Propagation of uncertainty in free vibration of Euler–Bernoulli nanobeam

  • Subrat Kumar Jena
  • S. ChakravertyEmail author
  • Rajarama Mohan Jena
Technical Paper


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.


Imprecisely defined parameter Material uncertainties Triangular fuzzy number Uncertainty propagation Vibration Euler–Bernoulli beam Rayleigh–Ritz method 



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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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  • Subrat Kumar Jena
    • 1
  • S. Chakraverty
    • 1
    Email author
  • Rajarama Mohan Jena
    • 1
  1. 1.Department of MathematicsNational Institute of Technology RourkelaRourkelaIndia

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