Smart devices have become an important entity for many applications in daily life activities. These devices have witnessed a rapid improvement in its technology to fulfill the increasingly diverse usage demands. In the meanwhile, rotating machinery vibration analysis based on low-cost sensors has gained a considerable attraction over the last few years. For a long time, the vibration analysis of machines has been accepted as an effective solution to detect and prevent failures in complex systems to avoid the sudden malfunction. The objective of this work is to use MEMS accelerometer measurements to monitor the different level of vibration of a machine. This work presents a new technique for rotating machinery vibration analysis. It uses Fast Fourier Transformation as a feature extraction algorithm and Fuzzy Logic System (FLS) as the classifier algorithm. A smartphone accelerometer is used to collect the data from the vibrating machine. The performance of the proposed technique is tested using data from different vibration resources at a different speed of operations. The results are discussed to illustrate the various vibration levels.
Vibration studying and analysis can help in identifying some of the major problems in the industrial rotating machines, vehicles, home appliances, and buildings. Technicians use meters or tools to monitor vibration at regular intervals and report vibration read- ings in real time [
Machine or structural vibration can take various forms. A particular machine part may vibrate over small or large distances, slowly or quickly, resulting or not resulting a noticeable heat or sound [
could happen due to numerous conditions which may act individually or in combination. These are some leading causes of vibration such as looseness, imbalance, misalignment, or wear. The consequences of unwanted vibration can be severe which it can speed the equipment damage rate and lead to reducing the performance conditions and production of the working place.
Detailed vibration analysis can be used to observe and make concerning about the health and performance of a rotating machine. If it is measured and analyzed correctly, vibration can be used as an indicator of machine condition, and help guide to take pre-emptive steps to maintain and prevent damages. Vibration is a continuous cyclic motion of a structure or a component that has some important attributes for developing a vibration analysis tool. The low cost MEMS accelerometers can measure object motion information. As any periodic signal, vibration can be analyzed using the frequency domain analysis. A 3D accelerometer is mounted at a point on the vibrating machine or structure to measure accelerations in three orthogonal directions. Then, the velocity and displacement are derived from the measured acceleration. Acceleration, velocity, and displacement are different ways to express the unit of vibration measuring.
In this paper, a novel technique is presented that used for rotating machinery vibration analysis. This method uses Fast Fourier Transformation as a feature extraction algorithm and Fuzzy Logic System (FLS) as the classifier algorithm. A smartphone accelerometer is used to collect the data from the vibrating machine. The rest of the paper is organized as follows. Section 2 explains the importance of the smartphone accelerometer. In Section 3 and Section 4, brief explanations of the FLS and FFT basics are explained, respectively. The detailed steps of the proposed technique are described in Section 5. Results and discussion are presented in Section 6 while Section 7 gives the conclusion of the work.
Sensor is a converter that measures a physical quantity and converts it into a signal which can be read by an observer or by an instrument. The vast majority of the smart devices contains a huge variety of sensors. Smart devices have become a favorite tool around the world and even an important part of the daily life activities. There were more than a billion smartphones were sold worldwide in 2014, a 23% increase in shipments for the full year 2013 and 2014 [
The earlier versions of smart devices, smartphone, models contained a single MEMS sensor, an accelerometer. The integration of 6 degrees of freedom inertial measurement unit (IMU) within the smart devices enabled the development of new methods dependent on the measurements from the IMU to estimate the human physical movement. The accelerometer is considered one of the most popular components in the smart devices. It can measure the earth gravity and the platform acceleration. In one application, it is used to sense enable the screen rotation of the device from one view to another view (for example from portrait to horizontal) based on the gravity measurements. In other application, it is used in falling protection application. A tri-ad accelerometer axes can be represented in the Cartesian coordinates where each accelerometer provides a measurement for one of the x, y, and z directions as shown in
There are several important specifications should be taken into account to choose the appropriate accelerometer for any application. The performance of the acceleration sensors are measured in range, accuracy, quantization, sampling rate, noise, and power
consumption [
Theoretically, the displacement can be calculated using Equation (1) [
where d(t) is the total displacement at time t, 𝑑0 is the initial displacement, v0 is the initial velocity, and a(t)is the measured acceleration.
Equation s1 is a continuous function while a(t) is discrete measurement due to sampling. To calculate the discrete displacement, Equation (2) has to be used [
where ai is 𝑖th sample, and Δ𝑡 is the time interval. Then, the velocity and displacement can be calculated as the following [
In Equations (3) and (4), i refers to the current value and i − 1 refers to the previous value.
The 3D accelerometer measures the acceleration in three dimensions where the total acceleration is represented by the vector summation of the three components in the three directions as in Equation (5).
Spec. | Bosch BMA280 | InvenSense MPU-6500 | Unit |
---|---|---|---|
ADC range | 14 | 16 | bit |
Range | ±2, ±4, ±8, ±16 | ±2, ±4, ±8, ±16 | g |
Output data rate | 2000 | 4000 | Hz |
Fuzzy logic [
Fuzzifier and defuzzifierare based on using a Membership Function (MF) to map the variables between crisp values and linguistic expressions. Each variable at the input side or the output side is assigned for a membership function. There are several possible choices of membership function shape (triangular, trapezoidal, singleton, and Gaussian) as shown in
Different characteristics are needed to be considered for the selection and design of the MF shape. First of all, the selection of the MF shape depends on the problem size and type. While the MF value is ranging from 0 to 1, the width of the MF can be varied and different from one MF to another based on the represented variable value ranges. Also, the selection of the suitable MF may consider the simplicity to implement and fast for computation such as triangle and trapezoidal shapes.
There are four basic elements of the FLS; Fuzzifier, Rule Base, Inference Engine, and Defuzzifier. A brief description of each part is given below.
A fuzzifier measures and converts crisp input variables into a fuzzy variable represented by membership grades in some fuzzy sets. To convert from crisp inputs to fuzzy inputs, fuzzy sets and associated membership functions must first be determined for each input.
A fuzzy system is characterized by a set of linguistic statements based on expert knowledge. The expert knowledge is usually in the form of IF-THEN rules, which are easily implemented by fuzzy conditional statements in fuzzy logic. These rules have the form of Equation (6):
where
In fuzzy inference engine, fuzzy logic principles are used to combine the fuzzy IF- THEN rules in the fuzzy rule base into a mapping from fuzzy sets in
There are several interpretations [
a) Mini-operation rule of fuzzy implication:
b) Product-operation rule of fuzzy implication:
where
Or according to the product-operation rule:
The defuzzifier performs a scale mapping from fuzzy sets in V to a crisp point
a) Maximum defuzzifier
The maximum method produces the point at which the membership function of the fuzzy output reaches a maximum value.
b) Center average defuzzifier
where y−l is the center of the fuzzy set G−l as a point in V at which
c) Modified center average defuzzifier
where
The signal behavior can be viewed and analyzed in both time domain (TD) and frequency domain (FD) where each domain provides remarkable and unique details about the measures that compose the signal. In TD, data is analyzed over a time period where the variable is always measured versus time. On the other hand, FD analysis refers to refers to analyzing a data with respect to the frequency. It is mostly useful to be used with signals or functions that are periodic over time. FD analysis is widely used in fields such as control systems engineering, electronics, and statistics. There are several transformations to convert from the time domain to the frequency domain and vice versa such as Fourier Transformation (FT) and Wavelet Transformation (WT). The frequency domain contains exactly the same information as the time domain, just in a different form. If the information about one domain is available, then the other domain can be calculated. Given the signal in the time domain, the process of calculating the frequency domain is called decomposition analysis or simply forward FT. If the signal is represented in frequency domain, the calculation of the time domain is called synthesis or simply the inverse FT (IFT). Both synthesis and forward analysis can be represented in equation form and computer algorithms.
Fourier analysis is a family of mathematical techniques, all based on decomposing signals into sinusoids. FT is used to convert a signal of any shape into a sum of an infinite number of sinusoidal waves based on the theory that all complex waves are made up of sine waves combined in various amplitudes, phases, and frequencies. All symmetrical waveforms, like triangle waves, and non-symmetrical waveforms, rectangular pulses with a skewed duty cycle, exhibit the same harmonic distribution with different amplitudes.
Fast Fourier Transform (FFT) is one of the most important algorithms in signal processing and data analysis. FFT transforms the sampled signal in the time domain to represent it as a series of spectral peaks in the frequency domain as shown in
where F(k)components are called the “Fourier Coefficients” or “harmonics”.
The sequence f(n) can be calculated from F(k) using the Inverse Discrete Fourier Transform (IDFT) as shown in Equation (15):
Usually, the sequences f(n) and F(k) are referred to as time domain data and frequency domain data, respectively where both f(n) and F(k) are complex. In most cases, the FFT size is chosen to be a power of 2 (N = 2p).
The proposed technique uses the accelerometer data to estimate the vibration level for the rotating machinery.
Pre-processing the low-cost accelerometer data is important to ensure the quality of the measured data. It is recommended to remove the gravity component from the measurements. This can happen by using leveled accelerometer data. The leveling process requires the knowledge of the device tilt angles, roll and pitch. These angles can be estimated using a static period of the data for about 30 seconds to be collected at the beginning of the test as shown in Equation (16).
where
In order to transfer observations into a horizontal plane, the direct cosine matrix is built by using Euler angle (roll (
(17)
Another component to be considered for removal is the total acceleration mean. The total acceleration is calculated using Equation (18).
where fx, fy, and fz are the acceleration vectors in the main directions.
From the signal processing perspective, vibration is defined as a cyclic mechanical motion. Frequency domain provides an idea about the signal main frequency and the different harmonics. The MEMS accelerometers collected data is sufficient for capturing motion information from a rotating machine. Therefore, a 3D accelerometer can offer a good opportunity for mounting flexibility than single accelerometer as the orthogonal orientation of the different axes enables calculating the total frequency regardless the vibration direction. The power spectral plot provides an appropriate way to describe the vibration shape (the relationship between vibration magnitude and frequency). The plot in
The effect frequencies of the frequency domain are used to analyze the vibration level of the signal. The primary frequency, main frequency, and the average of the harmonics are selected to be inputs for the fuzzy system. The average of the harmonics is calculated for the harmonics above acertain threshold. This threshold is selected based on the experience about the different dynamics and system behavior.
The proposed fuzzy system and its main parts are shown in
The different rule bases that used for the FIS are shown in
Number Summation | Very low | Low | Medium | Medium-high | High | Very high |
---|---|---|---|---|---|---|
Very low | Very weak | Very weak | Very weak | Very weak | Very weak | Very weak |
Low | Very weak | Very weak | Weak | Weak | Medium | Medium |
Medium | Very weak | Weak | Medium | Medium | High | High |
Medium-high | Weak | Medium | Medium | High | High | Very high |
High | Medium | Medium | High | High | Very high | Very high |
Very high | Medium | High | High | Very high | Super high | Super high |
reflect the relation between the different inputs with the output.
In this section, different tests are conducted for the purpose of evaluating the proposed algorithm performance in order to examine the capabilities of the smartphone sensors for measuring different levels of vibration.
The data is collected using a Galaxy s4 smartphone from different speeds of a small rotary machine. The accelerometer measurements are obtained and processed. The sliding window technique is used with window size of 1 second of data. The first step of processing is obtaining the roll and pitch angles from a static period at the beginning. The calculated angles are used to level the data to obtain the vertical and horizontal components of acceleration. The vertical acceleration is used where the gravity component is removed by subtracting 1 g (9.8) m/s2 from the vertical component. The next step is to convert the time domain vertical acceleration into its equivalent frequency domain using the FFT transformation. Using a threshold, all frequencies above the threshold are obtained to calculate the input variables values to the FIS. The first input is the summation of strengths above the threshold while the second input is the number of frequencies with strengths greater than or equal to the threshold. The value of the threshold is determined as a portion of the dc component of the frequency domain, approximately 1.4 of the dc component strength. The strength of all frequencies which are greater than this value is summed to provide input 1 while their number provide input 2 for the fuzzy system.
that the vertical data is provided after the removal of the gravity component. The acceleration values depict the change in the vibration level as the values are increased at different periods of time. The input to the FIS is shown in
The second dataset is somehow longer that the first dataset.
Vibration analysis is growing and developing and becomes an important characteristic to monitor the performance and condition of most of the machines. It could be used to indicate the need for an immediate maintenance or further assessment of causes. The new technology of low cost MEMS sensors is emerging new trends for vibration analysis where it is facilitating a wider usage for this type of measurements. This paper presents a new technique for vibration level estimation using the FFT feature extractions and fuzzy inference system. The data transforms into the frequency domain to select the appropriate features to identify the level of vibrations. The features are selected based on an adaptive threshold value which is a ratio of the primary frequency strength. The FIS based system estimates the vibration level using the different membership functions. The results show the ability of the algorithm to provide the different vibration levels which reflect the change in the input data correctly. Also, the adaptive selection of the threshold values helps in picking the appropriate vibration level corresponding to the strength of the input data.
Ali, A., El-Serafi, K., Mostafa, S.A.K. and El-Sheimy, N. (2016) Frequency Features Based Fuzzy System for Rotating Machinery Vibration Analysis Using Smartphones Low-Cost MEMS Sensors. Journal of Sensor Technology, 6, 56-74. http://dx.doi.org/10.4236/jst.2016.63005