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Stochastic response of a piezoelectric ribbon-substrate structure under Gaussian white noise

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Abstract

Due to the excellent piezoelectric and ferroelectric properties of lead zirconate titanate (PZT), the buckled PZT ribbon-substrate structure has been widely used in the design of wearable electronic devices. However, wearable electronic devices must work in a complex vibration environment and are subjected to random excitations such as irregular human body motion. Hence, the reliability of the buckled piezoelectric ribbon-substrate structure in a dynamic context needs to be ensured. In this paper, Gaussian white noise is introduced to describe the broadband random environment. A voltage is applied to the PZT ribbon to realize the desired wavy configuration of the ribbon to influence the dynamic responses of this structure. Based on the Euler–Bernoulli beam theory and the Lagrange equation, the governing equation of the buckled piezoelectric ribbon-substrate structure is derived. By using a stochastic averaging method, the stationary probability density of stochastic responses of the buckled piezoelectric ribbon-substrate structure is obtained. Several numerical examples are analysed to reveal the effects of the intensity of the Gaussian white noise and the voltage applied to the PZT ribbon on the stochastic responses of this buckled structure. Through these numerical results, it can be found that when the applied voltage is above the critical voltage value, the piezoelectric ribbon would wrinkle into multiple small waves on top of the soft substrate. By increasing the applied voltage while keeping the intensity of noise excitation constant, the static buckling amplitude increases, and the number of transitions between two equilibrium positions decreases, which implies that the stretchability and stability of the ribbon-substrate structure would be improved. The results of this paper can be helpful for the design of robust piezoelectric ribbon-based stretchable electronics.

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References

  1. Li, J., Zhao, J., Rogers, J.A.: Materials and designs for power supply systems in skin-interfaced electronics. Acc. Chem. Res. 52(1), 53–62 (2019)

    Article  Google Scholar 

  2. Chen, D., Pei, Q.: Electronic muscles and skins: a review of soft sensors and actuators. Chem. Rev. 117(17), 11239–11268 (2017)

    Article  Google Scholar 

  3. Dagdeviren, C., Joe, P., Tuzman, O.L., Park, K.I., Lee, K.J., Shi, Y., Huang, Y., Rogers, J.A.: Recent progress in flexible and stretchable piezoelectric devices for mechanical energy harvesting, sensing and actuation. Extreme Mech. Lett. 9, 269–281 (2016)

    Article  Google Scholar 

  4. Ma, Y., Zhang, Y., Cai, S., Han, Z., Liu, X., Wang, F., Cao, Y., Wang, Z., Li, H., Chen, Y., Feng, X.: Flexible hybrid electronics for digital healthcare. Adv. Mater. 32, 1902062 (2019)

    Article  Google Scholar 

  5. Li, G.Y., Xu, G., Zheng, Y., Cao, Y.: Non-leaky modes and bandgaps of surface acoustic waves in wrinkled stiff-film/compliant-substrate bilayers. J. Mech. Phys. Solids 112, 239–252 (2018)

    Article  Google Scholar 

  6. Wu, H., Huang, Y., Xu, F., Duan, Y., Yin, Z.J.A.M.: Energy harvesters for wearable and stretchable electronics: from flexibility to stretchability. Adv. Mater. 28, 9881–9919 (2016)

    Article  Google Scholar 

  7. Wang, Y., Ma, T., Yu, H., Jiang, H.: Random analysis on controlled buckling structure for energy harvesting. Appl. Phys. Lett. 102, 041915 (2013)

    Article  Google Scholar 

  8. Pillatsch, P., Yeatman, E.M., Holmes, A.S.: A piezoelectric frequency up-converting energy harvester with rotating proof mass for human body applications. Sens. Actuators A 206, 178–185 (2014)

    Article  Google Scholar 

  9. Jeon, S.B., Kim, D., Seol, M.L., Park, S.J., Choi, Y.K.: 3-Dimensional broadband energy harvester based on internal hydrodynamic oscillation with a package structure. Nano Energy 17, 82–90 (2015)

    Article  Google Scholar 

  10. Kang, Y., Wang, B., Dai, S., Liu, G., Pu, Y., Hu, C.: Folded elastic strip-based triboelectric nanogenerator for harvesting human motion energy for multiple applications. ACS Appl. Mater. Interfaces 7(36), 20469–20476 (2015)

    Article  Google Scholar 

  11. Bai, Y., Tofel, P., Hadas, Z., Smilek, J., Losak, P., Skarvada, P., Macku, R.: Investigation of a cantilever structured piezoelectric energy harvester used for wearable devices with random vibration input. Mech. Syst. Signal Process. 106, 303–318 (2018)

    Article  Google Scholar 

  12. Qiao, Y., Xu, W., Sun, J., Zhang, H.: Reliability of electrostatically actuated MEMS resonators to random mass disturbance. Mech. Syst. Signal Process. 121, 711–724 (2019)

    Article  Google Scholar 

  13. Ou, Z., Yao, X., Zhang, X., Fan, X.: Dynamic stability of flexible electronic structures under step loads. Eur. J. Mech. A. Solids 58, 247–255 (2016)

    Article  MathSciNet  Google Scholar 

  14. Wang, Y., Feng, X., Lu, B., Wang, G.: Surface effects on the mechanical behavior of buckled thin film. J. Appl. Mech. 80(2), 021002–021009 (2013)

    Article  Google Scholar 

  15. Wang, B., Bi, H., Wang, Y., Ouyang, H., Deng, Z.: Nonlinear vibration of buckled nanowires on a compliant substrate. Appl. Math. Model. 79, 230–242 (2020)

    Article  MathSciNet  Google Scholar 

  16. Wang, B., Bi, H., Ouyang, H., Wang, Y., Deng, Z.: Dynamic behaviour of piezoelectric nanoribbons with wavy configurations on an elastomeric substrate. Int. J. Mech. Sci. 182, 105 (2020)

    Google Scholar 

  17. Jin, Y., Zhang, Y.: Dynamics of a delayed Duffing-type energy harvester under narrow-band random excitation. Acta Mech. 232, 1045–1060 (2021)

    Article  MathSciNet  Google Scholar 

  18. Yang, T., Cao, Q.: Dynamics and energy generation of a hybrid energy harvester under colored noise excitations. Mech. Syst. Signal Process. 121, 745–766 (2019)

    Article  Google Scholar 

  19. Bonnin, M., Traversa, F.L., Bonani, F.: Analysis of influence of nonlinearities and noise correlation time in a single-DOF energy-harvesting system via power balance description. Nonlinear Dyn. 100, 119–133 (2020)

    Article  Google Scholar 

  20. Hu, R.C., Xiong, H., Jin, W.L., Zhu, W.Q.: Stochastic minimax semi-active control for MDOF nonlinear uncertain systems under combined harmonic and wide-band noise excitations using MR dampers. Int. J. Non-Linear Mech. 83, 26–38 (2016)

    Article  Google Scholar 

  21. Tabejieu, L.M.A., Nbendjo, B.R.N., Filatrella, G., Woafo, P.: Amplitude stochastic response of Rayleigh beams to randomly moving loads. Nonlinear Dyn. 89(2), 925–937 (2017)

    Article  Google Scholar 

  22. Su, M., Xu, W., Yang, G.: Stochastic response and stability of system with friction and a rigid barrier. Mech. Syst. Signal Process. 132, 748–761 (2019)

    Article  Google Scholar 

  23. Pei, B., Xu, Y., Wu, J.L.: Stochastic averaging for stochastic differential equations driven by fractional Brownian motion and standard Brownian motion. Appl. Math. Lett. 100, 106006 (2020)

    Article  MathSciNet  Google Scholar 

  24. Huan, R.H., Pu, D., Wei, X.Y.: Phase switch in the stochastic response of a micromechanical beam resonator. Acta Mech. 229(5), 2177–2187 (2018)

    Article  MathSciNet  Google Scholar 

  25. Lan, J., Wu, Y.J.: First-exit problem of MDOF strongly nonlinear oscillators under wide-band random excitations without internal resonances. Acta Mech. 228(1), 175–186 (2017)

    Article  MathSciNet  Google Scholar 

  26. Jia, W., Zhu, W.: Stochastic averaging of quasi-integrable and non-resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations. Nonlinear Dyn. 76(2), 1271–1289 (2014)

    Article  MathSciNet  Google Scholar 

  27. Zhang, C., Harne, R.L., Li, B., Wang, K.W.: Statistical quantification of DC power generated by bistable piezoelectric energy harvesters when driven by random excitations. J. Sound Vib. 442, 770–786 (2019)

    Article  Google Scholar 

  28. Mokem Fokou, I.S., Nono Dueyou Buckjohn, C., Siewe, M., Tchawoua, C.: Probabilistic behavior analysis of a sandwiched buckled beam under Gaussian white noise with energy harvesting perspectives. Chaos Solitons Fractals 92, 101–114 (2016)

    Article  MathSciNet  Google Scholar 

  29. Ge, G., Bo, Z.: Response of a cantilever model with a surface crack under basal white noise excitation. Comput. Math. Appl. 76(11), 2728–2743 (2018)

    Article  MathSciNet  Google Scholar 

  30. Qi, Y., Kim, J., Nguyen, T.D., Lisko, B., Purohit, P.K., McAlpine, M.C.: Enhanced piezoelectricity and stretchability in energy harvesting devices fabricated from buckled PZT ribbons. Nano Lett. 11(3), 1331–1336 (2011)

    Article  Google Scholar 

  31. Feng, X., Yang, B.D., Liu, Y., Wang, Y., Dagdeviren, C., Liu, Z., Carlson, A., Li, J., Huang, Y., Rogers, J.A.: Stretchable ferroelectric nanoribbons with wavy configurations on elastomeric substrates. ACS Nano 5(4), 3326–3332 (2011)

    Article  Google Scholar 

  32. Li, Y., Fang, B., Zhang, J., Song, J.: Surface effects on the wrinkling of piezoelectric films on compliant substrates. J. Appl. Phys. 110, 114303 (2011)

    Article  Google Scholar 

  33. Li, Y.S., Ren, J.H., Feng, W.J.: Bending of sinusoidal functionally graded piezoelectric plate under an in-plane magnetic field. Appl. Math. Model. 47, 63–75 (2017)

    Article  MathSciNet  Google Scholar 

  34. Ahmadi, M., Ansari, R., Darvizeh, M.: Free and forced vibrations of atomic force microscope piezoelectric cantilevers considering tip-sample nonlinear interactions. Thin-Walled Struct. 145, 106382 (2019)

    Article  Google Scholar 

  35. Yan, Z., Jiang, L.Y.: Vibration and buckling analysis of a piezoelectric nanoplate considering surface effects and in-plane constraints. Proc. R. Soc. A Math. Phys. Eng. Sci. 468(2147), 3458–3475 (2012)

    MathSciNet  MATH  Google Scholar 

  36. Dagdeviren, C., Hwang, S., Su, Y., Kim, S., Cheng, H., Gur, O., Haney, R., Omenetto, F., Huang, Y., Rogers, J.A.: Transient, biocompatible electronics and energy harvesters based on ZnO. Small 9, 3398–3404 (2013)

    Article  Google Scholar 

  37. Yin, S.F., Li, B., Cao, Y.P., Feng, X.Q.: Surface wrinkling of anisotropic films bonded on a compliant substrate. Int. J. Solids Struct. 141, 219–231 (2018)

    Article  Google Scholar 

  38. Song, J.: Mechanics of stretchable electronics. Curr. Opin. Solid State Mater. Sci. 19(3), 160–170 (2015) 

    Article  Google Scholar 

  39. Huang, Z.Y., Hong, W., Suo, Z.: Nonlinear analyses of wrinkles in a film bonded to a compliant substrate. J. Mech. Phys. Solids 53(9), 2101–2118 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Khan, M.B., Kim, D.H., Han, J.H., Saif, H., Lee, H., Lee, Y., Kim, M., Jang, E., Hong, S.K., Joe, D.J., Lee, T.I., Kim, T.S., Lee, K.J., Lee, Y.: Performance improvement of flexible piezoelectric energy harvester for irregular human motion with energy extraction enhancement circuit. Nano. Energy 58, 211–219 (2019)

    Article  Google Scholar 

  41. Fan, K., Tan, Q., Liu, H., Zhang, Y., Cai, M.: Improved energy harvesting from low-frequency small vibrations through a monostable piezoelectric energy harvester. Mech. Syst. Signal Process. 117, 594–608 (2019)

    Article  Google Scholar 

  42. Kumar, P., Narayanan, S., Gupta, S.: Bifurcation analysis of a stochastically excited vibro-impact Duffing-Van der Pol oscillator with bilateral rigid barriers. Int. J. Mech. Sci. 127, 103–117 (2017)

    Article  Google Scholar 

  43. Wang, Y., Yu, Z., Mao, G., Liu, Y., Liu, G., Shang, J., Qu, S., Chen, Q., Li, R.W.: Printable liquid-metal@PDMS stretchable heater with high stretchability and dynamic stability for wearable thermotherapy. Adv. Mater. Technol. 4(2), 1800435 (2019)

    Article  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the support from the National Natural Science Foundation of China (No. 11802319) and the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and astronautics No. MCMS-E-0221K01). The first author also would like to acknowledge the support from Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (No. CX202044). Part of this work is done during the second author’s visits to the University of Liverpool under the third author’s guidance.

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Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, China

    Haohao Bi

  2. Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an, 710072, China

    Bo Wang & Zichen Deng

  3. School of Engineering, University of Liverpool, Liverpool, L69 3GH, UK

    Huajiang Ouyang

  4. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics & Astronautics, Nanjing, 210016, China

    Yan Shi

  5. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an, 710072, China

    Zichen Deng

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  1. Haohao Bi
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Correspondence to Bo Wang or Huajiang Ouyang.

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Bi, H., Wang, B., Ouyang, H. et al. Stochastic response of a piezoelectric ribbon-substrate structure under Gaussian white noise. Acta Mech 232, 3687–3700 (2021). https://doi.org/10.1007/s00707-021-03026-0

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  • DOI: https://doi.org/10.1007/s00707-021-03026-0

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