Sage Journals: Discover world-class research

Abstract

This study presents nonlinear dynamic analysis and active control framework for an Euler–Bernoulli nanobeam modeled using the Nonlocal Strain Gradient Theory (NSGT). The governing sixth-order nonlinear partial differential equation is derived using Hamilton’s principle with von Kármán geometric strain assumptions and reduced to a Duffing-type single-mode model through the Galerkin method. Bifurcation analysis is conducted to examine the effects of nonlocal and strain-gradient length-scale parameters on resonance behavior, stability boundaries, and hysteresis. Two advanced nonlinear control strategies: Sliding Mode Control (SMC) and Feedback Linearization with Proportional–Derivative and Feed-Forward compensation (FBL-PD-FF) are designed and evaluated under deterministic harmonic excitation and stochastic white-noise disturbances. Numerical simulations demonstrate that SMC offers superior robustness, disturbance rejection, and overshoot suppression, while FBL-PD-FF achieves fast response but suffers from higher sensitivity to nonlinearities and noise. The results provide theoretical insight and practical guidance for designing stable, high-precision MEMS/NEMS resonators operating under nonlinear and stochastic conditions.

Keywords

Nonlinear controls non-local strain gradient theory sliding mode control feed-back linearization PD-stiffness FF control nonlinear analysis micro/nano-systems

Get full access to this article

View all access options for this article.

References

Eringen

. Nonlocal continuum field theories. New York: Springer, 2002.

Ghanbari

Babaei

Vakili

. Free vibration analysis of micro beams based on the modified couple stress theory, using approximate methods. Int J. Eng Technol Sci 2015; 3(2): 136–143.

Babaei

Ghanbari

Vakili-Tahami

. Size-dependent behavior of functionally graded micro-beams, based on the modified couple stress theory. Technology 2015; 3(5): 364–372.

Babaei

Noorani

MRS

Ghanbari

. Temperature-dependent free vibration analysis of functionally graded micro-beams based on the modified couple stress theory. Microsyst Technol 2017; 23(10): 4599–4610. https://doi.org/10.1007/s00542-017-3285-0

Rahmani

Babaei

. Vibration characteristics of functionally graded micro-beam carrying an attached mass. Mech Adv Compos Struct 2019.

Ghanbari

Babaei

. The new boundary condition effect on the free vibration analysis of micro-beams based on the modified couple stress theory. Int Res J Appl Basic Sci 2015; 9(3): 274–279.

Lim

. A unified nonlocal strain gradient theory for elastic nanobeams. Int J Eng Sci 2016; 109: 1–14.

. Nonlinear bending and vibration of functionally graded carbon-nanotube reinforced beams based on nonlocal strain gradient theory. Compos Struct 2017; 160: 185–200.

Şimşek

. Nonlinear free vibration of nanobeams based on strain gradient elasticity theory. J Vib Control 2016; 22(5): 1209–1223.

10.

Farajpour

Ghayesh

Hussain

, et al. Nonlinear dynamics of nanotubes and nano-scale beams. Int J Eng Sci 2018; 133: 92–106.

11.

Babaei

Yang

. Vibration analysis of rotating rods based on the nonlocal elasticity theory and coupled displacement field. Microsyst Technol 2018.

12.

Rahmani

Faroughi

Friswell

, et al. Eringen’s nonlocal and modified couple stress theories applied to vibrating rotating nanobeams with temperature effects. Mech Adv Mater Struct 2022; 29(26): 4813–4838. https://doi.org/10.1080/15376494.2021.1939468

13.

Babaei

Rahmani

. On dynamic-vibration analysis of temperature-dependent Timoshenko microbeam possessing mutable nonclassical length scale parameter. Mech Adv Mater Struct 2020; 27(16): 1451–1458. https://doi.org/10.1080/15376494.2018.1516252

14.

Babaei

Rahmani

Ahmadi

. Transverse vibration analysis of nonlocal beams with various slenderness ratios, undergoing thermal stress. Arch Mech Eng 2019; 66(1): 5–24. https://doi.org/10.24425/ame.2019.126368

15.

Babaei

. Forced vibration analysis of non-local strain gradient rod subjected to harmonic excitations. Microsyst Technol 2021; 27(3): 821–831. https://doi.org/10.1007/s00542-020-04973-9

16.

Babaei

. Forced vibrations of size-dependent rods subjected to: impulse, step, and ramp excitations. Arch Appl Mech 2021; 91(5): 2211–2223. https://doi.org/10.1007/s00419-020-01878-x

17.

Babaei

. Longitudinal vibration responses of axially functionally graded optimized MEMS gyroscope using Rayleigh–Ritz method, determination of discernible patterns and chaotic regimes. SN Appl Sci 2019; 1(8): 831. https://doi.org/10.1007/s42452-019-0867-8

18.

Babaei

Rahmani

. Vibration analysis of rotating thermally-stressed gyroscope, based on modified coupled displacement field method. Mech Base Des Struct Mach 2020.

19.

Babaei

Parker

Moshaver

. Adaptive neuro-fuzzy inference system (ANFIS) integrated with genetic algorithm to optimize piezoelectric cantilever-oscillator-spring energy harvester. Comput Eng Phys Model 2022; 5(4): 1–22.

20.

Babaei

Parker

Moshaver

. Free and forced vibrations of elastically restrained cantilever with lumped oscillator. Discov Appl Sci 2025; 6(5): 269. https://doi.org/10.1007/s42452-023-05564-9

21.

Babaei

. Active impulsive disturbance rejection and oscillation control of robot manipulator, using nonclassical rod models and proportional integral derivative controller. Noise Vib Worldw 2021; 52(11): 377–384. https://doi.org/10.1177/09574565211030708

22.

Slotine

J-JE

. Applied nonlinear control. Hoboken, NJ: Prentice Hall, 1991.

23.

Khalil

. Nonlinear systems. 3rd ed. Hoboken, NJ: Prentice Hall, 2002.

24.

Utkin

. Sliding mode control design principles and applications to electric drives. IEEE Trans Ind Electron 1993; 40(1): 23–36. https://doi.org/10.1109/41.184818

25.

Krstić

Kanellakopoulos

Kokotović

. Nonlinear and adaptive control design. Hoboken, NJ: Wiley, 1995.

26.

Babaei

Parker

Moshaver

. Energy resource for a RFID system based on dynamic features of Reddy-Levinson beam. In: ASME IMECE, 16–19 November 2020, p. 84553.

Non-linear dynamics and controls of non-local strain gradient beam with harmonic balance,sliding mode and feedback linearization methods

Abstract

Keywords

Get full access to this article

References