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Propagation of uncertainty in free vibration of Euler–Bernoulli nanobeam

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Abstract

In this paper, Euler–Bernoulli nanobeam based on the framework of Eringen’s nonlocal theory is modeled with material uncertainties where the uncertainties are associated with mass density and Young’s modulus in terms of fuzzy numbers. A particular type of imprecisely defined number, namely triangular fuzzy number, is taken into consideration. In this regard, double parametric-based Rayleigh–Ritz method has been developed to handle the uncertainties. Vibration characteristics have been investigated, and the propagation of uncertainties in frequency parameters is analyzed. Material uncertainties are considered with respect to three cases, viz. (1) Young’s modulus (2) mass density and (3) both Young’s modulus and mass density, as imprecisely defined. Frequency parameters and mode shapes are computed and presented for Pined–Pined (P–P) and Clamped–Clamped (C–C) boundary conditions. Accuracy and efficiency of the models are verified by conducting the convergence study for all the three cases. Lower and upper bounds of frequency parameters are computed with the help of the double parameter, and graphical results are plotted as the triangular fuzzy number showing the sensitivity of the models. Obtained results for frequency parameters are compared with other well-known results found in previously published literature(s) in special cases (crisp cases) witnessing robust agreement. The uncertainty modeling and the bounds of frequency parameters may serve as an effective tool for the designing and optimal quality enhancement of engineering structures.

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References

  1. Malik M, Dang HH (1998) Vibration analysis of continuous systems by differential transformation. Appl Math Comput 96(1):17–26

    MathSciNet  MATH  Google Scholar 

  2. Pradhan S, Murmu T (2010) Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever. Physica E 42(7):1944–1949

    Article  Google Scholar 

  3. Wang C, Zhang Y, He X (2007) Vibration of nonlocal Timoshenko beams. Nanotechnology 18:105–113

    Google Scholar 

  4. Jena SK, Chakraverty S (2018) Free vibration analysis of Euler–Bernoulli nanobeam using differential transform method International Journal of Computational. Mater Sci Eng 7:1850020

    Google Scholar 

  5. Chakraverty S, Jena SK (2018) Free vibration of single walled carbon nanotube resting on exponentially varying elastic foundation. Curved Layer Struct 5:260–272

    Article  Google Scholar 

  6. Jena SK, Chakraverty S (2018) Free vibration analysis of variable cross-section single layered graphene nano-ribbons (SLGNRs) using differential quadrature method. Front Built Environ 4:63

    Article  Google Scholar 

  7. Jena SK, Chakraverty S (2018) Free vibration analysis of single walled carbon nanotube with exponentially varying stiffness. Curved Layer Struct 5:201–212

    Article  Google Scholar 

  8. Jena RM, Chakraverty S (2018) Residual power series method for solving time-fractional model of vibration equation of large membranes. J Appl Comput Mech 5:603–615

    Google Scholar 

  9. Jena SK, Chakraverty S (2019) Differential quadrature and differential transformation methods in buckling analysis of nanobeams. Curved Layer Struct 6:68–76

    Article  Google Scholar 

  10. Jena SK, Chakraverty S, Jena RM, Tornabene F (2019) A novel fractional nonlocal model and its application in buckling analysis of Euler-Bernoulli nanobeam. Mater Res Express 6(055016):1–17

    Google Scholar 

  11. Jena SK, Chakraverty S, Tornabene F (2019) Vibration characteristics of nanobeam with exponentially varying flexural rigidity resting on linearly varying elastic foundation using differential quadrature method. Mater Res Express 6(085051):1–13

    Google Scholar 

  12. Jena SK, Chakraverty S, Tornabene F (2019) Dynamical behavior of nanobeam embedded in constant, linear, parabolic and sinusoidal types of winkler elastic foundation using first-order nonlocal strain gradient model. Mater Res Express 6(0850f2):1–23

    Google Scholar 

  13. Jena RM, Chakraverty S, Jena SK (2019) Dynamic response analysis of fractionally damped beams subjected to external loads using homotopy analysis method. J Appl Comput Mech 5:355–366

    Google Scholar 

  14. Behera L, Chakraverty S (2015) Application of differential quadrature method in free vibration analysis of nanobeams based on various nonlocal theories. Comput Math Appl 69(12):1444–1462

    Article  MathSciNet  Google Scholar 

  15. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2):288–307

    Article  MATH  Google Scholar 

  16. Aydogdu M (2009) A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E 41(9):1651–1655

    Article  Google Scholar 

  17. Eltaher MA, Alshorbagy AE, Mahmoud FF (2013) Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Appl Math Model 37(7):4787–4797

    Article  MathSciNet  MATH  Google Scholar 

  18. Özdemir Ö, Kaya MO (2006) Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method. J Sound Vib 289(1–2):413–420

    Article  MATH  Google Scholar 

  19. Özdemir Ö, Kaya MO (2006) Flapwise bending vibration analysis of double tapered rotating Euler–Bernoulli beam by using the differential transform method. Meccanica 41(6):661–670

    Article  MATH  Google Scholar 

  20. Challamel N, Wang CM (2008) The small length scale effect for a non-local cantilever beam: a paradox solved. Nanotechnology 19:345703

    Article  Google Scholar 

  21. Nguyen NT, Hui D, Lee J, Nguyen-Xuan H (2015) An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Methods Appl Mech Eng 297:191–218

    Article  MathSciNet  MATH  Google Scholar 

  22. Fernández-Sáez J, Zaera R, Loya JA, Reddy JN (2016) Bending of Euler–Bernoulli beams using Eringen’s integral formulation: a paradox resolved. Int J Eng Sci 99:107–116

    Article  MathSciNet  MATH  Google Scholar 

  23. Ansari R, Norouzzadeh A (2016) Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates: an isogeometric analysis. Physica E 84:84–97

    Article  Google Scholar 

  24. Norouzzadeh A, Ansari R (2017) Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity. Physica E 88:194–200

    Article  Google Scholar 

  25. Norouzzadeh A, Ansari R, Rouhi H (2017) Pre-buckling responses of Timoshenko nanobeams based on the integral and differential models of nonlocal elasticity: an isogeometric approach. Appl Phys A 123:330

    Article  Google Scholar 

  26. Norouzzadeh A, Ansari R, Rouhi H (2018) Isogeometric analysis of Mindlin nanoplates based on the integral formulation of nonlocal elasticity. Multidiscip Model Mater Struct 14:810–827

    Article  Google Scholar 

  27. Ansari R, Torabi J, Norouzzadeh A (2018) Bending analysis of embedded nanoplates based on the integral formulation of Eringen’s nonlocal theory using the finite element method. Phys B 534:90–97

    Article  Google Scholar 

  28. Norouzzadeh A, Ansari R (2018) Isogeometric vibration analysis of functionally graded nanoplates with the consideration of nonlocal and surface effects. Thin Walled Struct 127:354–372

    Article  Google Scholar 

  29. Norouzzadeh A, Ansari R, Rouhi H (2019) An analytical study on wave propagation in functionally graded nano-beams/tubes based on the integral formulation of nonlocal elasticity. Waves Random Complex Media 28:1–19

    Article  Google Scholar 

  30. Zhang DP, Lei Y, Shen ZB (2016) Vibration analysis of horn-shaped single-walled carbon nanotubes embedded in viscoelastic medium under a longitudinal magnetic field. Int J Mech Sci 118:219–230

    Article  Google Scholar 

  31. Zhang DP, Lei Y, Shen ZB (2016) Free transverse vibration of double-walled carbon nanotubes embedded in viscoelastic medium. Acta Mech 227(12):3657–3670

    Article  MathSciNet  Google Scholar 

  32. Zhang DP, Lei YJ, Adhikari S (2018) Flexoelectric effect on vibration responses of piezoelectric nanobeams embedded in viscoelastic medium based on nonlocal elasticity theory. Acta Mech 229(6):2379–2392

    Article  MathSciNet  MATH  Google Scholar 

  33. Chakraverty S, Behera L (2016) Static and dynamic problems of nanobeams and nanoplates. World Scientific Publishing Co., Singapore

    Book  Google Scholar 

  34. Zadeh L (1965) Fuzzy sets. Inf Control 8(3):338–353

    Article  MATH  Google Scholar 

  35. Hanss M, Turrin S (2010) A fuzzy-based approach to comprehensive modelling and analysis of systems with epistemic uncertainties. Struct Saf 32(6):433–441

    Article  Google Scholar 

  36. Rao MVR, Pownuk A, Vandewalle S, Moens D (2010) Transient response of structures with uncertain structural parameters. Struct Saf 32(6):449–460

    Article  Google Scholar 

  37. Farkas L, Moens D, Donders S, Vandepitte D (2012) Optimisation study of a vehicle bumper subsystem with fuzzy parameters. Mech Syst Signal Process 32(4):59–68

    Article  Google Scholar 

  38. Chang SL, Zadeh LA (1972) On fuzzy mapping and control. IEEE Trans Syst Man Cybern 2(1):30–34

    Article  MathSciNet  MATH  Google Scholar 

  39. Dubois D, Prade H (1982) Towards fuzzy differential calculus: Part 3 differentiation. Fuzzy Sets Syst 8(3):225–233

    Article  MATH  Google Scholar 

  40. Kaleva O (1990) The Cauchy problem for fuzzy differential equations. Fuzzy Sets Syst 35(3):389–396

    Article  MathSciNet  MATH  Google Scholar 

  41. Khastan A, Nieto JJ, Rodrıguez-López R (2011) Variation of constant formula for first order fuzzy differential equations. Fuzzy Sets Syst 177(1):20–33

    Article  MathSciNet  MATH  Google Scholar 

  42. Mikaeilvand N, Khakrangin S (2012) Solving fuzzy partial differential equations by fuzzy two dimensional differential transform method. Neural Comput Appl 21(1):307–312

    Article  Google Scholar 

  43. Khastan A, Nieto JJ, Rodrıguez-López R (2013) Periodic boundary value problems for first order linear differential equations with uncertainty under generalized differentiability. Inf Sci 222(10):544–558

    Article  MathSciNet  MATH  Google Scholar 

  44. Tapaswini S, Chakraverty S (2014) Dynamic response of imprecisely defined beam subject to various loads using Adomian decomposition method. Appl Soft Comput 24:249–263

    Article  Google Scholar 

  45. Lv Z, Liu H (2017) Nonlinear bending response of functionally graded nanobeams with material uncertainties. Int J Mech Sci 134:123–135

    Article  Google Scholar 

  46. Lv Z, Liu H (2018) Uncertainty modeling for vibration and buckling behaviors of functionally graded nanobeams in thermal environment. Compos Struct 184:1165–1176

    Article  Google Scholar 

  47. Liu H, Lv Z (2018) Uncertain material properties on wave dispersion behaviors of smart magneto-electro-elastic nanobeams. Compos Struct 202:615–624

    Article  Google Scholar 

  48. Chakraverty S, Tapaswini S, Behera D (2016) Fuzzy differential equations and applications for engineers and scientists. CRC Press, Boca Raton

    Book  MATH  Google Scholar 

  49. Chakraverty S, Tapaswini S, Behera D (2016) Fuzzy arbitrary order system: fuzzy fractional differential equations and applications. Wiley, Hoboken

    Book  MATH  Google Scholar 

  50. Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to acknowledge Defence Research & Development Organization(DRDO), Ministry of Defence, New Delhi, India (Sanction Code: DG/TM/ERIPR/GIA/17-18/0129/020) for the funding to carry out the present research work.

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Rourkela, Rourkela, 769008, India

    Subrat Kumar Jena, S. Chakraverty & Rajarama Mohan Jena

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  1. Subrat Kumar Jena
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  2. S. Chakraverty
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  3. Rajarama Mohan Jena
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Correspondence to S. Chakraverty.

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Technical Editor: José Roberto de França Arruda.

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Jena, S.K., Chakraverty, S. & Jena, R.M. Propagation of uncertainty in free vibration of Euler–Bernoulli nanobeam. J Braz. Soc. Mech. Sci. Eng. 41, 436 (2019). https://doi.org/10.1007/s40430-019-1947-9

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