# Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Fractional-Order Vibration Characterizing Viscoinertia

**Proposition**

**1.**

## 3. Distributed-Order Vibration Characterizing Viscoinertia

## 4. Results for the Weight Function in the Form of a Parametrized Exponential Function

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof**

**.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Scott-Blair, G.W. A Survey of General and Applied Rheology; Pitman: London, UK, 1949. [Google Scholar]
- Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity; Imperial College: London, UK, 2010. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Podlubny, I. Fractional Differential Equations; Academic: San Diego, CA, USA, 1999. [Google Scholar]
- Jiao, Z.; Chen, Y.; Podlubn, Y.I. Distributed-Order Dynamic Systems–Stability, Simulation, Applications and Perspectives; Springer: London, UK, 2012. [Google Scholar]
- Li, M. Theory of Fractional Engineering Vibrations; De Gruyter: Berlin, Germany; Boston, UK, 2021. [Google Scholar]
- Sheng, H.; Chen, Y.; Qiu, T. Fractional Processes and Fractional—Order Signal Processing–Techniques and Applications; Springer: London, UK, 2012. [Google Scholar]
- Magin, R.L. Fractional Calculus in Bioengineering; Begell House: Danbury, CT, USA, 2006. [Google Scholar]
- Can, N.H.; Jafari, H.; Ncube, M.N. Fractional calculus in data fitting. Alex. Eng. J.
**2020**, 59, 3269–3274. [Google Scholar] [CrossRef] - Torbati, M.M.; Hammond, J.K. Physical and geometrical interpretation of fractional operators. J. Frankl. Inst.
**1998**, 335, 1077–1086. [Google Scholar] [CrossRef] - Podlubny, I. Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal.
**2002**, 5, 367–386. [Google Scholar] - Koeller, R.C. Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech.
**1984**, 51, 299–307. [Google Scholar] [CrossRef] - Palade, L.I.; Verney, V.; Attané, P. A modified fractional model to describe the entire viscoelastic behavior of polybutadienes from flow to glassy regime. Rheol. Acta
**1996**, 35, 265–273. [Google Scholar] [CrossRef] - Pritz, T. Five-parameter fractional derivative model for polymeric dampling materials. J. Sound Vib.
**2003**, 265, 935–952. [Google Scholar] [CrossRef] - Duan, J.S.; Hu, D.C.; Chen, Y.Q. Simultaneous characterization of relaxation, creep, dissipation, and hysteresis by fractional-order constitutive models. Fractal Fract.
**2021**, 5, 36. [Google Scholar] [CrossRef] - Caputo, M. Linear models of dissipation whose Q is almost frequency independent. Ann. Geophys.
**1966**, 19, 383–393. [Google Scholar] [CrossRef] - Bagley, R.L.; Torvik, P.J. A generalized derivative model for an elastomer damper. Shock Vib.
**1979**, 49, 135–143. [Google Scholar] - Beyer, H.; Kempfle, S. Definition of physically consistent damping laws with fractional derivatives. ZAMM Z. Fur Angew. Math. Und Mech.
**1995**, 75, 623–635. [Google Scholar] [CrossRef] - Achar, B.N.N.; Hanneken, J.W.; Clarke, T. Response characteristics of a fractional oscillator. Phys. Stat. Mech. Appl.
**2002**, 309, 275–288. [Google Scholar] [CrossRef] - Li, M.; Lim, S.C.; Chen, S. Exact solution of impulse response to a class of fractional oscillators and its stability. Math. Probl. Eng.
**2011**, 2011, 657839. [Google Scholar] [CrossRef] [Green Version] - Li, M. Three classes of fractional oscillators. Symmetry
**2018**, 10, 40. [Google Scholar] [CrossRef] [Green Version] - Lim, S.C.; Li, M.; Teo, L.P. Locally self-similar fractional oscillator processes. Fluct. Noise Lett.
**2007**, 7, L169–L179. [Google Scholar] [CrossRef] - Huang, C.; Duan, J.S. Steady-state response to periodic excitation in fractional vibration system. J. Mech.
**2016**, 32, 25–33. [Google Scholar] [CrossRef] - Shen, Y.J.; Yang, S.P.; Xing, H.J. Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative. Acta Phys. Sin.
**2012**, 61, 110505. [Google Scholar] - Li, C.P.; Deng, W.H.; Xu, D. Chaos synchronization of the Chua system with a fractional order. Phys. Stat. Mech. Appl.
**2006**, 360, 171–185. [Google Scholar] [CrossRef] - Wang, Z.H.; Hu, H.Y. Stability of a linear oscillator with damping force of the fractional-order derivative. Sci. China Ser.
**2010**, 53, 345–352. [Google Scholar] [CrossRef] - Caputo, M. Mean fractional-order-derivatives differential equations and filters. Ann. Dell’Univ. Ferrara
**1995**, 41, 73–84. [Google Scholar] - Caputo, M. Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal.
**2001**, 4, 421–442. [Google Scholar] - Bagley, R.L.; Torvik, P.J. On the existence of the order domain and the solution of distributed order equations—Part I. Int. J. Appl. Math.
**2000**, 2, 865–882. [Google Scholar] - Atanackovic, T.M. A generalized model for the uniaxial isothermal deformation of a viscoelastic body. Acta Mech.
**2002**, 159, 77–86. [Google Scholar] [CrossRef] - Atanackovic, T.M.; Budincevic, M.; Pilipovic, S. On a fractional distributed-order oscillator. J. Phys. Math. Gen.
**2005**, 38, 6703–6713. [Google Scholar] [CrossRef] - Duan, J.S.; Baleanu, D. Steady periodic response for a vibration system with distributed-order derivatives to periodic excitation. J. Vib. Control
**2018**, 24, 3124–3131. [Google Scholar] [CrossRef] - Duan, J.S.; Chen, Y.Q. Mechanical response and simulation for constitutive equations with distributed order derivatives. Int. J. Model. Simul. Sci. Comput.
**2017**, 8, 1750040. [Google Scholar] [CrossRef] [Green Version] - Duan, J.S.; Xu, Y.Y. Vibration equation of fractional order describing viscoelasticity and viscous inertia. Open Phys.
**2019**, 17, 850–856. [Google Scholar] [CrossRef] - Smith, M.C. Synthesis of mechanical networks: The inerter. IEEE Trans. Autom. Control
**2002**, 47, 1648–1662. [Google Scholar] [CrossRef] [Green Version] - Chen, M.Z.Q.; Hu, Y.; Huang, L.; Chen, G. Influence of inerter on natural frequencies of vibration systems. J. Sound Vib.
**2014**, 333, 1874–1887. [Google Scholar] [CrossRef] [Green Version] - Chen, M.Z.; Hu, Y. Inerter and Its Application in Vibration Control Systems; Springer: Singapore, 2019. [Google Scholar]
- Papageorgiou, C.; Smith, M.C. Positive real synthesis using matrix inequalities for mechanical networks: Application to vehicle suspension. IEEE Trans. Control Syst. Technol.
**2006**, 14, 423–435. [Google Scholar] [CrossRef] [Green Version] - Evangelou, S.; Limebeer, D.J.N.; Sharp, R.S.; Smith, M.C. Control of motorcycle steering instabilities. IEEE Control Syst. Mag.
**2006**, 26, 78–88. [Google Scholar] - Hu, Y.; Chen, M.Z.Q.; Shu, Z.; Huang, L. Analysis and optimisation for inerter-based isolators via fixed-point theory and algebraic solution. J. Sound Vib.
**2015**, 346, 17–36. [Google Scholar] [CrossRef] [Green Version] - Liu, X.B.; Wang, Z.; Zhang, W. Active absorption measurement of panels using multichannel inertial absorbers in wideband frequencies. Noise Vib. Control
**2015**, 35, 213–216. [Google Scholar] - Wang, L.; Mao, M.; Lei, Q.; Chen, Y.; Zhang, X. Modeling and testing for a hydraulic inerter. J. Vib. Shock
**2018**, 37, 146–152. [Google Scholar]

**Figure 1.**The curves of $\tilde{c}-c$ versus $\lambda $ for $\eta =1$ and for $\omega =$ 0.2 (solid line), 0.5 (dotted line), 1 (dashed line), 2 (dot-dashed line), and 5 (dot-dot-dashed line).

**Figure 2.**The curves of $\tilde{m}-m$ versus $\lambda $ for $\eta =1$ and for $\omega =$ 0.2 (solid line), 0.5 (dotted line), 1 (dashed line), 2 (dot-dashed line), 5 (dot-dot-dashed line).

**Figure 3.**Frequency–amplitude curves for $k=m=1$, $c=0.5$, $\eta =0.5$, and $\lambda =1.1$ (solid line), 1.3 (dotted line), 1.5 (dashed line), 1.7 (dot-dashed line), or 1.9 (dot-dot-dashed line).

**Figure 4.**Frequency–amplitude curves for $k=m=1$, $c=0.5$, $\eta =5$, and $\lambda =1.1$ (solid line), 1.3 (dotted line), 1.5 (dashed line), 1.7 (dot-dashed line), or 1.9 (dot-dot-dashed line).

**Figure 5.**Frequency–phase curves for $k=m=1$, $c=0.5$, $\eta =0.5$, and $\lambda =1.1$ (solid line), 1.3 (dotted line), 1.5 (dashed line), 1.7 (dot-dashed line), or 1.9 (dot-dot-dashed line).

**Figure 6.**Frequency–phase curves for $k=m=1$, $c=0.5$, $\eta =5$, and $\lambda =1.1$ (solid line), 1.3 (dotted line), 1.5 (dashed line), 1.7 (dot-dashed line), or 1.9 (dot-dot-dashed line).

**Figure 8.**Curves of $W(\lambda ;p)$ versus $\lambda $ on $1\le \lambda \le 2$ when p takes different values: $p=0.01$ (solid line), $p=0.1$ (dotted line), $p=1$ (dashed line), $p=10$ (dot-dashed line), and $p=100$ (dot-dot-dashed line).

**Figure 9.**The damping contribution $\widehat{c}-c$ versus p for $\eta =1$ and for $\omega =0.2$ (solid line), 0.5 (dotted line), 1 (dashed line), 2 (dot-dashed line), and 5 (dot-dot-dashed line).

**Figure 10.**The mass contribution $\hat{m}-m$ versus p for $\eta =1$ and for $\omega =0.2$ (solid line), 0.5 (dotted line), 1 (dashed line), 2 (dot-dashed line), and 5 (dot-dot-dashed line).

**Figure 11.**Frequency–amplitude curves $\gamma \left(\omega \right)$ for $k=m=1$, $c=0.5$, $\eta =0.5$, and $p=0.01$ (solid line), $p=0.1$ (dotted line), $p=1$ (dashed line), $p=10$ (dot-dashed line), or $p=100$ (dot-dot-dashed line).

**Figure 12.**Frequency–amplitude curves $\gamma \left(\omega \right)$ for $k=m=1$, $c=0.5$, $\eta =5$, and $p=0.01$ (solid line), $p=0.1$ (dotted line), $p=1$ (dashed line), $p=10$ (dot-dashed line), or $p=100$ (dot-dot-dashed line).

**Figure 13.**Frequency–phase curves $\phi \left(\omega \right)$ for $k=m=1$, $c=0.5$, $\eta =0.5$, and $p=0.01$ (solid line), $p=0.1$ (dotted line), $p=1$ (dashed line), $p=10$ (dot-dashed line), or $p=100$ (dot-dot-dashed line).

**Figure 14.**Frequency–phase curves $\phi \left(\omega \right)$ for $k=m=1$, $c=0.5$, $\eta =5$, and $p=0.01$ (solid line), $p=0.1$ (dotted line), $p=1$ (dashed line), $p=10$ (dot-dashed line), or $p=100$ (dot-dot-dashed line).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Duan, J.-S.; Hu, D.-C.
Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia. *Fractal Fract.* **2021**, *5*, 67.
https://doi.org/10.3390/fractalfract5030067

**AMA Style**

Duan J-S, Hu D-C.
Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia. *Fractal and Fractional*. 2021; 5(3):67.
https://doi.org/10.3390/fractalfract5030067

**Chicago/Turabian Style**

Duan, Jun-Sheng, and Di-Chen Hu.
2021. "Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia" *Fractal and Fractional* 5, no. 3: 67.
https://doi.org/10.3390/fractalfract5030067