# Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}, J

_{2}—moment of inertia; M

_{1}, M

_{2}—torque; r1 and r2 are the roller radii; K, D—coefficient of elasticity and the damping constant of belt material; F—tension force; ω

_{1}and ω

_{2}—angular velocity of the first and second mass; M—electromagnetic torque; M

_{c}—load moment.

_{c}. Additional inertial links are also fixed at both ends of the shaft, J

_{EM}, J

_{N}. Ignoring the external viscous friction, the equation of the long shaft (Figure 3) will be as follows:

_{p}—polar moment of the shaft’s inertia; G—modulus of elasticity of the second type (shear modulus); ξ—coefficient of internal viscous friction.

_{j}—angular speed of j-th section of the shaft; φ

_{j}—rotation angle of j-th section of the shaft.

_{f}

_{1}and a

_{f}

_{2}—external viscous friction factors; ${J}_{1}={J}_{EM}+0.5\cdot \rho \cdot {J}_{\rho}\cdot L$ and ${J}_{2}={J}_{N}+0.5\cdot \rho \cdot {J}_{\rho}\cdot L$—moments of inertia of the first and second mass, respectively; L—shaft length; M—electromagnetic torque; M

_{c}—moment of the load; M

_{12}—elastic moment.

## 2. Model of a Two-Mass System with Concentrated Parameters

_{1}= J

_{2}= J, a

_{f}

_{1}= a

_{f}

_{2}= 0 and M = M

_{c}, equations for determining state variables were rewritten as:

^{3}; = 8.1 × 10

^{10}Nm; ξ = 0.5 Nm

^{2}; L = 4.5 m; D = 0.05 m; J

_{EM}= J

_{N}= 20 Nm

^{2}; Δx = 0.05 m. A comparative analysis of dependences of change in the velocity of the first mass (Figure 5) as a function of time, obtained using models with distributed and concentrated parameters, suggests that the model with concentrated parameters does not adequately reflect the processes in the system with an elastic shaft. The model with concentrated parameters inadequately addresses the velocity of the mechanical wave, which leads to inaccuracies in determining the actual parameters of the system. In particular, the calculation results in a delay angle δ, which reflects the difference between the real system and its prototype, obtained on the basis of a model with concentrated parameters.

_{12}(0) equals zero, it will become (10):

^{*}= β + (1 + α)c

_{12}in the classic two-mass system model. An equivalent model for determining state variables will take the following form (11).

_{1}= J

_{2}= J, a

_{f}

_{1}= a

_{f}

_{2}= 0 and M = M

_{c}, equations for determining state variables will become (12).

## 3. Analysis of the Two-Mass System Model

_{f1}= a

_{f2}= 0, the velocity transfer functions of the first and second mass electromagnetic torque and moment load will be:

_{1}= J

_{2}, between the traditional two-mass model of the system and the model in which the Caputo–Fabrizio operator is used to describe the fractional order integral in the elastic moment equations shows a slight difference between these models. Such a small distance between the models of the system allows for the assumption that choice of a model for the control system parameters synthesis will not have a decisive influence.

## 4. Synthesis of the Control System

_{1}= (1/J

_{1;}0; 0)

^{T}—vector of control influence; C = (0; 0; 1)—a vector that determines the generalized coordinate in the feedback linearization method; ρ—relative degree of the system; $\left[\begin{array}{ccc}{k}_{0}& {k}_{1}& \begin{array}{cc}\cdots & {k}_{\rho -1}\end{array}\end{array}\right]$ is a vector of coefficient that determines the desired location of the roots of a characteristic polynomial.

_{c}= const is defined [31]:

_{2}= (0; 0; −1/J

_{2})

^{T}—vector of control influence. In the traditional model, equations of the feedback coefficients for state variables were described as:

_{c}= const is equal to ${k}_{M}={J}_{1}/\beta \cdot \left({a}_{11}\cdot {\omega}_{0}-\beta /{J}_{2}\right)$ and the transfer function of the perturbation system will be as (33):

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

parameters and variables | |

A | matrix of the system |

a_{f1} | external viscous friction factor of first mass |

a_{f}_{2} | external viscous friction factor of second mass |

B_{1} | vector of control influence of the first mass |

B_{2} | vector of control influence of the second mass |

C | a vector that determines the generalized coordinate |

c_{12} | shaft stiffness factor |

D | the damping constant |

F | tension force |

G | modulus of elasticity of the second type (shear modulus) |

J_{1} | moment of inertia of first mass |

J_{2} | moment of inertia of second mass |

J_{EM} | inertial link of the first point of shaft |

J_{N} | inertial link of the end of shaft |

J_{p} | polar moment of inertia of the shaft |

K | coefficient of elasticity |

k_{M} | the perturbation control factor |

L | shaft length |

M_{1} | torque of first mass |

M_{2} | torque of second mass |

M | electromagnetic torque of the motor |

M_{c} | load moment |

M_{12} | elastic moment |

r1 | first roller radius |

r2 | second roller radius |

ω_{1} | angular velocity of the first mass |

ω_{2} | angular velocity of the second mass |

Δx | discrete of spatial derivative |

α | parameter in Caputo–Fabrizio representation |

β | coefficient of inertial viscous friction |

φ | shaft rotation angle |

ρ | density of the shaft material |

ξ | coefficient of internal viscous friction |

ρ_{s} | relative degree of the system |

Indices | |

j | number of the shaft section |

N | the number of discretization nodes |

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**Figure 5.**Speed dependencies: __ model with distributed parameters; __ traditional model of a two-mass system.

**Figure 6.**Block diagram of the two-mass system model when using a fractional order derivative to describe the elastic moment.

**Figure 7.**Speed dependencies: __ model with distributed parameters; __ traditional model of a two-mass system; __ model of a two-mass system using the Caputo–Fabrizio operator to describe of the fractional order derivative in the model of the elastic moment at α = 0.995.

**Figure 8.**Block diagram of a two-mass system model when using a fractional integral to describe the elastic moment.

**Figure 9.**Speed dependencies: __ model with distributed parameters;

_{ooo}model of a two-mass system using the Caputo–Fabrizio operator to describe the fractional order derivative in the model of the elastic moment at α = 0.984;

_{xxx}—speed calculation, according to the (12) when α = 0.984.

**Figure 11.**Transient characteristics and reaction of the system when the load moment changes at t = 0.35 s, by using feedback coefficients synthesized based on: __ traditional model; __ models with a fractional order integral to describe the elastic moment.

**Figure 12.**Bode diagrams of: __ open system; __ closed system by ω

_{0}= 25; __ closed system by ω

_{0}= 125.

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Lozynskyy, A.; Chaban, A.; Perzyński, T.; Szafraniec, A.; Kasha, L.
Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft. *Energies* **2021**, *14*, 1854.
https://doi.org/10.3390/en14071854

**AMA Style**

Lozynskyy A, Chaban A, Perzyński T, Szafraniec A, Kasha L.
Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft. *Energies*. 2021; 14(7):1854.
https://doi.org/10.3390/en14071854

**Chicago/Turabian Style**

Lozynskyy, Andriy, Andriy Chaban, Tomasz Perzyński, Andrzej Szafraniec, and Lidiia Kasha.
2021. "Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft" *Energies* 14, no. 7: 1854.
https://doi.org/10.3390/en14071854