{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:52:33Z","timestamp":1760151153929,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,2,28]],"date-time":"2022-02-28T00:00:00Z","timestamp":1646006400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>This paper proposes the procedure for minimising the dynamic error in the time and frequency domains, based on the example of a second-order sensor. Our procedure includes three main steps: modelling of the sensors using the Monte Carlo (MC) method; determination of the maximum value of the dynamic error using the integral-square criterion (ISC); and optimisation of the parameters of the sensor model by minimising the ISC. The uncertainties associated with the modelling procedure and the MC method are also considered. The mathematical formulae necessary for implementation in a given programming language (MathCad, MATLAB, C, etc.) are presented in detail. The proposed procedure was implemented in the frequency domain, using MathCad 15, and applied to the example of the Althen 731-207 accelerometer. Validation of the proposed procedure was carried out using a digital signal processor of type TMS320C6713. The proposed procedure can increase the accuracy of the signal processing obtained at the output of sensors applied to a wide range of measurements.<\/jats:p>","DOI":"10.3390\/s22051901","type":"journal-article","created":{"date-parts":[[2022,2,28]],"date-time":"2022-02-28T20:11:57Z","timestamp":1646079117000},"page":"1901","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Procedure Proposal for Minimising the Dynamic Error of Second-Order Sensors"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8922-6529","authenticated-orcid":false,"given":"Krzysztof","family":"Tomczyk","sequence":"first","affiliation":[{"name":"Faculty of Electrical and Computer Engineering, Cracow University of Technology, Warszawska 24, 31-155 Krakow, Poland"}]},{"given":"Ma\u0142gorzata","family":"Kowalczyk","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, Cracow University of Technology, Jana Paw\u0142a II 37 Avenue, 31-864 Krakow, Poland"}]},{"given":"Ksenia","family":"Ostrowska","sequence":"additional","affiliation":[{"name":"Faculty of Mechanical Engineering, Cracow University of Technology, Jana Paw\u0142a II 37 Avenue, 31-864 Krakow, Poland"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"62","DOI":"10.1016\/j.ijar.2019.03.001","article-title":"Learning Bayesian network parameters via minimax algorithm","volume":"108","author":"Gao","year":"2019","journal-title":"Int. J. Approx. Reason."},{"key":"ref_2","first-page":"1619","article-title":"Maximin Optimization Problem Subject to Min-Product Fuzzy Relation Inequalities with Application in Supply and Demand Scheme","volume":"29","author":"Pradhan","year":"2020","journal-title":"Int. J. Adv. Sci. Technol."},{"key":"ref_3","first-page":"1014","article-title":"Various Genetic Approaches for Solving Single and Multi-Objective Optimization Problems: A Review","volume":"3","author":"Punia","year":"2013","journal-title":"Int. J. Adv. Res. Comput. Sci. Softw. Eng."},{"key":"ref_4","unstructured":"Gorecki, H. (2018). Optimization and Control of Dynamic Systems, Springer."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1002\/nme.6038","article-title":"Material nonlinear topology optimization considering the von Mises criterion through an asymptotic approach: Max strain energy and max load factor formulations","volume":"118","author":"Zhao","year":"2019","journal-title":"Int. J. Num. Meth. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1088\/0957-0233\/17\/10\/028","article-title":"A Novel Method of Estimating Dynamic Measurement Errors","volume":"17","author":"Hessling","year":"2006","journal-title":"Meas. Sci. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1109\/19.930456","article-title":"Mapping Error of Linear Dynamic Systems Caused by Reduced-Order Model","volume":"50","author":"Layer","year":"2001","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1002\/nme.4932","article-title":"Fuzzy interval perturbation method for uncertain heat conduction problem with interval and fuzzy parameters","volume":"104","author":"Wang","year":"2009","journal-title":"Int. J. Num. Meth. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1109\/18.135644","article-title":"Maximizing the output energy of a linear channel with a time and amplitude limited input","volume":"38","author":"Honig","year":"1992","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"132592","DOI":"10.1109\/ACCESS.2020.3009982","article-title":"Identification of Dynamical Systems Usinga Broad Neural Network and Particle Swarm Optimization","volume":"8","author":"Han","year":"2020","journal-title":"IEEE Access"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Pintelon, R., and Schoukens, J. (2012). System Identification: A Frequency Domain Approach, John Wiley & Sons. [2nd ed.].","DOI":"10.1002\/9781118287422"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"758","DOI":"10.1016\/j.apm.2020.06.042","article-title":"Mathematical modeling and bifurcation analysis of pro- and anti-tumor macrophages","volume":"88","author":"Shua","year":"2020","journal-title":"Appl. Math. Model."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"64","DOI":"10.21014\/acta_imeko.v8i1.568","article-title":"Dynamic Measuring Methods: A Review","volume":"8","author":"Shestakov","year":"2019","journal-title":"Acta IMEKO"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2431","DOI":"10.1002\/nag.2988","article-title":"An intelligent response surface method for analyzing slope reliability based on Gaussian process regression","volume":"43","author":"Zhu","year":"2019","journal-title":"Int. J. Numer. Anal. Methods Geomech. Wiley"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S0219887815600270","article-title":"Time functions revisited","volume":"12","author":"Fathi","year":"2015","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"550","DOI":"10.1109\/9.280756","article-title":"The principle of matching: Practical Conditions for Systems with Inputs Restricted in Magnitude and Rate of Change","volume":"39","author":"Rutland","year":"1994","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Layer, E. (2002). Modelling of Simplified Dynamical Systems, Springer.","DOI":"10.1007\/978-3-642-56098-9"},{"key":"ref_18","first-page":"93","article-title":"Application of genetic algorithm to measurement system calibration intended for dynamic measurement","volume":"13","author":"Tomczyk","year":"2006","journal-title":"Metrol. Meas. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1080\/09720502.2019.1645399","article-title":"Fixed point algorithms using iteration technique","volume":"22","author":"Shukla","year":"2019","journal-title":"J. Interdiscip. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2528","DOI":"10.1002\/nme.6630","article-title":"Quantification and reduction of uncertainties in a wind turbine numerical model based on a global sensitivity analysis and a recursive Bayesian inference approach","volume":"122","author":"Hirvoas","year":"2021","journal-title":"Int. J. Num. Meth. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"414","DOI":"10.1016\/j.precisioneng.2016.03.021","article-title":"Validation model for coordinate measuring methods based on the concept of statistical consistency control","volume":"45","author":"Gromczak","year":"2016","journal-title":"Precs. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"107081","DOI":"10.1016\/j.buildenv.2020.107081","article-title":"Validation of dynamic hygrothermal simulation models for historical buildings: State of the art, research challenges and recommendations","volume":"180","author":"Leonforte","year":"2020","journal-title":"Build. Environ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"113487","DOI":"10.1016\/j.cma.2020.113487","article-title":"A general two-phase Markov chain Monte Carlo approach for constrained design optimization: Application to stochastic structural optimization","volume":"373","author":"Jensen","year":"2021","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Chan, V. (2013). Monte Carlo Simulations Applied to Uncertainty in Measurement. Theory and Applications of Monte Carlo Simulations, IntechOpen.","DOI":"10.5772\/45892"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"176","DOI":"10.1088\/0026-1394\/51\/4\/S176","article-title":"On a Monte Carlo Method for Measurement Uncertainty Evaluation and its Implementation","volume":"51","author":"Harris","year":"2014","journal-title":"Metrologia"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Tomczyk, K. (2020). Monte Carlo-based Procedure for Determining the Maximum Energy at the Output of Accelerometers. Energies, 13.","DOI":"10.3390\/en13071552"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Mahdiyar, A., Armaghani, D.J., Koopialipoor, M., Hedayat, A., Abdullah, A., and Yahya, K. (2020). Practical Risk Assessment of Ground Vibrations Resulting from Blasting, Using Gene Expression Programming and Monte Carlo Simulation Techniques. Appl. Sci., 10.","DOI":"10.3390\/app10020472"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"790","DOI":"10.1109\/JMEMS.2019.2930065","article-title":"Theoretical Modeling, Numerical Simulations and Experimental Study of Micro Thermal Convective Accelerometers","volume":"28","author":"Wang","year":"2019","journal-title":"J. Microelectromech. Syst."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Zhao, D., Gelman, L., Chu, F., and Ball, A. (2020). Novel Method for Vibration Sensor-Based Instantaneous Defect Frequency Estimation for Rolling Bearings Under Non-Stationary Conditions. Sensors, 20.","DOI":"10.3390\/s20185201"},{"key":"ref_30","first-page":"569","article-title":"The Structure Design and Analysis of Vibration Sensor","volume":"701","author":"Tian","year":"2014","journal-title":"Appl. Mech. Mater."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Xu, F., Li, X., Shi, Y., Li, L., Wang, W., He, L., and Liu, R. (2018). Recent Developments for Flexible Pressure Sensors: A Review. Micromachines, 9.","DOI":"10.3390\/mi9110580"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"829","DOI":"10.1109\/19.863933","article-title":"An Intelligent Pressure Sensor Using Neural Networks","volume":"4","author":"Patra","year":"2000","journal-title":"IEEE Trans. Instrum. Meas."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Morris, A.S., and Langari, R. (2021). Measurement and Instrumentation, Theory and Application, Elsevier Academic Press.","DOI":"10.1016\/B978-0-12-817141-7.00016-5"},{"key":"ref_34","unstructured":"Close, C.M., Frederick, D.K., and Newell, J.C. (2001). Modeling and Analysis of Dynamic Systems, Wiley."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Shan, X., Song, H., Zhang, C., Wang, G., and Fan, J. (2020). Linear System Identification and Vibration Control of End-Eector for Industrial Robots. Appl. Sci., 10.","DOI":"10.3390\/app10238537"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1214","DOI":"10.1002\/nme.5563","article-title":"Frequency response as a surrogate eigenvalue problem in topology optimization","volume":"113","author":"Andreassen","year":"2017","journal-title":"Int. J. Num. Meth. Eng."},{"key":"ref_37","unstructured":"(2008, November 01). BIPM, IEC, IFCC, ILAC, ISO, IUPAP, OIML, Guide to the Expression of Uncertainty in Measurement. Supplement 1\u2014Propagation of Distributions Using a Monte Carlo Method. Available online: https:\/\/www.iso.org\/standard\/50462.html."},{"key":"ref_38","unstructured":"(2011, November 01). BIPM, IEC, IFCC, ILAC, ISO, IUPAP, OIML, Guide to the Expression of Uncertainty in Measurement. Supplement 2\u2014Extension to any Number of Output Quantities. Available online: https:\/\/www.iso.org\/standard\/50463.html."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Sanchez-Sutil, F., Cano-Ortega, A., Hernandez, J.C., and Rus-Casas, C. (2019). Development and Calibration of an Open Source, Low-Cost Power Smart Meter Prototype for PV Household-Prosumers. Electronics, 8.","DOI":"10.3390\/electronics8080878"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1177\/0142331217753062","article-title":"Influence of Monte Carlo Generations Applied for Modelling of Measuring Instruments on Maximum Distance Error","volume":"41","author":"Tomczyk","year":"2019","journal-title":"Trans. Inst. Meas. Control"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1614","DOI":"10.1016\/j.csda.2006.05.019","article-title":"Generating Good Pseudo-Random Numbers","volume":"51","author":"Wichmann","year":"2006","journal-title":"Comput. Stat. Data Anal."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1145\/1287620.1287622","article-title":"Gaussian Random Number Generators","volume":"39","author":"Thomas","year":"2007","journal-title":"ACM Comput. Surv."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Dudzik, M., Tomczyk, K., and Jagie\u0142\u0142o, A. (2018, January 7\u20138). Analysis of the Error Generated by the Voltage Output Accelerometer Using the Optimal Structure of an Artificial Neural Network. Proceedings of the 19th International Conference on Research and Education in Mechatronics (REM\u20192018), Delft, The Netherlands.","DOI":"10.1109\/REM.2018.8421789"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Segers, J. (2014). Analysis Techniques for Racecar Data Acquisition, SAE International.","DOI":"10.4271\/R-408"},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Ponce-Cruz, P., and Ram\u00edrez-Figueroa, F.D. (2010). Intelligent Control Systems with LabVIEW, Springer.","DOI":"10.1007\/978-1-84882-684-7"},{"key":"ref_46","unstructured":"(2022, February 15). Texas Instruments TMS320C6713 Data Sheet. Available online: https:\/\/pdf1.alldatasheet.com\/datasheet-pdf\/view\/227375\/TI\/TMS320C6713.html."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/5\/1901\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:29:40Z","timestamp":1760135380000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/22\/5\/1901"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,28]]},"references-count":46,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,3]]}},"alternative-id":["s22051901"],"URL":"https:\/\/doi.org\/10.3390\/s22051901","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2022,2,28]]}}}