外文翻译(机械)(2)

in a mean force value close to zero, the signal to noise ratio is very small which can result in a poor least squares fit. To address these limitations, the paper demonstrates milling force modeling using the MCMC method for Bayesian inference. The advantage of using Bayesian inference is that experiments over multiple feed per tooth values, which can be time consuming and costly, are not necessary for determining the cutting force coefficient values. In addition, the uncertainty in the force coefficients can be evaluated by combining prior knowledge and experimental data. The main contribution of the paper is to propose Bayesian inference for milling force modeling and compare the results with the well-known linear regression approach. The remainder of the paper is organized as follows. First, generalized expressions for mechanistic cutting force coefficients are developed. Section 2 then introduces Bayes’ rule. Section 3 demonstrates the MCMC algorithm for Bayesian inference using a simple example followed by the application to milling force modeling in Sec. 4. Section 5 describes the experimental results and a comparison to the linear regression approach. Section 6 discusses the benefits of Bayesian inference followed by conclusions in Sec. 7.

In milling, the tangential, Ft, and normal, Fn, direction force components can be described using Eqs. (1) and (2), where b is the chip width (axial depth of cut), h is the instantaneous chip thickness, Kt is the tangential cutting force coefficient, Kte is the tangential edge coefficient, Kn is the normal cutting force coefficient, and Kne is the normal edge coefficient [1].

The chip thickness is time-dependent in milling and can be approximated using the feed per tooth, ft, and time-dependent cutter angle, , provided the ratio of the feed per tooth to cutter diameter is small [2]. See below equation

The forces in the x (feed) and y directions1, Fx and Fy, are determined by projecting the tangential and normal force components in the x and y directions using the cutter angle as shown in Fig. 1. See below equations

Expressions for the mean forces in the x and y directions, F¯¯¯x and F¯¯¯y, are provided in Eqs. (6) and (7), where Nt is the number of teeth on the cutter and s and

e are the cut start and exit angles, which are defined by the radial depth of cut [1]. In the Eqs. (6) and (7) average force expressions, the first term, which is a function of the feed per tooth, gives the slope of the linear regression to the average force values that correspond to the selected feed per tooth values. The second term, which does not include the feed per tooth, is the intercept of the linear regression. By rearranging Eqs.

(6) and (7), the four force coefficients are determined using Eqs. (8)–(11), where a1,x and a1,y are the slopes of the linear regressions to the x and y direction average force data and a0,x and a0,y are the intercepts.