Online analysis method to correlate the mode shape for forced vibration in milling thin-walled workpieces

Abstract

Forced vibration is a type of stable vibration different from chatter, which reduces the machining quality of a formed surface. The forced vibration characteristics of a thin-walled workpiece are difficult to capture due to varied milling excitation, cutter runout, and position dependence. Currently, insight into forced vibrations is still lacking without effective analysis methods. To solve these problems, a single-point online characterization method correlated to the mode shape was proposed. The modal contribution equation for a milled thin-walled workpiece was obtained by correlating the structural dynamics with the cutting force characteristics. Then, correlation factors were integrated to construct a state-space model, and the Kalman filter algorithm was used to estimate the modal contribution coefficients online based on a single-point response signal. Next, the principal mode shape was determined to characterize the forced vibration characteristics. The effectiveness of the proposed method was demonstrated by comparison with the operational deflection shape (ODS) results. These results revealed the contribution mechanism of each mode shape to the forced vibration under position-dependent excitation and varied cutting parameters, from which the vibration forms could be monitored.

Appendix

The derivation process of the modal contribution equation is provided here.

$$M\cdot \ddot{Q}\left(t\right)+C\cdot \dot{Q}\left(t\right)+K\cdot Q\left(t\right)=F\left(t\right)$$

(18)

where $M$ is the mass matrix of the thin-walled workpiece; $C$ is the damping matrix; $K$ is the stiffness matrix; $Q\left(t\right)$ is the displacement vector; $\dot{Q}\left(t\right)$ is the velocity vector; and $\ddot{Q}\left(t\right)$ is the acceleration vector. To decompose each mode from the system vibration, modal decoupling is needed. There is a set of modal vectors $\varphi =\left({\varphi }_{1}\text{, }{\varphi }_{2}...{\varphi }_{n}\right)$ in the modal space, where $n$ is the modal order. Assuming that $Q\left(t\right)=\varphi \cdot q\left(t\right)$, Eq. (18) can be expressed in modal space as:

$${\varphi }^{T}M\varphi \cdot \ddot{q}\left(t\right)+{\varphi }^{T}C\varphi \cdot \dot{q}\left(t\right)+{\varphi }^{T}K\varphi \cdot q\left(t\right)={\varphi }^{T}F\left(t\right)$$

(19)

where $q\left(t\right)$ is the linear independence modal displacement vector and ${\varphi }^{T}$ is the transpose matrix of $\varphi$. Based on the orthogonality of eigenvectors, Eq. (19) can be normalized as:

$$I\cdot \ddot{q}\left(t\right)+\Lambda \cdot \dot{q}\left(t\right)+L\cdot q\left(t\right)={\varphi }^{T}F\left(t\right)$$

(20)

$$I=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& ...& 0\\ 0& 0& 0& 1\end{array}\right],\Lambda =\left[\begin{array}{cccc}2{\xi }_{1}{\omega }_{1}& 0& 0& 0\\ 0& 2{\xi }_{2}{\omega }_{2}& 0& 0\\ 0& 0& ...& 0\\ 0& 0& 0& 2{\xi }_{n}{\omega }_{n}\end{array}\right],L=\left[\begin{array}{cccc}{\omega }_{1}^{2}& 0& 0& 0\\ 0& {\omega }_{2}^{2}& 0& 0\\ 0& 0& ...& 0\\ 0& 0& 0& {\omega }_{n}^{2}\end{array}\right]$$

(21)

where $I$ is the unit matrix, $\Lambda$ is the diagonal matrix related to the damping ratio ${\xi }_{n}$ and modal frequency ${\omega }_{n}$, and $L$ is the diagonal matrix related to the square of frequency ${\omega }_{n}^{2}$. To evaluate the effect of milling parameters on vibration, the cutting force $F\left(t\right)$ is modeled from the orthogonal direction:

$$\left\{\begin{array}{c}d{F}_{tj}=\left({K}_{tc}{s}_{t}+{K}_{te}\right)dz\\ d{F}_{rj}=\left({K}_{rc}{s}_{t}+{K}_{re}\right)dz\end{array}\right.$$

(22)

where $d{F}_{tj}$ is the tangential microelement cutting force; $d{F}_{rj}$ is the radial microelement cutting force; and ${K}_{tc}$ and ${K}_{rc}$ are the cutting force coefficients in relation to the shear component. The microelement equation for the milling force is:

$$dF=d{F}_{tj}\mathrm{sin}{\varphi }_{j}-d{F}_{rj}\mathrm{cos}{\varphi }_{j}$$

(23)

Combining Eqs. (22) and (23), the expansion of the milling force can be obtained:

$$F\left(t\right)=\sum_{j=0}^{N-1}g({\varphi }_{j})\left(\left({K}_{tc}{s}_{t}+{K}_{te}\right)\mathrm{sin}{\delta }_{j}-\left({K}_{rc}{s}_{t}+{K}_{re}\right)\mathrm{cos}{\varphi }_{j}\right)$$

(24)

where $g\left({\varphi }_{j}\right)$ is a window function to identify when the tool is in the cutting status, $j$ is the sequence number of the tool tooth, and ${s}_{t}$ is the milling chip thickness, which can be expressed as:

$${s}_{t}\left(s,v\right)={f}_{z}\mathrm{sin}{\varphi }_{j}\left(z\right)=\frac{v\mathrm{sin}{\varphi }_{j}\left(z\right)}{s*j}$$

(25)

where ${f}_{z}$ is the feed per tooth and ${\varphi }_{j}\left(z\right)$ is the instantaneous immersion angle. From Eqs. (20), (24), and (25), the following equation can be given:

$$I\cdot \ddot{q}\left(t\right)+\Lambda \cdot \dot{q}\left(t\right)+L\cdot q\left(t\right)={\varphi }^{T}F\left(s,v,t\right)$$

(26)

where F (s, v, t) is the instantaneous cutting force related to the spindle speed $s$, feed speed $v$, and machining time t. The equation for a certain arbitrary ${n}_{th}$ mode is:

$${\ddot{q}}_{n}\left(t\right)+2{\xi }_{n}{\omega }_{n}\cdot {\dot{q}}_{n}\left(t\right)+{\omega }_{n}^{2}\cdot {q}_{n}\left(t\right)={\varphi }_{n}^{T}{F}_{n}\left(s,v,t\right)$$

(27)

where ${F}_{n}\left(s,v,t\right)$ is the generalized force vector in modal coordinates and ${\varphi }_{n}^{T}$ is the transpose of the ${n}_{th}$ mode shape. The excitation force and modal displacement are converted into a complex exponential form:

$$\left\{\begin{array}{c}{F}_{n}\left(s,v,t\right)={f}_{n}\left(s,v\right)\cdot {e}^{j\omega t}\\ {q}_{n}={\eta }_{n}\cdot {e}^{j\omega t}\end{array}\right.$$

(28)

where ${f}_{n}\left(s,v\right)$ is the amplitude of the input force; ${\eta }_{n}$ is the amplitude contributed by the ${n}_{th}$ mode in the vibration response signal; and $\omega$ is the frequency of the input force. Equation (27) can be written as:

$$\left(-{\omega }^{2}+j\omega {\xi }_{n}+{\omega }_{n}^{2}\right){\eta }_{n}{e}^{j\omega t}={\varphi }_{n}^{T}{f}_{n}{e}^{j\omega t}$$

(29)