Task 1
The solution of this problem aims at determining the straightness error (Ezx) in the Z-direction of the X-axis by using a capacitance probe labelled as M1 and a metal flat mirror in an ultra-precision turning machine (Daul, L, et al. 2021). The difficulty is in separation of the straightness error e1 from mirror flatness error Eflat since both components contribute to the measured values.
In order to combat this, the capacitance probe is affixed to the X-slide so that it runs across the mirror which has an imperfection of some sort. Here, the total displacement as per the measurement includes both the Ezx and the Eflat components. In this work, based on the given data, a separation technique is used to accurately estimate both errors.
The method of least squares is employed to obtain a continuous function that estimates flatness error of the mirror. It also follows that if there is a general variation in capacitance probe readings, that it can be attributed to Eflat, while the remaining deviation from the true capacitance value is represented by Ezx. Removing the flatness error estimate from the total measurement will yield the straightness error.
Figure 1: Experimental setup for the measurement of X-axis straightness
The computed root mean square (RMS) values help in making the quantitative estimate of errors. Thus, the RMS value for Ezx is 0.00092358mm, and for Eflat it is 0.00039699mm. These values describe how much the straightness of the machine has shifted, as well as the mirror’s flatness.
Task 2: Sensors for manufacturing process condition monitoring and predictive maintenance (e.g. temperature, vibration, ultrasound and AE sensors)
Overview
The manufacturing industry is a very sensitive industry that depends on the rate of production, frequent breakdowns and high maintenance costs are some of the negatives which affect the production industry. Performance monitoring and the methods of predicting equipment failures have emerged key approaches towards these goals. Sensors, therefore, have the responsibility of timely data collection of these processes and providing alerts on early signs of failure to allow preventive measures to be taken. Some of the common sensors used in industries include the temperature, vibration, ultrasound as well as the acoustic emission (AE) sensors. These sensors measure various factors that are indicative of the health of the particular machine in order to avoid sudden breakdowns, which enhances the general performance of the machinery.
Temperature sensors measure a rise in temperature of the part of the machinery which may signal a condition like heat, friction or insulation failure. Vibration sensors therefore assist in detecting vibration or oscillation of any rotating structures which could be as a result of an imbalance, two bearings that are not parallel or perpendicular or a bearing that has developed a fault. Ultrasound sensors work at high frequency, and through the use of sound waves, the areas of the plate that have a crack, a leak or have inadequate lubrication can be identified, whether on the surface or beneath the surface. The AE or acoustic emission sensors utilized for monitoring and measurement collect high frequency waves which reflect the deformation of a material and any further failure sequences. Every type of sensor has its own roles, while application of all of them increases the dependability and the life cycle of manufacturing equipment.
Key Operating Principles
Temperature sensors generally measure temperature through the use of different techniques, namely resistance change (Resistance Temperature Detectors or RTDs), the use of thermoelectric effect (thermocouples) and lastly infrared sensing (National Instruments Corp 2025). RTDs are based on the principle of change in the electrical resistance that is proportional to the temperature, and thermocouples produce a voltage difference proportional to the difference in the temperature of junctions. Infrared temperature sensors are designed to be used by the sensing of thermal radiation emitted from the surface of objects, thus enabling contactless temperature measurement. These sensors are very important for measuring accelerated temperature rise in motors, bearings and electrical parts to avoid failure due to high temperatures.
These licenses help to identify anomaly vibrations in mechanical systems or parts. There are piezoelectric materials, capacitive elements or Micro Electro Mechanical System (MEMS) accelerometers (Hassan, et al. 2024). Piezoelectric vibration sensors measure electrical charge produced by mechanical stress on them and are used to obtain information on the state of vibration . Probably CAP Sensors measure changes of capacitance which occur due to movement while MEMS-Accelerometers use microstructures to measure changes of acceleration. Vibration sensors are used mainly for the initial recognition of lesions in cyclic equipment such as turbines, pumps, and motors.
Ultrasound sensors utilize sound waves with high frequencies capable of interaction with the material and structures, and identify deficiencies (Modsonic Instruments 2023). These are sensors that use wave reflection and attenuation, wherein sound waves pass through a given material and bounce off an area of variable density. These structures within the material appear as reflected waves so that through ultrasound sensors, analyses can simply have cracks, voids, or corrosion. It is widely applied in non-destructive testing (NDT) to inspect the weld, leakage and monitor the pipeline integrity, etc.
Acoustic emission (AE) sensors record the ultrasonic signals that occur as a result of stress in the material (NDE-Ed.org 2025). They operate on piezoelectricity and enable constant tracking of the material degradations, progression of cracks and structural failures. AE sensors rely on detecting vibrations that are produced during the loading phase and are effective at detecting even the small amount of strain change in the materials which makes them perfect for monitoring the health of Aerospace/aircraft, Automobile and Infrastructure components.
Characteristics
Sensors used in condition monitoring and predictive maintenance are described based on certain parameters such as sensitivity, response time, and operating conditions, an important application.
Temperature sensors guarantee high performance and quick initiations are appropriate for use in harshest atmospheric conditions characterizing industrial settings where heat can harm a machine’s components (Balakrishnan, et al. 2022). They are predominantly applied for temperature variation in electric motors, transformers and the mechanical bearing systems.
They are moderately to highly sensitive and have a fast response time thus making them good for rotating equipment. They are used in the detection of imbalance, misalignment, and mechanical deterioration in turbo sumps, gas turbines and motors etc with a view of mitigating sudden failure.
High sensitivity and fairly adequate response times due to the use of ultrasound sensors in structural contact or contactless detection of defects. They play a paramount role for inspection of cracks, leaks and lubrication status of various component sections such as pipelines, pressure vessels and welded joints.
Acoustic emission sensors are extremely sensitive and possess very short response time which makes them suitable for structural health monitoring. They integrate stress-generated acoustic emissions that help in the identification of micro-cracks, fatigue, and the general state of degradation of the material in question.
|
Sensor Type |
Sensitivity |
Response Time |
Operating Conditions |
Key Applications |
|
Temperature |
High |
Fast |
Harsh environments |
Overheating detection |
|
Vibration |
Moderate |
Fast |
Rotating machinery |
Imbalance and misalignment detection |
|
Ultrasound |
High |
Moderate |
Contact or remote |
Crack and leak detection |
|
AE |
Very High |
Fast |
Structural health |
Material defect analysis |
Example Applications
In IC engines, temperature sensors are put to use in the monitoring of motor windings, bearings and electrical connections in CNC machines. They are used to help identify potential problems in machinery that, if not tackled immediately, will result in blow up and reduced efficiency.
Vibration sensors are important tools towards condition-based monitoring of wind turbines as they can identify initial signs of bearing deterioration, gear and rotor party and other faults (Dwyer Instruments, LLC 2025). These devices also allow regular assessment of vibrations and thus scheduled maintenance, thereby limiting the time a turbine is out of service or would be rendered less efficient due to mechanical failure.
It is a fact that most of the automotive companies use ultrasound sensors specifically for the purposes of assessment of the quality of welds of the frames and other components in the automobiles. These sensors assist in detecting some of the unnoticeable areas in an assembly line that may affect its stability, making automotive assemblies safe and reliable.
It has been noted that aerospace industries make use of acoustic emission sensors for structural integrity assessment (DuBose, B., 2020). They identify very thin defects on aircraft elements and their effects so as to address them before they result in major breakdowns. Such sensors are useful not only for providing an increased life cycle and stability of aerospace structures, but also safety assurance.
Commercial Systems with Typical Performance and Limitations
Commercial temperature monitoring systems for various attached electronics thereto, include FLIR Infrared Thermal Cameras characterized by high-resolution thermal imaging for accurate heat recording. These systems are crucial in identifying cases of overheating of the industrial equipment (FLIR Systems AB 2025). However, their effectiveness could be smaller due to their expensive price tag as well as the fact that they are limited to line-of-sight distance measurements.
Figure 2: Infrared thermometers in one spot (L) and FLIR i3, temperature in 3,600 spots (R)
Measuring devices like the SKF Microlog Analyzer which aids in monitoring of live vibrations and frequency spectrum analysis (SKF 2025). These systems are very important in sensing faults that may exist in rotating machinery. However, there is a provision that they raise the danger and they strongly rely on the placement and calibration of the sensors used.
Currently, many industries utilize ultrasound solutions like the GE Mentor UT for weld, composite, or pipeline inspection. Despite the level of effectiveness in presenting highly accurate information, such systems have limitations that were designed to be operated by people, which puts them as labor-intensive.
MISTRAS AE Sensors, for instance, is a type of acoustic emission monitoring system, which is designed to respond to changes in the material state and any signs of the structural degradation. Such systems permit the detection of miniature cracks in materials relevant to structural parts at a very early stage. Nevertheless, there is always a possibility of getting interfered by the environmental noises leading to reduced accuracy.
Summary
Thus it can be deduced that condition monitoring and predictive maintenance are the indicators that the manifold manufacturing processes cannot afford to do without since this will lead to increased inefficiency and more frequent equipment failures. Temperature monitoring gives information about heat and vibration information gives idea of unbalance or misalignment while ultrasound and acoustic emission sensors give information of generation of different frequencies of sound from the machine which are dangerous for the health of the machinery. Modern commercial sensor systems are capable of real time monitoring and defect identification and the decision making remains easier with these systems despite some disadvantages like high cost, complicated installation and handling and difficulty in interpreting the extracted data. Continued development of artificial intelligence (AI) and Internet of Things (IoT) will add more precision into these sensors, hence improving the efficiency and effectiveness of the entire production process while at the same time cutting costs. It will also enable industries to adopt improved maintenance tactics, enhance the durability of equipment, enhance productivity of production in today’s manufacturing setting, and harness smart sensor technologies.
Task 3: Nonlinear Least Squares Algorithm
Development of a Nonlinear Least Squares Algorithm in MATLAB
In order to identify center and radius of circular structures from 2D microscopy measurements, an absolute nonlinear least squares algorithm was used. It enables a smallest error to be obtained between the actual and the estimated radius whilst maintaining the best fit to the given data points (Umar, et al. 2021). The first step of MATLAB implementation commences with the measurement data input in form of (X,Y) coordinates of the circular profile taken by the microscope. A first guess for the circle is the midpoint of the X and Y coordinates, the first radius is taken as the mean distance of the points from this first guess.
Figure 3: Plot for the coordinates from the ring gauge
Description of the Nonlinear Least Squares Algorithm
The Nonlinear Least Squares (NLS) is used here for better visualization of the nonlinearity. In contrast to least squares method where parameters appear linear, nonlinear models require recursive estimations to obtain the best values.
The optimization process requires the use of optimization techniques such as the Levenberg- Marquardt algorithm, which combines both gradient descent and Gauss-Newton optimization process to refine parameter estimates. The method optimizes the cost function with reference to the parameters until the value of the gradient approaches a minimum value. In order to control the number of iterations in the above algorithm, certain conditions arise: the step size is very small, parameter update does not help in improving the value of the function, or the norm of the gradient is lesser than a pre-specified value. When the parameters converge the final circle center and radius is achieved, the residuals are then used in the estimation of uncertainty.
Uncertainty Evaluation in Least Squares Methods
Uncertainty is a measure of precision of the estimated parameters and depends on measurement error, the distribution of data and residuals (Ding, et al. 2021). Measurement noise means that there is error in the readings taken by a microscope, while data distribution is the quality of the fit and the data points, a uniform data point is the best. Model residuals which are actually defined as the difference between the actual and the fitted values give information about the level of unreality of the parameters estimated.
Calculation of Uncertainties for the Estimated Circle Parameters
Under the condition that the standard uncertainty is equal to 10 µm (0.01 mm), the uncertainty of the centre coordinates will be determined by the following formula. Since there are total N = 13 data points, the uncertainty in x_c and y_c is:
Estimated Circle Center: (4.028, 5.029) mm
Estimated Radius: 80.001 mm
Uncertainty in Xc: ±0.00277 mm
Uncertainty in Yc: ±0.00277 mm
Uncertainty in Radius: ±0.01825 mm
Methods to Reduce Measurement Uncertainty
As for reducing the amount of measurement uncertainty, it is possible to collect more data points, use a more precise system that utilizes a microscope, improve the distribution of data, minimize the effects of environment, and calibration. Getting more points reduces the amount of error that can be made while estimating the true value of the population mean statistic. The variances of (x_c), (y_c) vary in 1/√N proportion hence, the more the number of measurements the higher the accuracy. What it means is that using a microscope that provides even higher detail of the sample, for instance by using one micrometer instead of ten, could result in lesser uncertainty in the respective coordinates and thus a more accurate least squares fit (Huang, et al. 2023).
Another way of increasing the amount of accuracy of the path traced by the cursor on the circular diagram is by placing the data points uniformly. This is because it will result in large variance that may lead to outliers and where the data majority are clustered, one can obtain a biased estimate of the center. Spacing of the points is even and thus, can give a better fit. These factors include temperature, shock and vibration, among others, also have an influence on the measurement system accuracy (Wang, et al. 2021). Other factors that may create errors include material expansion resulting from heat while mechanical vibrations may lead to small shifts in the readings. This plays a major point in making sure that the microscope they use is free from vibration and operates under an optimal temperature so as to minimize these impacts.
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Reference List
Journals
Baker Hughes Company 2025. Mentor UT Phased Array Flaw Detector, Waygate Technologies, [Online] Accessed from: https://www.bakerhughes.com/waygate-technologies/industrial-ultrasonic-testing/portable-ultrasound-equipment/ultrasonic-flaw-detectors/mentor-ut Accessed on: [3.3.2025]
Balakrishnan, GK, Yaw, CT, Koh, SP, Abedin, T, Raj, AA, Tiong, SK and Chen, CP 2022, “A Review of Infrared Thermography for Condition-Based Monitoring in Electrical Energy: Applications and Recommendations,” Energies, 15(16):6000, [Online] Accessed from: https://www.mdpi.com/1996-1073/15/16/6000?type=check_update&version=1 Accessed on: [3.3.2025]
Daul, L, Jin, T, Busch, I and Koenders, L 2021, “Influence of geometric properties of capacitive sensors on slope error and nonlinearity of displacement measurements,” Sensors, 21(13):4270, [Online] Accessed from: https://pmc.ncbi.nlm.nih.gov/articles/PMC8271541/ Accessed on: [3.3.2025]
Ding, F., Ma, H., Pan, J. and Yang, E., 2021. Hierarchical gradient-and least squares-based iterative algorithms for input nonlinear output-error systems using the key term separation. Journal of the Franklin Institute, 358(9), pp.5113-5135.
DuBose, B., 2020. U.S. Army develops sensing network to spot aircraft damage, [Online] Accessed from: https://www.materialsperformance.com/articles/coating-linings/2018/02/us-army-develops-sensing-network-to-spot-aircraft-damage Accessed on: [3.3.2025]
Dwyer Instruments, LLC 2025, Vibration sensors and their importance in vibration monitoring of rotating and/or reciprocating machines, [Online] Accessed from: https://www.dwyeromega.com/en-us/resources/vibration-sensors-in-vibration-monitoring Accessed on: [3.3.2025]
FLIR Systems AB 2025. An Informative Guide for the Use of Thermal Imaging Cameras in Industrial Applications Thermal Imaging Guidebook for Industrial Applications Content. [Online] Accessed from: https://www.flirmedia.com/MMC/THG/Brochures/T820264/T820264_EN.pdf Accessed on: [3.3.2025]
Hassan, IU, Panduru, K and Walsh, J 2024, “An In-Depth Study of Vibration Sensors for Condition Monitoring,” Sensors, 24(3):740, [Online] Accessed from: https://www.mdpi.com/1424-8220/24/3/740 Accessed on: [3.3.2025]
Huang, X., Cao, H. and Jia, B., 2023. Optimization of Levenberg Marquardt algorithm applied to nonlinear systems. Processes, 11(6), p.1794.
Modsonic Instruments Mfg. Co. (P) Ltd 2023, A complete guide to ultrasonic flaw detection technique, [Online] Accessed from: https://modsonic.com/ultrasonic-flaw-detection-guide/ Accessed on: [3.3.2025]
National Instruments Corp 2025. Measuring Temperature with Thermocouples, RTDs, and Thermistors 2023, [Online] Accessed from: https://www.ni.com/en/shop/data-acquisition/sensor-fundamentals/measuring-temperature-with-thermocouples-rtds-and-thermistors.html?srsltid=AfmBOopWFbzdRVT0LHTAlS33QG8c3INz1Qb_RieWo7b9Ug-879ycOmbc. Accessed on: [3.3.2025]
NDE-Ed.org 2025. Acoustic Emissions Applications. [Online] Accessed from: https://www.nde-ed.org/NDETechniques/AcousticEmission/AE_BarkhausenTechniques.xhtml Accessed on: [3.3.2025]
SKF 2025. SKF Microlog Analyzer DBX, SKF, [Online] Accessed from: https://www.skf.com/in/products/condition-monitoring-systems/portable-systems/microlog-analyzer-dbx. Accessed on: [3.3.2025]
Umar, A.O., Sulaiman, I.M., Mamat, M., Waziri, M.Y. and Zamri, N., 2021. On damping parameters of Levenberg-Marquardt algorithm for nonlinear least square problems. In Journal of Physics: Conference Series (Vol. 1734, No. 1, p. 012018). IOP Publishing.
Wang, X., Liu, X., Chen, L. and Hu, H., 2021. Deep-learning damped least squares method for inverse kinematics of redundant robots. Measurement, 171, p.108821.
