Keywords

1 Introduction

Rolling elements made from various thermoplastic polymers play a major role in modern bearing designs. Polymer rolls can not only be manufactured cost-effectively with large batch production methods like injection molding, but they also offer a variety of other advantages such as corrosion resistance, low weight and self-lubrication [1,2,3,4]. For these reasons, plastic bearings are widely used in many practical engineering applications from energy conversion systems to aerospace applications [5,6,7]. The materials used for polymer bearings are either high-performance polymers such as polytetrafluoroethylene (PTFE) and polyetheretherketone (PEEK) or engineering polymers such as polyamide (PA) and polyoxymethylene (POM). High-performance plastics are expensive and thus not suitable for mass production. However, POM offers good tribological properties at a moderate price. Therefore, it is widely used for mass production of injection molded components in many tribological applications.

While POM-H homopolymers initially show better short-term mechanical properties (higher stiffness) than copolymers (POM-C), they tend to degrade faster over time. For some applications this could lead to performance reduction fairly quickly. High stiffness and dimensional stability, favorable tribological properties (low friction and wear rate) and moderate creep make POM to an attractive bearing material [6, 8, 9]. Furthermore, it is assumed that POM reveals a pronounced self-lubricating effect which allows low friction coefficients at high loads over a long sliding or rolling time [10].

Naturally, a deeper understanding of the factors that influence the friction process is of prime theoretical and practical importance. Two specific factors that have been discussed intensively in the literature for polymers are surface roughness of the counterpart and normal force. Particularly interesting experimental investigations of these two factors for sliding were carried out by [11,12,13,14,15,16]. In terms of rolling Harrass et al. [17] provided a comprehensive experimental study of the tribological properties under rolling contact of various thermoplastic polymers. In further consequence, the contact as well as the global and local deformation characteristics of polymer rolls need to be concerned [18, 19].

The objective of this research was a macroscopic study of the effects of surface roughness Ra and normal force Fn on the sliding and rolling behaviour of POM-H rolls. A statistical design of experiments was carried out to analyze the coefficients of sliding friction and rolling resistance by means of polynomial regression models. The regression models help to establish a functional relationship and the interactions between Ra and Fn. The results of this contribution could be helpful in the further development and optimization of rolling bearings made of thermoplastic polymers.

The above experiments were designed distinctly as short-term experiments and we intentionally neglected the long-term effects such as transfer film formation on the counterpart and time dependent viscoelastic deformation of the rolls. However, due to the inherent viscoelasticity of POM-H the rolls reveal creep which results in a macroscopic flattening of the cylindrical surface [18, 19]. This flattening may significantly affect the rolling resistance. For the sake of simplicity this flattening was neglected in the recent study in our short-term experiments. The experimental results made the development of corresponding simulation models possible. We are going to analyze above additional effects in a forthcoming simulation study of the sliding and rolling behavior of the thermoplastic rolls.

2 Experimental

2.1 Material and Specimens

Injection molded cylindrical rolls made of the thermoplastic homopolymer polyoxymethylene POM-H were used in the current study. The elastoplastic material properties were measured according to DIN EN ISO 527-2 [20] with a Young’s modulus of 2439 ± 6.4 MPa, a Poisson’s ratio of 0.37 ± 0.01 and a yield strength of 83 ± 3.5 MPa. The process induced shrinkage and warpage resulted in a concave shape of the rolls (nominal diameter Ø= 5.90 mm, length = 6.00 mm) with a maximum difference in diameter of 0.04 mm in the middle of the specimen, shown in Fig. 1. The surface roughness value of the rolls was Ra 0.5 µm. The roughness was determined both tactically (MarTalk, Mahr GmbH, Göttingen) and optically (Alicona InfiniteFocus, Alicona Imaging GmbH, Graz) because of measurement difficulties due to the curved white surface. However, the two different methods gave the same results.

The counterparts of the rolls were flat friction tracks with different Ra values ranging from 0.01 to 0.5 µm. Again both tactical and optical methods were used in order to ensure an exact roughness measurement of the counterparts. The roughness values and the used materials for the counterparts are summarized in Table 1. The friction tracks made of tool steel (1.2343) were wire eroded to provide a surface with isotropic friction properties. To realize the glass and steel sheet friction tracks small pieces of these materials were glued onto racks made from tool steel.

Fig. 1
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The profile of the roll diameter shows the volumetric shrinkage caused by the injection molding process. The points correspond to the actual measured values with an approximated spline for better visualization

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Table 1 Materials and surface roughness values of the used friction tracks
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2.2 Test Setup

All tribological investigations were performed with a UMT Tribolab test rig (Bruker Corporation, Billerica) in a temperature-controlled room at RT 22 ± 1 \(^\circ {\mathrm{C}}\) and 50 % rel. humidity. The relative movement between the two friction partners was realized by a linear drive. The normal and tangential forces were measured using a 2D load cell of type DFH-100G with a resolution of 50 mN (Bruker Corporation, Billerica). A system with interchangeable inserts for the friction tracks was developed to enable a quick variation of the surface roughness. These can be mounted on the spring assembly as well as on the carriage by means of a bracket. The spring assembly serves as overload protection for the load cell which is sensitive to impact loads. The schematic structure is shown in Fig. 2 and was extended and modified depending on the test requirements.

Fig. 2
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Schematic structure of the test bench with interchangeable inserts to allow quick variation of the track surface roughness

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A customized measurement script with three main steps was used for the tests. In the first step the test setup moves fast in z direction and stops at 1 mm above the specimen. In the second step the measuring head slowly contacts the roll and builds up the desired radial load. With the parameters shown in Table 2 (left), a targeted overshoot of the contact force is applied to reach the desired load level more quickly as shown in Fig. 3. Since the application of the load is force-controlled, a break-off condition is introduced in case the test specimen is lost and to prevent a collision of the friction tracks. In the third step the carriage begins to move in x direction and the measurement starts. To exclude transient effects from the measurement the recording of data starts after the third cycle. The parameters of step 3 are shown in Table 2 (right). After finishing the measuring process the test rig moves into home position to enable the withdrawal of the specimen.

Table 2 Parameters of the measurement script
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Fig. 3
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Load (a) and velocity profile (b) for the targeted overshoot of the radial load. The time stamp \(t_1\) marks the end of step 2 and the beginning of step 3

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2.3 Sliding Test

In the sliding tests, the roll was clamped on one side to prevent the roll from rotating. Therefore, the lower friction track was replaced by a mounting for the roll which essentially consists of an undersized pocket. At the beginning of the sliding test the roll was pressed into this pocket as shown in Fig. 4a. The travel paths were adjusted so that the forward and backward movements do not overlap each other, see Fig. 4b. This was achieved by a lateral offset after the respective forward or backward movement, whereby the measuring head was lifted.

Fig. 4
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Roll pressed into an undersized pocket to prevent rolling (a), travel path of the measuring head with lateral offset (b)

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2.4 Rolling Test

For the rolling tests, a single roll was radially loaded between two equal friction plates while the lower plate was moved relatively to the upper one. The translational motion of the lower plate produced the desired rolling motion. To realize a full rotation of the roll the lower plate must move twice the period of the roll curve cycloid which equals to \(4 \pi R\). Since the actual radius R varies depending on the test specimen, a slightly larger travel distance of the theoretical value was selected for the tests. This ensured that at least one full rotation of the roll was made before the direction of movement was changed in the test.

The roll was aligned with a prism before the start of each experiment to guarantee reproducible results. In that way the roll was always aligned parallel to the rolling direction and the starting position remained the same for each experiment (Fig. 5). The radial load was force-controlled throughout the entire test. The translational movement of the friction tracks are displacement-controlled.

Fig. 5
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Roll on a friction track aligned by a prism

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2.5 Test Plan and Data Analysis

In order to be able to detect possible interactions between the different test parameters, a statistical design of experiment as shown in Table 3 was created. Although different materials were used as counter-faces for the friction tracks, they were not evaluated as a separate factor. For each setting (Ra: 0.01–0.5 µm, Fn: 150–350 N), six replicate tests were carried out in sliding. Each sliding test consisted of three partial measurements. For rolling 10 replicate tests were conducted for each testing configuration.

Table 3 Experimental design plan for sliding (left) and rolling (right)
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For the data analysis the slight offset of the force signal had to be corrected. In addition, the sections around the reversion of the testing direction were removed to guarantee a constant testing speed. Since the data evaluation works with points and their position, the number of points to be removed around the reversal point had to be calculated. Therefore Eq. 1 was used, where v is the testing speed, f the data rate and s the length of the reversal section.

$$\begin{aligned} Np_{removed}(f,s,v) = \frac{f s}{v} \end{aligned}$$
(1)

The distance required to overcome the inertia of the test setup after reversing the testing direction was determined experimentally and evaluated for every load level. The tangential force was than calculated as the arithmetic mean of its absolute values. In Fig. 6 a typical signal with its offset and the filtered sections can be seen.

With the values for the tangential and normal force the coefficient of friction \(\upmu \) and the rolling resistance \(\Pi \) were evaluated. With the corresponding values for Ra and Fn as factors a polynomial regression model was fitted (confidence interval 95 %). Polynomial regression functions were chosen due to the good fit in the examined measurement range and to analyze the data with the possible interactions of the parameters. An extrapolation beyond the investigated range is not possible due to the selected approach functions. In order to develop a sufficiently good model in the test range, polynomial functions up to degree three were used. In contrast, the sum of the powers of the crossed model terms was limited to three (e.g. \(f(x,y)=c_j \cdot x^n \cdot y^m| (n+m) \le 3\)).

Fig. 6
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Typical force signal of a rolling test with the corresponding velocity

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3 Results and Discussion

3.1 Sliding

The data of the friction coefficient \(\upmu \) depending on the normal force Fn with the roughness Ra as a group parameter are presented in Fig. 7. While the measurements for Ra 0.01 and 0.1 µm show a decreasing trend for \(\upmu \), the data for v indicates the opposite. In contrast, Fig. 8 shows the dependence on Ra with Fn as group parameter. The values of \(\upmu \) decrease at first but increase again with a distinct minimum at Ra 0.1 µm for all load levels of Fn. This trend is amplified at higher normal forces, possibily due to the deviation from the ideal cylindrical shape of the rolls. The concave shape introduces a concentration of contact at the roll ends. Less contact happens on the side surfaces with only local contacts at roughness peaks. Higher normal forces would lead to more deformation and thus a more even contact distribution along the roll. The same dependency of \(\upmu \) on Ra demonstrated in this work for POM-H was found by the authors of [11] for the polymers PTFE and PE.

Fig. 7
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The friction coefficient \(\upmu \) versus normal force Fn with the surface roughness Ra as group parameter

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Fig. 8
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The friction coefficient \(\upmu \) versus the surface roughness Ra with the normal force Fn as group parameter

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The polynomial regression functions (red curve) were used to fit the data and to establish a functional relationship between \(\upmu \) and the parameters Ra and Fn due to the good correlation in the measured test range. However, the local maximum between the measured Ra values of 0.25 and 0.5 µm does not necessarily exist and is rather an artifact of the polynomial functions. Alternatively, the blue line indicates a theoretical trend. We believe that the true trend converges towards a plateau value of \(\upmu \).

Figure 9 presents the regression model for \(\upmu \) in a contour plot. The polynomial regression model provides a good fit of R-sq. 83.55% for the coefficient of sliding friction. The low gradient in horizontal direction indicates an almost constant influence of Fn. In comparison the influence of Ra is far more pronounced and shows a distinct minimum at around 0.1 µm. The maximum in the contour plot might be an artefact of the used third order functions as discussed above. The gradient of \(\upmu \) in vertical direction changes with the applied load level and clearly indicates an interaction of the two parameters Ra and Fn. These findings offer compelling evidence that the roughness of the counter-face is the dominant factor in the sliding friction of POM-H rolls and that the normal force plays a subordinate role.

Fig. 9
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Contour plot of the regression model for the friction coefficient \(\upmu \) as a function of the normal force Fn and the surface roughness Ra

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3.2 Rolling

The data of the rolling resistance \(\Pi \) depending on Fn with Ra as a group parameter is presented in Fig. 10. It can be easily seen that \(\Pi \) increases strongly for higher values of Fn. The strong influence of the normal force can be traced back to the definition of the rolling resistance. It is strongly related to the contact radius \(r_c\), which is a function of the normal force Fn. The asymmetry of the contact forces during rolling is primarily a cause of the non-linearly increasing contact stresses on the contact surfaces in an ideal cylindrical body. The process-induced concave shape of the roll further complicates the contact problem with non-continuous contact surface and localized stress concentrations at the roll edges.

Fig. 10
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The coefficient of rolling resistance \(\Pi \) versus normal force Fn with the surface roughness Ra as group parameter

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Figure 11 shows a comparable behavior of the roughness dependency as in the sliding test but with a less distinctive influence of Ra. The curvature slowly decreases and the entire curve is shifted to higher values of \(\Pi \) as the load increases. This indicates an interaction between Ra and Fn as in sliding but in a reciprocal way (compare Figs. 8 and 11). With increasing deformation of the roll this interactional effect decreases and is superposed by the dominating effect of Fn.

Fig. 11
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The friction coefficient \(\Pi \) versus the surface roughness Ra with the normal force Fn as group parameter

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Fig. 12
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Contour plot of the regression model for the coefficient of rolling resistance \(\Pi \) as a function of the normal force Fn and the surface roughness Ra

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Figure 12 displays the polynomial regression model for \(\Pi \) as a contour plot. This regression provides a good fit of R-sq. 95.61%. Fn is the dominant factor with a big horizontal gradient for the rolling resistance. In terms of Ra, as with sliding, a minimum can be found at 0.1 µm which becomes weaker with rising Fn. The flattening of the curve at higher loads is due to the interaction terms of the regression model. This effect can be observed particularly well at the two extreme points for the surface roughness (Ra 0.1 µm and 0.4 µm). Contrary to the coefficient of sliding friction, the effect of the normal force dominates over the roughness in rolling. It is likely that the adhesive forces play a minor role than when sliding and thus the lower effect of the surface roughness.

4 Conclusions

The effects of the counter-face roughness Ra and the normal force Fn on the sliding friction \(\upmu \) and rolling resistance \(\Pi \) of POM-H were investigated. The experimental data was analyzed by means of a polynomial regression model to account for possible interactions. The experiments suggest that there is a fundamental difference between sliding and rolling regarding the influence of the roughness and the normal force. While \(\upmu \) was revealed to be strongly affected by Ra but only slightly by Fn, the opposite holds true for the rolling resistance \(\Pi \). However, the functional relationship of the roughness dependency with a distinct minimum at a Ra value of around 0.1 µm was found to be the same for sliding and rolling.

Furthermore, interactions between the two parameters exist in both cases but they express differently. Based on the curvature of the fits for \(\upmu \) and \(\Pi \) as a function of Ra the interaction with Fn can be illustrated well. On the one hand, the curvature of \(\upmu \) increases with increasing load level; while on the other hand, the curvature of \(\Pi \) decreases. In addition, the entire curve for \(\Pi \) is shifted to higher values when increasing Fn.

Lacking a model based on the underlying physics, it must be emphasized that the polynomial regression was used due to the good correlation in the observed range with a relatively high number of replicate tests (sliding R-sq. 85%, rolling R-sq. 95%). However, the polynomial functions introduce an artificial local maximum between the measured Ra values of 0.25 and 0.5 µm for both sliding and rolling. We believe that the true trend of the friction coefficient and the rolling resistance would converge to a plateau value for higher Ra. Further experiments or a model describing the physics for the influence of the roughness and normal force on the sliding and rolling behavior of polymers are necessary to establish the true functional relationship. Microstructural effects like the transfer film formation and kinetics might explain the interaction between the roughness and normal force on a physical basis.