Nonlinear Dynamics

, Volume 95, Issue 4, pp 3019–3039 | Cite as

Experimental investigation into the nonlinear dynamics of a bistable laminate

  • Samir A. EmamEmail author
  • Jared Hobeck
  • Daniel J. Inman
Original Paper


An experimental study of the single-well and twin-well, also referred to as intra-well and inter-well, respectively, nonlinear dynamics of a bistable composite laminate is presented. An asymmetric four-ply [0/90/0/90] carbon fiber laminate with two cylindrical stable equilibria supported at its center and free at all boundaries is used for the experimental testing. The mechanical bistability makes the laminate able to snap from one stable equilibrium to the other when displacement reaches a critical value. This snapthrough motion is highly nonlinear and associated with large-amplitude vibrations. This property opens chances for bistable laminates in morphing and energy-harvesting applications. An electromechanical shaker is used to excite the laminate at its center by a controlled amplitude and frequency excitation. The dynamic response of the laminate is measured using a Polytec laser vibrometer. In addition, four macro-fiber composite actuators are attached to the laminate to measure the output voltage due to vibration and hence assess its energy-harvesting capability. The laminate’s natural frequencies and damping under small-amplitude excitations that would match the natural frequencies of an underlying linear system are experimentally identified. A primary resonance excitation of the first bending mode is carried out using amplitude sweep and frequency sweep. It is shown that in both cases, the response starts as a small-amplitude single-well vibration around the static equilibrium position. Varying the excitation amplitude or frequency, the response exhibits a period-doubling bifurcation associated with higher levels of vibration. As the excitation conditions are varied further, the laminate snaps from one equilibrium position to the other. This snapthrough motion is characterized to be of a frequency that is half the natural frequency of the excited mode. It is shown that the period-doubling cascade is responsible for the escape from the single-well to the twin-well response. A secondary Hopf bifurcation that creates a period-three motion and a chaotic snapthrough were also identified. Two low-frequency interesting modes that are attributed to the elasticity of the midpoint support, referred to as rocking modes, were identified using a hammer. These modes got repeatedly excited via subharmonic resonance of order two. In terms of the energy harvesting, the snapthrough motion was found to greatly enhance the energy extraction capability. The frequency band of significant power generation was found to be about 25% of the excitation frequency, which is a good ratio for a single excitation frequency.


Nonlinear dynamics Experimental Bistable laminate Snapthrough Energy harvesting Period-doubling cascade 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.


  1. 1.
    Hyer, M.W.: Some observations on the cured shape of thin unsymmetric laminates. J. Compos. Mater. 15(eq2), 175–194 (1981)CrossRefGoogle Scholar
  2. 2.
    Hyer, M.W.: The room-temperature shapes of four-layer unsymmetric cross-ply laminates. J. Compos. Mater. 16(4), 318–340 (1982)CrossRefGoogle Scholar
  3. 3.
    Dano, M.L., Hyer, M.W.: Thermally induced deformation behavior of unsymmetric laminates. Int. J. Solids Struct. 35, 2101–2120 (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dano, M.L., Hyer, M.W.: Snap-through of unsymmetric fiber-reinforced composite laminates. Int. J. Solids Struct. 39(1), 175–198 (2002)CrossRefzbMATHGoogle Scholar
  5. 5.
    Diaconu, C.G., Weaver, P.M., Mattioni, F.: Concepts for morphing airfoil sections using bi-stable laminated composite structures. Thin Walled Struct. 46(6), 689–701 (2008)CrossRefGoogle Scholar
  6. 6.
    Cantera, M.A., Romera, J.M., Adarraga, I., Mujika, F.: Modelling and testing of the snap-through process of bi-stable cross-ply composites. Compos. Struct. 120, 41–52 (2015)CrossRefGoogle Scholar
  7. 7.
    Bowen, C.R., Giddings, P.F., Salo, A.I.T., Kim, H.A.: Modeling and characterization of piezoelectrically actuated bistable composites. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58(9), 1737–1750 (2011)CrossRefGoogle Scholar
  8. 8.
    Gude, M., Hufenbach, W., Kirvel, C.: Piezoelectrically driven morphing structures based on bistable unsymmetric laminates. Compos. Struct. 93(2), 377–382 (2011)CrossRefGoogle Scholar
  9. 9.
    Bowen, C.R., Betts, D.N., Giddings, P.F., Salo, A.I.T., Kim, H.A.: A study of bistable laminates of generic lay-up for adaptive structures. Strain 48(3), 235–240 (2012)CrossRefGoogle Scholar
  10. 10.
    Fernandes, F., Maurini, C., Vidoli, S.: Multiparameter actuation for shape control of bistable composite plates. Int. J. Solids Struct. 47(10), 1449–1458 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Schultz, M.R., Hyer, M.W.: Snap-through of unsymmetric cross-ply laminates using piezoelectric actuators. J. Intell. Mater. Syst. Struct. 14(12), 795–814 (2003)CrossRefGoogle Scholar
  12. 12.
    Bowen, C.R., Butler, R., Jervis, V., Kim, H.A., Salo, A.I.T.: Morphing and shape control using unsymmetrical composites. J. Intell. Mater. Syst. Struct. 18(1), 89–98 (2007)CrossRefGoogle Scholar
  13. 13.
    Betts, David N., Kim, H.Alicia, Bowen, Christopher R.: Modeling and optimization of bistable composite laminates for piezoelectric actuation. J. Intell. Mater. Syst. Struct. 22(18), 2181–2191 (2011)CrossRefGoogle Scholar
  14. 14.
    Hufenbach, W., Gude, M., Kroll, L.: Design of multistable composites for application in adaptive structures. Compos. Sci. Technol. 62(16), 2201–2207 (2002)CrossRefGoogle Scholar
  15. 15.
    Mattioni, F., Weaver, P.M., Potter, K.D., Friswell, M.I.: The application of thermally induced multistable composites to morphing aircraft structures. Proc. SPIE 6930, 693012 (2008)CrossRefGoogle Scholar
  16. 16.
    Nicassio, F., Scarselli, G., Pinto, F., Ciampa, F., Iervolino, O., Meo, M.: Low energy actuation technique of bistable composites for aircraft morphing. Aerosp. Sci. Technol. 75, 35–46 (2018)CrossRefGoogle Scholar
  17. 17.
    Diaconu, C.G., Weaver, P.M., Arrieta, A.F.: Dynamic analysis of bi-stable composite plates. J. Sound Vib. 322(4–5), 987–1004 (2009)CrossRefGoogle Scholar
  18. 18.
    Vogl, G.A., Hyer, M.W.: Natural vibration of unsymmetric cross-ply laminates. J. Sound Vib. 330(20), 4764–4779 (2011)CrossRefGoogle Scholar
  19. 19.
    Arrieta, A.F., Neild, S.A., Wagg, D.J.: Nonlinear dynamic response and modeling of a bistable composite plate for applications to adaptive structures. Nonlinear Dyn. 58(1), 259–272 (2009)CrossRefzbMATHGoogle Scholar
  20. 20.
    Arrieta, A.F., Wagg, D.J., Neild, S.A.: Dynamic snap-through for morphing of bistable composite plates. J. Intell. Mater. Syst. Struct. 22(2), 103–112 (2011)CrossRefGoogle Scholar
  21. 21.
    Arrieta, A.F., Neild, Simon A., Wagg, David J.: On the cross-well dynamics of a bi-stable composite plate. J. Sound Vib. 330(14), 3424–3441 (2011)CrossRefGoogle Scholar
  22. 22.
    Wu, Z., Li, H., Friswell, M.: Advanced nonlinear dynamic modelling of bi-stable composite plates. Compos. Struct. 201, 582–596 (2018)CrossRefGoogle Scholar
  23. 23.
    Shaw, A.D., Neild, S.A., Wagg, D.J., Weaver, P.M., Carrella, A.: A nonlinear spring mechanism incorporating a bistable composite plate for vibration isolation. J. Sound Vib. 332, 6265–6275 (2013)CrossRefGoogle Scholar
  24. 24.
    Taki, M.S., Tikani, R., Ziaei-Rad, S., Firouzian-Nejad, A.: Dynamic responses of cross-ply bi-stable composite laminates with piezoelectric layers. Arch. Appl. Mech. 86, 1003–1018 (2016)CrossRefGoogle Scholar
  25. 25.
    Borowiec, M., Rysak, A., Betts, D.N., Bowen, C.R., Kim, H.A., Litak, G.: Complex response of a bistable laminated plate: multiscale entropy analysis. Eur. Phys. J. Plus 129, 211 (2014)CrossRefGoogle Scholar
  26. 26.
    Thompson, J.M.T.: Chaotic phenomena triggering the escape from a potential well. Proc. R. Soc. Lond. A 421, 195–225 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Virgin, L., Plaut, R.H., Cheng, C.: Prediction of escape from a potential well under harmonic excitation. Int. J. Nonlinear Mech. 21, 357–365 (1992)CrossRefGoogle Scholar
  28. 28.
    Stewart, H.B., Thompson, J.M.T., Ueda, Y., Lansbury, A.N.: Optimal escape from potential wells—patterns of regular and chaotic bifurcations. Physica D 85, 259–295 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zarepoor, M., Bilgen, O.: Constrained-energy cross-well actuation of bistable structures. AIAA J. 54(9), 2905–2908 (2016)CrossRefGoogle Scholar
  30. 30.
    Udani, J.P., Arrieta, A.F.: Efficient potential well escape for bi-stable Duffing Oscillators. Nonlinear Dyn. 92, 1045–1059 (2018)CrossRefGoogle Scholar
  31. 31.
    Bilgen, O., Arrieta, A.F., Friswell, M.W., Hagedorn, P.: Dynamic control of a bistable wing under aerodynamic loading. Smart Mater. Struct. 22, 025020 (2013)CrossRefGoogle Scholar
  32. 32.
    Arrieta, A.F., Hagedorn, P., Erturk, A., Inman, D.J.: A piezoelectric bistable plate for nonlinear broadband energy harvesting. Appl. Phys. Lett. 97, 104102 (2010)CrossRefGoogle Scholar
  33. 33.
    Betts, D.N., Bowen, C.R., Kim, H.A., Gathercole, N., Clarke, C.T., Inman, D.J.: Nonlinear dynamics of a bistable piezoelectric composite energy harvester for broadband application. Eur. Phys. J. Spec. Top. 222(7), 1553–1562 (2013)CrossRefGoogle Scholar
  34. 34.
    Betts, D.N., Kim, H.A., Bowen, C.R.: Modeling and optimization of bistable composite laminates for piezoelectric actuation. J. Intell. Mater. Syst. Struct. 22(18), 2181–2191 (2011)CrossRefGoogle Scholar
  35. 35.
    Betts, D.N., Kim, H.A., Bowen, C.R., Inman, D.J.: Optimal configurations of bistable piezo-composites for energy harvesting. Appl. Phys. Lett. 100(11), 114104 (2012)CrossRefGoogle Scholar
  36. 36.
    Syta, A., Bowen, C.R., Kim, H.A., Rysak, A., Litak, G.: Experimental analysis of the dynamical response of energy harvesting devices based on bistable laminated plates. Meccanica 50, 1961–1970 (2015)CrossRefGoogle Scholar
  37. 37.
    Syta, A., Bowen, C.R., Kim, H.A., Rysak, A., Litak, G.: response of bistable piezoelectric-composite energy harvester by means of recurrences. Mech. Syst. Signal Process. 76–77, 823–832 (2016)CrossRefGoogle Scholar
  38. 38.
    Pellegrini, S.P., Tolou, N., Schenk, M., Herder, J.L.: Bistable vibration energy harvesters: a review. J. Intell. Mater. Syst. Struct. 24(11), 1303–1312 (2013)CrossRefGoogle Scholar
  39. 39.
    Harne, R.L., Wang, K.W.: A review of the recent research on vibration energy harvesting via bistable systems. Smart Mater. Struct. 22(2), 023001 (2013)CrossRefGoogle Scholar
  40. 40.
    Emam, S.A., Inman, D.J.: A review on bistable composite laminates for morphing and energy harvesting. Appl. Mech. Rev. 67(6), 060803 (2015)CrossRefGoogle Scholar
  41. 41.
    Tufillaro, N.B., Abbott, T., Reilly, J.: An Experimental Approach to Nonlinear Dynamics and Chaos. Addison-Wesley, Redwood City (1992)zbMATHGoogle Scholar
  42. 42.
    Carrella, A., Mattioni, A. F., Diaz, A.A., Friswell, M.I., Wagg, D.J., Weaver, P.M.: Static and dynamic analysis of a bistable plate for application in morphing structures. In: 7th European Conference on Structural Dynamics, EURODYN 2008, 7–9 July, Southampton, UK (2008)Google Scholar
  43. 43.
    Carrella, A., Friswell, M.I., Pirrera, A., Aglietti, G.S.: Numerical and experimental analysis of a square bistable plate. In: Proceedings of ISMA2008, Noise and Vibration Engineering Conference, 15–17 Sept., Leuven, Belgium (2008)Google Scholar
  44. 44.
    Samuel, S.C., McGehee, C.C., Mann, B.P.: Nonlinear dynamics for broadband energy harvesting: investigation of a bistable piezoelectric inertial generator. Phys. D Nonlinear Phenom. 239, 640–653 (2010)CrossRefzbMATHGoogle Scholar
  45. 45.
    Virgin, L.N.: Introduction to Experimental Nonlinear Dynamics. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  46. 46.
    Ewins, D.J.: Modal Testing: Theory, Practice and Application. Research Studies Press LTD, Baldock (2000)Google Scholar
  47. 47.
    Nayfeh, A.H.: Nonlinear Interactions, Analytical, Computational, and Experimental Methods. Wiley, New York (2000). ISBN-13: 978-0471175919zbMATHGoogle Scholar
  48. 48.
    Marsden, J.E., McCracken, M.: The Hopf Bifurcation and its Applications. Springer, New York (1976)CrossRefzbMATHGoogle Scholar
  49. 49.
    Harb, A.M.: Application of Bifurcation Theory to Subsynchronous Resonance in Power Systems, PhD dissertation, Virginia Tech, Blacksburg, VA, USA (1996)Google Scholar
  50. 50.
    Soliman, M.S.: Indeterminate secondary Hopf bifurcations in nonlinear oscillators. Phys. Rev. E 56, 4857 (1996)CrossRefGoogle Scholar
  51. 51.
    Caton, F., Hopfinger, E.: Primary and secondary Hopf bifurcations in stratified Taylor Couette flow. Phys. Rev. Lett. 82(23), 4647–4650 (1999)CrossRefGoogle Scholar
  52. 52.
    Emam, S.A.: A theoretical and experimental study of nonlinear dynamics of buckled beams, PhD dissertation, Virginia Tech, Blacksburg, VA, USA (2002)Google Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringAmerican University of SharjahSharjahUnited Arab Emirates
  2. 2.Faculty of EngineeringZagazig UnversityZagazigEgypt
  3. 3.Department of Mechanical and Nuclear EngineeringKansas State UniversityManhattanUSA
  4. 4.Department of Aerospace EngineeringUniversity of MichiganAnn ArborUSA

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