natural or angular frequency $ω_{n} = 2 rad / s$ and the damping ratio $η = 0.05, 0.5, 1, \sqrt{π}$ .

Numerical simulation of fractional continuously variable order mass–spring–damper system for forced oscillation with viscous–viscoelastic damping

It is important to mention here that in system in case 2, oscillation takes place with the small external impulse force $δ (t)$ with the continuous change in the fractional order viscous–viscoelastic oscillator q, with $q (x (t)) = 1 / 2 [1 + {(x (t) / L)}^{2}]$ which ranges between 0.5 and 1, that is, $0.5 \leq q \leq 1$ . Here, we have taken $0 \leq t \leq 6$ and 600 recursions for plotting the time response of the fractional continuous-order damping system with displacement $x (t)$ . Also in solution (45), we have taken mass $m = 1$ unit, natural or angular frequency $ω_{n} = 2 rad / s$ and the damping ratio $η = 0.05, 0.5, 1, \sqrt{π}$ .

Figures 5 and 7 represent the displacement–time graphs for cases 1 and 2, and Figure 6 presents the velocity–time graph (for case 1) of the fractional continuously variable order mass–spring–damper models for the different values of damping ratio $η$ . Figures 5 and 7 clearly bring out the difference in oscillatory character for the two different cases of continuously variable fractional derivative order in the system, for higher values of $η >> 0$ . Case 1 shows the presence of abrupt change in rate of position such as a kink, whereas case 2 has rather smooth decaying oscillation. It is noted here that the initial position state that is $x (t)$ with increase in time tends to asymptotic with the equilibrium state that is $x = 0$ . From the above figures, we observe the following:

When $η = 0.05$ or near to or equal zero, the system oscillates at its natural frequency $ω_{n}$ and the system is called ‘undamped’. That means the oscillation will continue almost forever like simple harmonic motion.

When $η = 0.5$ , 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 $ω_{d}$ to $ω_{n}$ in the case of continuously variable order damping case is to be developed; like we have in for integer-order systems, that is, given by $ω_{d} = ω_{n} \sqrt{1 - η^{2}}$ .

When $η = 1$ , 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.

When $η = \sqrt{π}$ , the system oscillates a little as 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, for which we have developed needs to be redefined with respect to the variable fractional order q of the system. But, here we have drawn similarity with the classical integer-order damped oscillator system.

Figure 5.

The displacement–time graph for fractional continuous-order spring–mass damper model for free oscillation with viscoelastic damping plots.

Figure 6.

The velocity–time graph for fractional continuous-order spring–mass damper model for free oscillation with viscoelastic damping plots.

Figure 7.

The displacement–time graph for fractional continuous-order mass–spring–damper model for forced oscillation with viscous–viscoelastic damping plots.

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 in time $Δ t$ with respect to both viscoelastic and viscous–viscoelastic oscillator of order q. We also find the analytical solutions of the fractional continuously variable order mass–spring–damper systems by the successive recursive 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, namely, undamped, under-damped, critical damping and over damping of the system, similar to those for integer-order classical damped oscillator system; however, 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 the study of various physical dynamic systems.

Footnotes

Academic Editor: Anand Thite

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: This research work was financially supported by BRNS, Department of Atomic Energy, Government of India vide Grant No. 2012/37P/54/BRNS/2382 for Research Project ‘Applications of Analytical Methods for the solutions of Generalized Fractional and Continuous order Differential Equations with the implementation in Computer Simulation’.

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

Abstract

Keywords

Introduction

Mathematical aspects of fractional calculus

Definition (Riemann–Liouville)

Definition (Mittag-Leffler function)

Basic principle of proposed successive recursive method

The problem formulation for mass–spring–damper system

Free oscillation with viscoelastic damping

Forced oscillation with viscous–viscoelastic or viscoelastic–viscous damping

Application of proposed successive recursive method for solution of fractional continuously variable order mass–spring–damper system

Implementation of successive recursive method for free oscillation of mass–spring viscoelastic damping system

Application of successive recursive method for forced oscillation of spring–mass viscous–viscoelastic damping system

Numerical simulations and discussion

Numerical simulation of fractional continuously variable order mass–spring–damper system for free oscillation with viscoelastic damping

Numerical simulation of fractional continuously variable order mass–spring–damper system for forced oscillation with viscous–viscoelastic damping

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References