Cross-axis and cross-point modal testing and parameter identification method for predicting the cutting stability

Abstract

The present invention provides a cross-axis and cross-point modal testing and parameter identification method for predicting the cutting stability, which is used to improve the accuracy of existing prediction methods of cutting stability. The method firstly installs a miniature tri-axial acceleration sensor at the tool tip, and conducts the cross-axis and cross-point experimental modal tests respectively. The measured transfer functions are grouped according to different measuring axes, and the dynamic parameters are separately identified from each group of transfer functions. Then, the contact region between the cutter and workpiece is divided into several cutting layer differentiators. After that, together with other dynamic parameters, all the parameters are assembled into system dynamic parameter matrices matching with the dynamic model. Finally, dynamic parameter matrices including the effects of cross-axis and cross-point model couplings are obtained. Moreover, the acceleration sensor in the method only needs to be installed once.

Claims

1. A cross-axis and cross-point modal testing and parameter identification method for predicting cutting stability, wherein the following steps are comprised: step 1 install a cutter in a handle, clamp the handle in a machine tool spindle, and establish cutter coordinate system: the origin of coordinates is set on the free end of the cutter, the feed direction of cutter is set as the X axis, the direction perpendicular to the surface to be machined is set as Y axis wherein the outward direction for down milling and inward direction for up milling, and the Z axis is set as the direction away from the free end of the cutter and along the cutter axis; step 2 starting from the free end of the cutter in a certain distance along the cutter axis, mark q nodes which will be impacted by a hammer, install a miniature tri-axial acceleration sensor at the tool tip, and impact at each node in two horizontally orthogonal X and Y directions with the hammer, to measure all the transfer functions of a spindle-handle-cutter system at each node; step 3 for the transfer functions measured via step 2, eliminate all transfer functions measured by Z axis of the acceleration sensor, and then divide the remaining transfer functions into two different transfer function groups according to the vibration response measured by X or Y axis of the acceleration sensor, and the two groups of transfer function are marked as {FRF.sub.x} and {FRF.sub.y}, respectively; step 4 identify the dynamic parameters respectively from the two groups of different transfer functions {FRF.sub.x} and {FRF.sub.y} obtained in Step 3; based on {FRF.sub.x}, the identified previous m order dynamic parameters are expressed as follows; natural frequencies are ω.sub.nx,1, ω.sub.nx,2, . . . , ω.sub.nx,m; damping ratios are ξ.sub.x,1, ξ.sub.x,2, . . . , ξ.sub.x,m; mode shape matrix is ψ.sub.x=[φ.sub.x,1φ.sub.x,2 . . . φ.sub.x,m].sub.2q×m, where the dimension of φ.sub.x,jj=1,2, . . . , m is 2q×1 and φ.sub.x,j represents the j-th order mode shape vector corresponding to each impact node in the principle vibration direction of X direction; based on {FRF.sub.y}, the identified previous m order dynamic parameters are expressed as follows; natural frequencies are ω.sub.ny,1, ω.sub.ny,2, . . . , ω.sub.ny,m; damping ratios are ξ.sub.y,1, ξ.sub.y,2, . . . , ξ.sub.y,m; mode shape matrix is ψ.sub.y=[φ.sub.y,1φ.sub.y,2 . . . φ.sub.y,m].sub.2q×m, where the dimension of φ.sub.y,j(j=1,2, . . . , m) is 2q×1 and φ.sub.y,j represents the j-th order mode shape vector corresponding to each impact node in the principle vibration direction of Y direction; step 5 divide a contact region between the cutter and a workpiece under a given axial cutting depth α.sub.p along the cutter axis into p cutting layer differentiators; according to the relative position between the center of each differentiator and above impact nodes, allocate these differentiators a value of a mode shape identified by step 4 through linear interpolation; step 6 according to different modal order, assemble different types of dynamic parameters into modal mass, damping, stiffness and mode shape matrices, and these matrices match a system dynamic model; after assembly, obtaining: model mass matrix $M={\left[\begin{array}{ccccccc}1& & & & & & \\ & 1& & & & & \\ & & 1& & & & \\ & & & 1& & & \\ & & & & \ddots & & \\ & & & & & 1& \\ & & & & & & 1\end{array}\right]}_{2m\times 2m};$ modal damping matrix $C={\left[\begin{array}{ccccccc}2{\xi }_{x,1}{\omega }_{\mathrm{nx},1}& & & & & & 0\\ & 2{\xi }_{y,1}{\omega }_{\mathrm{ny},1}& & & & & \\ & & 2{\xi }_{x,2}{\omega }_{\mathrm{nx},2}& & & & \\ & & & 2{\xi }_{y,2}{\omega }_{\mathrm{ny},2}& & & \\ & & & & \ddots & & \\ & & & & & 2{\xi }_{x,m}{\omega }_{\mathrm{nx},m}& \\ 0& & & & & & 2{\xi }_{y,m}{\omega }_{\mathrm{ny},m}\end{array}\right]}_{2m\times 2m};$ modal stiffness matrix $K={\left[\begin{array}{ccccccc}{\omega }_{\mathrm{nx},1}^2& & & & & & 0\\ & {\omega }_{\mathrm{ny},1}^2& & & & & \\ & & {\omega }_{\mathrm{nx},2}^2& & & & \\ & & & {\omega }_{\mathrm{ny},2}^2& & & \\ & & & & \ddots & & \\ & & & & & {\omega }_{\mathrm{nx},m}^2& \\ 0& & & & & & {\omega }_{\mathrm{ny},m}^2\end{array}\right]}_{2m\times 2m};$ mode shape matrix: ψ=[{tilde over (φ)}.sub.x,1 {tilde over (φ)}.sub.y,1 {tilde over (φ)}.sub.x,2 {tilde over (φ)}.sub.y,2 . . . {tilde over (φ)}.sub.x,m {tilde over (φ)}.sub.y,m].sub.2p×2m, where the dimension of {tilde over (φ)}.sub.d,j(j=1,2, . . . , m; d=x or y) is 2p×1 and {tilde over (φ)}.sub.d,j represents the j-th order mode shape vector corresponding to each cutting layer differentiator in the principle vibration direction of X or Y direction.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a graphical representation of cross-axis and cross-point modal testing of the present invention: (a) denotes the hammer impacting at the nodes one by one in X direction; (b) denotes the hammer impacting at the nodes one by one in Y direction.

(2) FIG. 2 is a graphical representation of the transfer function groups according to the vibration response measured by X or Y axis of the tri-axial acceleration sensor.

DETAILED DESCRIPTION

(3) Below, with the combination of attached drawings and technical solution, the concrete implementation process of the invention is explained in detail. As shown in FIGS. 1(a) and (b), the method firstly installs a miniature tri-axial acceleration sensor at the tool tip, and conducts cross-axis and cross-point experimental modal tests respectively in two horizontally orthogonal directions (X and Y directions) at preset nodes of the cutter axis using a force hammer The measured transfer functions are grouped according to different measuring axes, and the dynamic parameters (modal mass, damping, stiffness and mode shape) are separately identified from each group of transfer functions. Then, the contact region between the cutter and the workpiece is divided into several cutting layer differentials along the cutter axis under the condition of a given axial cutting depth, and the differentials of each layer are allocated with the value of the mode shape identified at preset nodes through linear interpolation. After that, together with other dynamic parameters, all the parameters are assembled into system dynamic parameter matrices matching with the dynamic model. Finally, dynamic parameter matrices including the effects of cross-axis and cross-point model couplings are obtained. Taking the cylindrical milling process as an example, the specific steps adopted are:

(4) Step 1 Install the cutter in the handle, clamp the handle in the machine tool spindle, and establish cutter coordinate system: the origin of coordinates is set on the free end of the cutter, the feed direction of cutter is set as the X axis, the direction perpendicular to the surface to be machined is set as Y axis (outward for down milling and inward for up milling), and the Z axis is set as the direction away from the free end of the cutter and along the cutter axis.

(5) Step 2 Starting from the free end of the cutter in a certain distance along the cutter axis, mark q nodes which will be impacted by the hammer, install a miniature tri-axial acceleration sensor at the tool tip, and impact at each node in two horizontally orthogonal X and Y directions with the hammer, to measure all the transfer functions of the spindle-handle-cutter system at each node.

(6) Step 3 For the transfer functions measured via Step 2, eliminate all transfer functions measured by Z axis of the acceleration sensor, and then divide the remaining transfer functions into two different transfer function groups according to the vibration response measured by X or Y axis of the acceleration sensor, and the two groups of transfer function are marked as {FRF.sub.x} and {FRF.sub.y}, respectively, as shown in FIG. 2.

(7) Step 4 Identify the dynamic parameters respectively from the two groups of different transfer functions {FRF.sub.x} and {FRF.sub.y} obtained in Step 3. Based on {FRF.sub.x}, the identified previous in order dynamic parameters are expressed as follows. Natural frequencies are ω.sub.nx,1, ω.sub.nx,2, . . . , ω.sub.nx,m. Damping ratios are ξ.sub.x,1, ξ.sub.x,2, . . . , ξ.sub.x,m. Mode shape matrix is ψ.sub.x=[φ.sub.x,1, φ.sub.x,2, . . . , φ.sub.x,m].sub.2q×m, where the dimension of φ.sub.x,j(j=1, 2, . . . , m) is 2q×1 and φ.sub.x,j represents the j-th order mode shape vector corresponding to each impact node in the principle vibration direction of X direction. Based on {FRF.sub.y}, the identified previous in order dynamic parameters are expressed as follows. Natural frequencies are ω.sub.ny,1, ω.sub.ny,2, . . . , ω.sub.ny,m. Damping ratios are ξ.sub.y,1, ξ.sub.y,2, . . . , ξ.sub.y,m. Mode shape matrix is ψ.sub.y=[φ.sub.y,1, φ.sub.y,2, . . . , φ.sub.y,m].sub.2q×m, where the dimension of φ.sub.y,j(j=1,2, . . . , m) is 2q×1 and φ.sub.y,j represents the j-th order mode shape vector corresponding to each impact node in the principle vibration direction of Y direction ψ.sub.x and ψ.sub.y can be expressed as follows, respectively:

(8) ${\Psi }_x={\left[\begin{array}{cccc}{u}_{x,x,1,1}& u_{x,x,1,2}& \mathrm{.Math.}& u_{x,x,1,m}\\ u_{x,y,1,1}& u_{x,y,1,2}& \mathrm{.Math.}& u_{x,y,1,m}\\ u_{x,x,2,1}& u_{x,x,2,2}& \mathrm{.Math.}& u_{x,x,2,m}\\ u_{x,y,2,1}& u_{x,y,2,2}& \mathrm{.Math.}& u_{x,y,2,m}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ u_{x,x,q,1}& u_{x,x,q,2}& \mathrm{.Math.}& u_{x,x,q,m}\\ u_{x,y,q,1}& u_{x,y,q,2}& \mathrm{.Math.}& u_{x,y,q,m}\end{array}\right]}_{2q\times m};$${\Psi }_y={\left[\begin{array}{cccc}{u}_{y,x,1,1}& u_{y,x,1,2}& \mathrm{.Math.}& u_{y,x,1,m}\\ u_{y,y,1,1}& u_{y,y,1,2}& \mathrm{.Math.}& u_{y,y,1,m}\\ u_{y,x,2,1}& u_{y,x,2,2}& \mathrm{.Math.}& u_{y,x,2,m}\\ u_{y,y,2,1}& u_{y,y,2,2}& \mathrm{.Math.}& u_{y,y,2,m}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ u_{y,x,q,1}& u_{y,x,q,2}& \mathrm{.Math.}& u_{y,x,q,m}\\ u_{y,y,q,1}& u_{y,y,q,2}& \mathrm{.Math.}& u_{y,y,q,m}\end{array}\right]}_{2q\times m}$
where u.sub.d,c,α,β (d=x or y, c=x or y, α=1, 2, . . . , q, β=1, 2, . . . , m) is the value of the β-th order mode shape of the α-th impact node in c direction with the principle vibration direction of d direction

(9) Step 5 Divide the contact region between the cutter and workpiece under a given axial cutting depth α.sub.p along the cutter axis into p cutting layer differentiators. According to the relative position between the center of each differentiator and above impact nodes, allocate these differentiators the value of the mode shape identified by Step 4 through linear interpolation.

(10) Step 6 According to different modal order, assemble different types of dynamic parameters into modal mass, damping, stiffness and mode shape matrices and these matrices should match the system dynamic model. After assembly, one can obtain:

(11) modal mass matrix:

(12) $M={\left[\begin{array}{ccccccc}1& & & & & & \\ & 1& & & & & \\ & & 1& & & & \\ & & & 1& & & \\ & & & & \ddots & & \\ & & & & & 1& \\ & & & & & & 1\end{array}\right]}_{2m\times 2m};$
modal damping matrix:

(13) $C={\left[\begin{array}{ccccccc}2{\xi }_{x,1}{\omega }_{\mathrm{nx},1}& & & & & & 0\\ & 2{\xi }_{y,1}{\omega }_{\mathrm{ny},1}& & & & & \\ & & 2{\xi }_{x,2}{\omega }_{\mathrm{nx},2}& & & & \\ & & & 2{\xi }_{y,2}{\omega }_{\mathrm{ny},2}& & & \\ & & & & \ddots & & \\ & & & & & 2{\xi }_{x,m}{\omega }_{\mathrm{nx},m}& \\ 0& & & & & & 2{\xi }_{y,m}{\omega }_{\mathrm{ny},m}\end{array}\right]}_{2m\times 2m};$
modal stiffness matrix:

(14) $K={\left[\begin{array}{ccccccc}{\omega }_{\mathrm{nx},1}^2& & & & & & 0\\ & {\omega }_{\mathrm{ny},1}^2& & & & & \\ & & {\omega }_{\mathrm{nx},2}^2& & & & \\ & & & {\omega }_{\mathrm{ny},2}^2& & & \\ & & & & \ddots & & \\ & & & & & {\omega }_{\mathrm{nx},m}^2& \\ 0& & & & & & {\omega }_{\mathrm{ny},m}^2\end{array}\right]}_{2m\times 2m};$
mode shape matrix:

(15) $\Psi ={\left[\begin{array}{ccccccc}{\tilde{u}}_{x,x,1,1}& {\tilde{u}}_{y,x,1,1}& {\tilde{u}}_{x,x,1,2}& {\tilde{u}}_{x,y,1,2}& \mathrm{.Math.}& {\tilde{u}}_{x,x,1,m}& {\tilde{u}}_{x,y,1,m}\\ {\tilde{u}}_{x,y,1,1}& {\tilde{u}}_{y,x,1,1}& {\tilde{u}}_{x,y,1,2}& {\tilde{u}}_{y,y,1,2}& \mathrm{.Math.}& {\tilde{u}}_{x,y,1,m}& {\tilde{u}}_{y,y,1,m}\\ {\tilde{u}}_{x,x,2,1}& {\tilde{u}}_{y,x,2,1}& {\tilde{u}}_{x,x,2,2}& {\tilde{u}}_{y,x,2,2}& \mathrm{.Math.}& {\tilde{u}}_{x,x,2,m}& {\tilde{u}}_{y,x,2,m}\\ {\tilde{u}}_{x,y,2,1}& {\tilde{u}}_{y,y,2,1}& {\tilde{u}}_{x,y,2,2}& {\tilde{u}}_{y,y,2,2}& \mathrm{.Math.}& {\tilde{u}}_{x,y,2,m}& {\tilde{u}}_{y,y,2,m}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\tilde{u}}_{x,x,p,1}& {\tilde{u}}_{y,x,p,1}& {\tilde{u}}_{x,x,p,2}& {\tilde{u}}_{y,x,p,2}& \mathrm{.Math.}& {\tilde{u}}_{x,x,p,m}& {\tilde{u}}_{y,x,p,m}\\ {\tilde{u}}_{x,y,p,1}& {\tilde{u}}_{y,y,p,1}& {\tilde{u}}_{x,y,p,2}& {\tilde{u}}_{y,y,p,2}& \mathrm{.Math.}& {\tilde{u}}_{x,y,p,m}& {\tilde{u}}_{y,y,p,m}\end{array}\right]}_{2p\times 2m}$

(16) where ũ.sub.{tilde over (d)},{tilde over (c)},{tilde over (α)},{tilde over (β)} (d{tilde over ( )}=x or y, c{tilde over ( )}=x or y, α{tilde over ( )}=1, 2, . . . , p, β{tilde over ( )}1, 2 . . . , m) is the value of the {tilde over (β)}-th order mode shape of the {tilde over (α)}-th cutting layer differentiator in {tilde over (c)} direction with the principle vibration direction of {tilde over (d)} direction.