Magnet alignment monitoring system with eigenfrequency-based wire sag correction

The MAMS, used to measure the relative positions of the fiducials of the magnets, is composed of a stretched wire, a series of sensor assemblies, and eigenfrequency measuring devices. Figure 1 is a schematic drawing showing the constitution of the system when setting up the prototype magnets of SPring-8-II. The instruments used in this system are listed in table 1. The fiducials were made on the top plates of the magnets. In addition to the fiducials, reference points for magnet pre-alignment using a laser tracker were made on the same plates.

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The selected CuBe wire has good linearity with a diameter of 0.2 mm. It is heat-treated and naturally straight [5], stretched with a knife-edged pulley (Fogale Nanotech), which maintains constant tension.

The sensor assembly embeds a capacitive sensor [15] in a ceramic ball (KYOCERA). The two-axis sensor measures the distances between the wire and the electrode plate in two transverse directions. A schematic drawing of the sensor assembly is shown in figure 2 (only the electrodes on the y-axis are shown). The sensor's measuring center and the ball center are identical. Therefore, the sensor assembly measures the wire position relative to the center of the ball. Calibration and testing of the sensor assembly are illustrated in section 3.1. We used a ceramic ball because of its high dimensional precision and stability. Moreover, it separates the electrical field of the sensor from the metal base.

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The acquisition of the sensor measurement was performed with a digital multiplexer (Keithley 2701) through the LabVIEW interface. The output of the sensor was acquired continuously at a sampling rate of up to 1 Hz. Furthermore, all sensors on a girder were acquired synchronously. An example of the measurement is illustrated in section 3.4.

The eigenfrequency measuring device was used to measure the eigenfrequency of the wire to derive the maximum sag. It consisted of a laser displacement sensor and an oscilloscope. The spot size of the laser was chosen to be larger than the wire diameter. To measure the wire eigenfrequencies, we drew the middle of the wire up and released it to excite free vibrations. A method to excite the wire automatically involves employing a dipole magnet and introducing a pulsed current to the wire going through the magnet to excite the wire with Lorentz force. Although it is unnecessary for us in this work, this methodology enables a system to perform remote measurement. The amplitude of wire oscillation was measured by the laser displacement sensor at a sampling rate of 1 kHz, and performed the FFT process by the oscilloscope. Figure 3 shows a typical signal (color) of the oscillation amplitude for a 5.5 m long wire, the maximum amplitude was ∼2 mm, and the frequency component of the FFT in white. The peaks indicate the eigenfrequencies of each vibration mode. Because the resolution of the FFT is inversely proportional to the sampling time, a long sampling time leads to better frequency resolution. However, experiments showed that the damping time of a wire that is a few meters long was a few seconds, and higher-order modes attenuated sooner. Thus, we set the recording time to 10 s; consequently, the FFT resolution was approximately 0.05 Hz.

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To examine the effect of nonlinear oscillation, concerning the magnitude of amplitude for instance, on the eigenfrequencies, a test was performed by dividing the collected data into two parts of 5 s each and performing Fourier transformation individually. The maximum difference of the eigenfrequencies between the first and second part was at the level of the frequency resolution (about 0.1 Hz for each case). The difference could be neglected and no effect of nonlinear oscillation was observed. Therefore, we processed the data of 10 s as a whole.

We calculated the maximum wire sag using eigenfrequencies and determined the wire cure with sag.

The curve of the hanging chain follows a catenary. If the lowest point is defined as the origin, the equation of a catenary has the form [16]

$$\begin{equation}y = a\left[ {\cosh \left( {\frac{x}{a}} \right) - 1} \right]{\text{.}}\end{equation} \tag{ 1 }$$

Taking the quadratic approximation of cosh (x/a) at the origin, equation (1) can be expressed by the parabola

$$\begin{equation*}y = \frac{{{x^2}}}{{2a}}.\end{equation*}$$

The chain stretched with a tension of T (N) has [17]

$$\begin{equation*}a = \frac{T}{{\rho g}}\end{equation*}$$

where ρ is the linear density (kg m⁻¹ ) of the wire and g is the gravity acceleration. The curve of the wire is then described as

$$\begin{equation}y = \frac{{\rho g}}{{2T}}{x^2}.\end{equation} \tag{ 2 }$$

Let the length of the wire be L, the sag in the middle of the wire is thus expressed in terms of ρ and T as

$$\begin{equation}S = \frac{{\rho g}}{{8T}}{L^2}.\end{equation} \tag{ 3 }$$

In the frequency domain, a uniform wire with two fixed ends vibrates in standing waves with discrete eigenfrequencies [18]:

$$\begin{equation}{f_n} = \frac{n}{{2L}}\sqrt {\frac{T}{\rho }} \,\,\,\,\,\,\,\,\end{equation} \tag{ 4 }$$

where ${ }n\,$ is the number of vibration modes.

Substituting equation (4) in equation (3), the maximum sag of the wire is derived in terms of the eigenfrequency as

$$\begin{equation}S = \frac{{{n^2}g}}{{32f_n^{\,\,2}}}.\end{equation} \tag{ 5 }$$

For n = 1 [4],

$$\begin{equation}S = \frac{g}{{32}}{f_1}^{ - 2}{} {} {} {} {} {} {} {} {} {} {} {} {} {} \end{equation} \tag{ 6 }$$

where ${f_n}$ is the eigenfrequency of n-order (Hz); that is, the maximum sag of the wire depends only on the eigenfrequency.

The maximum sag can be deduced from the eigenfrequency of any vibration mode. However, utilizing a higher mode eigenfrequency leads to a higher sag resolution. By differentiating the sag with respect to ${f_n}$ in equation (5), the resolution of the sag is given by that of the frequency as follows:

$$\begin{equation}\Delta S = - \frac{g}{{16nf_1^{\,\,3}}}\Delta {f_n}.\end{equation} \tag{ 7 }$$

That is, the resolution of the sag derived from the n-order eigenfrequency is n-times higher than the fundamental frequency. Moreover, the resolution of the sag is inversely correlated with the cube of the fundamental frequency. As the length of the wire increases, the resolution of the wire sag remarkably decreases because of the low natural frequency. Hence, it is important to use higher mode eigenfrequencies.

Substituting equation (3) in equation (2) and shifting y by the amount of the maximum sag, the curve of the wire is in the form of

$$\begin{equation}y = \frac{{4S}}{{{L^2}}}{x^2} - S.\end{equation} \tag{ 8 }$$

Note that the sag here represents the displacement of the wire from a straight line connecting its two endpoints. Even if the two endpoints have different heights, equation (8) is still applicable, as illustrated in appendix A.

3.1.1. Measurement repeatability of the sensor assembly.

The capacitive sensor measured the distances between the wire and four electrode plates (figure 2). It had a measurement range of ±5 mm and a high linearity region of ±1 mm at the central zone. The output voltage was directly proportional to the distance (V_out= kd, ±5 V). The center of the sensor was defined at 0 V. The offset of the center from the rotating center was calibrated by measuring the displacements of the wire from the 0 V point at two opposite positions in the direction of the roll. The central offset was obtained by averaging the displacements. The measuring center was thus the sensor's center compensated with the central offset.

The sensor is structurally symmetric to its center. The random tilt of the sensor does not affect the output if the wire passes through the ball center. Otherwise, it introduces a measuring error that approximately equals the product of the wire offset and the sensor's rotation angle (δ × θ). In most cases of short-range magnet alignment, the amount of wire sag is below 1 mm. The corresponding measuring error is within 1 μm if the random tilt of the sensor is less than 1 mrad. To control the tilt, we set the posture of the sensor assembly using a round bubble level (precision: 0.3 mrad) at the top to adjust the roll and pitch, and two indicators (figure 2(b)) to adjust the yaw by rotating the sensor and visually fitting the slits of indicators to the wire, making the sensor plates parallel to the wire. We confirmed the repeatability of the measurement of the sensor assembly at various wire offsets. We first performed five repetitive measurements with the wire at the ball center, resetting the sensor assembly at each measurement and shifting the wire in the vertical direction by 0.25 mm at each step. Figure 4 shows the measurement uncertainty in the x-axis against the wire offset in the y-direction; it is 0.5 μm in a standard deviation at a 1 mm offset. In addition, the uncertainty is small, approximately 0.2 μm, when the wire offset is below 0.5 mm, because it is comparable to the measuring noise of the sensor. Likewise, the wire offset in the x-direction brings measurement uncertainty in the y-axis.

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3.1.2. Reproducibility of the sensor assembly.

To measure the positions of a series of magnets on a common girder, it is crucial for a set of sensors (four to seven according to the number of magnets) to be mutually exchangeable and have an identical scale factor and central position. To calibrate the scale factor, we set the sensors on a calibration table at the same time and shifted the wire at the sensor's central zone with the standard linear gauges (Mitutoyo LGF-0110L). To ensure the identity of the central positions, we set up three points in a straight line at equal intervals, as shown in figure 5. We mounted each sensor on the middle point in turn and measured its displacement from the straight line connecting the centers of two reference sensors on both sides ($\delta {z_2} = {z_2} - \left( {{z_1} + {z_3}} \right)/2,{ }z:x,{ }y$). The displacements among the sensors were compared. The same wire was used in the calibration, and the two reference sensors were kept stationary. Thus, the measurement error was the repeatability of the sensor assembly. The displacements of the central positions from the nominal ball center in the x–y plane for a set of sensors are shown in figure 6. After compensating for the dimensional difference of the balls inspected by the manufacturer, we confirmed that the reproducibility of the central position for the sensors was within ±2 μm (maximum).

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We investigated the changes in the wire sag related to the eigenfrequency (equation (6)). To validate whether the system is sensitive to a minute change in sag, the standard Kevlar carbon wire for alignment (Fogale Ltd, ρ= 3.22 × 10⁻⁴ kg m⁻¹, 0.5 mm diameter) was used, considering it causes small sag because of high tension. Using the layout in figure 1, the 2.2 m long wire was initially stretched with a tension of 6 kg, corresponding to a fundamental frequency of approximately 100 Hz. We increased the tension by adding a nut of ∼15 g to the weight at each step and measured the change in the fundamental frequency. Each step corresponded to a 0.07 μm sag change. Figure 7 shows the sags derived from the frequency for each step and the positional movements of the wire measured by the sensor assembly; the two movements were consistent with each other at a sub-micron level. Moreover, the changes in the eigenfrequency (not shown) due to the increments of the tension exactly followed equation (4). Because the variation of the sag was small at each step, the wire position was measured at every second step (0.14 μm) and was averaged 50 times.

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As mentioned in section 2.2, a higher mode eigenfrequency has a high resolution for wire sag. Moreover, as the length of the wire increases, the resolution of the wire sag decreases. When the sag resolution by the fundamental mode is below the required value, it is important to use higher mode frequencies. This was experimentally demonstrated in appendix B.

To investigate the distribution of the wire sag, we used the same test bench described in Fukami et al [5], and compared the actual sag against the sag obtained from the eigenfrequency measurement.

Figure 8 shows a schematic drawing of the test bench. A WPS was mounted on a carriage and the height of the wire with respect to the carriage was measured. The carriage moved on a guide rail with a straightness of ±0.02 mm and 4 m in length. The height of the carriage was measured by a pair of HLS, with a movable HLS setting on the carriage connected to a fixed one beside the guide rail. The HLS provided an absolute height reference of the geoid for sag measurement. In addition, two electrical levels (WYLER AG) were mounted on the carriage to compensate for the errors of the roll and pitch. A ∼5.5 m long CuBe wire was stretched above the guide rail with a tension of 2 kg between the two endpoints at equal elevation. The parameters of the instrumentation are listed in table 2.

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Therefore, the height of the wire with respect to the hydrostatic surface of the fixed HLS was measured as

$$\begin{equation*}{Y_{{\text{wire}}}} = {Y_{{\text{wps}}}} + {Y_{{\text{hls}}}} + \delta {y_{{\text{roll}}{} {\text{correction}}}} + \delta {y_{{\text{pitch}}{} {\text{correction}}}}.\end{equation*}$$

In the measurement, the carriage moved in steps of 100 mm each and waited for 30 s until the water in the HLS reached equilibrium. It took 30 min for one measurement of the 4 m rail. A typical distribution of the wire sag is shown in figure 9, with s = 0 at the first point of measurement. The wire positions were fitted with equation (8) by the least-squares algorithm and extrapolated to two endpoints. The deviation of wire positions from the parabola was 0.6 μm in a standard deviation, which we believe is due to the linearity of the wire.

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The repeatability of the sag measurement was tested by repeating the above measurement ten times in three consecutive days. It was 0.5 μm (σ), as shown in figure 10. The mean sag was observed at 0.4953 mm. The temperature of the guide rail was monitored using thermocouples at six points in 80 cm intervals. It fluctuated by approximately 0.8 °C during the experiment. This caused a sag change ($\Delta s/s = - \Delta l/l$) of approximately 0.01 μm when considering the linear density change owing to wire expansion. Thus, the temperature did not affect the sag measurements.

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In contrast, the eigenfrequency was measured, and the maximum sag was calculated using the 5th-order eigenfrequency. We did not use higher modes than the 5th order because the deviations of the eigenfrequencies were increasing. This was considered to have happened because of the relatively small oscillation amplitude which leads to a worse SN ratio. In addtion, higher modes attenuate sooner; thus, the FFT resolution decreases. As illustrated in the Discussion section, we adopted an eigenfrequency of more than 100 Hz, which provides enough accuracy to calculate the sag. The measured 5th-order eigenfrequency was 124.440 Hz on an average, which is in contrast to the theoretical frequency (equation (4)) of 124.446 Hz. The corresponding sag was 0.4944 mm with a resolution of 0.4 μm. It differs from the wire sag measurement by −0.9 μm.

In addition, we compared the actual sag to the eigenfrequency measurement for 2.5 kg tension at the test bench. The results are listed in table 3. In this table, the titanium wire (Goodfellow) was tested on a common girder of the magnets, because they are often used in the magnetic measurement of multipole magnets [19].

The difference between the sag derived from the eigenfrequency and that of the real measurement (δs in the table) was approximately 1 μm. It is sufficiently small and acceptable when considering the following reasons restricting the accuracy of real measurement: (a) the measuring range was comparatively short for the wire length due to the spatial restriction, and (b) the straightness error of the wire was observed to be in the range of ±1–2 μm and was periodic along the length [5].

The MAMS was validated on the common girders for the prototype magnets of SPring-8-II. The wire was fixed on the two end magnets, and the sensor assemblies were mounted on the magnet fiducials. The system could be set up in 30 min and completed measurement in 10 min. It is a portable system and is suitable for periodical observations of magnet positions in the tunnel. Each sensor was compensated for the central offset, scale factor, and sag in the vertical plane. The output of the sensor was measured with a moving average for one minute and a standard deviation of approximately 0.1 μm. Figure 11(a) shows an example of the measurement of the magnet positions in the vertical direction, as shown in figure 1. The periodic variations of the magnet positions were due to the fluctuations of cooling water and room temperature. The transverse relative positions of the inner points relative to the two ends were calculated at every moment. The measurement repeatability of the MAMS was evaluated on the same girder by replacing the wire and resetting the sensors four times. The deviations of magnet positions from the first measurement are shown in figure 11(b). The maximum deviations were ±3 μm and ±2 μm for the horizontal and the vertical directions, respectively. The errors included the measurement uncertainty of the sensor assembly, the linearity error of the wire, and the measurement error of the sag.

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We constructed a test half-cell lattice of SPring-8-II. The multipole magnets on the girder were aligned using a vibrating-wire alignment system [5]. This system utilizes a wire excited with an AC passing through the magnets and detects the center of the magnetic field by scanning the vibrating amplitude of the wire at its resonant frequency. The total random error for the magnet alignment was tested at 1.7 μm (σ). Thereafter, the MAMS was used to record and monitor the positions of the magnets. We considered performing the magnet alignment of SPring-8-II on the common girder at an assembly area and then transport it to the tunnel. To simulate the transportation and confirm the alignment status, we used a truck and transported the precisely aligned girders as a whole from an experiment hall to an assembly building. The changes in relative positions of the magnet before and after transportation were measured using the two systems. The results for the girder in figure 1 are listed in table 4. It was demonstrated that the magnet positions monitored by the MAMS agreed with those measured by the vibrating-wire system for a maximum of 6 μm, which is within the maximum deviation (±6.4 μm) between the two systems by individual error estimation.