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

  • M. F. Lumentut
  • M. I. Friswell
Original Paper


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 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. 2.
    Housner, G.W.: Bending vibrations of a pipe line containing flowing fluid. J. Appl. Mech. 19, 205–208 (1952)Google Scholar
  3. 3.
    Niordson, F.I.: Vibrations of a cylindrical tube containing flowing fluid. Kungliga Tekniska Hogskolans Handlingar (Stockholm) No. 73 (1953)Google Scholar
  4. 4.
    Holmes, P.J.: Pipes supported at both ends cannot flutter. J. Appl. Mech. 45, 619–622 (1978)CrossRefzbMATHGoogle Scholar
  5. 5.
    Heinrich, G.: Schwingungen durchströmter Rohre (vibrations of pipes with flow). Z. Angew. Math. Mech. 36, 417–427 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benjamin, T.B.: Dynamics of a system of articulated pipes conveying fluid. I. Theory. Proc. R. Soc. Lond. A 261, 457–486 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benjamin, T.B.: Dynamics of a system of articulated pipes conveying fluid. II. Experiments. Proc. R. Soc. Lond. A 261, 487–499 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bottema, O.: On the stability of equilibrium of a linear mechanical system. Z. Angew. Math. Mech. 6, 97–104 (1955)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Smith, T.E., Herrmann, G.: Stability of circulatory elastic systems in the presence of magnetic damping. Acta Mech. 12, 175–188 (1971)CrossRefzbMATHGoogle Scholar
  10. 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)CrossRefzbMATHGoogle Scholar
  11. 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)CrossRefGoogle Scholar
  12. 12.
    Sokolnikoff, I.S.: Mathematical Theory of Elasticity, 2nd edn. McGraw-Hill, Maidenherd (1956)zbMATHGoogle Scholar
  13. 13.
    Païdoussis, M.P., Issid, N.T.: Dynamic stability of pipes conveying fluid. J. Sound Vib. 33, 267–294 (1974)CrossRefGoogle Scholar
  14. 14.
    Païdoussis, M.P., Li, G.X.: Pipes conveying fluid: a model dynamical problem. J. Fluids Struct. 7, 137–204 (1993)CrossRefGoogle Scholar
  15. 15.
    Ruta, G.C., Elishakoff, I.: Towards the resolution of the Smith–Herrmann paradox. Acta Mech. 173, 89–105 (2004)CrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)CrossRefGoogle Scholar
  18. 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)CrossRefGoogle Scholar
  19. 19.
    Lavooij, C.S.W., Tijsseling, A.S.: Fluid–structure interaction in liquid-filled piping systems. J. Fluids Struct. 5, 573–95 (1991)CrossRefGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 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)CrossRefGoogle Scholar
  22. 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)CrossRefGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. 24.
    Païdoussis, M.P.: Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 1, 2nd edn. Academic, London (2014)Google Scholar
  25. 25.
    Païdoussis, M.P.: Aspirating pipes do not flutter at infinitesimally small flow. J. Fluids Struct. 13, 419–425 (1999)CrossRefGoogle Scholar
  26. 26.
    Kuiper, G.L., Metrikine, A.V.: Dynamic stability of a submerged, free-hanging riser conveying fluid. J. Sound Vib. 280, 1051–1065 (2005)CrossRefGoogle Scholar
  27. 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)CrossRefGoogle Scholar
  28. 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)CrossRefGoogle Scholar
  29. 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)CrossRefGoogle Scholar
  30. 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)CrossRefzbMATHGoogle Scholar
  31. 31.
    Krommer, M.: On the correction of the Bernoulli–Euler beam theory for smart piezoelectric beams. Smart Mater. Struct. 10, 668–680 (2001)CrossRefGoogle Scholar
  32. 32.
    Fernandes, A., Pouget, J.: Analytical and numerical approach to piezoelectric bimorph. Int. J. Solids Struct. 40, 4331–52 (2003)CrossRefzbMATHGoogle Scholar
  33. 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)CrossRefGoogle Scholar
  34. 34.
    Irschik, H., Ziegler, F.: Eigenstrain without stress and static shape control of structures. AIAA J. 39, 1985–1990 (2001)CrossRefGoogle Scholar
  35. 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)CrossRefGoogle Scholar
  36. 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. 37.
    Irschik, H., Pichler, U.: Dynamic shape control of solids and structures by thermal expansion strains. J. Therm. Stresses 24, 565–576 (2001)CrossRefGoogle Scholar
  38. 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)CrossRefGoogle Scholar
  39. 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)CrossRefzbMATHGoogle Scholar
  40. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 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)CrossRefGoogle Scholar
  42. 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)CrossRefzbMATHGoogle Scholar
  43. 43.
    Krommer, M.: On the influence of pyroelectricity upon thermally induced vibrations of piezothermoelastic plates. Acta Mech. 171, 59–73 (2004)CrossRefzbMATHGoogle Scholar
  44. 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)CrossRefGoogle Scholar
  45. 45.
    Niederberger, D., Morari, M.: An autonomous shunt circuit for vibration damping. Smart Mater. Struct. 15, 359–364 (2006)CrossRefGoogle Scholar
  46. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 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)CrossRefGoogle Scholar
  48. 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)CrossRefGoogle Scholar
  49. 49.
    Goldschmidtboeing, F., Woias, P.: Characterization of different beam shapes for piezoelectric energy harvesting. J. Micromech. Microeng. 18, 104013 (2008)CrossRefGoogle Scholar
  50. 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)CrossRefGoogle Scholar
  51. 51.
    Dalzell, P., Bonello, P.: Analysis of an energy harvesting piezoelectric beam with energy storage circuit. Smart Mater. Struct. 21, 105029 (2012)CrossRefGoogle Scholar
  52. 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)CrossRefGoogle Scholar
  53. 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)CrossRefGoogle Scholar
  54. 54.
    Adhikari, S., Friswell, M.I., Inman, D.J.: Piezoelectric energy harvesting from broadband random vibrations. Smart Mater. Struct. 18, 115005 (2009)CrossRefGoogle Scholar
  55. 55.
    Ali, S.F., Friswell, M.I., Adhikari, S.: Piezoelectric energy harvesting with parametric uncertainty. Smart Mater. Struct. 19, 105010 (2010)CrossRefGoogle Scholar
  56. 56.
    Lumentut, M.F., Howard, I.M.: Intrinsic electromechanical dynamic equations for piezoelectric power harvesters. Acta Mech. 228(2), 631–650 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Friswell, M.I., Adhikari, S.: Sensor shape design for piezoelectric cantilever beams to harvest vibration energy. J. Appl. Phys. 108, 014901 (2010)CrossRefGoogle Scholar
  58. 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)CrossRefGoogle Scholar
  59. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 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)CrossRefGoogle Scholar
  61. 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)CrossRefGoogle Scholar
  62. 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)CrossRefGoogle Scholar
  63. 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)CrossRefGoogle Scholar
  64. 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)CrossRefGoogle Scholar
  65. 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)CrossRefGoogle Scholar
  66. 66.
    Wu, P.H., Shu, Y.C.: Finite element modeling of electrically rectified piezoelectric energy harvesters. Smart Mater. Struct. 24, 094008 (2015)CrossRefGoogle Scholar
  67. 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)CrossRefGoogle Scholar
  68. 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)CrossRefzbMATHGoogle Scholar
  69. 69.
    Hobbs, W.B., Hu, D.L.: Tree-inspired piezoelectric energy harvesting. J. Fluids Struct. 28, 103–114 (2012)CrossRefGoogle Scholar
  70. 70.
    Barrero-Gil, A., Alonso, G., Sanz-Andres, A.: Energy harvesting from transverse galloping. J. Sound Vib. 329(24), 2873–2883 (2010)CrossRefGoogle Scholar
  71. 71.
    Hémona, P., Amandolesea, X., Andriannec, T.: Energy harvesting from galloping of prisms: a wind tunnel experiment. J. Fluids Struct. 70, 390–402 (2017)CrossRefGoogle Scholar
  72. 72.
    Michelin, S., Doaré, O.: Energy harvesting efficiency of piezoelectric flags in axial flows. J. Fluid Mech. 714, 489–504 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  73. 73.
    Nye, J.F.: Physical Properties of Crystals: Their Representation by Tensors and Matrices. Clarendon Press, Oxford (1984)zbMATHGoogle Scholar
  74. 74.
    Ikeda, T.: Fundamentals of Piezoelectricity. Oxford University Press, New York (1990)Google Scholar
  75. 75.
    Tichý, J., Erhart, J., Kittinger, E., Prívratská, J.: Fundamentals of Piezoelectric Sensorics. Springer, Berlin (2010)CrossRefGoogle Scholar
  76. 76.
    Ritz, W.: Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. J. Reine Angew. Math. 135, 1–61 (1909)MathSciNetzbMATHGoogle Scholar
  77. 77.
    Courant, R., Hilbert, D.: Methoden der mathematischen Physik/English Ed.: Methods of Mathematical Physics. Interscience Publishers, New York (1953–1962)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Civil and Mechanical EngineeringCurtin UniversityBentleyAustralia
  2. 2.College of EngineeringSwansea UniversitySwanseaUK

Personalised recommendations