Abstract
In this work, a new method for solving a delay differential equation (DDE) with multiple delays is presented by using second- and third-order polynomials to approximate the delayed terms using the enhanced homotopy perturbation method (EMHPM). To study the proposed method performance in terms of convergency and computational cost in comparison with the first-order EMHPM, semi-discretization and full-discretization methods, a delay differential equation that model the cutting milling operation process was used. To further assess the accuracy of the proposed method, a milling process with a multivariable cutter is examined in order to find the stability boundaries. Then, theoretical predictions are computed from the corresponding DDE finding uncharted stable zones at high axial depths of cut. Time-domain simulations based on continuous wavelet transform (CWT) scalograms, power spectral density (PSD) charts and Poincaré maps (PM) were employed to validate the stability lobes found by using the third-order EMHPM for the multivariable tool.
Highlights
There are many phenomena in different fields of science and engineering where the physical response of a variable involves the value at time t and the effects that occur in an earlier state t − τ
Kuljanic et al [5] studied the incorporation of a chatter detection system based on multiple sensors to milling operations for industrial conditions, Zhuo et al [6] used a method based on fractal dimension for the flank milling of a thin-walled blade, which can reflect the chatter severity level through the morphological change in signal
Olvera and Elías-Zuñiga in [33] led to the development of the enhanced multistage homotopy perturbation method (EMHPM) to solve differential delay equations (DDEs) with constant and variable coefficients and this EMHPM was applied to predict the stability of a multivariate milling tool in which they consider the helix angle and the pitch angle variation of the cutting edges [34]
Summary
There are many phenomena in different fields of science and engineering where the physical response of a variable involves the value at time t and the effects that occur in an earlier state t − τ. Olvera and Elías-Zuñiga in [33] led to the development of the enhanced multistage homotopy perturbation method (EMHPM) to solve differential delay equations (DDEs) with constant and variable coefficients and this EMHPM was applied to predict the stability of a multivariate milling tool in which they consider the helix angle and the pitch angle variation of the cutting edges [34]. To demonstrate that one of the effective ways to suppress vibration in milling operations is to use tools with variable pitch and helix angle, Wang et al [12] proposed an improved semi-discretization method based on Floquet0 s theory. In order to study the proposed method performance in terms of convergency and computational cost, a multivariable milling tool with a variable pitch cutter and helix angle is used to determine milling process in stability domains.
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