Research Papers

Large and Infinite Mass–Spring–Damper Networks

[+] Author and Article Information
Kevin Leyden

Aerospace & Mechanical Engineering,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: kleyden@alumni.nd.edu

Mihir Sen

Professor Emeritus
Aerospace & Mechanical Engineering,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: msen@nd.edu

Bill Goodwine

Aerospace & Mechanical Engineering,
University of Notre Dame,
Notre Dame, IN 46556
e-mail: billgoodwine@nd.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received November 17, 2017; final manuscript received December 28, 2018; published online February 21, 2019. Assoc. Editor: Dumitru I. Caruntu.

J. Dyn. Sys., Meas., Control 141(6), 061005 (Feb 21, 2019) (8 pages) Paper No: DS-17-1570; doi: 10.1115/1.4042466 History: Received November 17, 2017; Revised December 28, 2018

This paper introduces mechanical networks as a tool for modeling complex unidirectional vibrations. Networks of this type have branches containing massless linear springs and dampers, with masses at the nodes. Tree and ladder configurations are examples demonstrating that the overall dynamics of infinite systems can be represented using implicitly defined integro-differential operators. Results from the proposed models compare well to numerical results from finite systems, so this approach may have advantages over high-order differential equations.

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Roemer, M. J. , and Kacprzynski, G. J. , 2000, “ Advanced Diagnostics and Prognostics for Gas Turbine Engine Risk Assessment,” IEEE Aerospace Conference, Big Sky, MT, Mar. 25, pp. 345–353.
Goumas, S. K. , Zervakis, M. E. , and Stavrakakis, G. S. , 2002, “ Classification of Washing Machines Vibration Signals Using Discrete Wavelet Analysis for Feature Extraction,” IEEE Trans. Instrum. Meas., 51(3), pp. 497–508. [CrossRef]
Altintas, Y. , Brecher, C. , Weck, M. , and Witt, S. , 2005, “ Virtual Machine Tool,” CIRP Ann.-Manuf. Technol., 54(2), pp. 115–138. [CrossRef]
Türkay, S. , and Akçay, H. , 2005, “ A Study of Random Vibration Characteristics of The Quarter-Car Model,” J. Sound Vib., 282(1–2), pp. 111–124. [CrossRef]
Murphy, K. , Hunt, G. , and Almond, D. P. , 2006, “ Evidence of Emergent Scaling in Mechanical Systems,” Philos. Mag., 86(21–22), pp. 3325–3338. [CrossRef]
Zhai, C. , Hanaor, D. , and Gan, Y. , 2017, “ Universality of the Emergent Scaling in Finite Random Binary Percolation Networks,” PLoS One, 12(2), p. e0172298.
Zemanian, A. H. , 1991, Infinite Electrical Networks, Cambridge University Press, Cambridge, UK.
Alastruey, J. , Parker, K. , Peiró, J. , and Sherwin, S. , 2009, “ Analysing the Pattern of Pulse Waves in Arterial Networks: A Time-Domain Study,” J. Eng. Math., 64(4), pp. 331–351. [CrossRef]
Flores, J. , Alastruey, J. , and Corvera, P. E. , 2016, “ A Novel Analytical Approach to Pulsatile Blood Flow in the Arterial Network,” Ann. Biomed. Eng., 44(10), pp. 3047–3068. [CrossRef] [PubMed]
Kelly, J. F. , and McGough, R. J. , 2009, “ Fractal Ladder Models and Power Law Wave Equations,” J. Acoust. Soc. Am., 126(4), pp. 2072–2081. [CrossRef] [PubMed]
Ionescu, C. M. , Tenreiro Machado, J. , and De Keyser, R. , 2011, “ Modeling of the Lung Impedance Using a Fractional-Order Ladder Network With Constant Phase Elements,” IEEE Trans. Biomed. Circuits Syst., 5(1), pp. 83–89. [CrossRef] [PubMed]
Cao, J. , Xue, S. , Lin, J. , and Chen, Y. , 2013, “ Nonlinear Dynamic Analysis of a Cracked Rotor-Bearing System With Fractional Order Damping,” ASME J. Comput. Nonlinear Dyn., 8(3) p. 031008.
Di Paola, M. , Failla, G. , and Zingales, M. , 2009, “ Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory,” J. Elast., 97(2), pp. 103–130. [CrossRef]
Di Paola, M. , Pinnola, F. P. , and Zingales, M. , 2013, “ Fractional Differential Equations and Related Exact Mechanical Models,” Comput. Math. Appl., 66(5), pp. 608–620. [CrossRef]
Goodwine, B. , 2014, “ Modeling a Multi-Robot System With Fractional-Order Differential Equations,” IEEE International Conference on Robotics and Automation (ICRA), Hong Kong, China, May 31–June 7, pp. 1763–1768.
Leyden, K. , and Goodwine, B. , 2016, “ Using Fractional-Order Differential Equations for Health Monitoring of a System of Cooperating Robots,” IEEE International Conference on Robotics and Automation (ICRA), Stockholm, Sweden, May 16–21, pp. 366–371.
Oldham, K. B. , and Spanier, J. , 1974, The Fractional Calculus, Academic Press, New York.
Ortigueira, M. D. , and Tenreiro Machado, J. , 2015, “ What is a Fractional Derivative?,” J. Comput. Phys., 293, pp. 4–13. [CrossRef]
Tarasov, V. E. , 2011, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer Science & Business Media, Berlin.
Heymans, N. , and Bauwens, J. , 1994, “ Fractal Rheological Models and Fractional Differential Equations for Viscoelastic Behavior,” Rheol. Acta, 33(3), pp. 210–219. [CrossRef]
Schiessel, H. , Friedrich, C. , and Blumen, A. , 2000, “ Applications to Problems in Polymer Physics and Rheology,” Applications of Fractional Calculus in Physics, World Scientific, Singapore, pp. 331–376.
Silva, M. F. , Machado, J. T. , and Lopes, A. , 2004, “ Fractional Order Control of a Hexapod Robot,” Nonlinear Dyn., 38(1–4), pp. 417–433. [CrossRef]
Delavari, H. , Lanusse, P. , and Sabatier, J. , 2013, “ Fractional Order Controller Design for a Flexible Link Manipulator Robot,” Asian J. Control, 15(3), pp. 783–795. [CrossRef]
Chen, Y. , and Moore, K. L. , 2002, “ Analytical Stability Bound for a Class of Delayed Fractional-Order Dynamic Systems,” Nonlinear Dyn., 29(1/4), pp. 191–200. [CrossRef]
Zhao, C. , Xue, D. , and Chen, Y. , 2005, “ A Fractional Order PID Tuning Algorithm for a Class of Fractional Order Plants,” IEEE International Conference on Mechatronics & Automation, Niagara Falls, ON, Canada, July 29–Aug. 1, pp. 216–221.
Monje, C. A. , Vinagre, B. M. , Feliu, V. , and Chen, Y. , 2008, “ Tuning and Auto-Tuning of Fractional Order Controllers for Industry Applications,” Control Eng. Pract., 16(7), pp. 798–812. [CrossRef]
Cao, Y. , and Ren, W. , 2010, “ Distributed Formation Control for Fractional-Order Systems: Dynamic Interaction and Absolute/Relative Damping,” Syst. Control Lett., 59(3–4), pp. 233–240. [CrossRef]
Cao, Y. , Li, Y. , Ren, W. , and Chen, Y. Q. , 2010, “ Distributed Coordination of Networked Fractional-Order Systems,” IEEE Trans. Syst., Man, Cybern., Part B: Cybern., 40(2), pp. 362–370. [CrossRef]
Machado, J. T. , Kiryakova, V. , and Mainardi, F. , 2011, “ Recent History of Fractional Calculus,” Commun. Nonlinear Sci. Numer. Simul., 16(3), pp. 1140–1153. [CrossRef]
Karsai, G. , and Sztipanovits, J. , 2008, “ Model-Integrated Development of Cyber-Physical Systems,” Software Technologies for Embedded and Ubiquitous Systems, Springer, Berlin, pp. 46–54.
Lee, E. A. , 2010, “ CPS Foundations,” 47th ACM/IEEE Design Automation Conference, Anaheim, CA, June 13–18, pp. 737–742.
Derler, P. , Lee, E. A. , and Vincentelli, A. S. , 2012, “ Modeling Cyber–Physical Systems,” Proc. IEEE, 100(1), pp. 13–28. [CrossRef]
O'Connor, W. , and Lang, D. , 1998, “ Position Control of Flexible Robot Arms Using Mechanical Waves,” ASME J. Dyn. Syst., Meas., Control, 120(3), pp. 334–339. [CrossRef]
Leyden, K. , Sen, M. , and Goodwine, B. , 2018, “ Models From an Implicit Operator Describing a Large Mass-Spring-Damper Network,” IFAC-PapersOnLine, 51(2), pp. 831–836. [CrossRef]


Grahic Jump Location
Fig. 1

Tree configuration. is input, is fixed. Components are indicated by .

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Fig. 2

Ladder configuration. is input, is fixed. Components are indicated by .

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Fig. 3

Single component of admittance Y. is input, is fixed.Admittance operator of a component is indicated by .

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Fig. 4

Single node of mass m. F1, F2, and F3 are forces on the node.

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Fig. 5

Tree configuration for n =1

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Fig. 6

Tree configuration for n =2

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Fig. 7

Tree configuration with no mass (n =3 shown); is input, is fixed

Grahic Jump Location
Fig. 8

Large tree approximation and elongation of frequency band suggesting half-order behavior for finite trees

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Fig. 9

Time-domain excitation at frequency π/2 rad/s showing agreement between tree and approximation

Grahic Jump Location
Fig. 10

Time-domain excitation at frequency 20π rad/s showing agreement between tree and adjusted approximation

Grahic Jump Location
Fig. 11

Frequency response showing large approximation and fractional-order behavior for ladders



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