{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,14]],"date-time":"2026-07-14T16:32:25Z","timestamp":1784046745143,"version":"3.55.0"},"reference-count":30,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T00:00:00Z","timestamp":1606089600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["106112017CDJPT280002"],"award-info":[{"award-number":["106112017CDJPT280002"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51675060"],"award-info":[{"award-number":["51675060"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Micromachines"],"abstract":"<jats:p>In this paper, we establish a dynamic model of a six-degrees-of-freedom (6-DOF) compliant positioning platform based on bridge-type amplifiers. Based on the elastic beam theory and energy relationship, we derived the bridge-type amplifier\u2019s dynamic model using the Lagrange equation. Then, we established a dynamic model of the compliant platform based on the equivalent mass and equivalent stiffness of the bridge-type amplifier, and the analysis formula of the natural frequency was derived. Finally, the analytical models of natural frequencies of the bridge-type amplifier and the compliant platforms were verified using the finite element analysis (FEA) method. Through modal experiments, the damping ratio and natural frequency were identified. Step response experiments in the X\/Y direction and Z direction were performed. The phenomenon that the experimental results appeared to match the theoretical calculations indicates that the dynamic model was accurate.<\/jats:p>","DOI":"10.3390\/mi11111024","type":"journal-article","created":{"date-parts":[[2020,11,23]],"date-time":"2020-11-23T11:50:34Z","timestamp":1606132234000},"page":"1024","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Dynamic Analysis and Experiment of 6-DOF Compliant Platform Based on Bridge-Type Amplifier"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5363-2365","authenticated-orcid":false,"given":"Chao","family":"Lin","sequence":"first","affiliation":[{"name":"State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shan","family":"Zheng","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mingdong","family":"Jiang","sequence":"additional","affiliation":[{"name":"State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400030, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,11,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2783","DOI":"10.1007\/s00707-014-1215-0","article-title":"A feasibility study of a post-buckled beam for actuating helicopter trailing edge flap","volume":"225","author":"Mallick","year":"2014","journal-title":"Acta Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"025014","DOI":"10.1088\/0960-1317\/26\/2\/025014","article-title":"A MEMSXY-stage integrating compliant mechanism for nanopositioning at sub-nanometer resolution","volume":"26","author":"Xi","year":"2016","journal-title":"J. Micromech. Microeng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.mechmachtheory.2011.11.012","article-title":"A millimeter-range flexure-based nano-positioning stage using a self-guided displacement amplification mechanism","volume":"50","author":"Kim","year":"2012","journal-title":"Mech. Mach. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"730","DOI":"10.1016\/j.sna.2005.12.028","article-title":"Analysis of the displacement amplification ratio of bridge-type flexure hinge","volume":"132","author":"Ma","year":"2006","journal-title":"Sens. Actuators A Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1826","DOI":"10.1016\/j.mechmachtheory.2009.03.007","article-title":"Optimized flexural hinge shapes for microsystems and high-precision applications","volume":"44","author":"Zelenika","year":"2009","journal-title":"Mech. Mach. Theory"},{"key":"ref_6","unstructured":"Dechev, N., Ren, L., Liu, W., Cleghorn, W.L., and Mills, J.K. (2006, January 15\u201319). Development of a 6-degree of freedom robotic micromanipulator for use in 3DMEMS micro assembly. Proceedings of the 2006 IEEE International Conference on Robotics and Automation, Orlando, FL, USA."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"035003","DOI":"10.1063\/1.3693345","article-title":"Development of flexure based 6-degrees of freedom parallel nano-positioning system with large displacement","volume":"83","author":"Kang","year":"2012","journal-title":"Rev. Sci. Instrum."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1320","DOI":"10.1109\/TMECH.2014.2342752","article-title":"A New Flexure-Based Y\u03b8 Nanomanipulator With Nanometer-Scale Resolution and Millimeter-Scale Workspace","volume":"20","author":"Tang","year":"2014","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1016\/j.mechmachtheory.2014.12.013","article-title":"Analysis of the displacement amplification ratio of bridge-type mechanism","volume":"87","author":"Qi","year":"2015","journal-title":"Mech. Mach. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"580","DOI":"10.1016\/j.ymssp.2014.05.047","article-title":"Rhombic micro-displacement amplifier for piezoelectric actuator and its linear and hybrid model","volume":"50","author":"Chen","year":"2015","journal-title":"Mech. Syst. Signal. Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1016\/j.ymssp.2008.06.007","article-title":"Development and dynamic modelling of a flexure-based Scott\u2013Russell mechanism for nano-manipulation","volume":"23","author":"Tian","year":"2009","journal-title":"Mech. Syst. Signal. Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1007\/s00542-015-2485-8","article-title":"A large-range compliant micropositioning stage with remote-center-of-motion characteristic for parallel alignment","volume":"22","author":"Qu","year":"2016","journal-title":"Microsyst. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"095109","DOI":"10.1063\/1.4930884","article-title":"Stiffness modeling of compliant parallel mechanisms and applications in the performance analysis of a decoupled parallel compliant stage","volume":"86","author":"Jiang","year":"2015","journal-title":"Rev. Sci. Instrum."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1109\/TMECH.2011.2105499","article-title":"Design and Control of a Three-Axis Serial-Kinematic High-Bandwidth Nanopositioner","volume":"17","author":"Kenton","year":"2011","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1016\/j.ymssp.2017.07.010","article-title":"A novel hybrid actuation mechanism based XY nanopositioning stage with totally decoupled kinematics","volume":"99","author":"Zhu","year":"2018","journal-title":"Mech. Syst. Signal. Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"234","DOI":"10.1109\/TNANO.2013.2242088","article-title":"Development of a Nanoprecision 3-DOF Vertical Positioning System With a Flexure Hinge","volume":"12","author":"Kim","year":"2013","journal-title":"IEEE Trans. Nanotechnol."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"536","DOI":"10.1016\/j.ymssp.2017.11.002","article-title":"Modeling and controller design of a 6-DOF precision positioning system","volume":"104","author":"Cai","year":"2018","journal-title":"Mech. Syst. Signal. Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1016\/j.sna.2017.12.030","article-title":"Kinematic characteristic analysis of a micro-\/nano positioning stage based on bridge-type amplifier","volume":"271","author":"Lin","year":"2018","journal-title":"Sens. Actuators A Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2019\/8501583","article-title":"Dynamic Analysis of a 5-DOF Flexure-Based Nanopositioning Stage","volume":"2019","author":"Shen","year":"2019","journal-title":"Math. Probl. Eng."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5223","DOI":"10.1007\/s12206-018-1020-0","article-title":"Dynamic analysis of a flexure-based compliant stage","volume":"32","author":"Du","year":"2018","journal-title":"J. Mech. Sci. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"352","DOI":"10.1016\/j.precisioneng.2015.06.002","article-title":"Dynamic analysis and experiment of a novel ultra-precision compliant linear-motion mechanism","volume":"42","author":"Hongzhe","year":"2015","journal-title":"Precis. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/j.mechmachtheory.2010.09.007","article-title":"Analytical modeling, optimization and testing of a compound bridge-type compliant displacement amplifier","volume":"46","author":"Xu","year":"2011","journal-title":"Mech. Mach. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/s00542-016-3048-3","article-title":"Design and analysis of a compliant micro-positioning platform with embedded strain gauges and viscoelastic damper","volume":"23","author":"Dao","year":"2016","journal-title":"Microsyst. Technol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"051011","DOI":"10.1115\/1.4040700","article-title":"A pseudo-static model for dynamic analysis on frequency domain of distributed compliant mechanisms","volume":"10","author":"Ling","year":"2018","journal-title":"J. Mech. Robot."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.jsv.2018.09.013","article-title":"Dynamic stiffness matrix for free vibration analysis of flexure hinges based on non-uniform Timoshenko beam","volume":"437","author":"Ling","year":"2018","journal-title":"J. Sound Vib."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/j.precisioneng.2019.04.004","article-title":"Kinetostatic and dynamic analyses of planar compliant mechanisms with a two-port dynamic stiffness model","volume":"57","author":"Ling","year":"2019","journal-title":"Precis. Eng."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1559","DOI":"10.1007\/s11012-015-0116-1","article-title":"Design and analysis of a compliant parallel pan-tilt platform","volume":"51","author":"Yu","year":"2016","journal-title":"Meccanica"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"502","DOI":"10.3390\/mi10080502","article-title":"Development of a Novel Three Degrees-of-Freedom Rotary Vibration-Assisted Micropolishing System Based on Piezoelectric Actuation","volume":"10","author":"Gu","year":"2019","journal-title":"Micromachines"},{"key":"ref_29","first-page":"1375","article-title":"Analysis and experiment of dynamic characteristics of micro\/nano positioning platform","volume":"46","author":"Lin","year":"2012","journal-title":"J. Zhejiang Univ. Eng. Science"},{"key":"ref_30","first-page":"254","article-title":"Analysis of guiding displacement of parallel four-bar mechanism with right angle flexible hinge","volume":"39","author":"Ye","year":"2010","journal-title":"J. Chi. Univ. Min. Technol."}],"container-title":["Micromachines"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-666X\/11\/11\/1024\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:36:15Z","timestamp":1760178975000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-666X\/11\/11\/1024"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,23]]},"references-count":30,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2020,11]]}},"alternative-id":["mi11111024"],"URL":"https:\/\/doi.org\/10.3390\/mi11111024","relation":{},"ISSN":["2072-666X"],"issn-type":[{"value":"2072-666X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,11,23]]}}}