Grinding is an abrasive machining process carried out in the last step of manufacturing due to its capability to obtain smooth surfaces and precise geometrical tolerances. Traverse grinding is an extended type of the cylindrical grinding process used for parts such as electric motor shafts, pneumatic cylinders and hydraulic cylinders and is based on the kinematics of plunge grinding with a cross-feed motion parallel to the workpiece axis. This process comes with several non-controlled variables that change during the operation, producing defects in the workpiece topography [
1], including shape errors [
1] and chatter [
1,
2,
3]. This study is focused on the shape error of slender parts because shape form for these parts must be very accurate. For this purpose, the grinding machine’s structural stiffness must be very high because the system flexibility will determine the part’s geometrical accuracy. The forces produced in the contact between tool and workpiece generate relative displacements between them, affecting the workpiece geometry [
4]. In the case of slender parts, the main stiffness problem is with the part itself, due to its high flexibility (which varies depending on its geometry along its length). Although, for slender parts, the part itself is the least rigid element of the system, system stiffness and headstock-center eccentricity must be considered to determine the shape error in slender parts [
5]. Several studies have demonstrated that the main cause of workpiece deformation during the traverse grinding cycle is the normal force [
6,
7]. The grinding wheel pushes the workpiece, and the part is not ground to the required diameter because of its elastic deformation, so the material removal rate decreases at the point where the flexibility is the highest [
8]. A representative scheme of this process is shown in
Figure 1a, where
vt is the traverse speed,
Ft is the tangential or axial force, and
Fn is the radial or normal force. Shape error is presented in
Figure 1b, where tolerances of diameter dimension, straightness, or cylindricity can be out of order. In other machining processes, such as turning, where the machine configuration is based on a cross-slide, a follower rest is a comprehensive solution for avoiding slender-part deformation [
9]. Nonetheless, in a machine configuration where the table is movable, a follower rest is not viable due to its complexity. In a movable-table machine configuration, the element that significantly controls the deformation of the workpiece is the steady rest.
This geometrical issue comes with a poor productivity index because the shape error must be corrected in the spark-out due to the shape quality requirements for ground workpieces. Depending on the shape-error magnitude, the number of spark-out strokes can be excessive and, in the case of slender workpieces, the wheel must translate long distances end-to-end, which involves an increase in machining time.
Some research has been conducted to clarify and predict the deformation behavior of slender parts during the traverse grinding process, based on the radial force induced by the wheel pushing the workpiece, both in external [
7,
10] and internal [
11] grinding processes. Having presented the prediction model, various solutions have been described to minimize the predicted shape error. Intelligent artificial monitoring of the traverse operation has been developed to correct the shape error, by stabilizing the motion trajectory of the grinding wheel [
1]. High-frequency oscillations are induced in the radial depth of the wheel in the workpiece. Modifying wheel transversal speed along the workpiece is another technique [
6] that has been proposed for turning operations [
12]. Due to a decrease in the zone of the workpiece, where the stiffness is most critical, the normal forces decrease, thus decreasing the shape error. Ding et al. [
13] presented an intelligent optimization control for the shape-error correction by varying the transversal speed, as reported by Onishi et al. [
6]. Based on the elastic deformation equation, variation in the traverse speed and workpiece speed were combined to optimally affect the normal force, and therefore the real depth of cut, decreasing the shape error. Automation for this solution is complicated, particularly as with a different transversal speed, topography and geometric parameters will change throughout the length of the workpiece. The same approach was presented by Fujii et al. [
14], but also took into consideration the use of steady rest, which leads to a more expensive- and difficult-to-set-up solution.
In this paper, a new model is presented to simplify the prediction of the shape error in a traverse grinding operation for long slender parts. The effect of steady rests on the deformation of the workpiece can be considered if required by the application. The position and the number of steady rests are considered, along with a proposal for the optimal position of the steady rests. This model is based on the computer simulation of the deformation of slender multi-diameter rollers developed by Gao [
10]; in this case, the model required normal-force data as an input. Other variables were analyzed, such as center eccentricity [
1]. However, the main reason for observing the shape error of the part during traverse grinding operations is the elastic deformation of the workpiece due to grinding normal force [
6,
7]. Thus, the depth of cut is affected due to the deflection of the workpiece, and this varies along the length of the ground workpiece. The present experimental study examines slender parts in the traverse grinding cycle without steady rests to validate the estimation of the shape error based on grinding normal force. Stiffness of headstock and tailstock was measured and considered within the model. An improvement was included by also considering a distributed load, corresponding to the grinding wheel width. A solution for correcting the shape error based on the normal grinding force was then presented by applying a cross-compensation of the grinding-wheel trajectory based on the part deformation predicted by the model. Automating this solution is easy because the only variable input needed from the process is the normal force, which can be readily measured. Thus, more complicated solutions based on steady rests can be avoided. This automatization will reduce excessive machine times due to the number of spark-outs needed to correct the shape error produced in the workpiece.