This study investigates the stochastic vibration of shape memory alloy (SMA) spring oscillators under Gaussian white noise, employing two hysteresis models (two-flag and piecewise polynomial). While piecewise models complicate stochastic averaging analysis, we propose an efficient solution via a radial basis function neural network (RBFNN)-based semi-analytic method—Gaussian RBFs approximate the steady-state joint PDF of displacement and velocity, optimized via Lagrange multipliers to minimize error. Simulations confirm identical results to Monte Carlo methods (MCS) while requiring significantly less computation, demonstrating the RBFNN method as a practical alternative for stochastic analysis of hysteretic systems.
AsnafiA (2024) Fully analytic approach to study bifurcation phenomenon for a shape memory alloy beam under temperature variation and lateral white-noise excitation. Journal of Vibration and Control30(23–24): 5431–5440.
2.
BishopC (1991) Improving the generalization properties of radial basis function neural networks. Neural Computation3(4): 579–588.
3.
BuMLiJGuoR, et al. (2023) Dynamical responses of a Gaussian colored noise-driven shape memory alloy oscillator with a periodic force. Chaos, Solitons & Fractals166: 112966.
4.
ChenZ (2023) Application of SMA materials in aerospace. Applied and Computational Engineering25: 22–29.
5.
ElmasET (2024) System design and development of a novel unique neuro-physical medical treatment method for SMA-spinal muscular atrophy disease and for similar neurological muscle diseases. Collective Journal of Neurology1: ART0037. parameters.
6.
ErochkoJChristopoulosCTremblayR, et al. (2013) Shake table testing and numerical simulation of a self-centering energy dissipative braced frame. Earthquake Engineering & Structural Dynamics42(11): 1617–1635.
7.
FamelisIDonasAGalanisG (2022) Comparative study of feedforward and radial basis function neural networks for solving an environmental boundary value problem. Results in Applied Mathematics16: 100344.
8.
FonsecaLMRodriguesGVSaviMA, et al. (2019) Nonlinear dynamics of an origami wheel with shape memory alloy actuators. Chaos, Solitons & Fractals122: 245–261.
9.
GeGHuY (2024) Response of Gaussian color noise excited oscillators with inertia nonlinearity based on the radial basis function neural network method. International Journal of Non-linear Mechanics163: 104731.
10.
GraesserEJCozzarelliFA (1991) Shape-memory alloys as new materials for aseismic isolation. Journal of Engineering Mechanics117(11): 2590–2608.
11.
HoneycuttRL (1992) Stochastic runge-kutta algorithms. i. white noise. Physical Review. A, Atomic, Molecular, and Optical Physics45(2): 600–603.
12.
ImranHYMajidDAbin Abdul HamidMF, et al. (2023) Protecting car engines and controlling their temperature by using shape memory alloy as an automatic mechanical cooling sensor. Journal of Mechanical Engineering and Sciences17(3): 9656–9662.
13.
KimMSHeoJKRodrigueH, et al. (2023) Shape memory alloy (SMA) actuators: the role of material, form, and scaling effects. Advanced Materials35(33): 2208517.
14.
LiYFengJWangY (2020) Probability response of shape memory alloy beam subjected to noise excitation [J]. Journal of Harbin Institute of Technology (New Series)27(1): 85–90.
15.
LiWGuanYHuangD, et al. (2023) Gaussian RBFNN method for solving FPK and BK equations in stochastic dynamical system with FOPID controller. International Journal of Non-linear Mechanics153: 104403.
16.
LiWLinMZhaoJ, et al. (2025) Stochastic reliability optimization of a controlled memristor-based Van der Pol circuit using a new intelligent algorithm. Engineering Applications of Artificial Intelligence154: 110921.
17.
LiuZPangJOuyangH, et al. (2023) A new general method for assigning frequencies with low spillover through structural modifications. Journal of Low Frequency Noise, Vibration and Active Control42(4): 1900–1918.
18.
LuZRongKTianL, et al. (2022) Studies on the damping mechanism of shape memory alloy-spring tuned vibration absorber attached to a multi-degree-of-freedom structure. Journal of Vibration and Control28(19-20): 2666–2677.
19.
Mai-DuyNTran-CongT (2005) An efficient indirect RBFN-based method for numerical solution of PDEs. Numerical Methods for Partial Differential Equations21(4): 770–790.
20.
ParkJSandbergIW (1993) Approximation and radial-basis-function networks. Neural Computation5(2): 305–316.
21.
PiccirilloVBalthazarJMPontesBR (2010) Analytical study of the nonlinear behavior of a shape memory oscillator: part I—primary resonance and free response at low temperatures. Nonlinear Dynamics59: 733–746.
QiuJStiharuI (2022) Vibration and noise reduction of pipelines using shape memory alloy. Science and Engineering of Composite Materials29(1): 165–175.
24.
RoyAGovilSMirandaR (1997) A neural-network learning theory and a polynomial time RBF algorithm. IEEE Transactions on Neural Networks8(6): 1301–1313.
25.
SongJDer KiureghianA (2006) Generalized Bouc–Wen model for highly asymmetric hysteresis. Journal of Engineering Mechanics132(6): 610–618.
26.
SpanosPDCacciolaPRedhorseJ (2004) Random vibration of SMA systems via Preisach formalism. Nonlinear Dynamics36: 405–419.
27.
TabrizikahouABiałasikJBorysiakS, et al. (2024) Shear strengthening of damaged reinforced concrete beams with iron-based shape memory alloy (Fe-SMA) strips: numerical and parametric analysis. Archives of Civil and Mechanical Engineering24(3): 189.
28.
VietNVZakiW (2021) Bending model for functionally graded porous shape memory alloy/poroelastic composite cantilever beams. Applied Mathematical Modelling97: 398–417.
29.
WangXJiangJHongL, et al. (2022) Random vibration analysis with radial basis function neural networks. International Journal of Dynamics and Control10: 1–10.
30.
WilliamsKAChiuGTCBernhardRJ (2005) Dynamic modelling of a shape memory alloy adaptive tuned vibration absorber. Journal of Sound and Vibration280(1–2): 211–234.
31.
XuYGuRZhangH, et al. (2011) Stochastic bifurcations in a bistable Duffing--Van der Pol oscillator with colored noise. Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics83(5): 056215.
32.
XuLHFanXWLiZX (2017) Cyclic behavior and failure mechanism of self-centering energy dissipation braces with pre-pressed combination disc springs. Earthquake Engineering & Structural Dynamics46(7): 1065–1080.
33.
YangFChenLYuanZ, et al. (2023) Transient response of energy harvesting systems with multi-well potential under Poisson white noise excitations. International Journal of Non-linear Mechanics155: 104463.
34.
YuYZhangCChenZ, et al. (2020) Relaxation and mixed mode oscillations in a shape memory alloy oscillator driven by parametric and external excitations. Chaos, Solitons & Fractals140: 110145.
35.
ZhaoDRuanBChenG (2016) Validation of the modified irregular unloading-reloading rules based on Davidenkov skeleton curve and the implementation in ABAQUS software. In: Seventh China-Japan-us Trilateral Symposium on Lifeline Earthquake Engineering. Reston, VA: American Society of Civil Engineers, 449–455.
36.
ZhuZWZhangQXXuJ (2014) Nonlinear dynamic characteristics and optimal control of a giant magnetostrictive film-shaped memory alloy composite plate subjected to in-plane stochastic excitation. Chinese Physics B23(8): 088201.
