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A smart pipe energy harvester excited by fluid flow and base excitation

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Abstract

This paper presents an electromechanical dynamic modelling of the partially smart pipe structure subject to the vibration responses from fluid flow and input base excitation for generating the electrical energy. We believe that this work shows the first attempt to formulate a unified analytical approach of flow-induced vibrational smart pipe energy harvester in application to the smart sensor-based structural health monitoring systems including those to detect flutter instability. The arbitrary topology of the thin electrode segments located at the surface of the circumference region of the smart pipe has been used so that the electric charge cancellation can be avoided. The analytical techniques of the smart pipe conveying fluid with discontinuous piezoelectric segments and proof mass offset, connected with the standard AC–DC circuit interface, have been developed using the extended charge-type Hamiltonian mechanics. The coupled field equations reduced from the Ritz method-based weak form analytical approach have been further developed to formulate the orthonormalised dynamic equations. The reduced equations show combinations of the mechanical system of the elastic pipe and fluid flow, electromechanical system of the piezoelectric component, and electrical system of the circuit interface. The electromechanical multi-mode frequency and time signal waveform response equations have also been formulated to demonstrate the power harvesting behaviours. Initially, the optimal power output due to optimal load resistance without the fluid effect is discussed to compare with previous studies. For potential application, further parametric analytical studies of varying partially piezoelectric pipe segments have been explored to analyse the dynamic stability/instability of the smart pipe energy harvester due to the effect of fluid and input base excitation. Further proof between case studies also includes the effect of variable flow velocity for optimal power output, 3-D frequency response, the dynamic evolution of the smart pipe system based on the absolute velocity-time waveform signals, and DC power output-time waveform signals.

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References

  1. Feodos’ev, V.P.: Vibrations and stability of a pipe when liquid flows through it. Inzh. Sb. 10, 1013–1024 (1951)

    Google Scholar 

  2. Housner, G.W.: Bending vibrations of a pipe line containing flowing fluid. J. Appl. Mech. 19, 205–208 (1952)

    Google Scholar 

  3. Niordson, F.I.: Vibrations of a cylindrical tube containing flowing fluid. Kungliga Tekniska Hogskolans Handlingar (Stockholm) No. 73 (1953)

  4. Holmes, P.J.: Pipes supported at both ends cannot flutter. J. Appl. Mech. 45, 619–622 (1978)

    Article  Google Scholar 

  5. Heinrich, G.: Schwingungen durchströmter Rohre (vibrations of pipes with flow). Z. Angew. Math. Mech. 36, 417–427 (1956)

    Article  MathSciNet  Google Scholar 

  6. Benjamin, T.B.: Dynamics of a system of articulated pipes conveying fluid. I. Theory. Proc. R. Soc. Lond. A 261, 457–486 (1961)

    Article  MathSciNet  Google Scholar 

  7. Benjamin, T.B.: Dynamics of a system of articulated pipes conveying fluid. II. Experiments. Proc. R. Soc. Lond. A 261, 487–499 (1961)

    Article  MathSciNet  Google Scholar 

  8. Bottema, O.: On the stability of equilibrium of a linear mechanical system. Z. Angew. Math. Mech. 6, 97–104 (1955)

    MathSciNet  MATH  Google Scholar 

  9. Smith, T.E., Herrmann, G.: Stability of circulatory elastic systems in the presence of magnetic damping. Acta Mech. 12, 175–188 (1971)

    Article  Google Scholar 

  10. Gregory, R.W., Païdoussis, M.P.: Unstable oscillation of tubular cantilevers conveying fluid. I. Theory. Proc. R. Soc. Lond. A 293, 512–527 (1966)

    Article  Google Scholar 

  11. Gregory, R.W., Païdoussis, M.P.: Unstable oscillation of tubular cantilevers conveying fluid. II. Theory. Proc. R. Soc. Lond. A 293, 528–542 (1966)

    Article  Google Scholar 

  12. Sokolnikoff, I.S.: Mathematical Theory of Elasticity, 2nd edn. McGraw-Hill, Maidenherd (1956)

    MATH  Google Scholar 

  13. Païdoussis, M.P., Issid, N.T.: Dynamic stability of pipes conveying fluid. J. Sound Vib. 33, 267–294 (1974)

    Article  Google Scholar 

  14. Païdoussis, M.P., Li, G.X.: Pipes conveying fluid: a model dynamical problem. J. Fluids Struct. 7, 137–204 (1993)

    Article  Google Scholar 

  15. Ruta, G.C., Elishakoff, I.: Towards the resolution of the Smith–Herrmann paradox. Acta Mech. 173, 89–105 (2004)

    Article  Google Scholar 

  16. De Bellis, M.L., Ruta, G.C., Elishakoff, I.: A contribution to the stability of an overhanging pipe conveying fluid. Contin. Mech. Thermodyn. 27, 685–701 (2015)

    Article  MathSciNet  Google Scholar 

  17. Gorman, D.G., Reese, J.M., Zhang, Y.L.: Vibration of a flexible pipe conveying viscous pulsating fluid flow. J. Sound Vib. 230, 379–392 (2000)

    Article  Google Scholar 

  18. Lee, U., Pak, C.H., Hong, S.C.: The dynamics of a piping system with internal unsteady flow. J. Sound Vib. 180(2), 297–311 (1995)

    Article  Google Scholar 

  19. Lavooij, C.S.W., Tijsseling, A.S.: Fluid–structure interaction in liquid-filled piping systems. J. Fluids Struct. 5, 573–95 (1991)

    Article  Google Scholar 

  20. Zhang, Y.L., Gorman, D.G., Reese, J.M.: Analysis of the vibration of pipes conveying fluid. J. Mech. Eng. Sci. 213, 849–60 (1999)

    Article  Google Scholar 

  21. Wadham-Gagnon, M., Païdoussis, M.P., Semler, C.: Dynamics of cantilevered pipes conveying fluid, part 1: nonlinear equations of three dimensional motion. J. Fluids Struct. 23, 545–567 (2007)

    Article  Google Scholar 

  22. Païdoussis, M.P., Semler, C., Wadham-Gagnon, M.: Dynamics of cantilevered pipes conveying fluid. Part 2: dynamics of the system with intermediate spring support. J. Fluids Struct. 23, 569–587 (2007)

    Article  Google Scholar 

  23. Modarres-Sadeghi, Y., Semler, C., Wadham-Gagnon, M., Païdoussis, M.P.: Dynamics of cantilevered pipes conveying fluid. Part 3: three dimensional dynamics in the presence of an end-mass. J. Fluids Struct. 23, 589–603 (2007)

    Article  Google Scholar 

  24. Païdoussis, M.P.: Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 1, 2nd edn. Academic, London (2014)

    Google Scholar 

  25. Païdoussis, M.P.: Aspirating pipes do not flutter at infinitesimally small flow. J. Fluids Struct. 13, 419–425 (1999)

    Article  Google Scholar 

  26. Kuiper, G.L., Metrikine, A.V.: Dynamic stability of a submerged, free-hanging riser conveying fluid. J. Sound Vib. 280, 1051–1065 (2005)

    Article  Google Scholar 

  27. Païdoussis, M.P., Semler, C., Wadham-Gagnon, M.: A reappraisal of why aspirating pipes do not flutter at infinitesimal flow. J. Fluids Struct. 20, 147–156 (2005)

    Article  Google Scholar 

  28. Kuiper, G.L., Metrikine, A.V.: Experimental investigation of dynamic stability of a cantilever pipe aspirating fluid. J. Fluids Struct. 24, 541–558 (2008)

    Article  Google Scholar 

  29. Giacobbi, D.B., Rinaldi, S., Semler, C., Païdoussis, M.P.: The dynamics of a cantilevered pipe aspirating fluid studied by experimental, numerical and analytical methods. J. Fluids Struct. 30, 73–96 (2012)

    Article  Google Scholar 

  30. Krommer, M., Irschik, H.: An electromechanically coupled theory for piezoelastic beams taking into account the charge equation of electrostatics. Acta Mech. 154, 141–158 (2002)

    Article  Google Scholar 

  31. Krommer, M.: On the correction of the Bernoulli–Euler beam theory for smart piezoelectric beams. Smart Mater. Struct. 10, 668–680 (2001)

    Article  Google Scholar 

  32. Fernandes, A., Pouget, J.: Analytical and numerical approach to piezoelectric bimorph. Int. J. Solids Struct. 40, 4331–52 (2003)

    Article  Google Scholar 

  33. Moita, J.M., Correia, I.F.P., Soares, C.M.M.: Active control of adaptive laminated structures with bounded piezoelectric sensors and actuators. Comput. Struct. 82(17–19), 1349–58 (2004)

    Article  Google Scholar 

  34. Irschik, H., Ziegler, F.: Eigenstrain without stress and static shape control of structures. AIAA J. 39, 1985–1990 (2001)

    Article  Google Scholar 

  35. Irschik, H., Krommer, M., Belyaev, A.K., Schlacher, A.K.: Shaping of piezoelectric sensors/actuators for vibrations of slender beams: coupled theory and inappropriate shape functions. J. Intell. Mater. Syst. Struct. 9, 546–554 (1998)

    Article  Google Scholar 

  36. Irschik, H., Krommer, M., Pichler, U.: Dynamic shape control of beam-type structures by piezoelectric actuation and sensing. Int. J. Appl. Electromagn. Mech. 17, 251–258 (2003)

    Google Scholar 

  37. Irschik, H., Pichler, U.: Dynamic shape control of solids and structures by thermal expansion strains. J. Therm. Stresses 24, 565–576 (2001)

    Article  Google Scholar 

  38. Krommer, M., Zellhofer, M., Heilbrunner, K.-H.: Strain-type sensor networks for structural monitoring of beam-type structures. J. Intell. Mater. Syst. Struct. 20, 1875–1888 (2003)

    Article  Google Scholar 

  39. Krommer, M., Vetyukov, Yu.: Adaptive sensing of kinematic entities in the vicinity of a time-dependent geometrically nonlinear pre-deformed state. Int. J. Solid. Struct. 46, 3313–3320 (2009)

    Article  Google Scholar 

  40. Kapuria, S., Yasin, M.Y.: Active vibration control of smart plates using directional actuation and sensing capability of piezoelectric composites. Acta Mech. 224, 1185–1199 (2013)

    Article  MathSciNet  Google Scholar 

  41. Tzou, H.S., Tseng, C.I.: Distributed vibration control and identification of coupled elastic/piezoelectric systems: finite element formulation and applications. Mech. Syst. Signal Process. 5, 215–231 (1991)

    Article  Google Scholar 

  42. Krommer, M., Irschik, H.: A Reissner–Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect. Acta Mech. 141, 51–69 (2000)

    Article  Google Scholar 

  43. Krommer, M.: On the influence of pyroelectricity upon thermally induced vibrations of piezothermoelastic plates. Acta Mech. 171, 59–73 (2004)

    Article  Google Scholar 

  44. dell’Isola, F., Maurini, C., Porfiri, M.: Passive damping of beam vibrations through distributed electric networks and piezoelectric transducers: prototype design and experimental validation. Smart Mater. Struct. 13, 299–308 (2004)

    Article  Google Scholar 

  45. Niederberger, D., Morari, M.: An autonomous shunt circuit for vibration damping. Smart Mater. Struct. 15, 359–364 (2006)

    Article  Google Scholar 

  46. Schoeftner, J., Krommer, M.: Single point vibration control for a passive piezoelectric Bernoulli–Euler beam subjected to spatially varying harmonic loads. Acta Mech. 223, 1983–1998 (2012)

    Article  MathSciNet  Google Scholar 

  47. Vasques, C.M.A.: Improved passive shunt vibration control of smart piezo-elastic beams using modal piezoelectric transducers with shaped electrodes. Smart Mater. Struct. 21, 125003 (2012)

    Article  Google Scholar 

  48. Liao, Y., Sodano, H.: Modeling and comparison of bimorph power harvesters with piezoelectric elements connected in parallel and series. J. Intell. Mater. Syst. Struct. 21, 149–159 (2010)

    Article  Google Scholar 

  49. Goldschmidtboeing, F., Woias, P.: Characterization of different beam shapes for piezoelectric energy harvesting. J. Micromech. Microeng. 18, 104013 (2008)

    Article  Google Scholar 

  50. Kim, M., Hoegen, M., Dugundji, J., Wardle, B.L.: Modeling and experimental verification of proof mass effects on vibration energy harvester performance. Smart Mater. Struct. 19, 045023 (2010)

    Article  Google Scholar 

  51. Dalzell, P., Bonello, P.: Analysis of an energy harvesting piezoelectric beam with energy storage circuit. Smart Mater. Struct. 21, 105029 (2012)

    Article  Google Scholar 

  52. Lumentut, M.F., Howard, I.M.: Analytical and experimental comparisons of electromechanical vibration response of a piezoelectric bimorph beam for power harvesting. Mech. Syst. Signal Proc. 36, 66–86 (2013)

    Article  Google Scholar 

  53. Lumentut, M.F., Howard, I.M.: Parametric design-based modal damped vibrational piezoelectric energy harvesters with arbitrary proof mass offset: numerical and analytical validations. Mech. Syst. Signal Process. 68–69, 562–586 (2016)

    Article  Google Scholar 

  54. Adhikari, S., Friswell, M.I., Inman, D.J.: Piezoelectric energy harvesting from broadband random vibrations. Smart Mater. Struct. 18, 115005 (2009)

    Article  Google Scholar 

  55. Ali, S.F., Friswell, M.I., Adhikari, S.: Piezoelectric energy harvesting with parametric uncertainty. Smart Mater. Struct. 19, 105010 (2010)

    Article  Google Scholar 

  56. Lumentut, M.F., Howard, I.M.: Intrinsic electromechanical dynamic equations for piezoelectric power harvesters. Acta Mech. 228(2), 631–650 (2017)

    Article  MathSciNet  Google Scholar 

  57. Friswell, M.I., Adhikari, S.: Sensor shape design for piezoelectric cantilever beams to harvest vibration energy. J. Appl. Phys. 108, 014901 (2010)

    Article  Google Scholar 

  58. Lumentut, M.F., Howard, I.M.: Electromechanical finite element modelling for dynamic analysis of a cantilevered piezoelectric energy harvester with tip mass offset under base excitations. Smart Mater. Struct. 23, 095037 (2014)

    Article  Google Scholar 

  59. Tang, L., Wang, J.: Size effect of tip mass on performance of cantilevered piezoelectric energy harvester with a dynamic magnifier. Acta Mech. 228(11), 3997–4015 (2017)

    Article  MathSciNet  Google Scholar 

  60. Karami, A., Inman, D.J.: Electromechanical modeling of the low frequency zigzag micro energy harvester. J. Intell. Mater. Syst. Struct. 22(3), 271–282 (2011)

    Article  Google Scholar 

  61. Zhou, S., Hobeck, J.D., Cao, J., Inman, D.J.: Analytical and experimental investigation of flexible longitudinal zigzag structures for enhanced multi-directional energy harvesting. Smart Mater. Struct. 26, 035008 (2017)

    Article  Google Scholar 

  62. Lumentut, M.F., Francis, L.A., Howard, I.M.: Analytical techniques for broadband multielectromechanical piezoelectric bimorph beams with multifrequency power harvesting. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 59, 1555–68 (2012)

    Article  Google Scholar 

  63. Zhang, H., Afzalul, K.: Design and analysis of a connected broadband multi-piezoelectric-bimorph-beam energy harvester. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 61, 1016–1023 (2014)

    Article  Google Scholar 

  64. Lien, I.C., Shu, Y.C., Wu, W.J., Lin, H.C.: Revisit of series-SSHI with comparisons to other interfacing circuits in piezoelectric energy harvesting. Smart Mater. Struct. 19, 125009 (2010)

    Article  Google Scholar 

  65. Lin, H.C., Wu, P.H., Lien, I.C., Shu, Y.C.: Analysis of an array of piezoelectric energy harvesters connected in series. Smart Mater. Struct. 22, 094026 (2013)

    Article  Google Scholar 

  66. Wu, P.H., Shu, Y.C.: Finite element modeling of electrically rectified piezoelectric energy harvesters. Smart Mater. Struct. 24, 094008 (2015)

    Article  Google Scholar 

  67. Lumentut, M.F., Howard, I.M.: Effect of shunted piezoelectric control for tuning piezoelectric power harvesting system responses—analytical techniques. Smart Mater. Struct. 24, 105029 (2015)

    Article  Google Scholar 

  68. Lumentut, M.F., Howard, I.M.: Electromechanical analysis of an adaptive piezoelectric energy harvester controlled by two segmented electrodes with shunt circuit networks. Acta Mech. 228(4), 1321–1341 (2017)

    Article  Google Scholar 

  69. Hobbs, W.B., Hu, D.L.: Tree-inspired piezoelectric energy harvesting. J. Fluids Struct. 28, 103–114 (2012)

    Article  Google Scholar 

  70. Barrero-Gil, A., Alonso, G., Sanz-Andres, A.: Energy harvesting from transverse galloping. J. Sound Vib. 329(24), 2873–2883 (2010)

    Article  Google Scholar 

  71. Hémona, P., Amandolesea, X., Andriannec, T.: Energy harvesting from galloping of prisms: a wind tunnel experiment. J. Fluids Struct. 70, 390–402 (2017)

    Article  Google Scholar 

  72. Michelin, S., Doaré, O.: Energy harvesting efficiency of piezoelectric flags in axial flows. J. Fluid Mech. 714, 489–504 (2013)

    Article  MathSciNet  Google Scholar 

  73. Nye, J.F.: Physical Properties of Crystals: Their Representation by Tensors and Matrices. Clarendon Press, Oxford (1984)

    MATH  Google Scholar 

  74. Ikeda, T.: Fundamentals of Piezoelectricity. Oxford University Press, New York (1990)

    Google Scholar 

  75. Tichý, J., Erhart, J., Kittinger, E., Prívratská, J.: Fundamentals of Piezoelectric Sensorics. Springer, Berlin (2010)

    Book  Google Scholar 

  76. Ritz, W.: Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. J. Reine Angew. Math. 135, 1–61 (1909)

    MathSciNet  MATH  Google Scholar 

  77. Courant, R., Hilbert, D.: Methoden der mathematischen Physik/English Ed.: Methods of Mathematical Physics. Interscience Publishers, New York (1953–1962)

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Lumentut, M.F., Friswell, M.I. A smart pipe energy harvester excited by fluid flow and base excitation. Acta Mech 229, 4431–4458 (2018). https://doi.org/10.1007/s00707-018-2235-y

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