#### List of Symbols

F_{R}

F_{X}

F_{Y}

F_{Z}

F_{N}

F_{f}

F_{T}

ΔF

_{R}and F

_{T})

Δd

Off_{x}

Off_{z}

$AA_OFF

SYG_IU

Δt

K_{P}

K_{I}

K_{D}

v_{d/t}

f_{w}

v_{w}

d_{r}

d_{d}

σ

### 1 Introduction

1) Smoothing of the workpiece surface, which reduces the notch effect.

2) Strain hardening in the subsurface of the workpiece.

3) Generation of residual compressive stresses on the workpiece surface, which reduce tensile stresses under load.

### 2 Rolling Force Control for Mechanical Deep Rolling

### 2.1 Basic Considerations for a Rolling Force Control for Mechanical Deep Rolling Tools

1) Variations of the workpiece diameter within the tolerance limits due to inaccuracy of the turning process.

2) Deflection of the workpiece caused by the deep rolling force F

_{R}.3) Positioning errors caused by an imprecise calibrated deep rolling tool or by inaccuracies of the machine tool axes.

_{R}.

_{R}is a three-dimensional force consisting of the force components in x-, y- and z-direction, as shown in Fig. 1. The force F

_{Y}is the tangential force on the roller. This force points in the rolling direction of the roller and it therefore corresponds to the rolling resistance. As the rolling resistance is minimal, this force component is not taken into account here. The main part of the rolling force acts in the x-z plane, therefore it is assumed that F

_{R}only consists of the force components F

_{X}and F

_{Z}. The rolling force normal to the surface (F

_{N}) of the workpiece is particularly important for deep rolling. In order to determine the normal force F

_{N}from the force components F

_{X}and F

_{Z}, the orientation of the workpiece contour must be known. For a cylindrical workpiece, F

_{N}is equal to F

_{X}and F

_{Z}is equal to the feed force F

_{f}. In a measurement on a cylindrical workpiece, F

_{X}= F

_{N}= 1,000 N and F

_{Z}= F

_{f}= 115 N were measured. This means that the rolling force is F

_{R}= √F

_{X}

^{2}+ F

_{Z}

^{2}= 1,007 N. Thus, during deep rolling the feed force F

_{f}is low in relation to F

_{N}so that F

_{R}≈ F

_{N}. It is consequently not necessary to differentiate between the value of the normal force F

_{N}and the rolling force F

_{R}. Therefore, only F

_{R}is considered in this paper.

_{R}has to stay within defined limits. An unwanted variation of the rolling force is thus a problem for mechanical deep rolling. A rolling force control enables a reliable manufacturing within the desired limits and to compensate for the geometric disturbance Δd. The deep rolling force control (DRFC) considered here, is based on offsets (Off

_{x}and Off

_{z}) that are applied to the machine tool axes. By setting these offsets to the axes, the deflection of the leaf springs of the deep rolling tool is changed. This in turn changes the resulting rolling force. Thus, by controlling the axes offsets, the deep rolling force can be controlled.

### 2.2 Implementation of the DRFC

_{R}is controlled to match the target force F

_{T}. The DRFC was realized on a machine tool of the type DMG MORI CTX beta 800 TC. The machine control on this machine tool is a Siemens Sinumerik 840D sl. Within the Sinumerik, an adjustment of the machine tool axis offset is allowed. Via the Sinumerik variable $AA_OFF offset can be set to the machine tool axes. In process, the variable $AA_OFF can be changed by synchronized actions. Within a synchronized action the axes offset $AA_OFF is set to the value of a Sinumerik GUD (Global User Data) variable of type SYG_IU. This type of GUD variable is capable of synchronized actions. In addition, the GUD variable can be changed via OPC UA from an external PC to control the axes offset. On the other hand, the rolling force was measured by the sensory deep rolling tool. The force measurement of the tool is based on strain gauges. The tool has integrated electronics that transmit the strain gauge data wirelessly via MQTT (Message Queuing Telemetry Transport) to the external control PC. There the rolling force is calculated from the strain gauge data. Since the rolling force is available and the axes offset can be written, a force control can be realized. For the rolling force control a PID-controller was used. It controls the axes offsets corresponding to the measured rolling force. The parameters of the PID-controller were adjusted experimentally on the machine tool, taking into account the step response of the controller (Further details in section 3.2). After setting the parameters of the PID-controller, a deep rolling force control (DRFC) was available.

### 3 Investigation of the DRFC

### 3.1 Dead Time of the DRFC

_{R}was therefore measured over 100 repetitions. In addition to the dead time of OPC UA, the dead time of the wireless data transmission of the tools force data via MQTT is included in the overall dead time Δt. For the machine tool CTX 800 used here, an average dead time of 0.45 s was determined and at maximum a dead time of up to 3 s occurred. Considering the high dead time, control operations are no typical use case for OPC UA. Thus, the potentials of an OPC UA based force control for mechanical deep rolling are not known. The DRFC was therefore initially investigated independently of the actual machining of a workpiece.

### 3.2 Setting the PID-Controller

_{P}, the integral gain factor K

_{I}and the derivative gain factor K

_{D}. In Fig. 4, the response of the controller to a step in the target force F

_{T}is shown. For this experiment, the deep rolling tool was positioned directly above the workpiece surface. This means that the measured rolling force F

_{R}without additional offset is 0 N. The target force was then changed from F

_{T}= 0N to F

_{T}= 400 N and the resulting rolling force was measured. Different parameter settings of the controller were tested and compared in the experiment. Regardless of the parameters selected, the rolling force F

_{R}increases with a delay to the target force F

_{T}. The delay corresponds to the dead time of about 0.45 s measured in section 3.1. First K

_{I}and K

_{D}were kept at 0 and the proportional gain K

_{P}was gradually increased. Measurement M01 shows the response of the controller for K

_{P}= 0.2. With this setting, there is a clear tendency to oscillate and a further increase in K

_{P}is not possible since the controller tends to become unstable at a higher K

_{P}. Nevertheless, the distance to the target force is still very large at over 200 N. The reason for the instability of the controller at a higher K

_{P}is the high dead time measured in section 3.1. To avoid overshooting, K

_{P}was reduced to 0.1 and an integral gain of K

_{I}= 0.005 was added (M02). The integral gain leads to a slowly increasing rolling force until the target force is reached. To reach the target force faster, K

_{I}was increased further until the target force was overshot (M03). Overshooting is not desirable in deep rolling, as a rolling force higher than the target force can damage the surface of the workpiece. Therefore, K

_{I}was reduced until overshoot no longer occurred (M04). Finally, the derivative gain K

_{D}was added (M05). The derivative gain K

_{D}dampens the response. This allows K

_{I}to be increased without the controller overshooting. M05 shows the system response for this final setting, which was then used for all further experiments. With this parameter setting, the target force is reached after about 2.5 s without overshooting. To summarize, it can be said that the main part of the control deviation is compensated by the integrating term of the controller. This means that the PID controller used here essentially behaves like an I-controller.

### 3.3 Ramp Response of the DRFC

_{d/t}. As shown in Fig. 5 a geometric deviation Δd occurs between the displaced contour of the workpiece and the non-displace contour. This type of deviation can be caused by a deflection of the workpiece as a result of the rolling force. During machining, the geometric deviation decreases as the deflection of the workpiece is less close to the chuck. V

_{d/t}thus depends on the machining time Δt and the geometric deviation Δd. To investigate the influence of V

_{d/t}on the residual tracking error ΔF of the deep rolling force F

_{R}, different V

_{d/t}were applied to the deep rolling tool while using the DRFC. In an experiment, a defined V

_{d/t}was achieved by applying a ramp function to the deep rolling tool through the movement of the machine tool axis. V

_{d/t}is thereby equivalent to the gradient of the ramp. The resulting ΔF depending on V

_{d/t}is show in Fig. 5. A linear correlation was determined. At a change of the deviation of 1 mm/min the remaining deep rolling force error ΔF is about 35 N. This error is relatively high considering the low change rate of the deviation of only 1 mm/min. The reason for that is the high dead time within the DRFC. To achieve a stable behaviour of the DRFC despite the dead time, the parameters of the PID-controller must be set conservative. This leads to a slow response time of the controller. However, deep rolling does not require a high dynamic of the controller. A typical machining time is approx. 1 min and the deviations that actually occur on the workpiece in production are small (<< 1 mm). That means the V

_{d/t}in production is usually much lower than 1 mm/min and the tracking error of the force is thus less than 35 N. The sensory deep rolling tool is suitable for deep rolling forces up to 4,000 N meaning that the relative tracking error for a ramp response of the DRFC is less than 1%.

### 3.4 Step Response of the DRFC

_{R}. The DRFC reduces this initial force error to 56 N. Additionally the DRFC reduces the remaining mean error of 83 to 3 N after a settling time of 3 s. Only a residual force noise of ±15 N around the target force remains. Furthermore, the step response allows to evaluate the stability of the DRFC. The step disturbance covers a broad frequency band. Thus, potential natural frequencies of the DRFC are excited by the step disturbance. Since the system response of the DRFC has no dominant oscillation and the oscillations that occur are quickly dissipated, the behaviour of the DRFC can be classified as stable.

_{T}. In this work the target force is assumed to be constant. Thus, the reference frequency response wasn’t investigated here.

### 3.5 DRFC in Two Dimensions

_{R}. The rolling force changes from almost complete x-direction to an equally distributed force in x- and z-direction and back to complete x-direction of the force. To achieve a two-dimensional force control, the returned offset from the PID-controller is separated into the x- and z-direction according to the direction of the currently acting rolling force. Thereby the target force F

_{T}is achieved as a combination of the forces F

_{X}and F

_{Z}. The measurement results in Fig. 7 show a maximum deviation of 65 N from the target force during the uncontrolled deep rolling process. During the controlled process the deviation is reduced to 54 N. Here the effect of the DRFC appears to be small. The reason for this is that vibrations occur in the radius during deep rolling. These vibrations are caused by a change of the contact geometry and the surface quality within the radius. The vibrations have a frequency much higher than the cutoff-frequency of the DRFC. Thus, they cannot be compensated. Despite this, the mean force for the uncontrolled process is 274 N and for the controlled process 299 N. The target force is 300 N. The error in the mean force is reduced from 26 to 1 N. Thus, the DRFC is capable to control a two-dimensional force as well.

### 4 Effect of DRFC on Resulting Residual Stress in the Workpiece

_{d}= 46 to 46.2 mm. Thereby an increasing disturbance Δd is achieved. As in section 3.3, this kind of disturbance can be caused by the bending of the Workpiece. The measurement results without DRFC show an increasing deep rolling force corresponding to the disturbance Δd. The rolling force is rising from 650 to 1,030 N over the length of the workpiece. After deep rolling the residual stress σ at the surface (information depth max. 5.5 μm) was measured with an x-ray diffractometer of type Seifert XRD 3003 TT. Three measurement points were selected along the machined path. The measured residual stress is decreasing from σ

_{ax}= −331 MPa (axial) at the first measurement point, over σ

_{ax}= −481MPa at the second point to σ

_{ax}= −525 MPa. Overall, a small geometric disturbance of Δd = 0.1 mm leads to a change in the achieved residual stress of Δσ

_{ax}= 194 MPa. For components with a high dynamic load this can significantly reduce the service life. Thus, to ensure a certain service life, the residual stress must be kept within defined limits. A constant residual stress level is potentially achieved by the DRFC. To prove this, the conical workpiece was deep rolled a second time with the DRFC.

_{w}, the rolling speed v

_{w}and the workpiece diameter of d

_{d}= 46 mm the processing time for a length of 90 mm is 0.58 min. Thus, the calculated V

_{d/t}is 0.17 mm/min and the corresponding tracking error is 6.6 N. The estimation of the tracking error according to Fig. 5 therefore provides good results. In comparison to the uncontrolled deep rolling process the force error is reduced considerable by more than 90%. The measurement of the residual stress shows that the variation of the achieved residual stress is reduced as well. The previously measured variation of 194 MPa is reduced to 28 MPa. It can therefore be concluded that the effectiveness of the DRFC in terms of constant residual stresses is achieved.