Spectral Analysis of Temporal Velocity Changes as an Approach to Determining the Properties of Positioning Systems
We describe a method of monitoring of the instantaneous velocity of moving elements
in positioning systems. It is shown that various internal or external sources (engines,
transmissions, carriages, etc.) cаn give contribution into the velocity variation. A comparison of several drives is given, including the case of a sharp velocity reduction, namely, the defect of non-critical “jamming” when the amplitude of the vibration velocity components decreases to zero. The developed method is of interest for monitoring the occurrence and development of defects in positioning systems during its operation.
O. S. Bolshakov, A. V. Kirsanov, Cand. of Eng. Sc., V. V. Chernov, Cand. of Phys.-Math. Sc., Institute of Applied Physics RAS (IAP RAS), Nizhny Novgorod, Russia
Roughness of the positioning speed or feed stopping in experiments and technological processes leads to data distortion or to reduction in the quality of the final product. In the last few decades, IAP RAS has developed a technology for growing and processing KDP – DKDP crystals, and accordingly, research facilities have been created to performed a number of experiments on developing technologies for growing crystals and its coating [1,2]. Velocity deviations during such processes lead to defects appearance at surfaces of nonlinear crystals. One of the finishing process in the production of water-soluble KDP-DKDP crystals is the sole-gel coating. It has several functions. First, the film formed on the surface of the crystal has properties similar to those of ceramics, which bleaches the surface after diamond micro-milling and, second, it protects the water-soluble crystal from the influence of moisture presenting in the atmospheric air. The sole-gel coating technology realizes in the process of a uniform lifting of a non-linear element from a specially prepared liquid. Velocity deviations during such process lead to non-uniform drying of the protective film and subsequent non-uniformity of reflection and transmission properties of the nonlinear crystal. Correspondingly, the technical task arises for moving a sample in the processing area with a constant velocity over time. Acceleration and deceleration areas should not be included in the processing area.
of the specific implementation
of the positioning system
In this work, we investigated the positioning system of the crystal coating, namely, a translation stage , assembled according to a common scheme (see Fig. 1) based on commercially available elements [4, 5] including two HIWIN20 rails which are located on the base plate 1. On rails 2, four carriages 3 are moved, on which the object table 4 is placed. A comparison of velocities measurement data for three different drive implementation systems was carried out based on a spectral analysis of velocities variations. First, pair of steel screw with a trapezoidal thread with a pitch of 4 mm per revolution and a caprolon nut (realization 1) was used. A stepper motor (SM) of the NEMA23 standard was used as an actuator. Since the velocity of the process is relatively low (feedrate of 1.6–1.65 mm / s), a 10 : 1 reduction gear is installed between the stepper motor and the shaft. In the control system step division by 32 (microstepping) and software suppression of resonances of the SM with l oad were used for smooth operation. Position control was provided using the Ditron optical linear glass scale DC‑11 . The velocity is recorded by a special “Speedometer” controller with the appropriate software. After a certain amount of experimental testing in the system described above, there was some wear and, accordingly, a backlash in the shaft-nut friction pair. This was the reason for replacing the sliding friction pair of the lead screw-nut with a ball-screw transmission (realization 2) with some design adjustments to increase stability of velocity. Separately, a direct drive system was tested as a drive on the same rails and carriage supports (realization 3). A commercially available positioning direct drive system is presented in .
of the “Speedometer” controller
To analyze data from the optical scale, a speedometer microcontroller based on the ATmega328 microprocessor with the appropriate software has been developed . In the Speedometer microcontroller, its own quartz frequency generator forms its own timescale, independent of computer interrupts. With certain selected frequency F, the microcontroller polls and writes to the buffer the coordinate value from the optical scale for further data transfer via the USB interface to the computer, in which the Speedometer software unit records the time and coordinate of the moving part in a “csv” file for further analysis. The recording frequency F can vary widely up to 10 000 times per second (10 kHz), which, according to Kotelnikov’s theorem , allows analyzing the spectrum up to 5 000 Hz. The amount of movement data and velocity is huge even for relatively small displacement, but due to the rapid development of computer technology, arrays of millions of rows can be completely processed by mathematical programs (MATLAB, MathCAD, ORIGIN, etc.). Experience of operation with mechanical systems shows that the main resonances of the mechanical system is located below 200 Hz. However the frequency of the stepper motor with its resonances are in the range of 300–600 Hz, which is useful to monitoring. Thus, the high frequency of recording coordinates and velocities F = 10 kHz is unnecessary, and, besides, it leads to a significant increase in the processed array. Therefore, the recording frequency F = 2 kHz was chosen, i. e. one measurement every 500 ms.
Accordingly, the velocity variation spectrum up to 1 kHz was analyzed. For measurements, DC‑11 series scales of different lengths were used. This series of scales has a 1µ coordinate sampling. Hence, data sampling for the velocity was δv = ±2mm / s.
To visualize the velocity temporal dependence, it should be averaged over some window. After considering several options for the size of the averaging window, we selected such window for 500 points or 0.25 seconds at the recording frequency. The recording start and finish can be recorded incorrect for the motion stop of 0.25 s. Therefore these points as well as acceleration and deceleration areas were not taken into consideration. Choosing the recording range more than technologically necessary, it is possible to resolve this problem programmatically, averaging either on the next 500 points for the motion start or on the previous points for the finish with a certain time of stitching in the middle of motion.
Figure 2 shows the temporal dependence of the instantaneous velocity of the object table of the positioning system in the realization 1 (lead screw with a pitch of 4mm per revolution with a caprolon nut) normalized to the maximum for two ranges of consideration, including the whole technological process from the beginning to the 200th second (Fig. 2a) and the time range from the 10th to the 12th second (Fig. 2b). In Fig.2a, one can recognized two types of velocity fluctuations including slow amplitude modulation with a frequency of 0.041 Hz and high-frequency modulation with a frequency of 0.413 Hz (the frequencies are easily calculated from the graph). In Fig. 2b, a fragment of the previous graph for 2 seconds (from the 10th to the 12th seconds) are shown. It is seen, that the temporal velocity dependence additionally includes rough more high-frequency fluctuations. Thus, two conclusions can be drawn:
The periodical dependence of the average frequency on time v (t) is determined by a period Т = S / vср., where S = 4 mm is the screw pitch for the system realization 1. For a given average speed vср. = 1,65 mm / s, we can find Т ≈ 2.42 s;
An additional consideration shows the presence of vibrations. Actually, vibrations of the object table mean that the system has the free stroke and that there are some sources of vibration in the system. Free stroke in mechanical systems is necessary for their normal functioning, for example, the presence of thermal gaps is essential.
Detection of different types of velocity fluctuations can be done based on consideration of the spectrum of its temporal dependence both at ultra-low frequencies (from 0 to 1 Hz) as well as at frequencies up to 200 Hz.
The velocity spectrum normalized to the maximum spectral density is shown in Fig. 3. It is obtained as a result of Fourier analysis of the velocity-time dependence over the entire time of the technological process. Figure 3a demonstrate the spectrum from 0 to 300 Hz, while the oval 1 marks the spectral components at a frequency of 0.413 Hz, which corresponds to the frequency of rotation of the drive shaft In the system realization 1 for the processing average velocity of 1.65 mm / s and a the screw pitch of 4mm per revolution with a 1 : 10 reducer between the stepper motor and the running shaft. This component is presented in more detail in Fig. 3b. The oval 2 highlights a group of components at an average frequency of 82.5 Hz, shown in more detail in Fig. 3c. It can be seen that the spectrum is discrete with a discreteness of 0.413 Hz. The oval 3 marks the spectrum component at a frequency of approximately 206 Hz which is 1 / 4 of the frequency of the stepper motor steps for this average speed of 1.65 mm / s.
There are no selected spectrum components above 300 Hz, if resonances of the stepper motor are suppressed and the step division is used (in our case by 32). The long recording time, and corresponding large number of recording points, makes it possible to consider ultra-low frequencies in the spectral dependence. Thus, the visible velocity beats in Fig.2a with a frequency of 0.041 Hz is represented as a frequency component in the spectral density. A component with a frequency of 0.041 Hz occurs as 1 / 10 of the frequency of the running shaft beats. Since all velocity fluctuations are a negative phenomenon for users to deal with, the tasks of analyzing the velocity spectrum arises aimed to identification of all spectrum components and determination of the causes of their occurrence.
Sufficiently large velocity deviations for the realization 1 (6% at the beginning and 7.4% at the finish of a technological process) and appeared wear-out caused the development of the realization 2 for this positioning system. In this realization, the sliding friction pair, namely, “running shaft – screw nut” was replaced with a pair of rolling friction, namely, the ball-and-screw unit with a step of 5 mm per revolution. For minimization of velocity fluctuations, we tried to minimize the defects caused by angular and coordinate misalignment of the spindle and drive mechanisms.
Let us consider the resulting velocity characteristic and its spectrum in Fig.4. One can see that the peak-to-peak velocity deviations are about 4%. In the spectrum, in contrast with the case 1, a narrow (about 0.07 Hz at half maximum) component at the frequency of 64 Hz with the same intensity as the previous components is added to the blurred widened components with frequencies of 80 Hz and 160 Hz. In the area of ultra-low frequencies, there is a component of 0.32 Hz which corresponds to the beat of the main drive shaft.
We tested also similar positioning system (see Fig. 5) which is based on the same rails and support carriages but with linear stepper motor drive (realization 3). One can see that the same components with frequencies of 80 Hz, 160 Hz, and harmonics of 240 Hz and 320 Hz are presented in the velocity spectrum shows. The velocity deviation in this case is less than 1%.
Thus, for all three analyzed realizations of the object table movement drive, we can identify common characteristic features of the spectra. For a speed of about 1.6 mm / sec, there are components with average frequencies close to 80, 160, and 240 Hz. In addition, in the case of ball-and-screw unit, there is another component with a frequency of 64 Hz. If we change the velocity, the average frequencies of all these components change according to the formulas:
fcarriages = n A v, (1)
fballsrewnut = 0.8 A v, (2)
where n = 1, 2, 3... is the harmonic number, A = 50 [mm–1] is a coefficient typical for this type of HIWIN carriages, v [mm / s] is the velocity of the object table.
Since the velocity changes according to (1), (2), we have a mechanical frequency broadening of the spectral components, i. e. the associated components changes together with the velocity value. It can be seen on the spectra of realizations 1 and 2. In the realization 3, the velocity changes by less than 1% and, correspondingly, the broadening is not noticeable in this case. The source of these types of vibrations are systems consisting of the carriage bodies and balls crew nuts, on the one side, and recirculating rolling elements-balls on the other side (see Fig. 1b).
Rolling elements enter / exit the pre-tensioning zone with a certain impact which excites vibrations of the carriage body or ball screw nut. In some sense, this is the downside of using pre-tensioning systems. During the operation of the positioning system, it must be serviced and lubricated according to the regulations. It should be noted that after smearing the elements, for example all the carriages 3 (Fig. 1), the amplitude of the vibrations excited by them decreases and gradually increases for the next service, because the part of the lubricant inevitably leaves the contact zone of the rolling elements. As conclusions about the use of this type of positioning system, it should be noted that, obviously, the drive type shaft-nut ball screw works better than the drive based on a pair of running shaft and nut on sliding friction. In all systems with a motion source based on rotation motors, it is necessary to minimize angular and coordinate misalignments of the drive shaft and the SM. In this case, it is possible to achieve velocity deviations of less than 1%, as in the case of a linear stepper motor type drive. Velocity deviations of less than 1% have almost no effect on the technological process.
Influence of external sources
In order to test the capabilities of the Speedometer system, the following experiment was performed. A vertical radiator was installed on the object table. In the process of moving, it radiated at a low frequency of 66 Hz. The geometry of the radiator and positioning system was chosen so that the main impact of the radiator was co-directed with the direction of movement. The translation system represented a conventional translation stage shown in Fig. 1 and includes two guides with carriages and circulating balls. The object table was fixed to them. When the vibration source was switched on, several new narrow spectral components (less than 1%) appear in the velocity spectrum. These frequencies correspond to the source vibration frequency of 66 Hz, at the second and the third harmonics of 132 Hz and of 198 Hz, respectively. The typical for a given velocity spectrum components remain in this system, since the sources of vibration are not related to each other.
Short-term speed decreases
During the operation of positioning systems, there are situations in which the velocity of the object table drops fast. An example of a short-term velocity drop is shown in Fig. 6. Figure 6a shows the velocity normalized to its maximum for 26 seconds of movement, the circle indicates the case of a short-term reduction in speed by 5% at the 242s second of recording. Without going into a consideration of the reasons for this “jamming”, we can say that the system missed the defect. In this case, the defect was not critical for the positioning system. Figure 6b shows in detail the velocity in this case. Times T1 and T2 represent the moments when the amplitude of the velocity vibrations is zero, i. e. the moments of jamming. Such one-time bursts strongly distort the velocity spectrum at ultra-low frequencies, which can be used for diagnostics of such defects. Weak components such as 1 / 10 of the drive shaft velocity are not visible against their background.
A new approach to the analysis of the instantaneous velocity of moving objects in experiments or technological processes is proposed and implemented. The developed approach to the spectral analysis of temporal changes in the velocity allows controlling the occurrence and development of negative defects in moving systems during their operation. In this paper, we considered the case of using this method of velocity analysis for linear motion systems.
However, it may be more interesting for control of rotation systems. Instead of a signal from an optical linear scale, the Speedometer system can use an angular position signal from an axial incremental encoder, for example, with 1000 counts per revolution. The angular velocity should be calculated. Then, after the averaging procedure, one can observe the dynamics of the angular velocity or perform a Fourier analysis of the angular velocity data array in order to consider the spectrum of the velocity dependence in which the features or defects of the mechanical system are reflected.