### ORS MEMS accelerometer results

To study the results of the ORS placed on the spindle rotor system, three Cartesian coordinate systems were defined, as shown in Figure 4. \(XYZ\) is the coordinate of the stationary frame of the rotor system, which acts as a reference for any rotating object. The \({X}_{O}{Y}_{O}{Z}_{O}\) is the rotating coordinate due to strong static vibration of the rotating shaft under the influence of shear force, as shown in Figure 4a. The center of the rotating shaft was moved to \({O}_{0}\)and dynamic vibration is expressed as \(\ddot{x}

(1)

where \({in}\) and \({a}_{c}\) can be expressed as:

$$\begin{sorted} & a_{c} = r\omega ^{2} \\ & a_{t} = r\dot{\omega } \\ \end{sorted}$$

(2)

Then the matrix format of Eq. (1) is:

$$\left[\begin{array}{c}\ddot{u}

(3)

Equation (3) shows that the measured waves include acceleration, which reflects the characteristic rotational properties of the rotor system due to frictional action and gravitational acceleration components. The latter is not a desired signal and should be removed in order to increase the rotor dynamic signals for detecting the cutting condition.^{21}.

### Acceleration signal reconstruction

Assuming that the rotor rotates with the angular velocity of time \(\omega\)like \(\omega ={\omega }_{0}+{\omega }^{^{\prime}}\)where \({\omega }_{0}\) is the velocity of the angular momentum and \({\omega }^{^{\prime}}\) is the variable component of the velocity. Then, centripetal \({a}_{c}\) and tangential acceleration \({in}\) can be written as:

$$\begin{sorted} & a_{c} = \left( {\omega _{0} + \omega ^{\prime}} \right)^{2} = r\omega _{0}^{2 } + 2r\omega _{0} \omega ^{\prime} + r\omega ^{{\prime 2}} \\ & a_{t} = r\frac{{d\omega ^{\prime}} {{dt}} \\ \end{aligned}$$

(4)

Because the dynamic change of the angular velocity of the rotor \({\omega }^{^{\prime}}\) is very small compared to the angular velocity \({\omega }_{0}\)quadratic term \(r{{\omega }^{^{\prime}}}^{2}\) it’s a little. Therefore, a strong centripetal acceleration \({a}_{c}^{^{\prime}}\) can be estimated as:

$${a}_{c}^{^{\prime}}\approx 2r{\omega }_{0}{\omega }^{^{\prime}}$$

(5)

Furthermore, the variable angular velocity can be assumed to be constant and expanded as a Fourier series as follows:

$${\omega }^{^{\prime}}=\sum_{n=1}^{\infty}{A}_{n}\mathrm{sin}(n{\omega }_{0}t+ {\varphi }_{n})$$

(6)

where \({A}_{n}\) and \({\varphi }_{n}\) is the amplitude and phase of *n*th harmonic, respectively.

Finally, centripetal force \({a}_{c}^{^{\prime}}\) and tangential acceleration \({a}_{t}^{^{\prime}}\) It can be expressed as a combination of harmonic components as follows:

$$\begin{aligned} & a_{c}^{\prime } = \sum\limits_{{n = 1}}^{\infty } 2 r\omega _{0} A_{n} {\text{ sin}}(n\omega _{0} t + \varphi _{n} ) \\ & a_{t}^{\prime } = \sum\limits_{{n = 1}}^{\infty } n rA_{n} {\text{cos}}(n\omega _{0} t + \varphi _{n} ) \\ \end{planned}$$

(7)

Inside of \({X}_{O}{Y}_{O}{Z}_{O}\) coordinate system, \(\omega t\) can be expressed as follows:

$$\omega t={\theta }_{0}+{\omega }_{0}t+{\int }_{0}^{t}{\omega }^{^{\prime}}dt$ $

(8)

where \({\theta }_{0}\) is the initial phase, and the third part can be neglected compared to the first two components. Subsequently, the time-varying vibration, as shown in Eq. (3) can be rearranged as

$$\left[ {\begin{array}{*{20}l} {\ddot{u}\left( t \right)} \\ {\ddot{v}\left( t \right)} \\ \end{array} } \right] = \ left[ {\begin{array}{*{20}c} { – {\text{sin}}\left( {\theta _{0} + \omega _{0} t} \right)} & {{\text{cos}}\left( {\theta _{0} + \omega _{0} t} \right)} \\ {{\text{cos}}\left( {\theta _{0} + \omega _{0} t} \right)} & {{\text{sin}}\left( {\theta _{0} + \omega _{0} t} \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\ddot{x}\left( t \right)} \\ {\ddot{y}\left( t \right) + g} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\sum\nolimits_{{n = 1}}^{\infty } {nrA_{n} \;{\text{cos}}\;(n\omega _{0} t + \varphi _{n} )} } \\ {\sum\nolimits_{{n = 1}}^{\infty } {2r\omega _{0} A_{n} \;{\text{sin}}\;(n\omega _{0} t + \varphi _{n} )} } \\ \end{array} } \right]$$

(9)

From Eq. (9), the reconstructed force acceleration \(\ddot{u}\left(t\right)\) and \(\ddot{v}\left(t\right)\), \(\mathrm{respectively},\) is projected on *U*-axis and *V*-axis, they consist of two components: the power vibration of \(\ddot{x}\left(t\right)\) and \(\dddot{y}

Calculate and determine the rotational position of the rotor system after Fourier transformation.

Subtract 1.0 g from the amplitude in complex domains, in the approximation of *X*– orientation and *Y*– direction in the rotation cycle.

Reconstruct the time domain signal using an inverse Fourier transform.

### Modal analysis of a spindle rotor system

Given that the spindle-chuck assembly has a non-negligible effect on the dynamics of machined workpieces.^{22,23}, a multi-degree-of-freedom system consisting of a spindle, gear, chuck, and workpiece was established through the finite element method (FEM), as shown in Figure 5. The front bearing group was composed of two fixed DBBs. NSK 51214 and NSK 32014 bearings. The rear bearing is a double row roller bearing type NSK NN3019K. The main bearing parameters are presented in Table 2. Hardness was calculated using the theoretical method described in the previous study.^{24} and is presented in Table 3.

Table 4 summarizes the modal results of the spindle rotor system. Four frequency bands were observed at approximately 46 Hz, 350-450 Hz, 750-900 Hz, and 1000-1200 Hz.

### Vibration model of rotor system under shear force

As shown in Figure 5, the spindle rotor system is axially symmetrical, and its dynamic characteristics are assumed to be equal in *X*-and *Y*– instructions. \({F}_{x}\) and \({F}_{y}\) is an estimate of the shear strength *X*-and *Y*-ax, respectively. Tangential force \({F}_{x}\) is the primary shear force, accounting for more than 95% of the resultant shear force, and the radial force \({F}_{y}\) account less than 10%. Therefore, the center of the spindle rotor shows a slight lateral swing in *X*-direction. The dynamic rotations of the spinning rotor system then exhibit significant changes compared to those of conventional circular or circular rotations.

The corresponding dynamic equation of the spindle rotor system exciting and cutting force is expressed as^{4}

$$\ddot{x}\left(t\right)+2{\omega }_{n}\zeta \dot{x}\left(t\right)+{\omega }_{n}^{2 }x\left(t\right)=\frac{1}{m}F

(10)

where \(m,\) \({\omega }_{n},\) and \(\zeta\) are equal mass, natural frequency, and damping ratio, respectively, of the system. \(F

(11)