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This article is an Open Access article distributed under the terms and conditions of the Creative Commons Attribution license (

This paper presents a computational method for detecting vibrations related to eccentricity in ultra precision rotation devices used for nano-scale manufacturing. The vibration is indirectly measured via a frequency domain analysis of the signal from a piezoelectric sensor attached to the stationary component of the rotating device. The algorithm searches for particular harmonic sequences associated with the eccentricity of the device rotation axis. The detected sequence is quantified and serves as input to a regression model that estimates the eccentricity. A case study presents the application of the computational algorithm during precision manufacturing processes.

The growth in recent decades of the nanotechnology area has led to the emergence of new challenges for researchers and engineers, due to the need for the development of sensors and devices to characterize physical phenomena or quantify the properties and characteristics of materials at the nano-scale [

Electro-mechanical devices that are usually employed in precision manufacturing processes typically have nonlinear behavior for most representative physical variables, low signal-to-noise ratio, strong influence of environmental factors, the high presence of uncertainty and a huge volume of data generated at high frequencies. Therefore, conventional methods often cannot be applied for the characterization of physical phenomena in these devices. However, the use of advanced signal processing strategies, and experimental modelling techniques are useful and feasible ways for studying physical processes at these devices.

Recent researches on precision manufacturing are focused on the development of rotary actuators for positioning with high accuracy [

The main contribution of this paper is the development of a method, based on a computational algorithm for signal analysis in the frequency domain combined with a regression model, to detect nano-scale vibration, and to estimate the eccentricity at the spinning axis of ultra precision rotation devices. This knowledge can be applied to reducing systemic errors, thus reducing manufacturing time.

This paper is organized as follows: Section 2 presents an introduction about the use of ultra precision rotation devices in manufacturing operations and a mathematical model of vibrations in rotation devices; Section 3 describes an experimental analysis to study the relationship between vibrations and shaft eccentricity in an ultra precision rotation device; Section 4 explains the implementation of an algorithm to find sequences of harmonics in the frequency spectrum of a vibration signal and, as an example, introduces a regression model to estimate the eccentricity of the rotation device. Finally, some conclusions about the work results are shown in Section 5.

In precision manufacturing processes, the use of rotation devices with ultra precision requirements [

In spite of the state-of-the-art mechanical and computational technology, inadequate dynamic behavior of a positioning system affects the dimensional accuracy of manufactured parts. The appearance of vibrations can cause unwanted motion in any axis. The dynamic forces that arise during the rotation of devices, such as the spindle of an air bearing [

Eccentricity in the shaft of a rotating device occurs when its center of mass differs from its geometric center [

These forces constitute a harmonic excitation to the rotating device, causing vibrations in the same direction and frequency of the excitation force [

If _{m} of the unbalanced mass

The general equation of motion is represented by:

The excitation input to the system is the unbalance force component in the _{n} its natural frequency.

From the second derivative of

The above equations represent the relationship between the eccentricity, caused by the imbalanced mass, and vibrations that take place in a rotating device. The amplitude of both vibration and its acceleration is proportional to the unbalance mass amount and its eccentricity.

In order to experimentally study the relationship between vibrations and shaft eccentricity, an experimental platform has been installed on a spindle model SP-150 from Precitech Inc, mounted on an ultra precision lathe. These types of machines are employed for finishing operations in curved and flat surfaces of both brittle and ductile materials, with very low error tolerances. Components (e.g., an optical lens) with arithmetic average surface roughness below 10 nm and few hundred nanometers of form accuracy can be manufactured.

Vibration signals are measured with two accelerometer sensors rigidly attached to the spindle housing (see

The ultra precision lathe, model Nanoform 200 from Precitech Inc, is located within an industrial environment and part of a functioning production line, necessitating that the experimental platform does not interfere with the manufacturing process.

The eccentricity reference’s value is obtained from a measurement system embedded into the lathe’s computer numerical control (CNC). Amplitude and phase of this value correspond to the maximum eccentricity position of spindle shaft, which are depicted on the graphical user interface of the CNC. These values can only be obtained prior to each manufacturing operation. For the experimental analysis only the amplitude of eccentricity is used as reference value.

In order to analyze the relationship between vibration level and shaft eccentricity, different operation conditions of the spindle have been considered: not rotating, rotating at different speeds and different eccentricity values. Some of the operating conditions for the experiments are shown in

From expert operator criteria, the acceptable tolerance for spindle shaft eccentricity is 50 nm, and values lower than this number, are insignificant.

Only the X-axis spindle vibration signal is analyzed in this study. The fast Fourier transform (FFT) is applied to this signal, generating a frequency spectrum. A sample size of 50,000 samples is used for each transform, thus the frequency step in the spectrum is equal to 1 Hz.

The spectra are quite similar at first glance, except in a region around 5 kHz where there are harmonics related to the rotation. New harmonics also appear near the frequency of the main harmonics.

The above analysis in the frequency domain is the basis for formulating the following hypothesis: if a harmonics sequence separated by the rotation frequency of the device exists in a frequency range around one of the main harmonics of the spectrum, then the device shaft has an eccentricity due to its mass imbalance.

For a better understanding of the formulated hypothesis,

An algorithm to find sequences in the spectrum of the vibration signal is designed from the previous study and hypothesis. The main parameters of the algorithm are the rotation frequency of the device and the desired frequency search range.

The main steps of the algorithm depicted in

The DC component of the measured signal is removed by subtracting the mean value and then transformed into the frequency domain by applying an FFT,

Find the amplitude of the main harmonic component, _{max}, and its corresponding frequency, _{max}.

Set the desired frequency range around _{max} to perform the search, taking into account the sampling frequency of the signal (

To detect the harmonics within the frequency range, any peak detector function can be applied.

Calculate the distance in frequency between the harmonics up to a maximum window given by the spindle rotation frequency.

Harmonics separated to the spindle rotation frequency, are counted and grouped into the corresponding sequence.

In order to calculate the Fourier transform, the algorithm uses a number of samples (

For each harmonic sequence found, the total number (

Furthermore, on each sequence, the relationship between the main harmonic and the rest is calculated, obtaining a measure of the relative power between the harmonics:

The eccentricity in the spindle shaft can be estimated on the basis of the information obtained from the frequency analysis of the vibration signal. The next subsection proposes a simple model based on regression techniques to estimate the eccentricity in the spindle shaft.

A direct model that is represented by a hyperbolic tangent function with a correlation coefficient (R^{2}) equal to 0.98 is adjusted from an experimental set of data by applying regression techniques and the HSD algorithm. The model [see

In order to adjust both model equations, several values of spindle rotation frequency and its corresponding shaft eccentricity, are taken into account. The experimental dataset is detailed in

In the above equations _{min}_{min}_{min}

In order to evaluate the model performance, the absolute error between the model estimation and the measured eccentricity is calculated by the following equation:

The last column of

In this work a computational method that incorporates a signal processing strategy is proposed to estimate the eccentricity in ultra precision rotation devices, due to its inertial mass imbalance. The eccentricity is estimated from steady state vibrations caused on the device structure, during the rotary movement. These vibrations are measured employing piezoelectric accelerometer sensors. The use of piezoelectric accelerometer sensors instead of displacement sensors, such as capacitive and inductive sensors is justified due to the versatility of this sensor type under several working environment conditions.

The harmonic components related with these vibrations are identified by applying advanced spectral analysis algorithms to the vibration signals. The quantification of the harmonics power, serves to estimate the eccentricity of the rotating device. This procedure enables the design of new control systems in order to compensate for nano-scale vibrations, and thus improving accuracy and precision of rotation devices.

This work was supported by the industrial research project CIT-420000-2008-0013 NANOCUT-INT. We acknowledge the collaboration of the Rubén González.

Schematic of a flat rotor and imbalanced mass.

Physical model of a rotary device with an imbalanced mass [

(a) A workpiece attached to the spindle and machine axes. (b) Ultra precision spindle model SP-150. (c) Piezoelectric accelerometer sensors model 352C15.

Magnitude spectra of the X-axis vibration signal. (a) Spindle not rotating. (b) Spindle rotating at 1000 r/min, eccentricity of 39 nm. (c) Spindle rotating at 1000 r/min, eccentricity of 205 nm.

Frequency spectrum expanded around the main harmonic components.

Frequency spectrum for six cases and harmonics sequences.

Block diagram of the harmonic sequences detection algorithm.

Experimental data and fitted curves, (a) direct model,

Technical specifications of spindle model SP-150 from Precitech Inc.

7000 r/min | |

175 N/μm | |

87 N/μm | |

Axial/Radial ≤ 25 nm | |

0.13 arc-sec | |

+/−2 arc-sec |

Operating conditions of the experiments for the study of spindle vibrations.

1000 | 14, 192, 205, 309 |

2000 | 17, 635 |

3000 | 17, 39, 98, 162, 1358 |

5000 | 56, 2874 |

Experimental data, eccentricity estimation and error of regression model.

_{M}^{(1)} |
_{E}^{(2)} |
|||
---|---|---|---|---|

16.67 | 0.031 | 0.013 | 0.010 | 23.08 |

0.032 | 0.016 | 0.010 | 37.50 | |

0.013 | 0.021 | 0.010 | 52.38 | |

| ||||

33.33 | 0.053 | 0.049 | 0.071 | 45.20 |

0.078 | 0.149 | 0.162 | 8.87 | |

| ||||

50.00 | 0.039 | 0.018 | 0.010 | 44.44 |

0.055 | 0.077 | 0.078 | 1.69 | |

0.116 | 0.304 | 0.263 | 13.53 | |

| ||||

66.67 | 0.013 | 0.017 | 0.010 | 41.18 |

0.084 | 0.155 | 0.182 | 17.30 | |

0.266 | 0.478 | 0.491 | 2.66 | |

| ||||

83.33 | 0.021 | 0.019 | 0.010 | 47.37 |

0.053 | 0.114 | 0.069 | 39.65 | |

0.074 | 0.215 | 0.152 | 29.47 | |

0.446 | 0.665 | 0.670 | 0.78 | |

| ||||

Measured eccentricity.

Estimated eccentricity.