{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T23:00:15Z","timestamp":1778281215158,"version":"3.51.4"},"reference-count":35,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,25]],"date-time":"2019-03-25T00:00:00Z","timestamp":1553472000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U1537215"],"award-info":[{"award-number":["U1537215"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U1737207"],"award-info":[{"award-number":["U1737207"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Assembly accuracy and accuracy stability prediction are significant research directions for improving the reliability and efficiency of precision assembly. In this study, an improved method for assembly accuracy stability prediction, based on the contact-pressure distribution entropy, is presented. By using the contact-pressure distribution as the evaluation parameter instead of the strain-energy distribution, the improved method can not only predict the assembly accuracy of precision assembly more efficiently, but also predict the stability of the assembly accuracy with variations in the ambient temperature. The contact pressure has a clearer mechanical significance than strain energy density in the assembly process, which can be used to distinguish the actual contact area from the contact surface. Hence, the improved method is more efficient and accurate than the original. This study utilizes the same case used in the original method and an additional case from the actual production process to verify the improved method. The correctness and validity of the improved method are proved by these case studies.<\/jats:p>","DOI":"10.3390\/e21030322","type":"journal-article","created":{"date-parts":[[2019,3,29]],"date-time":"2019-03-29T03:50:21Z","timestamp":1553831421000},"page":"322","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Entropy-Based Method to Evaluate Contact-Pressure Distribution for Assembly-Accuracy Stability Prediction"],"prefix":"10.3390","volume":"21","author":[{"given":"Xiao","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xin","family":"Jin","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ke","family":"Shang","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhijing","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Liu, C.H. (2016). Tolerance Redistributing of the Reassembly Dimensional Chain on Measure of Uncertainty. Entropy, 18.","DOI":"10.3390\/e18100348"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1115\/1.1497362","article-title":"A new mathematical model for geometric tolerances as applied to round faces","volume":"124","author":"Davidson","year":"2002","journal-title":"J Mech. Des"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.cirp.2014.03.103","article-title":"From solid modelling to skin model shapes: Shifting paradigms in computer-aided tolerancing","volume":"63","author":"Anwer","year":"2014","journal-title":"CIRP Ann-Manuf. Technol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.cad.2014.01.001","article-title":"Skin model shapes: A new paradigm shift for geometric variations modelling in mechanical engineering","volume":"50","author":"Schleich","year":"2014","journal-title":"Comput. Aided Des."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.cad.2017.04.007","article-title":"Tolerance analysis\u2014Form defects modeling and simulation by modal decomposition and optimization","volume":"91","author":"Homri","year":"2017","journal-title":"Comput. Aided Des."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0007-8506(07)61457-7","article-title":"Mode-based decomposition of part form error by discrete-cosine-transform with implementation to assembly and stamping system with compliant parts","volume":"51","author":"Huang","year":"2002","journal-title":"CIRP Ann-Manuf. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5708","DOI":"10.1080\/00207543.2016.1158881","article-title":"Integration of geometric variation and part deformation into variation propagation of 3-D assemblies","volume":"54","author":"Guo","year":"2016","journal-title":"Int. J. Prod. Res."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1177\/0954405414565140","article-title":"An innovative method of modeling plane geometric form errors for precision assembly","volume":"230","author":"Zhang","year":"2016","journal-title":"Proc. Inst. Mech. Eng. B J. Eng. Manuf."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"528","DOI":"10.1007\/s00170-005-0222-7","article-title":"The modeling and analysis of a butting assembly in the presence of workpiece surface roughness and part dimensional error","volume":"31","author":"Lee","year":"2006","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1057","DOI":"10.1016\/j.ijmecsci.2004.07.007","article-title":"Stability analysis of a rotor bearing system due to surface waviness and number of balls","volume":"46","author":"Harsha","year":"2004","journal-title":"Int. J. Mech. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Fang, Y., Jin, X., Huang, C., and Zhang, Z. (2017). Entropy-Based Method for Evaluating Contact Strain-Energy Distribution for Assembly Accuracy Prediction. Entropy, 19.","DOI":"10.3390\/e19020049"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"279","DOI":"10.1016\/j.procir.2018.04.023","article-title":"Shape Error Modelling and Analysis by Conditional Simulations of Gaussian Random Fields for Compliant Non-Ideal Sheet Metal Parts","volume":"75","author":"Babu","year":"2018","journal-title":"Procedia CIRP"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Bolotov, M., Grachev, I., and Kudashov, E. (2018, January 30). Investigation of Parts Assembly Error, Taking into Account the Deviation of the Shape of Their Surfaces. Proceedings of the International Conference on Modern Trends in Manufacturing Technologies and Equipment (ICMTMTE 2018), Les Ulis, France.","DOI":"10.1051\/matecconf\/201822401098"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1139","DOI":"10.1007\/s00170-017-0936-3","article-title":"A novel modelling method of geometric errors for precision assembly","volume":"94","author":"Zhang","year":"2018","journal-title":"Int. J. Adv. Manuf. Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"694","DOI":"10.1016\/j.triboint.2005.11.021","article-title":"The influence of surface roughness and the contact pressure distribution on friction in rolling\/sliding contacts","volume":"40","author":"Xiao","year":"2007","journal-title":"Tribol. Int."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"809","DOI":"10.1016\/S0890-6955(00)00103-6","article-title":"Prediction of thermo-elastic behavior in a spindle\u2013bearing system considering bearing surroundings","volume":"41","author":"Kim","year":"2001","journal-title":"Int. J. Mach. Tool. Manuf."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"676","DOI":"10.1016\/j.optlaseng.2010.01.010","article-title":"3D mechanical analysis of aeronautical plain bearings: Validation of a finite element model from measurement of displacement fields by digital volume correlation and optical scanning tomography","volume":"48","author":"Germaneau","year":"2010","journal-title":"Opt. Lasers Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1835","DOI":"10.1016\/S0020-7403(99)00072-7","article-title":"Design of interference fits via finite element method","volume":"42","author":"Zhang","year":"2000","journal-title":"Int. J. Mech. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4213","DOI":"10.1016\/j.jpowsour.2009.12.113","article-title":"Effect of assembly error of bipolar plate on the contact pressure distribution and stress failure of membrane electrode assembly in proton exchange membrane fuel cell","volume":"195","author":"Peng","year":"2010","journal-title":"J. Power Sources"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4178","DOI":"10.1143\/JJAP.41.4178","article-title":"Evaluation of pressure uniformity using a pressure-sensitive film and calculation of wafer distortions caused by mold press in imprint lithography","volume":"41","author":"Deguchi","year":"2002","journal-title":"Jpn. J. Appl. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Yang, G., Hong, J., Wang, N., Zhu, L., Ding, Y., and Yang, Z. (2011, January 25\u201327). Member stiffnesses and interface contact characteristics of bolted joints. Proceedings of the 2011 IEEE International Symposium on Assembly and Manufacturing (ISAM), Tampere, Finland.","DOI":"10.1109\/ISAM.2011.5942304"},{"key":"ref_22","first-page":"330","article-title":"Interface pressure distribution in a bolt-flange assembly","volume":"101","author":"Ito","year":"1979","journal-title":"J. Mech. Des."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1080\/05698190590899967","article-title":"Bearing friction torque in bolted joints","volume":"48","author":"Nassar","year":"2005","journal-title":"Tribol. T."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.1016\/S0890-6955(00)00010-9","article-title":"Error compensation in machine tools\u2014A review: Part II: Thermal errors","volume":"40","author":"Ramesh","year":"2000","journal-title":"Int. J. Mach. Tool. Manu."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1115\/1.3183848","article-title":"Importance of contact pressure distribution on heat transfer in structural joints of machine tools","volume":"102","author":"Attia","year":"1980","journal-title":"J. Eng. Industry"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/0043-1648(96)06978-5","article-title":"Deterioration of rolling contact fatigue life of pearlitic rail steel due to dry-wet rolling-sliding line contact","volume":"197","author":"Tyfour","year":"1996","journal-title":"Wear"},{"key":"ref_27","unstructured":"Johnson, K.L. Contact Mechanics, Cambridge University Press."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Laskowski, R., Smyk, A., Rusowicz, A., and Grzebielec, A. (2016). Determining the optimum inner diameter of condenser tubes based on thermodynamic objective functions and an economic analysis. Entropy, 18.","DOI":"10.3390\/e18120444"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Mihelich, M., Faranda, D., Paillard, D., and Dubrulle, B. (2017). Is Turbulence a State of Maximum Energy Dissipation?. Entropy, 19.","DOI":"10.3390\/e19040154"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Mu\u00f1oz-Cobo, J.L., Mendiz\u00e1bal, R., Miquel, A., Berna, C., and Escriv\u00e1, A. (2017). Use of the Principles of Maximum Entropy and Maximum Relative Entropy for the Determination of Uncertain Parameter Distributions in Engineering Applications. Entropy, 19.","DOI":"10.3390\/e19090486"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Baggenstoss, P. (2018). Beyond Moments: Extending the Maximum Entropy Principle to Feature Distribution Constraints. Entropy, 20.","DOI":"10.3390\/e20090650"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Krantz, R., Gemmetto, V., and Garlaschelli, D. (2018). Maximum-Entropy Tools for Economic Fitness and Complexity. Entropy, 20.","DOI":"10.3390\/e20100743"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2530","DOI":"10.3390\/e16052530","article-title":"Measuring Instantaneous and Spectral Information Entropies by Shannon Entropy of Choi-Williams Distribution in the Context of Electroencephalography","volume":"16","author":"Melia","year":"2014","journal-title":"Entropy"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Xiao, J., and He, Z.Y. (2016). A Concept Lattice for Semantic Integration of Geo-Ontologies Based on Weight of Inclusion Degree Importance and Information Entropy. Entropy, 18.","DOI":"10.3390\/e18110399"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/3\/322\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:40:33Z","timestamp":1760186433000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/3\/322"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,25]]},"references-count":35,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["e21030322"],"URL":"https:\/\/doi.org\/10.3390\/e21030322","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,25]]}}}