**Experimental Analysis of the Effect of Vibration Phenomena on Workpiece Topomorphy Due to Cutter Runout in End-Milling Process †**

**Constantine David <sup>1</sup> , Dimitrios Sagris 1,\* ID , Evlampia Stergianni <sup>2</sup> , Christos Tsiafis <sup>2</sup> and Ioannis Tsiafis <sup>2</sup>**


Received: 4 April 2018; Accepted: 13 June 2018; Published: 1 July 2018

**Abstract:** Profile end-milling processes are very susceptible to vibrations caused by cutter runout especially when it comes to operations where the cutter diameter is ranging in few millimeters scale. At the same time, the cutting conditions that are chosen for the milling process have a complementary role on the excitation mechanisms that take place in the cutting area between the cutting tool and the workpiece. Consequently, the study of milling processes in the case that a cutter runout exists is of special interest. The subject of this paper is the experimental analysis of the effect of cutter runout on cutter vibration and, by extension, how this affects the chip removal and, thereby, the workpiece topomorphy. Based on cutting force measurements correlated with the workpiece topomorphy under various cutting process parameters, such as the cutting speed, feed rate, and the axial cutting depth, some useful results are extracted. Hence, the effect of vibration phenomena, caused by cutter runout, on the workpiece topomorphy in end milling can be evaluated.

**Keywords:** end-milling; cutter runout; cutting force; workpiece topomorphy

#### **1. Introduction**

During the cutting process, it is very important to know, as precisely as possible, the relationship between the vibration behavior of the machine tool-cutting tool-workpiece system and the cutting conditions (cutting speed, feed rate and axial depth of cut) in order to achieve high workpiece surface quality [1,2].

This is especially the case when it comes to the use of contemporary computer numerical control (CNC) machine tools in micromachining processes [3], for which it is crucial to define a group of optimal cutting conditions aiming at:


Finding an appropriate combination of process parameters has been such a challenging problem that a great deal of research has been devoted to the topic. Generally, it is difficult to define optimal cutting conditions to fit every cutting process. However, it is feasible to find out proper cutting parameters for a specific combination of machine tool, cutting tool and workpiece material.

The aim of the paper is to investigate experimentally the effect of the tool runout both on the cutting forces and the resulting workpiece surface. The paper follows a previous one [7], which was dealing with the effect of vibration phenomena on the workpiece topomorphy in general. The current research puts particular emphasis on the effect of vibrations caused by cutter runout on the workpiece surface quality. This is even more important when it comes to micromachining [3,8,9].

Tool clamping is usually accompanied by cutter runout. Although the cutter runout may be reduced by careful clamping performed by skilled machinists, a residual runout exists in the micro scale.

In the case of a residual cutter runout, unequal cutting forces are developed on the tool flutes during each cutter revolution. One of the flutes, where the eccentricity appears, undertakes the main cutting work and, therefore, is over-loaded, whereas the other flutes remain under-loaded. Because of that higher stresses are developed on the rake surface and the principal cutting edge of the particular flute. Such a situation leads to early flute wear, which deteriorates the whole cutting process. Moreover, the process is susceptible to becoming unstable [1].

Additionally, the effect of the cutter runout should be taken into consideration in order to secure the surface quality in profile end milling [10]. In this case, the enhanced cutting forces induce tool bending, thereby worsening the surface finish.

To investigate the effect of cutter runout on the cutting forces and the workpiece surface texture, dedicated experiments were conducted evaluating the cutting forces and the workpiece surface topomorphy. In particular, by means of experimental results, the cutting conditions which affect the cutting process outcome were studied. In this sense, the purpose of this experimental study was to find out the effect of the cutter runout on the cutting mechanism, on the tool vibration, and sequentially on the workpiece surface quality.

#### **2. Experimental Procedure**

To measure the cutting forces, specific milling experiments were conducted on the 5-axis CNC machining center DECKEL MAHO MH600C (DECKEL MAHO GmbH, Pfronten, Germany) shown in Figure 1. Before the measurement of the cutting forces, the workpiece was roughened in order to achieve the necessary flatness thus ensuring a constant depth of cut through the entire machined surface. For the roughing process, under the usage of coolant, an end-mill cutting tool of 40 mm diameter was selected.

The cutting tool used in the experiments was manufactured by the OSAWA Company (Osaka, Japan) with type G2CS4, and is shown in Figure 2. It is a 4-flute end-mill cutter of 2 mm diameter. The cutting tool material is micrograin carbide with multilayer physical vapor deposition (PVD) high performance coating of 3500 HV hardness. Due to its excellent abrasion resistance and the fact that it can withstand higher temperatures, this cutting tool is suitable for working with or without lubrication at high cutting speeds. All experiments of the research were performed under dry machining conditions.

The cutting tool was mounted on the ISO40 Tool holder by the aim of a ER32 collet. By a typical tool clamping procedure, the cutter was set up with a radial eccentricity leading to cutter runout of 7 µm. The resulted cutter runout was measured using a specific tool measurement device (DMG Violinear) with 1 µm resolution. The result of the eccentric tool mounting is schematically explained in Figure 3a.

**Figure 1.** The experimental setup.

**Figure 2.** The cutting tool used in the experiments.

The cutting kinematics considering the cutter runout is depicted on the surface texture with a varied waviness profile (Figure 3c). Specifically, the height of the waviness profile is increased as well as the waviness length. The theoretical maximum roughness R<sup>t</sup> of the roughness profile is expressed by Equation (1) when the tool is ideally axisymmetric and, correspondingly, by Equation (2) in the case that a tool eccentricity exists [11,12]. From these equations it can be clearly seen that an increase in feed per tooth f<sup>z</sup> causes surface roughness growth, whereas an increase in tool diameter D, causes a decline in surface roughness.

$$\mathbf{R\_{to}} = \frac{\mathbf{D}}{2} - \sqrt{\frac{\mathbf{D}^2 - \mathbf{f\_z^2}}{4}} \simeq \frac{\mathbf{f\_z^2}}{4 \cdot \mathbf{D}} \tag{1}$$

$$\mathbf{R\_{le}} = \mathbf{R} - \frac{1}{2} \cdot \sqrt{4\mathbf{R}^2 - \mathbf{f\_z^2} - 2 \cdot \mathbf{e\_r} \cdot \mathbf{f\_z} - \mathbf{e\_r^2}} \simeq \frac{\left(\mathbf{f\_z + e\_r}\right)^2}{4 \cdot \mathbf{D}} \tag{2}$$

It is also worth noticing that the surface roughness parameters in the above equations often vary from the real surface roughness values, especially for low tool feed speed. One of the reasons of these discrepancies are cutter displacements related to the cutter's runout. Moreover, the tool's vibrations have an additional effect also.

**Figure 3.** Representation of the workpiece waviness profile (**c**) due to tool radial eccentricity (**a**,**b**).

The parameter e<sup>r</sup> depicted in Figure 3b denotes the radial displacement between the cutting edges related to cutter runout. The cutter's static runout is mainly due to the offset between the position of the tool rotation axis and the spindle rotation axis. It is worth mentioning that in an end-milling process this radial cutter runout affects only the wall surface of the machined workpiece.

The material of the workpieces used for the experiments was aluminum alloy of class 7075 T651 (AlZn5.5MgCu). In Figure 4 the metallographic microstructure is shown. In the micrograph there are visible various phases granulation formed (Al-MgZn2, MgSi2, and phase contains Fe) in the Al matrix. This material has very high strength and, therefore, is used widely in applications in aerospace and in military industry.

In order to measure the cutting forces, metrological equipment was used, which consists of the stationary dynamometer type KISTLER 9257B (Kistler Group, Winterthur, Switzerland), the charge amplifier type KISTLER 5233A and an Analog/Digital converter of the National Instruments type NI PCI-MIO-16E (National Instruments, Austin, TX, USA) which was connected to a computer.

The three cutting force components Fx, Fy and Fz were measured in real time, whereby Fx component represents the feed force, Fy component represents the back force and Fz component represents the main cutting force (Figure 1). The measurements were taken with the aid of the inbuilt piezoelectric sensors. The output of the sensors is an electric charge (Q) linearly proportional to the force acting to the sensor. The charge amplifier intensifies this charge into a normalized voltage signal, which can then be digitized and acquired in the signal-processing system. The digital signals are subjected to a further processing in order to be evaluated both in time and frequency domain. The cutting parameters of the conducted experiments are shown in Table 1.

**Figure 4.** Metallographic structure of the aluminum alloy used in the experiments.


**Table 1.** Cutting conditions.

#### **3. Experimental Results and Discussion**

The cutting force components, after undergoing a low-pass filtering of 200 Hz, were recorded within a time frame of 0.04 s by a sampling rate of 25,000 samples/s. In Figures 5 and 6, typical measurement graphical interfaces are illustrated, showing the cutting parameters and the three cutting force components both in the time and frequency domains.

From all measurements, it is obvious that there is a direct correlation between the feed rate and the amplitude of the main cutting force (Fz) and the feed force (Fx). Regarding the passive force component (Fy), it is negligible.

Furthermore, taking into account that the cutting force is proportional to the axial cutting depth, it is confirmed that by working at higher material removal rates (MRR) more intense tool excitation is expected, leading to vibration phenomena capable of affecting the workpiece surface quality.

The effect of the tool runout on the cutting force and thereby on the vibration phenomena is clearly depicted in the Fast Fourier Transform (FFT) spectra of the cutting-force components. The tool rotation frequency amounts to 91.67 s−<sup>1</sup> . From the cutting force spectra, we can see that only one cutter flute contributes to the cutting process at this frequency. This is more evident for the main cutting force (Fz) in relation to the feed force (Fx). Generally, the primer cutter flute located at the offset direction of the tool undertakes the main cutting work and the next flutes have less material to remove. This is demonstrated with the lower magnitude spectra peaks at 2×, 3× and 4× of the main tool-rotation frequency.

**Figure 5.** Cutting force components during milling process (feed rate 0.01 mm/tooth, axial depth of cut 0.4 mm).

**Figure 6.** Cutting force components during milling process (feed rate 0.02 mm/tooth, axial depth of cut 0.4 mm).

In manufacturing, it is important to verify the topomorphy of the final workpiece in order to produce high-quality products. The influence of the cutting parameters in cutting processes and the workpiece surface is evaluated through the workpiece topomorphy and the roughness profile. The combination of the cutting parameters has a direct impact on the cutting forces. This is due to the modification of the cutting process kinematics. As a result, the cutting process dynamics changes and, therefore, tool vibration excitations emerge.

In order to correlate the cutting parameters with the cutting force and thus the cutting dynamics the workpiece topomorphy was studied. For the quantitative and qualitative assessment of both workpiece topomorphy and the roughness profile, an optical profilometer of type VEECO NT1100 (Veeco, Tucson, AZ, USA) was used, which operates on the principle of white light interferometry. The measurement resolution in the depth direction of the workpiece topomorphy (perpendicular to the surface) amounted to less than 1 nanometer, while the resolution in the plane of the surface was about 0.4 microns. The resolution of the scanned areas by the white light interferometer was about 1 mm<sup>2</sup> (0.92 mm × 1.2 mm) for the ×10 magnification and 0.045 mm<sup>2</sup> (0.1853 mm × 0.2436 mm) for the ×50 magnification, captured by a 0.3 Mpixel sensor. The measurements were taken on the floor as well as on the wall of the machined surfaces. In Figure 7 the cutting tool paths in end milling are shown.

**Figure 7.** Cutting tool path and machined surfaces.

In Figure 8 the workpiece topomorphy and the roughness profile of the machined floor surface for feedrate 0.01 mm/tooth under various axial cutting depths are presented. The cutting tool traces are clearly depicted and according to the tool feedrate (0.04 mm/rev) the distance between two successive traces is approximately equal to 40 µm. In the digitized roughness profile, which refers to a length of 0.92 mm, a total of 23 tracks can be identified. This means that at every tool revolution just only one cutter flute path was imprinted on the surface texture. Apparently, each cutting flute takes part in the cutting process, but the eccentric one demonstrates higher force amplitude, which means that the specific flute cuts much more material. The traces of lower depth generated by the other flutes were removed because of the dominating flute path, leaving on the surface only one trace per tool revolution. Additionally, the tool deflection yielded an "axial runout", where the dominating flute was touching the workpiece more than the others, thus generating the observed bottom marks. Respectively, in Figure 9 the corresponding surface characteristics for feedrate 0.02 mm/tooth are shown. In this case, as expected, the distance between the successive tool traces amounted to 80 µm.

Table 2 shows the results of the maximum roughness value R<sup>t</sup> of the roughness profile concerning the machined floor surface under three axial cutting depths. Each of the results represents the mean value of R<sup>t</sup> (Rt) for 10 profiles measured in the feed direction.

By increasing the axial cutting depth, the workpiece roughness becomes higher. At the same time an enhancement of the tool feedrate results in an additional increment of the workpiece roughness. This is expected since by increasing the volume of removed material the main cutting force increases too, leading to cutting tool deflection, which yields higher roughness.

**Figure 8.** Topomorphy (zoom ×10) and roughness profile of the workpiece floor surface for 0.01 mm/tooth feedrate and 0.2 mm (**a**) and 0.4 mm (**b**) axial cutting depth.

**Figure 9.** Topomorphy (zoom ×50) and roughness profile of the workpiece floor surface for 0.02 mm/tooth feedrate and 0.4 mm (**a**) and 0.6 mm (**b**) axial cutting depth.


**Table 2.** Maximum roughness profile value Rt of the machined floor surface.

In Figure 10 the topomorphy of the machined workpiece floor surface is presented in detail (zoom ×50). It can be observed that waviness is formed on the workpiece surface inside the cutting tool path (see figure detail). This waviness has occurred due to the tool vibration during the cutting process. The tool was subjected to high vibrations, as a result of the relatively high cutting forces, because of the tough cutting conditions of the specific experiment in relation to the others.

**Figure 10.** Workpiece topomorphy detail (zoom ×50) at the workpiece floor surface for tool feedrate of 0.02 mm/tooth and axial cutting depth of 0.6 mm.

In Figure 11, the workpiece topomorphy of the machined wall surface is illustrated for a tool feedrate of 0.02 mm/tooth and axial cutting depth of 0.2, 0.4 and 0.6 mm. The cutting tool traces due to the tool kinematics are shaped in the form of waves at each tool revolution. Observing the number of the waves at the examined length, the cutting tool feedrate can be validated. That means that in a length of about 0.9 mm, 11 waves appears with a wavelength of 80 µm, which is equal to the cutter feed per revolution (0.08 mm/rev). This means that just one cutting edge per tool revolution dominates in the cutting process because of the cutter runout effect. The waviness profile height depends on the axial depth of cut because of the superimposed cutter declination and cutter runout.

**Figure 11.** Workpiece topomorphy (zoom ×10) at the machined wall surface for tool feedrate of 0.02 mm/tooth.

#### **4. Conclusions**

In profile end-milling processes, the excitation mechanisms that take place in the cutting area between the cutting tool and the workpiece need to be carefully studied in order to ascertain the factors that affect the cutting operation. The present paper examines the effect of cutter runout on cutting forces and, thereby, on the workpiece topomorphy in cases where cutting tools of a few millimeters scale diameter are implemented in end-milling operations.

Specific experiments under various cutting conditions were conducted, whereby the cutting forces were recorded and appropriately processed. The results revealed that the cutting forces increased with the tool feedrate. Moreover, a rise of the axial cutting depth which resulted in a higher metal removal rate can cause more intense tool excitation, thus leading to vibration phenomena capable of affecting the workpiece surface finish.

In the case of the use of cutters with a residual runout, the topomorphy of the machined workpiece surface was also investigated. Increasing the axial cutting depth led to a higher surface roughness. Moreover, the surface roughness became worse with a higher tool feedrate. The experimental results have revealed that a direct correlation between the cutting conditions and both the workpiece topomorphy and the surface roughness exists.

In the case that a residual cutter runout is present, it has been verified that one cutting flute per tool revolution mostly takes part in the cutting process. Furthermore, evaluation of the machined workpiece topomorphy in end-milling process provides useful information about the effect of the vibration phenomena caused by cutter runout. To this end, the findings of this research can serve as useful suggestions for machinists who are confronting problems caused by tool runout.

**Author Contributions:** Conceptualization, C.D. and D.S.; Methodology, C.D. and D.S.; Software, D.S. and C.T.; Validation, C.D., D.S. and I.T.; Formal Analysis, C.D. and I.T.; Investigation, C.D., D.S. and E.S.; Resources, C.D., D.S., E.S., C.T. and I.T.; Data Curation, C.D. and D.S.; Writing-Original Draft Preparation, C.D., D.S. and E.S.; Writing-Review & Editing, C.D. and D.S.; Visualization, D.S.; Supervision, C.D.; Project Administration, C.D.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

1. Altintas, Y. *Manufacturing Automation—Metal Cutting Mechanics, Machine Tool Vibrations and CNC Design*, 2nd ed.; University of British Columbia: Vancouver, BC, Canada; Cambridge University Press: Cambridge, UK, 2012.


© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Prediction of Thrust Force and Cutting Torque in Drilling Based on the Response Surface Methodology**

**Panagiotis Kyratsis 1,\* ID , Angelos P. Markopoulos <sup>2</sup> ID , Nikolaos Efkolidis <sup>1</sup> , Vasileios Maliagkas <sup>1</sup> and Konstantinos Kakoulis <sup>1</sup> ID**


Received: 25 April 2018; Accepted: 31 May 2018; Published: 4 June 2018

**Abstract:** In this research, experimental studies were based on drilling with solid carbide tools and the material used was Al7075. The study primarily focused on investigating the effects of machining parameters (cutting speed, feed rate, diameter of the tool) on the thrust force (*Fz*) and the cutting torque (*Mz*) when drilling an Al7075 workpiece. The experimental data were analyzed using the response surface methodology (RSM) with an aim to identify the significant factors on the development of both the *F<sup>z</sup>* and *Mz*. The application of the mathematical models provided favorable results with an accuracy of 3.4% and 4.7%, for the *F<sup>z</sup>* and the *Mz*, respectively. Analysis of variance (ANOVA) was applied in order to examine the effectiveness of the model, and both mathematical models were established and validated. The equations derived proved to be very precise when a set of validation tests were executed. The importance of the factors' influence over the responses was also presented. Both the diameter of the cutting and the feed rate were found to be the factors of high significance, while cutting speed did not affect considerably the *F<sup>z</sup>* and *M<sup>z</sup>* in the experiments performed.

**Keywords:** Al7075; thrust force; cutting torque; response surface methodology

#### **1. Introduction**

A majority of products are directly or indirectly related to the metal cutting processes during their production processes. From the most commonly used processes, such as milling, turning, tapping and so forth, drilling constitutes the most commonly used. A significant number of holes are necessary for the assembly of different parts in order to be connected each other and develop the product itself.

Many researchers have developed empirical methodologies based on statistics, which can be used to learn the interactions between manufacturing factors. Response surface methodology (RSM) is one of the most widely used. RSM is a collection of an empirical modeling approach for defining the relationship between a set of manufacturing parameters and the responses. Different criteria are used and the significance of these manufacturing parameters on the coupled responses are examined. This methodology uses a number of techniques based on statistics, graphics, and mathematics, for both improving and optimizing a manufacturing process. Kumar and Singh [1] used ANOVA to investigate the material removal rate (MRR) and surface roughness of optical glass BK-7, which was drilled by a rotary ultrasonic machine. A number of manufacturing parameter combinations were used in order to effectively investigate the drilling process itself. The experimental process showed that the most

influential factor was the feed rate, while the developed prediction models kept the error within 5% at a 95% confidence level. RSM was used by Balaji et al. [2] on the drilling process of Ti-6Al-4V alloy. Drilling parameters (i.e., spindle speed, helix angle, feed rate) were tested and the surface roughness, flank wear, and acceleration of drill vibration velocity (ADVV) were measured. A multi-response optimization was performed with a view to optimize the drilling parameters for minimizing the output in each case (surface roughness, flank wear, ADVV). Similarly, Balaji et al. [3] investigated the effects of helix angle, spindle speed, and feed rate on surface roughness, flank wear, and ADVV during the drilling process of AISI 304 steel with twist drill tools. The significant parameters were retrieved based on the RSM, while the optimum drilling parameters were identified based on a multi-response surface optimization method.

Furthermore, Nanda et al. [4] used RSM to measure the responses, such as material removal rate (MRR), surface roughness, and flaring diameter, with the three input parameters—pressure, nozzle tip distance, and abrasive grain size—on a borosilicate-glass workpiece with zircon abrasives. Boyacı et al. [5] developed a fuzzy mathematical model using a multi-response surface methodology based on the drilling process of PVC samples in an upright drill. Cutting parameters, such as cutting speed, feed rate, and material thickness, were tested for the minimization of surface roughness and cutting forces. RSM was also used by Ramesh and Gopinath [6] during drilling of sisal-glass fiber reinforced polymer composites to predict the influence of cutting parameters on thrust force. The adequacy of the models was checked using the analysis of variance (ANOVA). The results showed that the feed rate was the most influencing parameter on the thrust force, followed by the spindle and the drill diameter. Jenarthanan et al. [7] developed a mathematical model in order to predict how the input parameters (tool diameter, spindle speed, and feed rate) influence the output response (delamination) in machining of the ARALL composite using different cutting conditions. RSM was used for the analysis of the influences of entire individual input machining parameters on the delamination.

Rajamurugan et al. [8] developed empirical relationships to model thrust force in the drilling of GFRP composites by a multifaceted drill bit. The empirical relationships were developed by the response surface methodology, incorporating cutting parameters such as spindle speed (N), feed rate (f), drill diameter (d), and fiber orientation angle (q). The result was that the developed model can be effectively used to predict the thrust force in drilling of GFRP composites within the factors and their limits of the study. Rajkumar et al. [9] used RSM to predict the input factors influencing the delamination and thrust force on drilled surfaces of carbon-fiber reinforced polymer composite at different cutting conditions with the chosen range of 95% confidence intervals. Ankalagi et al. [10] used response surface methodology to analyze the machinability and hole quality characteristics in the drilling of SA182 steel with an HSS drill. They investigated the effects of machining parameters, such as cutting speed, feed rate, and point angle, on (a) thrust force, (b) specific cutting coefficient, (c) surface roughness, and (d) circularity error. The experimental methodology showed that thrust force, specific cutting coefficient, and surface roughness decrease with the increase in cutting speed, whereas the circularity error increases with increased cutting speed and feed, on micro-EDM drilling process parameters. Natarajan et al. [11] analyzed the effect of the manufacturing parameters (i.e., feed rate, capacitance, voltage) on machining a stainless-steel shim with a tungsten electrode. In all cases, the surface roughness, the MRR, and tool wear were measured. RSM helped the development of statistical models for multi-response optimization, based on the desirability function with an aim to determine the optimum manufacturing parameters. Finally, Kyratsis et al. [12] investigated the relationship of three manufacturing parameters (diameter of the tools, cutting speed, feed rate) on the cutting forces developed when drilling an Al7075 workpiece. The response surface methodology was the main tool for establishing the appropriate mathematical prediction models.

This paper presents a study of the Al7075 drilling process when using solid carbide tools with different diameters. A series of appropriately selected cutting speeds and feed rates were applied and different mathematical models for the prediction of the thrust force (*Fz*) and the cutting torque (*Mz*) were calculated. Analysis of variance (ANOVA) was used in order to validate the adequacy of the

mathematical models and depict the significance of the manufacturing parameters. The novelty of this work is the development of mathematical models that can be used with high levels of confidence in order to predict the thrust force and torque expected during the drilling of Al7075, within a wide range of cutting parameters (cutting speed 50~150 m/min, feed rate 0.15~0.25 mm/rev). The optimum usage of cutting tools is something crucial for CNC users as it can affect the whole machining process.

#### **2. Materials and Methods**

#### *2.1. Selection of Materials*

Nowadays, the metal cutting process is a very important issue for the manufacturing sector. Al7075 is an aluminum alloy having zinc as the main alloying element. Due to its excellent properties (strength, density, thermal behavior), it is widely used in the manufacturing industry for vehicles. In addition, it provides the sector of aviation with a lighter, stronger aluminum component, as well as enabling safer, lighter, and more fuel-efficient vehicles. Furthermore, its ability to be highly polished makes it suitable for the mold tool manufacturing industry. The aforementioned reasons have led a number of researchers to focus their studies on these factors [13–15]. In this study, an Al7075 plate (150 mm × 150 mm × 15 mm) was selected for performing the experimental work. The material's mechanical properties and chemical composition is depicted in Table 1.


**Table 1.** Mechanical properties and chemical composition of Al7075.

#### *2.2. Response Surface Methodology*

RSM is a popular method for deriving mathematical models of manufacturing systems, based on the principles of statistics. The same method can be the basis for applying optimization as well. The input parameters are identified together with their value ranges in the application under study. Then, the experimental process is set. Finally, mathematical models are calculated for the responses measured in each case. The effects of all input parameters and their interactions on the response are analyzed and their importance is derived. Different statistical methodologies (i.e., analysis of means and variances) are developed in order to establish the accuracy and adequacy of the derived mathematical model.

#### *2.3. Experimental Details*

In this research, an Al7075 block was used as a stock material. A HAAS VF1 CNC machining center was used to perform all the drilling tests. The machining center is able to perform with continuous speed and feed control within its operational limits. During the tests, three cutting speeds (*V*) and three feed rates (*f*) were applied in combination with three cutting tools' diameters (*D*). Table 2 depicts the factors used in this research together with their levels.


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**Table 2.** Machining factors and their levels.

A mechanical vise was used for clamping the workpiece and a Weldon clamping device with high rigidity was used to clamp the drill. The type of cutting fluid was KOOLRite 2270 which was provided by the delivery system near to the cutting tool. A Kistler dynamometer type 9123 (capable of measuring four components at the same time) measured the thrust force (*Fz*) and torque (*Mz*) values in each case of a set of twenty-seven experiments, which were executed by using 27 different (one tool for each hole) non-through coolant solid carbide drill tools (Kennametal B041A, flute helix angle of 30 degrees). All the possible combinations of the manufacturing parameters were used (cutting speed, feed rate, tool diameter). Figure 1 depicts the steps of the research conducted. The main shape of the cutting tools and all the dimension details are illustrated in Figure 2. A mechanical vise was used for clamping the workpiece and a Weldon clamping device with high rigidity was used to clamp the drill. The type of cutting fluid was KOOLRite 2270 which was provided by the delivery system near to the cutting tool. A Kistler dynamometer type 9123 (capable of measuring four components at the same time) measured the thrust force (*Fz*) and torque (*Mz*) values in each case of a set of twenty-seven experiments, which were executed by using 27 different (one tool for each hole) non-through coolant solid carbide drill tools (Kennametal B041A, flute helix angle of 30 degrees). All the possible combinations of the manufacturing parameters were used (cutting speed, feed rate, tool diameter). Figure 1 depicts the steps of the research conducted. The main shape of the cutting tools and all the dimension details are illustrated in Figure 2.

**Figure 1.** The workflow used for the research. **Figure 1.** The workflow used for the research.

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**Figure 2.** Cutting tool geometry details. **Figure 2.** Cutting tool geometry details. **Figure 2.** Cutting tool geometry details.

Figure 3 demonstrates a sample graph of the measurements of thrust force and cutting torque.

As mentioned, Kistler 9123 dynamometer has been used to measure the thrust force and cutting torque during the drilling process. Dynoware software (type 2825D-02) was used in order to monitor and record the values through a three-channel charge amplifier with a data acquisition system. During the tests, the thrust force and cutting torque were displayed graphically on the computer monitor and analyzed, enabling early error detection, and ensuring a steady state condition. As the sampling rate was high (approximately 10 kHz), the mean value was used as the final data in order to secure the value accuracy. The thrust force and cutting torque monitoring was developed as a means of enhancing the process monitoring capabilities. Figure 3 demonstrates a sample graph of the measurements of thrust force and cutting torque. As mentioned, Kistler 9123 dynamometer has been used to measure the thrust force and cutting torque during the drilling process. Dynoware software (type 2825D-02) was used in order to monitor and record the values through a three-channel charge amplifier with a data acquisition system. During the tests, the thrust force and cutting torque were displayed graphically on the computer monitor and analyzed, enabling early error detection, and ensuring a steady state condition. As the sampling rate was high (approximately 10 kHz), the mean value was used as the final data in order to secure the value accuracy. The thrust force and cutting torque monitoring was developed as a means of enhancing the process monitoring capabilities. Figure 3 demonstrates a sample graph of the measurements of thrust force and cutting torque. As mentioned, Kistler 9123 dynamometer has been used to measure the thrust force and cutting torque during the drilling process. Dynoware software (type 2825D-02) was used in order to monitor and record the values through a three-channel charge amplifier with a data acquisition system. During the tests, the thrust force and cutting torque were displayed graphically on the computer monitor and analyzed, enabling early error detection, and ensuring a steady state condition. As the sampling rate was high (approximately 10 kHz), the mean value was used as the final data in order to secure the value accuracy. The thrust force and cutting torque monitoring was developed as a means of enhancing the process monitoring capabilities.

**Figure 3.** Experimental value (mean value) of thrust force and torque for a step drill with tool diameter of 8 mm, cutting speed of 50 m/min, and feed rate of 0.15 mm/rev. **Figure 3.** Experimental value (mean value) of thrust force and torque for a step drill with tool diameter of 8 mm, cutting speed of 50 m/min, and feed rate of 0.15 mm/rev.

Figure 4 illustrates all the experimental values derived from the dynamometer. It is obvious from the figure that when the diameter of the tool increases, both the *F<sup>z</sup>* and *M<sup>z</sup>* values increase. The same happens in the case of the feed rate. As the feed rate values increase, the *F<sup>z</sup>* and the *M<sup>z</sup>* increase, respectively. On the other hand, the different values of cutting speed do not noticeably affect their value in both cases. Figure 4 illustrates all the experimental values derived from the dynamometer. It is obvious from the figure that when the diameter of the tool increases, both the *Fz* and *Mz* values increase. The same happens in the case of the feed rate. As the feed rate values increase, the *Fz* and the *Mz* increase, respectively. On the other hand, the different values of cutting speed do not noticeably affect their value in both cases.

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**Figure 4.** Experimental values derived from Kistler 9123. **Figure 4.** Experimental values derived from Kistler 9123.

#### **3. Results 3. Results**

RSM was employed with a view to developing mathematical models for the *Fz* and the *Mz* in terms of cutting speed, feed rate, and cutting tool diameters. Least square fitting was used with the mathematical models in order to offer reliability. A full factorial strategy was applied and twenty-seven drilling experiments were performed, and both the *Fz* and *Mz* were modelled separately using polynomial mathematical models. RSM was employed with a view to developing mathematical models for the *F<sup>z</sup>* and the *M<sup>z</sup>* in terms of cutting speed, feed rate, and cutting tool diameters. Least square fitting was used with the mathematical models in order to offer reliability. A full factorial strategy was applied and twenty-seven drilling experiments were performed, and both the *F<sup>z</sup>* and *M<sup>z</sup>* were modelled separately using polynomial mathematical models.

#### *RSM-Based Predictive Models RSM-Based Predictive Models*

is indicated by the following equation:

The RSM is a precise tool used in order to examine the influence of a series of input variables on the response, when studying a complex phenomenon. The analysis of variance (ANOVA) was performed to check the adequacy and accuracy of the fitted models. MINITAB was used for the statistical analysis. The models produced used least square fitting and provided reliable mathematical modeling. The 2nd order nonlinear model with linear, quadratic, and interactive terms The RSM is a precise tool used in order to examine the influence of a series of input variables on the response, when studying a complex phenomenon. The analysis of variance (ANOVA) was performed to check the adequacy and accuracy of the fitted models. MINITAB was used for the statistical analysis. The models produced used least square fitting and provided reliable mathematical

modeling. The 2nd order nonlinear model with linear, quadratic, and interactive terms is indicated by the following equation:

$$Y = b\_0 + b\_1 X\_1 + b\_2 X\_2 + b\_3 X\_3 + b\_4 X\_1^2 + b\_5 X\_2^2 + b\_6 X\_3^2 + b\_7 X\_1 X\_2 + b\_8 X\_1 X\_3 + b\_9 X\_2 X\_3 \tag{1}$$

where:

*Y* is the response,

*X<sup>i</sup>* stands for the coded values, and

*b<sup>i</sup>* stands for the model regression coefficients.

According to the experimental values derived from Kistler 9123 (Figure 2) and the aforementioned mathematical model, the following equations were derived for the thrust force (in N) and the cutting torque (in Nm) respectively:

$$F\_2 = -79 + 51.4D + 1.22V - 504f - 2.65D^2 - 0.0102V^2 + 2038f^2 + 0.128DV + 187Df + 0.07Vf(\text{N})$$

and

$$M\_{\Xi} = 1.51 - 0.309D + 0.00236V + 1.06f + 0.016D^2 - 0.000037V^2 - 12.5f^2 + 0.000208DV + 1.33Df$$
 
$$+ 0.0213Vf \text{(Nm)}$$

where:

*D* is the diameter of the tool in mm,

*f* is the feed rate in mm/rev, and

*V* is the cutting speed in m/min.

The adequacy of the models is secured at a 0.05 level of significance. The validity of the developed models is proved by the use of ANOVA. The R-sq(adj) is very high in both cases (99.4% for the *F<sup>z</sup>* and 98.9% for the *Mz*). In addition, when a 0.05 level of significance is used, the main contributors of the models are those with a *p*-value less than 0.05. In the case of the *Fz*, these contributors are: *D* (*p* = 0.026), *V* (*p* = 0.022), *D*<sup>2</sup> (*p* = 0.017), *V* 2 (*p* = 0.000), *DxV* (*p* = 0.000) and *Dxf* (*p* = 0.000), while in the case of the *M<sup>z</sup>* the main contributors are: *D*<sup>2</sup> (*p* = 0.048), *V* 2 (*p* = 0.007), *Dxf* (*p* = 0.000), *Vxf* (*p* = 0.023).

The validity of the models is also proved from the values of *F* and *p*. They indicate the significance of the mathematical model. The quality of the entire mathematical model can be proved by the *F* value which considers all the manufacturing parameters at a time. The *p* value depicts the probability of the manufacturing parameters having insignificant effect on the response. A large *F* value signifies better fit of the mathematical model with the acquired data from the experiments. The derived values of F-ratio for the models of *F<sup>z</sup>* and *M<sup>z</sup>* (Tables 3 and 4) are calculated equal to 467.15 and 261.36, respectively. They are both higher than the standard tabulated values. A mathematical model is considered reliable when the *F* value is high and the *p*-value is low (below 0.05 at a 0.05 level of significance).

Figures 5 and 6 both depict the main effect and interaction plots for the *F<sup>z</sup>* and *Mz*. In the interaction plot the interaction between diameter of the tool versus the cutting speed, and diameter of the tool versus the feed rate in all cases has been highlighted. As can be seen in Table 3, *V* 2 (*p* = 0.000), which implies that the square of the cutting speed affects the value of *F<sup>z</sup>* and *Mz*. This is more obvious when observing the main effect plots for *F<sup>z</sup>* and *M<sup>z</sup>* where the three means of the V depict a second order shape, compared to the rest of the mean effects plots that form almost straight lines. The same happens in the case of the cutting torque.


**Table 4.** ANOVA table for *Mz* (torque).


Residual analysis was performed in order to test the models' accuracy in both cases. The result was that the residuals appear to be normally distributed. They follow the indicated straight lines (almost linear pattern) while at the same time they are distributed almost symmetrically around the zero residual value proving that the errors are normally distributed. For both cases, there is not considerable variation of the observed values around the fitted line. The discrepancy of a particular value from its predicted value is called the residual value. When the residuals around the regression line are small, it means that the mathematical model is of high accuracy. All the scatter diagrams of the *F<sup>z</sup>* and *M<sup>z</sup>* residual values versus the fitted values provide enough evidence that the residual values are randomly distributed on both sides of the provided graph. The same is true for the residual values versus the order that the experiments were conducted (Figure 7). There is no evidence of any particular pattern in the residual values and, thus, the models proved their efficiency.


*Machines* **2018**, *6*, x FOR PEER REVIEW 8 of 12

Constant −78.9 135.1 0.58 0.567 *D* 51.36 21.06 2.44 0.026 *V* 1.2224 0.4867 2.51 0.022 *f* −503.7 712.0 −0.71 0.489 *D\*D* −2.651 1.002 −2.64 0.017 *V\*V* −0.010202 0. −6.36 0.000 *f\*f* 2038 1604 1.27 0.221 *D\*V* 0.12850 0.02835 4.53 0.000 *D\*f* 186.67 28.35 6.58 0.000 *V\*f* 0.067 1.134 0.06 0.954

**Table 4.** ANOVA table for *Mz* (torque).

Regression 9 12.7870 1.4208 261.36 0.000

Constant 1.505 1.014 1.48 0.156 *D* −0.3086 0.1581 −1.95 0.068 *V* 0.002357 0.003654 0.65 0.527

**Standard Error** 

**Square** *f***-Value** *p***-Value** 

**Coefficient** *t***-Value** *p***-Value** 

**Coefficient Coefficient** 

**Source Degree of Freedom Sum of Squares Mean** 

Residual Error 17 0.0924 0.0054

Total 26 12.8794

**Coefficient** 

**Predictor Parameter Estimate** 

R-Sq(adj) = 98.9%

**Figure 5.** Main effects plot for *Fz* and *Mz*. **Figure 5.** Main effects plot for *<sup>F</sup><sup>z</sup>* and *<sup>M</sup>z*. *Machines* **2018**, *6*, x FOR PEER REVIEW 9 of 12

**Figure 6.** Interaction plot for *Fz* and *Mz*. **Figure 6.** Interaction plot for *Fz* and *Mz*.

Residual analysis was performed in order to test the models' accuracy in both cases. The result was that the residuals appear to be normally distributed. They follow the indicated straight lines (almost linear pattern) while at the same time they are distributed almost symmetrically around the zero residual value proving that the errors are normally distributed. For both cases, there is not considerable variation of the observed values around the fitted line. The discrepancy of a particular value from its predicted value is called the residual value. When the residuals around the regression That is the characteristic of well suited mathematical modeling and reliable data derived from the equations provided. The derived mathematical models can be considered as very accurate and can be used directly for predicting both the *F<sup>z</sup>* and the *M<sup>z</sup>* within the limits of the manufacturing parameters used (diameter of tool, feed rate, cutting speed). The accuracy achieved is very high when comparing the measured values and those calculated from the mathematical models (3.4% and 4.7%, respectively).

line are small, it means that the mathematical model is of high accuracy. All the scatter diagrams of the *Fz* and *Mz* residual values versus the fitted values provide enough evidence that the residual values are randomly distributed on both sides of the provided graph. The same is true for the residual values versus the order that the experiments were conducted (Figure 7). There is no evidence of any particular pattern in the residual values and, thus, the models proved their efficiency. A set of experiments was conducted with a view to validate the given mathematical models for the prediction of the *F<sup>z</sup>* and *Mz*. Values of feed rate and cutting speed, within the range of experiments, were randomly selected (*V*: 70 m/min, *f*: 0.2 mm/rev) and tested with the three different cutting tools (*D*: 8-10-12 mm). The produced results can be considered satisfactory with less than 5.4% variation (Table 5). Especially, for the thrust force derived with the values (*D*: 10 mm, *V*: 70 m/min, *f*: 0.2 mm/rev), the variation was 0%.

That is the characteristic of well suited mathematical modeling and reliable data derived from the equations provided. The derived mathematical models can be considered as very accurate and can be used directly for predicting both the *Fz* and the *Mz* within the limits of the manufacturing

A set of experiments was conducted with a view to validate the given mathematical models for the prediction of the *Fz* and *Mz*. Values of feed rate and cutting speed, within the range of experiments, were randomly selected (*V*: 70 m/min, *f*: 0.2 mm/rev) and tested with the three different cutting tools (*D:* 8-10-12 mm). The produced results can be considered satisfactory with less than 5.4% variation (Table 5). Especially, for the thrust force derived with the values (*D*: 10 mm, *V*: 70

and 4.7%, respectively).

m/min, *f*: 0.2 mm/rev), the variation was 0%.

*Machines* **2018**, *6*, x FOR PEER REVIEW 10 of 12

**Figure 7.** Residuals analyses for the *Fz* and *M<sup>z</sup>*. **Figure 7.** Residuals analyses for the *Fz* and *Mz*.


## **4. Conclusions**

The aim of this study was the generation of mathematical models for the prediction of the thrust force (*Fz*) and cutting torque (*Mz*) related to the diameter of the drilling tool and other crucial manufacturing parameters (feed rate, cutting speed) during the drilling process. Research shows clearly that the prediction models sufficiently explain the relationship between the thrust force and cutting torque with the independent variables. A full factorial experimental process was executed under different conditions of the aforementioned parameters using an Al7075 workpiece and a full set of solid carbide tools. Response surface methodology was used as the base for the prediction of both the *F<sup>z</sup>* and the *M<sup>z</sup>* in a series of drilling operations with Al7075 as the material under investigation. The developed models were considered as very accurate for the prediction of the *F<sup>z</sup>* and *M<sup>z</sup>* within the range of the manufacturing parameters used. The outcomes proved that when using these models, the accuracy achieved was 3.4% and 4.7%, respectively. People working with CNC machines very often are faced with dilemmas about which cutting parameters are the most appropriate for the available cutting tools in order to treat materials, such as aluminum alloy 7075, which is suitable for a variety of specific applications in aerospace and chemical industries. Prediction of the thrust force and the cutting torque can lead to higher productivity when selecting manufacturing parameters due to the reduced wear effect on the drilling tool and the related energy consumption.

**Author Contributions:** Data curation, A.P.M.; formal analysis, N.E. and V.M.; investigation, P.K. and K.K.; methodology, P.K., A.P.M. and K.K.; validation, N.E. and V.M.

**Conflicts of Interest:** The authors declare no conflicts of interests

#### **References**


© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*

## **Influence of Hub Parameters on Joining Forces and Torque Transmission Output of Plastically-Joined Shaft-Hub-Connections with a Knurled Contact Surface**

#### **Lukáš Suchý 1,\*, Erhard Leidich <sup>1</sup> , Thoralf Gerstmann <sup>2</sup> ID and Birgit Awiszus <sup>2</sup>**


Received: 27 February 2018; Accepted: 4 April 2018; Published: 9 April 2018

**Abstract:** A knurled interference fit is a machine part connection made by a plastic joining, which includes the advantages of commonly-used shaft-hub-connections. The combination of the friction and form fit, which are responsible for torque transmission, results in a higher power density than conventional connections. In this paper, parameter gaps are bridged with the aim of enhance the design calculation of the knurled interference fit. Experimental investigations on the shaft chamfer angle (100Cr6) and hub-diameter-ratio (AlSi1MgMn) were performed. The analytical approaches are developed for calculating the joining force and maximal torque capacity by accounting for experimentally investigated loss of load transmission at high hub-diameter-ratios and high shaft chamfer angles. The presented calculation approach is an accurate tool for the assessment of early machine designs of the knurled interference fit and helps to save from having to perform time-extensive tests.

**Keywords:** knurled interference fit; shaft-hub-connection; joining by plastic forming; torque capacity; joining force; drive train

## **1. Introduction**

Turning drives are commonly used and important actuators in machines. Several principles for transmitting rotatory drive torque through machine parts from the source of power to the output side are established. Well-known principles are the key fit joint or the splined shaft fit, which belong to the group of so-called "positive fits" or "form fits". By contrast, the interference fit ("non-positive fit" or "friction fit"), where an oversized cylindrical shaft is pressed into the undersized cylindrical hub, transmits torque by joint pressure and is used in numerous applications, such as train axles. Combining these two well-known joining methods enables assembly of a splined shaft and hub with a cylindrical hole by axial plastic forming, bringing together the advantages of both principles and increasing the maximal transmissible torque (Figure 1). This so-called knurled interference fit (KIF) is fabricated by axial pressing the knurled shaft into a slightly undersized hole in the softer hub. The plastic forming of the counter profile in the hub eliminates tolerance errors and guarantees a homogenous force distribution over all tappets during operation. Furthermore, axial loads can be transmitted by the groove pressure in the connection.

The joining process of well-known interference push-in-connection can be estimated both according to the German standard for calculating interference fit DIN 7190 [1] as well as numerical methods from several investigations (e.g., [2,3]). In contrast to this investigation dealing global elastic-plastic relations with low deformation degree, a KIF connection is formed mainly locally at the contact groove with very high local deformation. Therefore the joining process of KIF cannot be reproduced with the presented methods.

In contrast to the conventional interference fit, the groove pressure decreasing separation frequency [4] in high-speed applications does not lead to transmission break down due to the form fit of the tappets. The knurl profile is understood according to German standard DIN 82 [5].

**Figure 1.** (**a**) Fusing of principles for torque transmission; and (**b**) geometry of knurled interference fit according to [5].

Equations describing this joint were recently published. Extensive recent overviews of past research can be found in [6–8], where the joining process and torsional load capacity are investigated. Nonetheless, the state of the knowledge is not sufficient for design of a universal KIF for a broad range of applications. Additionally, the valid patents in Europe lack detailed instructions for design and scientific explanations.

Lätzer [6] recently described the influence of different geometric parameters on joining and operating behavior, including important consequences of joining by cutting and forming. Fits joined by forming with shaft chamfer angles (SCA) *φ <* 60 ◦ are able to transmit 40% more torque than fits joined by cutting, where *φ =* 90 ◦ . Additionally, an analytical approach for designing KIF based on operational state is introduced. Equations (1) and (2) shows Lätzer's approach [6] for estimation of necessary joining force and maximal transmittable torque:

$$T\_{\mathsf{T\_8}} = D\_{\mathsf{S}} / 2 \cdot t \cdot l\_{\mathfrak{j}} \cdot i \cdot \pi\_{\mathsf{S}} \left( \varepsilon\_{\mathrm{KIF}}^{pl} \right) \tag{1}$$

$$\mathbf{F}\_{\text{I}} = \mathbf{A}\_{\text{contact}} \cdot \boldsymbol{\mu} \cdot \mathbf{p} \left(\boldsymbol{\varepsilon}\_{\text{KIF}}^{pl}\right) \cdot \mathbf{K}\_{QH} \tag{2}$$

With the exception of the hub-diameter-ratio (HDR), this formulation includes the geometry parameters of KIF (Figure 1). The terms *τ<sup>S</sup> ε pl K IF* and *p ε pl K IF* , the equivalent plastic strain and the strain hardening, can be calculated by the Ludwik approach. Additionally, differentiation between the cutting and forming joining methods is performed. The results of this study are valid only for thick walled hubs of ductile aluminum material. The value of the hub-diameter ratio is the thickness rate of the hub and can be calculated for steady hub outer diameters using Equation (3):

$$Q\_H = \frac{D\_{iH}}{D\_{oH}}\tag{3}$$

The interference *Igeo* is the difference between the shaft diameter *D<sup>S</sup>* and the hub diameter *DiH*:

$$I\_{\text{geo}} = D\_{\text{S}} - D\_{\text{iH}} \tag{4}$$

Bader investigated the self-cutting knurling shaft-hub-connection [9] and determined the necessary hardness-ratio of the hub and shaft, which is calculated from different combinations of steel, brass, and aluminum. The minimal hardness ratio was 1.8. Furthermore, the first applicable empirical approaches were derived, but they lacked the universal validity for forming KIF.

Against this background, the central question that motivates the present research is the influence of other parameters on the maximal transmittable torque of the knurled interference fit, such as the hub thickness in combination with the shaft chamfer angle.

Further studies investigate forming KIF connections with different, and not standardized, geometries and tappet orientations [8]. Similar increasing load capacity tendencies were recorded in dependence of growing interference. Due to high influence of the different tappet geometry on the plastic forming during joining process, the quantitative comparability is not possible. Additionally, the shape development of the tappets is not introduced.

The lack of complete investigations on the influence of hub outside-diameter on joining and torsional load force make this research necessary. The loss of stiffness of thin walled hubs will change the maximal possible load capacity of the KIF. The validity condition of this statement is the failure of the knurls.

First, experiments investigating joining and operating state are performed. The present analytical approaches will then be extended.

Table 1 shows the parameters that influence joining forces and maximal transmittable torsional load in the design of a KIF. The most important factors are the shaft diameter, the joining length, and the geometric interference (Figure 1).


**Table 1.** Parameters that influence knurled interference fits: ր represents a larger influence with an increasing parameter; ց represents an opposed influence with an increasing parameter, ( ) represents a hypothesized influence.

#### **2. Materials and Methods**

Investigations on hub outside-diameter are performed on specimens with flange geometries as shown in Figure 2. The figured hub specimens are manufactured from rods of AlMgSi1, EN AW-6082, with heat treatment T6. The shaft material is bearing steel 100Cr6. The basic shaft geometry is manufactured in the untreated state. Subsequently, the knurling is fabricated on the specimen by recursive axial forming. Finally, the specimen is hardened to a hardness of approximately 758 HV. The resulting hardness ratio of joined specimens is 1:7 (Table 2).

**Figure 2.** (**a**) Aluminum hub, forging alloy AlMgSi1; (**b**) steel knurled shaft bearing steel 100Cr6; and (**c**) experimental parameters.



\* calculated from hardness [6].

Figure 2 shows the specimen geometry as well as the chosen parameter values for the present study. The influence of different hub-diameter-ratios *Q<sup>H</sup>* on joining forces and maximal torque capacity differed by cutting and forming joining method is examined by varying the shaft chamfer angle *φ* for each HDR.

The joining process is performed in a special guiding appliance that guarantees the coaxial position of hub and shaft during the assembly in hydraulic press. The joining velocity *v<sup>j</sup>* was 0.5 mm/s due to comparability within the presented literature. The surfaces were cleaned before a dry joining process. The maximal joining force is derived from the gained force-stroke-signal.

After a one-day rest period the assembled connection is tested in a hydraulic torsional test bench, where the torque progression over the torque angle is measured until the failure of KIF occurs (stress-controlled test). At this value, the maximal torque capacity is obtained.

Material properties (Table 2) were estimated in tensile testing according to ISO 6892 [10] for the hub material AlMgSi1. In the case of the hardened steel shaft the hardness was measured according to ISO 6507 [11] and the tensile strength was calculated according to [6]. All tests were performed at room temperature.

#### **3. Results**

#### *3.1. Experimental Results*

Joining force plots shown in Figure 3a summarize the maximal forces of tested specimens. Every point represents a mean value of two repetitions with a maximal standard deviation of maximally 3.4% (joining force) and 2.9% (torque), respectively. At the lower HDRs of *Q<sup>H</sup>* = 0.65 and *Q<sup>H</sup>* = 0.5, the maximum joining force increases with decreasing SCA, peaking at *α* = 5 ◦ . 12% lower values for *Q<sup>H</sup>* = 0.65 are observed at this point.

**Figure 3.** Results from hub-diameter ratio and shaft chamfer angle testing, (**a**) maximum joining force; and (**b**) maximal torque capacity.

In contrast, the thin walled hub with *Q<sup>H</sup>* = 0.83 shows a plateau at lower joining forces. While all HDR forces are the same at *α* = 90 ◦ , values diverge nonlinearly with decreasing SCA.

The forming process causes redirection of the axial force to a hub-expending radial force, which results in higher groove pressure and higher joining forces than the cutting process for hubs with higher wall thicknesses. In contrast, the thin walled hub does not show this effect due to high hub expansion during joining and the related lower plastic deformation and shorter height of teeth in the hub. The joining forces of forming KIF are about 20% lower than for cutting KIF.

Figure 3b shows the corresponding maximal torsional load, which is the load where the KIF fails. Similar trends of joining force (Figure 3a) can be observed. Except for the thin-walled hub, where the values at the plateau are lower, the forming connection also achieves a higher maximal torque. The higher torque capacity of forming KIF is rationalized by strain hardening of the ductile material AlMgSi1.

Additionally, the maximal torsional load of a thin walled KIF *Q<sup>H</sup>* = 0.83 joined by cutting (*α* = 90 ◦ ) is about 40% lower than KIF with *Q<sup>H</sup>* = 0.5 and *Q<sup>H</sup>* = 0.65. This difference can be explained by radial expansion of the hub caused by the torque applied during testing. Enlargement of the hub inner diameter, caused by the force split of the torque on the sloping side of the formed/cut knurling, leads to decreased contact area. Interrelated reduction of the teeth shear area results in a smaller torque capacity for the hub with thin walls with *Q<sup>H</sup>* = 0.83.

#### *3.2. Analytical Approach for Calculation of Joining Force and Torque Capacity*

The following description involves including the tested parameter in the available calculation method, thereby expanding the approach of Lätzer.

The first set of analysis highlights the impact of HDR on the formation of the counter profile in the hub. Figure 4 shows the different groove heights *hgroove* dependency on SCA and HDR. The green area represents the neglected area in the field of thin hub and forming joining process. At this point, the contact area is reduced as a result of radial hub expansion.

**Figure 4.** (**a**) Hub groove height formation after KIF joining; and (**b**) calculation areas for equivalent plastic strain *ε pl RPV* .

When low radial forces are initiated in the cutting process, groove formation is due to material removal resulting in equal groove heights (Figure 4a, yellow area).

Knurl forming on the thick hub, which is at lower HDR, leads to constant groove heights. In contrast to the cutting process, the material undergoes strain hardening, which is calculated in Equations (5) and (6) in the equivalent plastic strain term *ε pl K IF* .

The intersection of the described areas border the crossing parameter area, where the joining processes are mixed.

In addition to Lätzer's approach, the following equations are used for calculation of joining force *F<sup>J</sup>* and maximum torque *Ttorque*,*max* considering the tested parameters:

$$F\_{\rm I} = A\_{\rm contact} \cdot \mu \cdot p \left( \varepsilon\_{\rm KIF}^{pl} \right) \cdot \mathcal{K}\_{\rm I} \tag{5}$$

$$T\_{torque,max} = \left(\frac{D\_S}{2} - h\_{group}(Q\_H)\right) \cdot l\_f \cdot i \cdot t \cdot \tau\_S \left(\varepsilon\_{KIF}^{pl}\right) \cdot \mathcal{K}\_T \tag{6}$$

Here the coefficient of friction *µ* is determined by a standardized tube friction method from [12]. The groove pressure *p ε pl RPV* is calculated according to [6] to be equal to the flow stress *k <sup>f</sup> ε pl* .

The modification to the original Lätzer approach [6] consists of new reference areas *A*<sup>0</sup> and *A*<sup>1</sup> for calculation of equivalent plastic strain *ε pl RPV* , which is defined as:

$$
\varepsilon\_{KIF}^{pl} = \left| \ln \left( \frac{A\_1}{A\_0} \right) \right| \tag{7}
$$

$$\mathbf{A}\_{0} = \mathbf{t} \cdot \mathbf{h}\_{\text{grove}}(\mathbf{Q}\_{H}) \tag{8}$$

*Machines* **2018**, *6*, 16

$$\mathbf{A}\_{1} = \frac{h\_{\text{knurl,shaft}} \cdot \mathbf{t}}{\mathbf{2}} - \left(h\_{\text{knurl}} - h\_{\text{groove}}(Q\_{H})\right)^{2} \ast \tan(a/2) \tag{9}$$

The adjusted calculation of the areas *A*<sup>0</sup> and *A*<sup>1</sup> (Figure 4b) leads to a more accurate equivalent plastic strain which allows the numerical modelling of flow stress with the Ludwik approach [13]. With varying hub thickness, counter profile geometry also has to be considered. According to Figure 4, the required hub groove height *hgroove*(*QH*) can be established by:

$$h\_{\text{group}}(Q\_H) = \begin{cases} \frac{I\_{\text{go}}}{2}; Q\_H \le 0.65; \forall \varphi\\ \frac{I\_{\text{go}}}{2} - \left(\frac{4}{5}Q\_H + \frac{1}{2}\right)mm; 0.85 > Q\_H \ge 0.65; \ \varphi < 15^{\circ} \\\ \frac{I\_{\text{go}}}{2}; 0.85 > Q\_H > 0.55; \ \varphi > 60^{\circ} \end{cases} \tag{10}$$

The height of the knurl *hknurl* results from the groove angle and the pitch:

$$h\_{\text{knurl}} = \frac{t}{2 \cdot \tan(a/2)}\tag{11}$$

Furthermore, the contact area *Acontact* which influences the contact normal force is calculated by:

$$A\_{\text{contact}} = i \cdot l\_{\text{j}} \frac{2 \cdot h\_{\text{group}}(Q\_H)}{\cos(\infty / 2)} \tag{12}$$

The influence of HDR and SCA is accounted for with the empirical-analytical functions *KJ*(*QH*, *ϕ*) and *KT*(*QH*, *ϕ*), which include expansion of the hub and differentiate between the cutting and forming processes:

$$\begin{array}{l} \mathbb{K}\_{f}(Q\_{H'}\,\boldsymbol{\varrho}) = \mathbb{C}\_{j1} + \mathbb{C}\_{j2} \cdot Q\_{H} + \mathbb{C}\_{j3} \cdot \boldsymbol{\varrho} + \mathbb{C}\_{j4} \cdot Q\_{H}^{2} + \mathbb{C}\_{j5} \cdot \boldsymbol{\varrho}^{2} + \mathbb{C}\_{j6} \cdot Q\_{H} \cdot \boldsymbol{\varrho} \\ \text{with } \mathbb{C}\_{j1} = 0.7759, \mathbb{C}\_{j2} = 1.6146, \ \mathbb{C}\_{j3} = -0.0102, \\ \mathbb{C}\_{j4} = -2, 1717, \mathbb{C}\_{j5} = -0.0000145, \ \mathbb{C}\_{j6} = 0.00868 \end{array} \tag{13}$$

$$\begin{aligned} K\_T(Q\_{H'}\boldsymbol{\varphi}) &= \frac{\mathbb{C}\_{T1} + \mathbb{C}\_{T2}\cdot Q\_H + \mathbb{C}\_{T3}\cdot \boldsymbol{\varphi}}{1 + \mathbb{C}\_{T4}\cdot Q\_H + \mathbb{C}\_{T5}\cdot Q\_H^2 + \mathbb{C}\_{T6}\cdot \boldsymbol{\varphi}} \\ \text{with } \mathbb{C}\_{T1} = 0.3017, \mathbb{C}\_{T2} = -0.1986, \text{ } \mathbb{C}\_{T3} = 0.00158, \\ \mathbb{C}\_{T4} = -2.878, \text{ } \mathbb{C}\_{T5} = 2.5396, \text{ } \mathbb{C}\_{T6} = 0.003107 \end{aligned} \tag{14}$$

Figure 5 shows the cross-section curves for several HDR depending on SCA according to Equations (13) and (14). Corresponding to the experimental results, a decrease of load can be registered with decreasing SCA (cutting process) and increasing HDR (thin-walled hub).

**Figure 5.** Factors influencing joining force (**a**) and maximal torque (**b**).

#### **4. Discussion**

As stated in the Introduction, the goal of the present research was to bridge knowledge gaps in the parameter range of the knurled interference fit. Joining force, as well as torsional load tests, were performed with varying hub-diameter ratios with the cutting and forming joining processes of shafts and hubs.

To conclude the experimental results, the present study supports previous research in this area. A similar influence of shaft chamfer angle was determined as found in [7,8]. As expected, including the new parameters in the known analytical approaches led to calculation errors, which overestimated the load capacity of KIF at higher HDR. Therefore, modifications of Lätzer's approach were made with the goal of minimizing calculation error. Considering the described points, Figure 6 shows a comparison of experimental results between "Lätzer-original" und "Lätzer-modified". Both show a low the deviation of calculated joining force of maximal 7% in case of forming joining (experiments no. 1–6, Figure 6a). For cutting connections (*ϕ* = 90 ◦ ) deviations up to 38% of Lätzer-original" and 46% of "Lätzer-modified" are possibly caused by the unknown strain hardening formation on the knurl cutting tip during cutting process.

**Figure 6.** Comparison of joining force (**a**) and torque (**b**) calculations to experiments, *D<sup>S</sup>* = 30 mm, *α* = 103 ◦ , *Igeo* = *t* · 2 ⁄3, *t* = 1 mm, *µ* = 0.3.

The accuracy of the new calculation approach is apparent with increasing HDR, where deviation of maximally 17% and −3% on average is reached (Figure 6b). In contrast, the "Lätzer-original" approach reaches deviations up to 155% (experiment no. 7). Acceptable values for maximal transmitting torque are achieved only for thick-walled hubs (experiment no. 1–3).

The described approach considers the specific specimen geometry. In practice different part geometries can lead to different load capacity. Therefore, a numerical approach for calculation of KIF is investigated additionally [14].

Furthermore, an axial load can be transmitted by the KIF connection due to remaining elastic portion of radial forming of the hub. The transmittable axial loading can be calculated regarding the axial strength described in [6]. The typical axial loads are about 30–45% of the joining force for cutting process and 60–90% of the joining force for forming process depending on the geometric interference of the KIF [6]. In dependency of rotational speed of the connection, the maximal transmittable force can differ according to [4].

#### **5. Conclusions**

Innovative design engineers often face novel problems, where missing knowledge leads to underachieving the structure potential. The knurl interference fit has a long record of use in different rotating applications, but the accurate calculation of possible transmittable torque is not fully known.

An example application can be found in modular structures where global structure should stay ductile (lower risk of crack) and contact parts should keep high hardness (wear-resistance). The connection of parts with these different properties are carried out by the KIF. As a result, increased functional integration of the part design is reached.

The results in present paper have been very promising for solving advanced design questions and minimizing experimental costs at the early construction phase of drive systems. Especially, the influence of the strain hardening, which corresponds to an improved material strength, is presently well described for aluminum hubs in dependence of different SCA and different HDR.

Future work concentrates on investigations with more material combinations, extending the analytic approach for more geometry variations. Additionally, a numerical model is built with the goal of investigating an expanded range of parameter. The validity of the present calculation approach for different knurled shapes, such as trapeze or inner knurling, is currently being investigated.

**Acknowledgments:** This research was made possible by a grant from the German Research Foundation DFG, for which are the authors very grateful.

**Author Contributions:** Lukas Suchy and Erhard Leidich conceived and designed the experiments; Lukas Suchy performed the experiments; Lukas Suchy and Thoralf Gerstmann analyzed the data; Thoralf Gerstmann and Birgit Awiszus contributed analysis tools; and Lukas Suchy wrote the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**


#### **References**


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