Stochastic modeling of nonlinear dynamics of the machine-tool-workpiece system and its influence on the formation of surface topography during finishing
DOI:
https://doi.org/10.15276/opu.2.72.2025.01Keywords:
stochastic modeling, nonlinear dynamics, machining, surface topography, Stochastic Differential Delay Equation, Monte Carlo simulation, regenerative effectAbstract
Ensuring stable surface quality during finishing operations is a critical task in mechanical engineering; however, traditional deterministic models of machining dynamics fail to fully capture the statistical nature of surface topography formed under real-world conditions. This limitation arises from neglecting random factors such as microstructural material inhomogeneity, tool wear fluctuations, and external vibrational disturbances. This paper proposes a novel nonlinear stochastic model of the “machine-tool-workpiece” dynamic system to bridge this gap between theory and practice. Mathematically, the system is formulated as a Stochastic Differential Delay Equation, which comprehensively incorporates the regenerative effect of cutting forces, nonlinear cubic structural stiffness, and additive stochastic perturbations modeled as white noise. The numerical implementation of the model was performed using the Euler-Maruyama scheme within a Monte Carlo framework (N=50). Simulation results demonstrated that the stable system, under the influence of noise, forms a stochastic attractor, generating bounded non-periodic oscillations. The primary contribution of this study is the derivation of a full Probability Density Function for the predicted Root Mean Square surface roughness, with a mean value of μ=14.14 μm. This enables a shift from single-point deterministic predictions to probabilistic forecasting of surface quality. A rigorous model adequacy validation was conducted, yielding a near-perfect coefficient of determination (R2=0,9999) between the input noise variance and the output displacement variance, confirming the physical consistency of the proposed approach. The developed methodology provides a robust framework for predicting process uncertainty, assessing machining reliability, and minimizing scrap rates in high-precision finishing operations.
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