Vol.:(0123456789) 1 3 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:436 https://doi.org/10.1007/s40430-019-1947-9 TECHNICAL PAPER Propagation of uncertainty in free vibration of Euler–Bernoulli nanobeam Subrat Kumar Jena 1  · S. Chakraverty 1  · Rajarama Mohan Jena 1 Received: 25 March 2019 / Accepted: 9 September 2019 © The Brazilian Society of Mechanical Sciences and Engineering 2019 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 defned 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 defned. Frequency parameters and mode shapes are computed and presented for Pined–Pined (P–P) and Clamped–Clamped (C–C) boundary conditions. Accuracy and efciency of the models are verifed 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 mod- els. 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 efective tool for the designing and optimal quality enhancement of engineering structures. Keywords Imprecisely defned parameter · Material uncertainties · Triangular fuzzy number · Uncertainty propagation · Vibration · Euler–Bernoulli beam · Rayleigh–Ritz method 1 Introduction Nowadays, uncertainty is growing as a concern in the area of designing and manufacturing of various engineering struc- tures. The uncertainty associated with the various proper- ties of structural elements such as mass density, Young’s modulus, external loads and edge conditions arises due to the imprecise or incomplete data about the scaling param- eters because of human errors, environmental condition, wear–tear, faulty experiment, etc. This imprecision or vagueness leads to uncertainty in the static and dynamical behavior of structures. Uncertainty imposes a severe threat to safety, reliability, productivity, the efectiveness of per- formance and so on. As regards, these issues need to be dealt with by proposing new models that explicitly incor- porate imprecisely defned parameters of the structure and efciently solve it. In the last decades, several factors have been introduced to ensure the factor of safety. However, the craving of substantial efciency and reliability and upgraded performance and minimized costs compel for designing and developing computationally efcient improved method or model. The main aim is to apply a computationally efcient method or model to produce structures which are safe, con- sistently good in quality and should have admissible noise and vibration. Impreciseness or uncertainties in the prop- erties of structural members propagate through the system resulting uncertainties in static and dynamic response such as frequency parameters and frequency response. Technical Editor: José Roberto de França Arruda. * S. Chakraverty sne_chak@yahoo.com Subrat Kumar Jena sjena430@gmail.com Rajarama Mohan Jena rajarama1994@gmail.com 1 Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India