A curved edge diffraction-utilized displacement sensor for spindle metrology Free

Spindles represent a key part of any machine tool. They determine the quality of the final product produced and the overall productivity and efficiency of the machine tool itself.^1–4 In state of the art tooling, the spindle can take up a considerable share of the total cost of the machine. In its most general form, a spindle is a device that provides power transmission via rotation between two components. Examples can be found in machine tools, metrology instruments, and rotary tables.^3–8 Relative motion is provided by one or more bearings that provide stiffness, load capacity, and accurate, repeatable rotation. The demand for advancement of the accuracy and productivity in production systems continues to increase. Spindle technology also continues to increase in variety, maximum speed, and number of spindles per machine. Therefore, standardization of accurate and efficient tests of spindle performance is in high demand.^9–14 Despite simple static measurements being easily performed, the largest demand is for tests at full operating speed.

The spindle rotation error is an important index to measure when assessing the machining accuracy of the tool. The accuracy of the spindle will greatly affect the shape, precision, and surface roughness of the part.^10–12 Consequently, the measuring technology and the rotation error separation technique of precision spindles have recently become a research focus. Any movement in the remaining five degrees of freedom is undesired and may be classified as either spindle error (i.e., motion that results from the spindle’s design, manufacturing, or operating conditions) or as a response to an external influence, such as thermal gradients, applied force, external vibration, or some other unintended load.¹² 

The spindle motion is typically measured with a capacitive sensor (CS) mounted in a precision fixture. A CS is applied to measure the dynamic displacement of the spindle shaft or master ball targets. The CS is best suited because of its high bandwidth and its independence from material properties or inhomogeneity (as seen in inductive sensors).¹⁵ A standard setup consists of three probes mounted in the x, y, and z directions, respectively. The sensors simultaneously measure the motion of the spindle shaft. This allows for a real-time dynamic radial and axial measurement. With a total of five probes, the tilt of the spindle can also be determined in the x, y, and z directions.^16,17 The high performance software collects the readings from the probe while the spindle is rotating and analyzes the results. However, it is standard practice for a commercial CS to be factory calibrated with flat target surfaces. Because the probes are calibrated this way, measuring a target with a curved surface will cause errors.¹⁶ The gap at zero volts will be different than when the system was calibrated because the probe will measure the average distance to the target. For a curved surface the average distance will be greater than the true distance away from the spindle. Other error sources will also be introduced because of the different behavior of the electric field between a flat surface and a curved surface.¹ In cases where a curved target must be measured, the system can be factory calibrated to the final target shape or the CS has to be custom-made for a certain shape. Overall, CS targeting spindle surfaces will lead to four sources of error if they are calibrated with flat targets. First, the sensitivity of the sensor increases resulting in exaggerated displacement measurements. Furthermore, the sensing range is both decreased and shifted towards the target, and the otherwise linear output of the sensor system becomes increasingly nonlinear for targets of decreasing diameter. The sensing noise issues also become significant due to interference between probes in the case of multi-probe methods. Since these effects lead to measurement error in precision manufacturing and metrology applications requiring the highest accuracy, new dimensional sensing methods for a curved surface metrology must be investigated.

This paper presents a new dimensional sensing principle for a curved surface metrology based on curved edge diffraction (CED) applied to precision spindle metrology. Assuming that an electromagnetic (EM) wave is incident on a smooth, curved, and perfectly conducting surface surrounded by an isotropic homogeneous medium, such fields produced by the EM wave incident on the spindle surface may be excited at shadow boundaries of the curved surface.^18–20 In this article, the proposed curved edge sensor (CES) is applied to the displacement measurement of the curved surface, especially for the precision spindle system, and is compared with the commercially available CS in terms of resolution, linearity, bandwidth, drift, and sensor positioning effects under both static and dynamic spindle conditions.

The measurement limits of a CS due to the target surface’s shape (in this case a spindle shaft) were investigated. The CS used for this experiment has the following specifications: an effective sensing area of ϕ5.6 mm, a measuring range of 500 μm, and a resolution of 10 nm.²¹ Cylindrical target samples (R1.0, R1.5, R2.0, R2.5, R6.0, R16.0 mm) were used, where R represents the radius of curvature of the sample rods. A digital indicator (1 μm resolution) was used as a comparison sensor. As seen in Figure 1, the CS output was measured by slowly moving a sample along the vertical direction by using a manual stage. The output was compared with that of the digital indicator. It was found that the sensitivity of the CS (manufacturer setting 40 V/mm) exponentially increases as the radius of curvature decreases, which indicates that the CS’s characteristics vary with the target surface’s shape. The errors induced on the CS measurements by spindle shafts smaller than R16.0 mm require additional means to compensate. This makes the CS a poor choice for taking measurements on such tools.

In this research, a new measurement tool based on CED is used for curved surface metrology. As seen in Figure 2, an EM wave incident on the curved edge gives rise to a transmitted wave, reflected wave, and edge diffracted wave, and an edge excited wave which propagates along a surface ray. Such surface ray fields may also be excited at shadow boundaries of the curved surface. The total electric field may be represented by a sum of the four waves. This sum predicts the scattering of the optical large platform and involves the use of spatial domain Fresnel integrals for the scattered or diffracted fields. There exist a few methods which provide a convenient and efficient computational method for curved surface diffraction problems in the transition region adjacent to shadow and reflection boundaries.^18–20 However, unlike knife-edge diffraction,²² an analytical approach to diffraction by a curved edge is very difficult because CED can be understood as an asymptotic solution of several canonical problems, which involve the illumination of the edge by different wavefronts.²⁴ In this work, an experimental approach to a CED-based displacement sensor making use of CED characteristics has been focused because of the ambiguity and difficulty in mathematical and computational approaches.

In previous research, a sensing method utilizing knife edge diffraction was introduced.^22–24 The transmitted light and the diffracted light created from the knife edge are superimposed to generate interference fringes, or a so-called interferogram. The diffracted light contributes to an increase in the peak power of the 1st fringe more than 40% of the transmitted light power, which optically amplifies the light signal due to knife edge diffraction. In a similar way, illustrated in Figure 2, the laser light incident to the cylindrical surface of the spindle shaft of some radius of curvature, R, with a detector placed some distance, L, away from the spindle, collects the total field of the transmitted wave, reflected wave, edge diffracted wave, and edge excited wave. Due to the interference of the multiple waves, the interferogram will be measured at the detector according to the motion of the spindle system along the y_s direction. The intensity obtained from the single cell detector can be calibrated with the position (y_s direction) of the spindle system. As a baseline study, the edge diffraction from the knife edge and the curved edge (R16.0 mm) was compared under the same experiment condition. The He–Ne laser is incident on the edge surface and the total intensity was measured by the photodetector (PD) while the edge moves along the axis perpendicular to the laser propagation axis as shown in Figure 3. The PD with a small effective area (0.2 × 0.2 mm²) compared to the laser beam diameter (ϕ0.9 mm) was chosen to obtain the detailed comparison between the knife and curved edges. The clear edge diffraction fringe was observed in case of the knife edge compared to the curved edge. Because the edge diffraction is associated with the surface roughness and sharpness at the edge, the fringe pattern from the curved edge is weaker than that of the knife edge. It was thought that the curved edge with good surface quality may increase the chance of interference.

The sensing system with such small detector sizes is limited to the measuring range. In this work, the detector with an effective sensing area (2.0 × 2.0 mm²) was used to design the sensing range ∼500 μm (the same as CS measuring range). As shown in Figure 4, the fringe patterns were not found in two cases because the light intensity is integrated over the sensing area. Here, it was observed that the detector size and edge shape affect the sensor outputs. The comparison of two edge diffraction effects was performed at a given experimental condition in this study. Currently, the edge diffraction effects due to edge shapes, beam sizes, and detector sizes to obtain both high sensitivity and large measuring range are under investigation.

The cylindrical surface of a ball bearing spindle was used as the measurement target, and an artifact (R16.0 mm) was attached at the front facet of the spindle. As seen in Figure 5, a He–Ne laser (beam diameter α ϕ0.9 mm, optical power 0.5 mW) travelled through an optical chopper incident to the cylindrical surface of the spindle. A single cell photodetector was placed 50 mm away from the spindle. The CS (with an effective sensing area of ϕ5.6 mm) was placed on the top surface of the spindle to calibrate the CES. A digital indicator was also installed on the top surface of the spindle housing to measure the y_s displacement. The optical chopper driven by a function generator (F/G) was used to modulate the intensity of the laser light. A lock-in amplifier was used to improve the signal-to-noise ratio.²⁵ The CS and CES were real-time monitored at the same time using NI LabView hardware and software. The photodetector electronics were designed with bandwidth 14 kHz.

The CES was calibrated using the CS across a 500 μm range by slowly moving the spindle along the y_s direction while the spindle was not rotating. As shown in Figure 6, the CES output showed good agreement, in both sensitivity and linearity, with that of the CS. When compared to the CS output, the CES output showed a standard deviation of 3.33 mV. This corresponds to a displacement of approximately 84.1 nm (0.017% of the full range). This discrepancy is too small to be considered critical, and comparatively a CS typically has 0.2% nonlinearity in full scale.

The displacement of the rotating spindle was then measured within a range of 500 μm by, again, slowly moving the spindle system along the y_s direction. As shown in Figure 7, the two sensors showed similar results, and a maximum discrepancy of 5.514 μm was found at approximately 4.2 s. This can be understood as a characteristic of the sensor itself, similar to the nonlinearity of a CS.

The dynamic characteristics of the spindle system were investigated by a hammering test. As seen in Figure 8, the two sensors showed nearly identical outputs and could successfully measure the vibration signals of the spindle system: a natural frequency of 181.8 Hz and a damping ratio of 0.042 (calculated by logarithm decrement method).

As discussed earlier, the CS is sensitive to the target’s surface shape. As seen in Figure 9, the sensing position effect of the two sensors was investigated. The outputs were simultaneously measured by slowly moving the spindle system (not in motion) along the z_s direction. It was found that the CS is highly sensitive to the offset distance along the z_s direction, and the measurement error in the CS increases by approximately 90 μm at a 2 mm offset. On the other hand, the CES shows an approximate fluctuation of 10–15 μm in the ±2.5 mm interval. This range of deviation may be considered to be an out-of-plane motion error of the manual stage or installation error along the y_s direction.²⁶ This result indicates that the CES is less sensitive to motion in the y_s direction.

Additionally, the stability of the two sensors was tested for 7 min while the spindle was not in motion. The low pass filter (LPF) with cutoff frequency 1 kHz was applied to both sensors. As seen in Figure 10, the resolution of the CS and CES was determined to be 11 nm and 14 nm, respectively. On the other hand, the drift of the CS, 18 nm, was slightly larger than that of CES (<10 nm). These results indicate that the CES is robust and exhibits both high resolution and stability, making it comparable to the CS. Similar to the earlier studies regarding the knife edge sensors,^22,23,27 edge diffraction-utilized displacement sensors show a good sensing performance in terms of resolution, measuring range, and bandwidth as well as cost-effectiveness.

A spindle system supported by ball bearings was chosen as the subject of the measurement. An experiment for comparing the new sensor, CES, with a conventional sensor, CS, was performed at different spindle speeds to verify the validity of the new sensor under dynamic conditions. Radial runout of the spindle system was measured along the y_s axis at a single point by CES. NI LabView data acquisition unit was used to simultaneously collect the CS and CES outputs. The CES results were compared with the results obtained by the CS. (Due to bandwidth limits of the optical chopper, the signal modulation technique was not used in the spindle runout measurement experiment.)

Both sensors showed nearly identical outputs, 22.5 μm at 750 rpm (Figure 11(a)) and 22.5 μm at 2000 rpm (Figure 11(b)). This indicates that the CES can measure a spindle runout which contains a spindle synchronous error and part form error.^12,13 The large sine waveform error typically results from imbalance of the spindle system, artifact alignment error, and/or ball bearing alignment error. The small peaks in the sine waveform are due to the surface quality. Also, it was found that the noise of the CES was smaller than that of the CS under dynamic spindle conditions, even though the measurement bandwidth of the CES (14 kHz) is higher than that of CS (10 kHz). It is well known that CS is sensitive to environmental conditions. This is because the dielectric constant of air varies with temperature (5 ppm/°C), relative humidity (1.4 ppm/%RH), and pressure (100 ppm/atm).²⁸ Thus, it was considered that the mechanical vibration, runout error of the spindle system, and the air pressure distribution between the CS and the spindle affect the measurement results of the CS. The offset distance effects of the CS due to spindle motion error shown in Figure 6 could also be one of reasons. The CES output was clean and showed good agreement with the CS output. Therefore, it can be said that the CES is comparable to CS in terms of performance; it is less sensitive to the spindle dynamic characteristic-induced errors and is appropriate for precision spindle metrology as a displacement sensing tool.

As a part of uncertainty estimation of the proposed sensor, the displacement error could be calculated according to each parameter, respectively. As discussed earlier, an analytical approach to diffraction by a curved edge is so difficult and vague that the deterministic relationships between the uncertainty sources and the sensor performance may not be clearly explained. However, because the knife edge and curved edge diffraction behaviors are similar, this sensor uncertainty can be estimated based on the mathematical model of the knife edge sensor earned in the previous work,²³ assuming that the curved edge diffraction can be related with the knife edge diffraction at some points. Under the condition that the distance L is long enough and the beam diameter is relatively bigger than the detector size, the displacement error was calculated with respect to the parametric errors in percentage (wavelength, distance (L), beam diameter) as seen in Figure 12. The displacement error was calculated by multiplying V_out (Eq. 4) by an amplification gain, which produces ±10 V as a final output. It can be seen that the parameters, distance and wavelength, are significant uncertainty sources. The standard deviation of each uncertainty source is summarized in Table I. The combined standard uncertainty known as root sum of the squares was estimated 85.9 nm under current experiment conditions. It was found that the linearity error was the biggest uncertainty source. This indicates that intensity-stabilized laser system is required in this system. The uncertainty sources of the edge surface roughness and the laser parallelism were not included in this work.

The proposed sensor was constructed in a simple configuration consisting of a He–Ne laser, optical chopper, and PD. It was calibrated, tested, and compared with a CS in terms of resolution, linearity, bandwidth, and drift under both static and dynamic conditions. It was found that the sensitivity of CS varies depending on target surface curvature, and that a CS is not suitable for miniature spindle systems. The CES showed a nonlinearity, 0.017% in full scale, and a 14 nm resolution within a 500 μm measurement range with less than 10 nm of drift, and the measurement uncertainty of CES was estimated 85.9 nm under current experiment conditions. It was found that CES is capable of analyzing dynamic characteristics and runout of a precision ball bearing spindle system. It is the first attempt to apply the CED principle to displacement sensors. This research promises to create new instrumentation principles related to multi-axis CES as well as an ultra-miniaturized precision spindle error separation and characterization paradigm in an accurate, convenient, and low cost manner. For the future work, convenient and efficient computational methods will be investigated to solve CED problems in the transition region adjacent to shadow and reflection boundaries.

The research was supported by NSF (Award No. CMMI 1463502) through Tennessee Technological University. Similarly, this work was supported by the Center for Manufacturing Research and the Center for Energy Systems Research at Tennessee Technological University.