## 2 Grinding Surface Topography of the Screw Raceway
### Grinding Kinematics
Assume that the surface topography of the workpiece surface consists of an array of points with different surface heights, as shown in Figure 2. The surface of the workpiece is uniformly discretized into grids, and all grid points form a matrix \(g_{m,n}\). The element \(g(m,\,n)\) of the matrix \(g_{m,n}\) represents the height of the point (\(m\),\(n\)) on the surface of the workpiece relative to the global coordinate system. The subscripts m and n are the positions of the workpiece surface height g(\(m\),\(n\)) in the x and y directions, respectively. In the same way, all grid points on the surface of the grinding wheel are characterized by a matrix \(h_{i,j}\), and element \(h(i,j)\) of the matrix \(hi,j\) stands for the height of the cutting edge of the abrasive grain at the point (\(i,j\)). To convert the local coordinates into the global coordinates, the distance from the origin o to the origin \(o\) has to be determined. The trajectory of the first abrasive grain is used as a reference, and the origin of the local coordinate system of the other abrasive grains is shifted to the global coordinate system. The offset distance of the local coordinate system in the \(x\)-direction is equal to the movement distance of the workpiece, \(\Delta O_{ij}=v_{w}t\). The arc  
Figure 2: Mapping of the grain trajectories to the workpiece surface. length of the rotational movement of the abrasive cutting edge is calculated, \(L_{ij}=v_{s}t\). Due to the small spacing of the abrasive grains in the circumferential direction of the grinding wheel, the arc length is approximated as a straight line. \(L_{i,j}=(i-1)\Delta x_{s}\). The offset distance of the local coordinate system in the \(x\)-direction is obtained as follows:  
\[\Delta O_{i,j}=(i-1)\Delta x_{s}v_{w}/v_{s} \tag{7}\]  
The translation distance of the local coordinate system relative to the global coordinate system in the \(z\)-direction is written as follows:  
\[\Delta z_{i,j}=a-\left(h_{\max}-h_{i,j}\right) \tag{8}\]  
where \(a\) is the grinding depth of the wheel. \(h_{\max}\) is the maximum height of the cutting edge of the abrasive grain, \(h_{\max}=max(h_{i,j})\). In the global coordinate system, the trajectories of all the abrasive grains are therefore described as follows:  
\[z_{i,j}=\frac{\left(m\Delta x_{w}-\Delta O_{i,j}\right)^{2}}{2r_{i,j}(1+v_{w}/ v_{s})^{2}}-\Delta z_{i,j} \tag{9}\]  
To acquire the grinding surface topography of the workpiece, it is necessary to map the trajectory of all abrasive grains onto the surface of the workpiece according to the grinding kinematics between the trajectory of all abrasive grains and the workpiece. The trajectory of several abrasive grains is recorded at a point \((m,n)\) on the surface of the workpiece, and the trajectory with the lowest point is selected as the final trajectory of the abrasive grains. The final trajectory of the abrasive grains on the workpiece surface is described as follows:  
\[z(m,n)=\min\left\{\frac{\left(m\Delta x_{w}-\Delta O_{i,j}\right)^{2}}{2r_{i,j }(1+v_{w}/v_{s})^{2}}-\Delta z_{i,j}\right\} \tag{10}\]  
The topography model of the grinding surface is established based on the grinding kinematics of all the abrasive grains. The height matrix of the surface topography model of the workpiece is obtained as follows:  
Figure 1:0  
where the subscript \(k\) is the surface profile produced by the \(k\)-th abrasive grain. To simulate a multi-pass grinding operation, the generated surface is fed back into the programmer as the initial surface texture of the workpiece.