1 Introduction

Meeting the rising requirements for machine tool main spindles regarding manufacturing accuracy represents a challenge for research and industry. The substitution of the standard angular contact ball bearings, so called spindle bearings (SBs), with tapered roller bearings (TRBs) in the machine tool main spindle is a promising approach due to its enhanced stiffness behavior [1]. In contrast, TRBs have a lower speed parameter compared to SBs [2]. Nevertheless, it could already be demonstrated that TRBs can be operated with sufficient lubrication under spindle relevant speed parameters [3]. In other approaches TRBs are designed with ceramic rollers and a direct lubricated rib on the outer ring, which reached a speed parameter of \(1.25\times 10^{6}\) mm/min [4]. Besides this, there are already high speed TRBs available on the market. These TRBs are high precision bearings with an elastically or hydraulically adjustable rib, which avoids a thermal clamping of this critically loaded contact [5]. Furthermore, a TRB with enlarged rollers and cage diameter to increase the bearing stiffness for automotive applications was presented in [6]. Moreover, a hybrid TRB with lubricated rib on the outer ring was developed and successfully tested up to a speed parameter of \(1 \times 10^{6}\) mm/min [7]. Another approach based on a geometrical optimisation of the TRB with smaller rollers did not improve the speed capability [1]. The causes of the early failure of the bearing were explained by excessive skewing of the rollers, anisotropic deformation of the rib and excessive deviation between the nominal and actual geometry. Other authors identified the rib-roller-contact as the speed limiting rolling contact due to its high sliding friction in the bearing and therefore designed and tested a new TRB neglecting the rib with four contact points, two on the outer ring and two on the inner ring [8]. However, only low speed parameters up to \(0.24\times 10^{6}\) mm/min were investigated with the bearing and the comparison to a standard bearing showed no significant friction reduction. Current works demonstrated, that spindle-relevant speed parameters of \(0.9\times 10^{6}\) mm/min already can be achieved with standard TRBs with suitably high lubrication [9, 10]. Therefore, the focus of this paper is to characterise the stiffness increase of TRBs compared to SBs in main spindles.

2 Static stiffness tests

In the following, two main spindles, which are externally driven, will be investigated on a test rig and in machine tests. Their set-ups are depicted in Fig. 1 with the bearing and force application positions in a simplified simulation model in Mesys [11]. Additionally, the radial displacements are measured in the test. Therefore, the distances c between the bearing and the measurement positions (MP) are also given. The first spindle, which is used in industry, is rigidly arranged with a spindle bearing package SBP (Type B7016-E) and a single SB (B7014-E) with 500 N preload, shown in 1a. Based on the first spindle, a second spindle was redesigned with an elastically arranged TRB (Type 32016-X) with the necessary preload of 2600 N, which is given in 1b.

Fig. 1
figure 1

a Rigidly arranged SBP Spindle (\(F_{P}=500 N\)). b Elastically arranged TRB spindle (\(F_{P}=2 600 N\))

Full size image

2.1 Test rig and measurement set-up

Prior to the machine tests, both spindle types were investigated on an isolated test rig to determine their load–displacement characteristics. This test rig with its radial loading system, consisting of the radial load unit and a hydraulic cylinder as well as the temperature and acceleration sensors are shown in Fig. 2a. Besides this, a spindle integrated force measuring system was used based on the contactless measurement of the shaft displacement [12]. The system consists of three axial and three radial eddy current displacement sensors positioned at measurement point (see Fig. 1). The arrangement of the sensors on the measuring ring is illustrated in Fig. 2b. The displacement signals are directly evaluated during the measurement in X-, Y- and Z-axis direction, the coordinate system is also depicted in in Fig. 2b. With known load–displacement characteristics and the measuring system in the main spindle, the process forces during machining can be estimated [13]. The radial force is applied by the hydraulic cylinder controlled by a feedback signal of a force sensor between the load unit and the cylinder. During the tests, the spindles are cyclically loaded and unloaded with a maximum force of 3.8 kN for constant rotational speeds of 2000 rpm, 4000 rpm, and 6000 rpm. Since an additional lubrication of the TRB on the side of the rib-roller contact proved to be more beneficial, this was also implemented in the investigated TRB spindle [9]. The spindle was lubricated according to these findings with a lubrication quantity of 480 µl/h per nozzle, one on the loading side and one on the motor side (see also oil injection in Fig. 1).

Fig. 2
figure 2

a Test rig and b measurement set-up

Full size image

The measured load–displacement characteristics of the SBP spindle (a.) and the TRB spindle (b.) are depicted in Fig. 3. Additionally, the radial stiffnesses \(k_{i}\) at the MPs of the two spindles are given for the different rotational speeds. Since non-linear behaviour occurs in the SBP starting from 2 kN, the stiffness is only evaluated up to this point, which is also marked in the diagram. The radial stiffnesses are obtained by linearisation of the load–displacement curves, using the derivation of the straight line slope in Eq. (1).

$$\begin{aligned} k_{i}=\dfrac{dF_{rad}}{ds_{rad}} \end{aligned}$$
(1)

The curves are evaluated for 10 loading cycles. Especially at 2000 rpm, the curve of the spindle bearing is characterised by a non-linear behaviour from a radial load of about 2 kN. A possible reason for this can be, that the single balls of the front SB lose contact due to the radial load. With higher rotational speeds, this effect is minimised by the inner ring expansion and the higher centrifugal forces. Especially at the maximum radial force of 3.8 kN and 2000 rpm, the conventional spindle shows greater displacements by a factor of 2 compared to the TRB spindle. For 6000 rpm, only a radial stiffness of 293.1 N/µm can be achieved with the conventional spindle, whereas the radial stiffness of the TRB spindle is 541.3 N/µm. With the TRB, the higher speed also contributes to increased stiffness, due to the expansion on the inner ring and the higher centrifugal forces that press the rolling elements against the outer ring and thus increase the rolling contact area. Furthermore, hysteresis effects can be observed with the TRB spindle in the low load range up to 0.5 kN. Besides the measurements, additional calculations were performed for 6000 rpm with the spindle models depicted in Fig. 1. In general, the calculations show good agreement with the experimental results. In case of the conventional spindle, deviations can be attributed to the inaccurate adjustment of the preload force for rigidly arranged spindle bearing systems. The lower calculated displacements for the TRB spindle may be related to thermal effects.

Fig. 3
figure 3

Load–displacement characteristics: a conventional SBP spindle b TRB spindle

Full size image

In general, the tests show, that the TRB spindle can reach a speed parameter of \(0.6\times 10^{6}\) mm/min. Therefore, tests in the machine tool are possible.

The potential for increasing the stiffness of the main spindle by using TRBs instead of SBs is illustrated in the Fig. 4. A simplified simulation model of the spindle with a general support representing the front bearing was used. For increasing front bearing stiffness, the stiffness at the tool center point (TCP) and at the MPs were calculated. Additionally, the measured stiffnesses with the SBP in blue and with the TRB in red were added in the figure. The stiffness of the bearing can be determined by projecting the measuring point stiffness onto the corresponding characteristic curve. Applying the bearing stiffness to the TCP curve enables the determination of TCP stiffnesses. The comparison reveals that by using TRBs instead of SBs, the TCP stiffness can be increased by \(25 \%\). Nevertheless, the potential gain in stiffness is limited by the stiffness of the shaft. In case the bearing stiffness exceeds that of the shaft, the relative stiffness increase at the TCP becomes smaller and approaches a limiting value.

Fig. 4
figure 4

Stiffness increase potential of TRBs compared with SBPs

Full size image

3 Machine tests

This chapter presents the milling tests of the two spindle types in the machine. First, the dynamic behaviour of the two spindles are investigated by frequency response measurements. Then the results of the cutting tests with different tools and cutting parameters are analysed.

3.1 Frequency response functions of the spindles

To characterize the dynamic behaviour of the two spindles, the Frequency Response Functions (FRF) are measured. Since the end mill is not suitable for impulse excitation under rotational speeds due to its helical blades, a comparable tool (End mill SF25 reference) with a rod is used. The measurement setups are depicted in Fig. 5a for the end mill SF25 reference tool and (b.) the cutter head MK50. The spindle is excited by an automated impulse hammer in X-, Y- and Z-direction for different rotational speeds up to 6000 rpm. For the measurements, the transmission behavior between the excitation point at the impulse hammer and the displacement sensor system must be considered. The FRFs are evaluated by rectangular windowing of the displacement signal in the corresponding direction.

Fig. 5
figure 5

Measurement setup for a the end mill SF25 reference tool and b the cutter head MK50

Full size image

In the following, the FRF measurements of the SF25 end mill reference in the radial X and axial Z directions and the FRF of the cutter head MK50 in the radial Y direction are discussed as representative curves of this test series. In Fig. 6, the FRFs of an end mill SF25 reference tool in radial X-direction is shown for different rotational speeds, on the left side the FRF for the SBP and on the right side for the TRB spindle. In general, the measured FRFs show little deviation at standstill and under rotational speed. Also in this case, the improved dynamic behaviour of the TRB spindle compared to the conventional spindle becomes evident. For the SBP, a resonance point occurs at 520 Hz with an amplitude of 0.09 µm/N. Several more peaks can be observed at 700 Hz, 850 Hz, and 1400 Hz. On the other hand, the resonance peaks for the TRB spindle are rather low over the considered frequency spectrum. The highest amplitude is at 830 Hz with 0.02 µm/N. Furthermore, resonance frequencies occur at 570 Hz and 1550 Hz. The larger rolling contact of the TRB in combination with a bigger bearing load leads to a higher lubrication film and higher damping of the spindle, which has a potential positive influence on machining. Furthermore an instable operational behaviour of the spindle is expected, if the phase drops below \(-90^{\circ }\). In the X-direction, the SBP spindle exceeds this value at 550 Hz, while the TRB spindle reaches this value at 1050 Hz under rotation.

Fig. 6
figure 6

FRF of an end mill SF25 reference tool in X-axis for both spindles

Full size image

The FRF spectrum for the cutter head MK50 in Y-direction with both spindles is similar, which is depicted in Fig. 7. Deviations between the radial frequency spectra can be attributed to the different tools, the different excitation points and minor geometric differences in the spindle housing. For the SBP, there are significant peaks at 550 Hz with 0.039 µm/N and 830 Hz with 0.045 µm/N. A further resonance can be observed at about 1480 Hz. For the TRB spindle, significantly lower resonances appear at 560 Hz, 940 Hz, and 1530 Hz with amplitudes below 0.02 \(\upmu\)m/N. In general, the FRF measurements with the cutter head also confirm the improved damping in radial direction of the TRB spindle compared to the SBP spindle. Regarding the phase measurements, for the SBP spindle an instable behaviour can occur at a frequency of 550 Hz, whereas the TRB spindle the critical phase value is reached at 950 Hz. Therefore, a much more stable behaviour for the TRB spindle can be expected in radial direction.

Fig. 7
figure 7

FRF of a cutter head MK50 in Y-axis for both spindles

Full size image

In the axial Z-direction, the FRFs of the two spindles appear to be different for the SF25 end mill reference tool, which are presented in Fig 8. For the conventional spindle a resonance peak can be noticed at 600 Hz with an amplitude of 0.024 µm/N, for the TRB spindle a peak of 0.02 µm/N occurs at 670 Hz. The different FRFs of the TRB spindle at standstill and at speed can be attributed to misalignment of the rollers at standstill, which leads to clamping and significantly increases the static stiffness. Similar behaviour could also be observed in the frequency spectrum of the TRB spindle with the MK50 cutter head. Due to the increased static stiffness, the bearing is rarely affected in the considered frequency spectrum. In the axial direction, the stability performance is only slightly better, the SBP spindle reaches the critical \(-90^{\circ }\) phase at 550 Hz, whereas the TRB spindle exceeds the critical value at 650 Hz.

Fig. 8
figure 8

FRF of of an end mill SF25 reference tool in Z-axis for both spindles

Full size image

In this case, the damping advantage of the TRB spindle is not so significant. This can be attributed to the fact, that the axial stiffness of the conventional spindle adds up to the single bearing stiffnesses of the rigidly arranged spindle. This can be illustrated by the equivalent spring circuit of both spindles shown in Fig. 9. The total axial stiffness of the SBP spindle is thus comparable to the stiffness of the single elastically arranged TRB spindle.

Fig. 9
figure 9

Axial equivalent spring circuit for both spindles

Full size image

For the rigidly mounted SBP spindle, the front and back bearing are parallel connected to each other leading to equation (2).

$$\begin{aligned} k_{tot,1}=k_{SBP}+k_{SB} \end{aligned}$$
(2)

However, for the TRB spindle the back bearing is connected in series with the spring package. The total axial stiffness thus results from equation (3). Assuming that the stiffness of the back bearing is much higher than that of the spring package \(c_{tot}\ll k_{SB}\), equation (3) can be simplified as follows. This theoretical consideration explains the comparable dynamic behaviour of both spindles in axial direction of the TRB spindle compared to the SBP spindle due to the reduced stiffness attributed to the elastic preloaded sliding seat.

$$\begin{aligned} {k_{tot,2}=k_{TRB}+\left( \dfrac{1}{k_{SB}}+\dfrac{1}{c_{tot}}\right) ^{-1} \approx k_{TRB}+c_{tot}} \end{aligned}$$
(3)

Alongside the static measurements on the test rig, the FRF measurements also demonstrate the reduced dynamic vibration behaviour of the TRB spindle. In the following, a comparative analysis is carried out during machining using full slot cuts. Subsequently, the machine tests with both spindles are analysed in comparison.

3.2 Machine set-up and test plan

The machine tool for the cutting tests has an externally driven spindle, with a maximum rotational speed of 10000 rpm and a maximum power of 38 kW. The machine can be used for 4-axis machining and has a spindle with a HSK 63-A tool interface.

The tools used for the investigations with the respective test parameters are shown in Fig. 10. On the one hand, a three-edged end mill SF25 with 25 mm diameter and helix was used with a tool length of 135 mm till face contact. On the other hand, a cutter head MK50 was tested with a 50 mm diameter, a length of 150 mm and three cutting edges. Full groove cuts were manufactured on an aluminum block (EN AW-7075, 300 mm x 300 mm x 100 mm), for varying rotational speeds up to 8000 rpm, varying cutting depth up to 8 mm and a feed up to 0.35 mm.

Fig. 10
figure 10

Tools and test plan

Full size image

3.3 Results of the cutting tests

In the following, the test results for the high feed per tooth f, for the end mill of 0.23 mm and for the cutter head of 0.35 mm are presented exemplary. During the cuts, the displacement of the shaft was measured and analysed in the X-, Y- and Z- spatial directions. The displacement shares of a full slot manufactured with the conventional spindle an end mill SF25 at 6000 rpm are shown as an example together with the evaluated mean values in Fig. 11. Based on the measurements, the mean values \(\delta _{m}\) were calculated, which represent the static share of the displacement signals. Furthermore, the root mean square error \(\delta _{RMSE}\) is visualised by the scatter bars, which characterises the energy content of an oscillation and in this case represents the dynamic share of the displacement.

Fig. 11
figure 11

Example of the evaluation method by machining full slots with the SF25 end mill at 6000 rpm with the SBP, feed rate 0.23mm per tooth

Full size image

This characteristic value is formed based on the deviations of the measured time-dependent displacement values \(\delta _{t,i}\) from the mean value, summarised as the root mean square error in Eq. (4).

$$\begin{aligned} \delta _{RMSE}=\sqrt{\dfrac{1}{n}\sum ^{n}_{t=1}(\delta _{t,i}-\delta _{m})^2} \end{aligned}$$
(4)

Here, n stands for the number of samples, i for the respective spatial direction and t for the measurement value. To illustrate the displacements in more detail, the magnitude of the mean displacement for the tests performed with the end mill is depicted in Fig. 12. In general, the static displacement shares of the TRB spindle are smaller than those of the conventional spindle until the bearing speed limit of 5000 rpm is exceeded. This can be explained by the higher stiffness of the TRB spindle due to its larger rolling contact area and therefore higher stiffness in radial direction. The increased mean displacements at 6000 rpm and 8000 rpm in the Z-direction are due to a friction-induced greater thermal expansion of the TRB spindle compared to the SBP spindle. Furthermore, the higher effective displacements with increasing cutting depth, especially with the conventional spindle, can be attributed to the higher process forces. This effect is less pronounced with the TRB spindle. Generally, higher rotational speeds lead to higher displacements due to the rising centrifugal forces. Nevertheless, this effect is not so strong for both spindle types, since the tested rotational speeds are low compared to e. g. grinding spindles. The TRB spindle is also much more stable regarding the dynamic process behaviour, with significantly lower root mean square error values in all cutting tests. Overall, the TRB spindle with the SF25 tool is characterised by stable static and dynamic process behaviour even at the highest speeds of 8000 rpm. Nevertheless, the greater thermal expansion of the TRB spindle has to be considered during manufacturing.

Fig. 12
figure 12

Displacements during the groove cuts for an SF25 tool and a feed rate of 0.23 mm per tooth

Full size image

The influence of a different tool is investigated in the following tests with the MK50 cutter head, as shown in Fig. 13. The cuts with the conventional spindle tend to show higher mean displacement values up to 6000 rpm compared to the TRB spindle, which can be attributed again to the higher radial stiffness of the TRB. At 8000 rpm the process becomes unstable for the TRB spindle, resulting in increased displacement values. The increased displacement values of the TRB spindle may be related to the critical cutting speeds of the tipped tool. The displacements in X- and Y- direction are the largest, because the end mill moves in Y-direction and remains constant in Z-direction. The dynamic excitation, represented by the scattering bar, is again higher for the SBP in all test cases. Apart from the speed limitation due to the tipped tool and thermal influences, in these tests, the static and dynamic stiffness of the TRB spindle is again higher.

Fig. 13
figure 13

Displacements during the groove cuts for an MK50 tool and a feed rate of 0.35 mm per tooth

Full size image

4 Conclusion

In this paper, a modified spindle with an elastically arranged TRB was validated on the test rig and in the machine tool and compared with a conventional rigidly arranged spindle. The conventional spindle consists of a SBP (B7016-E) in the front and a spindle bearing type (B7014-E) in the back. The TRB spindle consists of a bearing of the type 32016-X in the front, which is additionally lubricated at the rib-roller side, and a spindle bearing type 7014 in the back. For all tests, a contactless displacement measurement system was used, which detects the relative displacement between the shaft and housing. The measured load–displacement characteristics of the TRB spindle exhibit a \(50\%\) reduction in displacement behavior compared to the SBP spindle under radial loads up to 3.8 kN and various rotational speeds up to 6000 rpm. Based on the measurements in the test rig, the TCP stiffness can be increased by \(25 \%\) by using a TRB. Furthermore, the significantly improved dynamic characteristics of the TRB spindle compared to the conventional spindle could be demonstrated by FRF at standstill and for rotational speeds of up to 6000 rpm. Also in the machine tests, the TRB spindle shows, compared to the conventional spindle, significantly reduced displacements in X-, Y- and Z-direction. These results can be confirmed by cutting tests with an end mill SF25 and a cutter head MK50 for different feeds, cutting depths and rotational speeds up to 8000 rpm. Only cutting tests with the cutter head became critical at 8000 rpm, which corresponds to a speed parameter of \(0.8\times 10^{6}\) mm/min. The maximum cutting speeds of the tipped tool can also have a limiting influence here. Nevertheless, it should be noted that the results were achieved with standard TRBs, whose limiting speed parameter is \(0.5\times 10^{6}\) mm/min according to the manufacturer. With a geometrically optimised TRB with reduced friction, much higher rotational speeds can be achieved, making industrial use in the machine tool main spindle increasingly likely. The results presented in this paper show the great potential of TRBs for main spindles to increase stiffness and the manufacturing accuracy. Further research can focus, for example, on machining high-alloy steels such as titanium. Moreover, with regard to the application in the main spindle, service life considerations of TRBs are also becoming more important for further research work.