In-process surface quality monitoring of the slender workpiece machining with digital twin approach

Abstract

In-process monitoring of production quality plays a significant role in intelligent manufacturing. Both part deformation and vibration happen simultaneously in machining processes. They are two prominent issues that can affect the surface quality of machined parts, especially those with low rigidity. The purpose of this study is to explore a hybrid modeling solution to simultaneously monitor the diametrical errors and early chatter vibrations when turning a slender workpiece. A generic analytical model of the slender workpiece turning is formulated based on dynamics of machining. It is proved that the system complies with the principle of superposition. Accordingly, the explicit expressions free from cutting force modeling that is widely used in literature are derived for characterizing the finished surface quality with analytical and finite element methods, respectively. A data-driven model is also developed using the wavelet packet transform to the displacement signals. The independent decompositions of the displacement signals are then correlated with both the dimensional error model and the turning chatter model. Interconnecting the dynamics-based model and the data-driven model contributes to a digital-twin prototype, which allows for in-process detection of the geometrical distortion and the onset of chatter on the part surface. Finally, two different machining cases were performed to verify the proposed methodology. The results show that the developed model consisting of the formulated deflection correlation and chatter indicator is capable of simultaneously evaluating and detecting the dimensional error prediction and the early-onset chatter. By comparison with the analytical modeling, the high fidelity digital twin using the finite element modeling could exhibit higher prediction accuracy. The proposed monitoring strategy could provide a pragmatic approach to online quality control for intelligent machining of flexible workpieces on the shop floor.

Appendix A

For turning slender workpieces, the governing equation of motion of the SDOF system is expressed as:

$$ \begin{aligned} &m\ddot{x}(t) + c\dot{x}(t) + kx(t)\\ &\quad = K_{r} df_{0} + K_{r} d\left[ {x(t - \tau ) - x(t)} \right] + F_{0} \sin \omega t \end{aligned}$$

(A.1)

Assume that $x_{1} (t)$ is a particular solution to the equation

$$ m\ddot{x}_{1} (t) + c\dot{x}_{1} (t) + kx_{1} (t) = K_{r} df_{0} $$

(A.2)

and $x_{2} (t)$ is a particular solution to the equation

$$ m\ddot{x}_{2} (t) + c\dot{x}_{2} (t) + kx_{2} (t) = K_{r} d\left[ {x_{2} (t - \tau ) - x_{2} (t)} \right] $$

(A.3)

and $x_{3} (t)$ is a particular solution to the equation

$$ m\ddot{x}_{3} (t) + c\dot{x}_{3} (t) + kx_{3} (t) = F_{0} \sin \omega t $$

(A.4)

Let $x(t)$ be equal to the sum of the above responses $x_{i} (t){\kern 1pt}$(i = 1,2,3), so that

$$ x(t){\kern 1pt} = x_{1} (t){\kern 1pt} + x_{2} (t){\kern 1pt} + x_{3} (t){\kern 1pt} $$

(A.5)

Suppose $x(t){\kern 1pt}$ is a solution to the differential equation (A.1), we substitute it into the left-hand side of Eq. (A.1) and obtain,

$$ \begin{aligned} & m[\ddot{x}_{1} (t) + \ddot{x}_{2} (t) + \ddot{x}_{3} (t)] + c[\dot{x}_{1} (t) + \dot{x}_{2} (t) + \dot{x}_{3} (t)]\\ &\qquad + k[x_{1} (t) + x_{2} (t) + x_{3} (t)] \hfill \\ &\quad = [m\ddot{x}_{1} (t) + c\dot{x}_{1} (t) + kx_{1} (t)]\\ &\qquad + [m\ddot{x}_{2} (t) + c\dot{x}_{2} (t) + kx_{2} (t)] \\ &\qquad+ [m\ddot{x}_{3} (t) + c\dot{x}_{3} (t) + kx_{3} (t)] \hfill \\ \end{aligned} $$

(A.6)

Applying Eqs. (A.2), (A.3), and (A.4) to Eq. (A.6) yields

$$\begin{aligned} & m\ddot{x}(t) + c\dot{x}(t) + kx(t)\\ &\qquad = K_{r} df_{0} + K_{r} d\left[ {x_{2} (t - \tau ) - x_{2} (t)} \right] + F_{0} \sin \omega t \end{aligned}$$

(A.7)

On the other hand, substituting Eq. (A.5) into the right-hand side of Eq. (A.1), we have

$$ \begin{aligned} & K_{r} df_{0} + K_{r} d\left[ {x(t - \tau ) - x(t)} \right] + F_{0} \sin \omega t\\ &\quad = K_{r} df_{0} + K_{r} d\{ \left[ {x_{1} (t - \tau ) - x_{1} (t)} \right] \\ &\qquad+ \left[ {x_{2} (t - \tau ) - x_{2} (t)} \right] \\ &\qquad + \left[ {x_{3} (t - \tau ) - x_{3} (t)} \right]\} + F_{0} \sin \omega t \hfill \\ \end{aligned} $$

(A.8)

Generally speaking, the differential equation does not satisfy the superposition principle due to the existence of the delayed term in it. But in this given case, according to fundamentals of mechanical vibrations the responses of $x_{1} (t){\kern 1pt}$ and $x_{3} (t){\kern 1pt}$ have the forms with the initial conditions equal to zero, respectively

$$ x_{1} (t) = {{K_{r} df_{0} } \mathord{\left/ {\vphantom {{K_{r} df_{0} } k}} \right. \kern-0pt} k} $$

(A.9)

$$ x_{3} (t) = A\sin (\omega t - \varphi ) $$

(A.10)

where $A = {{F_{0} } \mathord{\left/ {\vphantom {{F_{0} } {[(k - m\omega^{2} )^{2} + c^{2} \omega^{2} ]}}} \right. \kern-0pt} {[(k - m\omega^{2} )^{2} + c^{2} \omega^{2} ]}}^{0.5}$ and $\varphi = \tan^{ - 1} [{{c\omega } \mathord{\left/ {\vphantom {{c\omega } {(k - }}} \right. \kern-0pt} {(k - }}m\omega^{2} )]$ denote the amplitude and phase angle of the response, respectively.

Clearly, the response $x_{1} (t){\kern 1pt}$ is a constant. Thus, we can obtain

$$ x_{1} (t - \tau ) - x_{1} (t) = 0 $$

(A.11)

Meanwhile, since the time delay is equal to the period of the spindle rotation, that is, $\tau = \frac{60}{N} = \frac{2\pi }{\omega }$, we apply Eq.(A.10) and also derive

$$ x_{3} (t - \tau ) - x_{3} (t) = 0 $$

(A.12)

Finally, inserting Eqs. (A.11) and (A.12) into Eq. (A.8) produces the same expression as Eq. (A.7). As a result, it is proved that the combination of the response $x_{i} (t){\kern 1pt}$ under each individual excitation is a solution to Eq. (A.1) and thereby the given system complies with the principle of superposition.