Experimental investigation into the implications of transmission errors for rack-and-pinion drives

The TE of the system is firstly measured quasi-statically in no-load operation. Therefore, the table is moved with a velocity of 5 \(\hbox { mm}/\hbox {s}\). In this way, dynamic effects can be excluded to solely observe the geometric deviations in the drive train. The table only has to overcome the friction that occurs in the system, which in this case is about 500 \(\hbox { N}\). The measurement is carried out in both directions of travel.

Figure 4 shows the TE curves acquired. Basic characteristica are equally observed for both curves, which are composed of different independent sources of error. These individual effects have differing impact and are superimposed, thereby reinforcing or balancing each other out. Therefore, the specific composition of the cumulative error is highly individual and differs for each installation. Thus, the following discussion focuses on the quantification of the separate error components to evaluate their impact. For this purpose, different sources of error were measured and their influence on the TE was analyzed. The difference between the drive and the table is firstly subject to a drift. With constant motion of the motor, the table deviates from the corresponding ideal path. One reason for this, is the distance between the individual teeth of the rack being subject to manufacturing inaccuracies. During motion, these inaccuracies add up and result in a drift between the rotational movement of the pinion and the linear table movement. For the examined example, the accumulated pitch error over the measuring range amounts for up to 23 \({\upmu }\hbox {m}\) in positive direction and 44 \({\upmu }\hbox {m}\) in negative direction and thus contributes significantly to the position drift. Additional deviations are caused by tolerances in the alignment between racks and guide rails. Specifically, center distance and vertical alignment are subject to variation. For the system under consideration, the variation in center distance \(\varDelta \,d\) was acquired by measuring the horizontal displacement of the racks perpendicular to the direction of movement. The observed deviation ranges up to 47 \({\upmu }\hbox {m}\). Based on the gearing geometry, the resulting path deviation \(\text {TE}_d\) through shifting of the contact point is estimated to 18 \({\upmu }\hbox {m}\) byusing the pressure angle \(\alpha _n\) analog to the calculation of the radial gearing force [18]. The vertical alignment \(\varDelta \,h\), acquired by measuring the vertical displacement of the racks perpendicular to direction of movement, deviates up to 110 \({\upmu }\hbox {m}\), resulting in path deviations \(\text {TE}_d\) of up to 39 \({\upmu }\hbox {m}\), as given byThe cause here likewise is the displacement of the contact point. Besides these deviations within the drive train, the measurement of the table position itself is also subject to inaccuracies. Tolerances in the alignment of the scale, pitch errors of the table measuring system as well as temperature variations can lead to deviations in the measurement of the distance travelled [19]. Due to the table measuring system being integrated in the guide rail, only the misalignment between it and the rack is a concern. While potentially relevant for long travel distances, over a distance of 2000 \(\hbox { mm}\) even a significant parallelism error of 2 \(\hbox { mm}\) only leads to a position deviation of 1 \({\upmu }\hbox {m}\) [19]. This source of error can therefore be neglected when referring to the magnitude of the observed drift. The accuracy of the measuring system itself is given by the manufacturer with \(\pm \,\)5 \({\upmu }\hbox {m}\) per 1000 \(\hbox { mm}\) of travel for a temperature range of 0 – 70 \(^{\circ }\hbox { C}\) [20]. In summary, it can be stated that pitch errors of the rack, tolerances in the center distance and vertical alignment contribute to the total TE to a similar extent. The accuracy of the linear measuring system, albeit not negligible, is of minor significance. While pitch errors and measurement accuracy are predetermined through manufacturing, special attention should be paid to the alignment of the racks in order to minimize drift, especially when using an indirect position control. If the pitch errors of the individual rack elements are known, it is possible to arrange them in an order that keeps the total pitch error minimal.

Superimposed on the drift is a distinctly recognizable low-frequency oscillation in both runs. Its amplitude ranges from 20 to 40 \({\upmu }\hbox {m}\) and the period of 266.66 \(\hbox { mm}\) corresponds to one complete revolution of the pinion as per \(d_p \cdot \pi\). The measurement results therefore suggest, that the errors are attributable to deviations in the rotary motion of the pinion respectively the gearbox output. An examination of the planetary gearbox revealed a periodic angular error of 11 \(\hbox {m}^{\circ }\) between input and output shaft over one full revolution of the pinion in both directions of motion. Coupled with the pitch diameter \(d_p\), this results in a deviation of 8.1 \({\upmu }\hbox {m}\) in terms of the linear motion. In addition, alignment and manufacturing tolerances affect the rotational motion of the pinion and thus contribute to the TE. In the case under consideration, the pinion is welded onto the gear shaft and exhibits a radial runout of 12 \({\upmu }\hbox {m}\), measured on the tooth tip at half the gear width. Complementary to this, the axial runout deviation, measured on the underside at the pitch circle, totals 25 \({\upmu }\hbox {m}\). To estimate the corresponding effect on the TE, the vertical displacement caused by the axial runout is substituted into Eq. (4), resulting in a deviation of up to 9 \({\upmu }\hbox {m}\). It is subsequently evident that both, the gearing in the gearbox and the runout characteristics of the pinion, have significant effects on the TE of the drive. It is therefore advisable to use high-quality gearboxes and to ensure that attention is paid to the alignment of the pinion on the output shaft. The established runout measurement in one plane should be supplemented by a second measurement in another plane in order to cover the rotational properties more precise.

In addition to the low-frequent periodic error, there is another, higher-frequency one superimposed. Its amplitude is in the single-digit \({\upmu }\hbox {m}\) range and its period corresponds to the meshing frequency of the rack and pinion, which, corresponding to the number of teeth z, is higher by a factor of 20 than that of the pinion revolutions. This error component can therefore be attributed to deviations of the tooth flank geometry from the ideal involute shape. One reason for such deviations are unavoidable tolerances in the manufacturing process of the gearing components. The typical magnitude of these tolerances corresponds with the observed errors [21]. However, there are also specifically intended modifications of the tooth flank shape [22]. Such tooth flank modifications are primarily used to reduce stress peaks under load and consequently improve the load capacity. The gearing thus is designated for a specific load condition. The tooth flanks that deviate from the ideal shape in the load-free state then deform in such a way that optimal load distribution and motion uniformity are achieved under nominal load [12]. This effect can be observed for the meshing of the RPD under investigation, which becomes clear in Sect. 3.2. For operating conditions with loads differing from the optimized range, this results in an added TE for tooth meshing, which typically falls within the range of the deviations observed in the example [23].

While the error components just discussed are directly evident from the TE graphs, there are several others. While their effect on the TE of the system is negligible in terms of the resulting position deviations, they nevertheless represent an excitation of the system, especially when they occur periodically. For further investigation, the recorded TEs are transformed into the frequency domain. Figure 5 shows the resulting diagram.

The x-axis is not scaled by the frequency f in Hz, but has been normalized to full pinion revolutions n using the velocity v according toIn this way, the error spectrum is detached from the velocity and the individual components can be assigned to their sources. Thus, the distinct peak at the value 1 corresponds with the full pinion revolutions and the the one at factor 20 correlates with the number of teeth z. Moreover, for both directions of travel, peaks appear at factors 4.27 and 6.67. The odd multiples of the pinion revolutions indicate that the causes are to be found in the translational movement of the drive. The corresponding conversion using the pitch circle of the pinion as in Eq. (1) results in a period of 62.5 \(\hbox { mm}\) and 40 \(\hbox { mm}\), which corresponds to the bore spacing of the rack and the guide rails, respectively. Due to the punctual mounting of the components, their stiffness fluctuates, leading to varying degrees of deformation and thus positional oscillations [13].

The approach to the investigations under load largely complies with the prior measurements. The only difference are constant forces applied by the LDD contrary to the direction of travel. The results are presented in Fig. 6.

The diagrams show the resulting TEs in load-free condition and with \({1000}{\hbox { N}}\), \({2000}{\hbox { N}}\) and \({3000}{\hbox { N}}\) counterforce on top of the friction. It becomes apparent that there is a constant offset between the individual curves, which corresponds to the static deformation of the mechanical system under the constant load. The corresponding average stiffness for the observed range of forces is 48.5 \(\hbox { N}/{\upmu }\hbox {m}\) in positive and 64.1 \(\hbox { N}/{\upmu }\hbox {m}\) in negative direction of movement. In addition, there is a considerable alteration of the TEs with increasing load, particularly in the negative direction of movement. While the basic shape including drift and low-frequency oscillation remains unchanged, the amplitude of the errors with tooth meshing frequency decreases significantly. This is due to the deformations of the tooth flanks in conjunction with the profile modifications described in Sect. 3.1. The geometric deviations in the load-free state converge to the ideal involute shape under load and the uniformity of motion improves.

A relevant consideration at this point is, that counterforce on the gearing does not inevitably arise from external sources. As mentioned in Sect. 1, in practical implementations two drives are typically preloaded against each other to minimize backlash. In terms of the individual drives the origin of the counterforce is irrelevant and a constant preload results in an additional force offset to the load. However, the extent to which the TEs of the individual drives interfere with each other is subject of future research.

While a substantial decrease in the amplitude can be observed for the negative direction of travel, the difference is more subtle for the positive direction. This and other noticeable direction-dependent characteristics are a specific property of RPDs [2]. The reason for this is to be sought in the meshing of the rack and the pinion. As mentioned before, most gears are designed with helical teeth to achieve smoother motion and higher contact ratio. Such helical toothing leads to considerable axial forces along the gearbox shaft. For the present helix angle \(\beta\), a tangential load \(F_{load}\) of \({3000}{\hbox { N}}\) according toleads to an axial force \(F_{ax}\) of \({1064}{\hbox { N}}\) acting on bearings and supports. In order to be capable of withstanding these forces, the output shaft of the gearbox is commonly carried by angular contact bearings. Figure 7 illustrates such an assembly by the example of a back to back arrangement of tapered roller bearings.

It is evident that the complementary inclinations of the bearing planes lead to a differing distribution of forces depending on the direction of the axial force. As a consequence, the resulting tangential and axial forces are largely absorbed by one of the two bearings in each case, depending on the direction of travel. The mounting arrangement of the bearings results in different leverage and hence differing stiffness and tilting rigidity. In conjunction with the inevitable hysteresis and potential backlash of the bearing, a direction-dependent vertical shift of the pinion occurs [24]. The resulting change of the contact conditions of the gearing produces the observed discrepancies between the two directions of travel of the feed drive [25].

The phenomenon just described does not only apply to the rack and pinion, but can also be transferred to the gearbox, which utilizes helical gearing as well. Thus, the direction-dependent occurrence of the motor revolutions observed in the frequency spectrum in Fig. 5 can also be traced back to changed contact conditions. As the preceding considerations indicate, the heavily direction-dependent characteristics of RPDs are intrinsically linked to the design and functional principle of such drives. However, constructive efforts can be made to reduce consequences for the accuracy. Stiffer or preloaded bearings can decrease the resulting displacements. In addition, consideration should be made regarding the direction-dependent behavior of the drive, when defining the helix angle. For high accuracy, the resulting axial forces should be kept as low as feasible.

The measurement run for the velocity of \(0.75\, v_{ref}\) in Fig. 8 reveals an unexpected response. A significant high-frequency oscillation dominates the control error. The velocity dependent occurrence suggests the excitation of a natural frequency of the system. To verify this, the position errors for several velocities are transformed into the frequency domain, as shown in Fig. 9.

This diagram illustrates how the error components of the TE shift in relation to the velocity, the corresponding dominant peaks move along the frequency axis proportionally. In the proximity of 50 Hz, however, excitations appear for all measurement runs. This fixed frequency correlates with the first mechanical eigenfrequency of 53 \(\hbox { Hz}\) as shown in the frequency response of the mechanical system in Fig. 10. The distinct oscillations for the considered velocity \(v\!=\!0.75\, v_{ref}\!=\!{135}\hbox { mm}/\hbox {s}\) in Fig. 8 are due to this eigenfrequency as it is evident by the dominant peak in Fig. 9. To investigate the correlation with the TEs, Eq. (5) is inverted. Thus, the error components identified in Fig. 5 can be assigned to the corresponding frequencies for the specified velocity. For the factor \(n \! = \!108\) of the gearing within the output stage of the gearbox, this frequencycoincides with the mechanical eigenfrequency. The two excitations subsequently reinforce each other and cause the observed position oscillations.

It is evident that the periodically occurring parts of the TE cannot only result in respective position deviations. Due to the varying frequency determined by the velocity, they coincide and interfere with natural frequencies of the system in certain cases. However, the affected states of operation can be identified and avoided, if the natural frequency of the system is known. The dominant TE components can be identified through transformation into the frequency domain as in Fig. 5 and the corresponding frequencies can be derived for all drive velocities utilizing Eq. (5). The velocity ranges in which one of these falls within the scope of a natural frequency should be avoided in operation. In addition, this issue should be taken into account during the design phase. In particular, the use of multi-stage gearboxes multiplies the number of excitations induced in the system and should therefore be carefully considered.