The missing mem-inerter and extended mem-dashpot found


Anyone who ever gave attention to advances in circuit elements will be familiar with the three elementary two-terminal passive elements: resistor, inductor, capacitor and their memory counterparts: memristor, meminductor and memcapacitor, respectively. Similarly, it is reported that the mem-dashpot and the mem-spring were viewed as the memory counterparts of the damper and the spring in the field of mechanical engineering. However, what is the memory counterpart of the inerter as a new two-terminal element? In 2018, Zhang predicted the existence of such a mechanical element, which he called a mem-inerter. Although he postulated this element, until now no one offered either a useful physical realization or an example of a mem-inerter. Here, a displacement-dependent fluid inerter device is found to be a physical realization of a mem-inerter, though with a parasitic element called the extended mem-dashpot, which is viewed as the mechanical counterpart of an extended memristor. Experimental results confirm this finding and the existence of the mem-inerter and the extended mem-dashpot in the real world. This work is very helpful in finding and designing mem-inerter and extended mem-dashpot devices, and modeling some important nonlinear hysteretic devices and systems, which is a fundamental branch of research in nonlinear dynamics.

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  1. 1.

    For the details on the bend radius of the helical channel and the hydraulic diameter of the channel, see [29] and references therein.

  2. 2.

    Equations (9)–(13) have been justified by using fluid-based arguments and confirmed by the experiment in [29]. To avoid repeating the work done, here they are directly cited because the devices in [29] and Fig. 3 have the same operating principle and fluid-based parameters.

  3. 3.

    As a matter of fact, Newton’s original law of motion is represented in terms of momentum and velocity.


  1. 1.

    Biolek, D., Biolek, Z., Biolkova, V.: Memristors and other higher-order elements in generalized through-across domain. In: 2016 IEEE International Conference on Electronics, Circuits and Systems (ICECS), pp. 604–607. IEEE (2016)

  2. 2.

    Brzeski, P., Pavlovskaia, E., Kapitaniak, T., Perlikowski, P.: The application of inerter in tuned mass absorber. Int. J. Non-Linear Mech. 70, 20–29 (2015)

    Article  Google Scholar 

  3. 3.

    Chen, M.Z., Hu, Y.L., Huang, L.X., Chen, G.R.: Influence of inerter on natural frequencies of vibration systems. J. Sound Vib. 333(7), 1874–1887 (2014)

    Article  Google Scholar 

  4. 4.

    Chen, M.Z., Papageorgiou, C., Scheibe, F., Wang, F.C., Smith, M.C.: The missing mechanical circuit element. IEEE Circuits Syst. Mag. 9(1), 10–26 (2009)

    Article  Google Scholar 

  5. 5.

    Chua, L.: Resistance switching memories are memristors. Appl. Phys. A 102(4), 765–783 (2011)

    Article  Google Scholar 

  6. 6.

    Chua, L.: If it’s pinched it’s a memristor. In: Memristors and Memristive Systems, pp. 17–90. Springer (2014)

  7. 7.

    Chua, L.: Everything you wish to know about memristors but are afraid to ask. Radioengineering 24(2), 319–368 (2015)

    Article  Google Scholar 

  8. 8.

    Chua, L.O.: Memristor-the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507–519 (1971)

    Article  Google Scholar 

  9. 9.

    Chua, L.O.: Nonlinear circuit foundations for nanodevices, Part I: the four-element torus. Proc. IEEE 91(11), 1830–1859 (2003)

    Article  Google Scholar 

  10. 10.

    Chua, L.O., Kang, S.M.: Memristive devices and systems. Proc. IEEE 64(2), 209–223 (1976)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Di Ventra, M., Pershin, Y.V., Chua, L.O.: Circuit elements with memory: memristors, memcapacitors, and meminductors. Proc. IEEE 97(10), 1717–1724 (2009)

    Article  Google Scholar 

  12. 12.

    Gonzalez-Buelga, A., Lazar, I.F., Jiang, J.Z., Neild, S.A., Inman, D.J.: Assessing the effect of non-linearities on the performance of a tuned inerter damper. Struct. Control Health Monit. 24(3), e1879 (2017)

    Article  Google Scholar 

  13. 13.

    Hu, Y.L., Chen, M.Z.: Performance evaluation for inerter-based dynamic vibration absorbers. Int. J. Mech. Sci. 99, 297–307 (2015)

    Article  Google Scholar 

  14. 14.

    Jeltsema, D., Doria-Cerezo, A.: Port-hamiltonian formulation of systems with memory. Proc. IEEE 100(6), 1928–1937 (2011)

    Article  Google Scholar 

  15. 15.

    Lewis, T.D., Jiang, J.Z., Neild, S.A., Gong, C., Iwnicki, S.D.: Using an inerter-based suspension to improve both passenger comfort and track wear in railway vehicles. Veh. Syst. Dyn. 10, 1–22 (2019)

    Article  Google Scholar 

  16. 16.

    Li, Y., Lombardi, L., De Luca, F., Farbiarz, Y., Blandon, J.J., Lara, L.A., Rendon, J.F., Jiang, J.Z., Neild, S.: Optimal design of inerter-integrated vibration absorbers for seismic retrofitting of a high-rise building in Colombia. In: Journal of Physics: Conference Series, vol. 1264, p. 012031. IOP Publishing (2019)

  17. 17.

    Liu, X.F., Jiang, J.Z., Titurus, B., Harrison, A.J.L., McBryde, D.: Testing and modelling of the damping effects for fluid-based inerters. Proc. Eng. 199, 435–440 (2017)

    Article  Google Scholar 

  18. 18.

    Luo, J.N., Macdonald, J., Jiang, J.Z.: Identification of optimum cable vibration absorbers using fixed-sized-inerter layouts. Mech. Mach. Theory 140, 292–304 (2019)

    Article  Google Scholar 

  19. 19.

    Marian, L., Giaralis, A.: Optimal design of a novel tuned mass-damper-inerter (TMDI) passive vibration control configuration for stochastically support-excited structural systems. Probab. Eng. Mech. 38, 156–164 (2014)

    Article  Google Scholar 

  20. 20.

    Oster, G.F., Auslander, D.M.: The memristor: a new bond graph element. J. Dyn. Syst. Meas. Contr. 94(3), 249–252 (1972)

    Article  Google Scholar 

  21. 21.

    Pei, J.S., Wright, J.P., Todd, M.D., Masri, S.F., Gay-Balmaz, F.: Understanding memristors and memcapacitors in engineering mechanics applications. Nonlinear Dyn. 80(1–2), 457–489 (2015)

    Article  Google Scholar 

  22. 22.

    Pershin, Y.V., Di Ventra, M.: Experimental demonstration of associative memory with memristive neural networks. Neural Netw. 23(7), 881–886 (2010)

    Article  Google Scholar 

  23. 23.

    Pershin, Y.V., La Fontaine, S., Di Ventra, M.: Memristive model of amoeba learning. Phys. Rev. E 80(2), 021926 (2009)

    Article  Google Scholar 

  24. 24.

    Sah, M.P., Yang, C.J., Kim, H., Muthuswamy, B., Jevtic, J., Chua, L.: A generic model of memristors with parasitic components. IEEE Trans. Circuits Syst. I Regul. Pap. 62(3), 891–898 (2015)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Shen, Y.J., Chen, L., Liu, Y.L., Zhang, X.L., Yang, X.F.: Optimized modeling and experiment test of a fluid inerter. J. VibroEng. 18(5), 2789–2800 (2016)

    Article  Google Scholar 

  26. 26.

    Smith, M.C.: Synthesis of mechanical networks: the inerter. IEEE Trans. Autom. Control 47(10), 1648–1662 (2002)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Smith, M.C., Wang, F.C.: Performance benefits in passive vehicle suspensions employing inerters. Veh. Syst. Dyn. 42(4), 235–257 (2004)

    Article  Google Scholar 

  28. 28.

    Stulov, A.: Hysteretic model of the grand piano hammer felt. J. Acoust. Soc. Am. 97, 2577–2585 (1995)

    Article  Google Scholar 

  29. 29.

    Swift, S.J., Smith, M.C., Glover, A.R., Papageorgiou, C., Gartner, B., Houghton, N.E.: Design and modelling of a fluid inerter. Int. J. Control 86(11), 2035–2051 (2013)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Wang, F.C., Hong, M.F., Chen, C.W.: Building suspensions with inerters. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 224, 1605–1616 (2010)

    Article  Google Scholar 

  31. 31.

    Wang, F.C., Liao, M.K.: The lateral stability of train suspension systems employing inerters. Veh. Syst. Dyn. 48(5), 619–643 (2010)

    Article  Google Scholar 

  32. 32.

    Wang, F.C., Liao, M.K., Liao, B.H., Su, W.J., Chan, H.A.: The performance improvements of train suspension systems with mechanical networks employing inerters. Veh. Syst. Dyn. 47(7), 805–830 (2009)

    Article  Google Scholar 

  33. 33.

    Wang, F.Z.: A triangular periodic table of elementary circuit elements. IEEE Trans. Circuits Syst. I Regul. Pap. 60(3), 616–623 (2013)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Wang, F.Z., Chua, L.O., Yang, X., Helian, N., Tetzlaff, R., Schmidt, T., Li, C., Carrasco, J.M.G., Chen, W.L., Chu, D.: Adaptive neuromorphic architecture (ANA). Neural Netw. 45, 111–116 (2013)

    Article  Google Scholar 

  35. 35.

    Wang, F.Z., Li, L., Shi, L.P., Wu, H.Q., Chua, L.O.: \(\phi \) memristor: real memristor found. J. Appl. Phys. 125(5), 054504 (2019)

    Article  Google Scholar 

  36. 36.

    Zhang, S.Y., Jiang, J.Z., Neild, S.: Optimal configurations for a linear vibration suppression device in a multi-storey building. Struct. Control Health Monit. 24(3), e1887 (2017)

    Article  Google Scholar 

  37. 37.

    Zhang, X.L., Gao, Q., Nie, J.M.: The mem-inerter: a new mechanical element with memory. Adv. Mech. Eng. 10(6), 1–13 (2018)

    Article  Google Scholar 

  38. 38.

    Zhang, X.L., He, H., Nie, J.M., Chen, L.: A semi-active skyhook-inertance control strategy based on continuously adjustable inerter. Shock Vib. 2018, 1–8 (2018)

    Google Scholar 

  39. 39.

    Zhang, X.L., Nie, J.M., Huang, Z.X., He, H., Chen, L.: Hydraulic mem-inerter device and its application (2019). US Patent App. 16/158,857

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This study was funded by the National Natural Science Foundation of China (Grant Nos. 51875257 and 51805223), the Jiangsu Government Scholarship for Overseas Studies (Grant No. JS-2019-192) and the Six Talent Peaks Program of Jiangsu Province of China (Grant No. 2016-GDZB-097)

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Correspondence to Xiao-Liang Zhang.

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Zhang, X., Geng, C., Nie, J. et al. The missing mem-inerter and extended mem-dashpot found. Nonlinear Dyn 101, 835–856 (2020).

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  • Nonlinear element
  • Mem-inerter
  • Extended mem-dashpot
  • Memory effect
  • Displacement-dependent Mechanism
  • Pinched hysteresis loop