{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,13]],"date-time":"2026-05-13T18:58:15Z","timestamp":1778698695180,"version":"3.51.4"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2020,2,23]],"date-time":"2020-02-23T00:00:00Z","timestamp":1582416000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>When analyzing the dynamic behavior of multi-body elastic systems, a commonly used method is the finite element method conjunctively with Lagrange\u2019s equations. The central problem when approaching such a system is determining the equations of motion for a single finite element. The paper presents an alternative method of calculation theses using the Gibbs\u2013Appell (GA) formulation, which requires a smaller number of calculations and, as a result, is easier to apply in practice. For this purpose, the energy of the accelerations for one single finite element is calculated, which will be used then in the GA equations. This method can have advantages in applying to the study of multi-body systems with elastic elements and in the case of robots and manipulators that have in their composition some elastic elements. The number of differentiation required when using the Gibbs\u2013Appell method is smaller than if the Lagrange method is used which leads to a smaller number of operations to obtain the equations of motion.<\/jats:p>","DOI":"10.3390\/sym12020321","type":"journal-article","created":{"date-parts":[[2020,2,26]],"date-time":"2020-02-26T04:18:29Z","timestamp":1582690709000},"page":"321","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":72,"title":["Energy of Accelerations Used to Obtain the Motion Equations of a Three- Dimensional Finite Element"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8679-2579","authenticated-orcid":false,"given":"Sorin","family":"Vlase","sequence":"first","affiliation":[{"name":"Department of Mechanical Engineering, Faculty of Mechanical Engineering, Transilvania University of Bra\u0219ov, B-dul Eroilor 29, 500036 Bra\u0219ov, Romania"},{"name":"Technical Sciences Academy of Romania; B-dul Dacia 26, 030167 Bucharest, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0018-5183","authenticated-orcid":false,"given":"Iuliu","family":"Negrean","sequence":"additional","affiliation":[{"name":"Technical Sciences Academy of Romania; B-dul Dacia 26, 030167 Bucharest, Romania"},{"name":"Department of Mechanical Systems Engineering, Technical University of Cluj-Napoca, Str. Memorandumului 28, 400114 Cluj-Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1552-3763","authenticated-orcid":false,"given":"Marin","family":"Marin","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Transilvania University of Bra\u0219ov, B-dul Eroilor 29, 500036 Bra\u0219ov, Romania"}]},{"given":"Maria Lumini\u021ba","family":"Scutaru","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Faculty of Mechanical Engineering, Transilvania University of Bra\u0219ov, B-dul Eroilor 29, 500036 Bra\u0219ov, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2020,2,23]]},"reference":[{"key":"ref_1","first-page":"1193","article-title":"A General Method for Kineto-Elastodynamic Analysis and Synthesis of Mechanism. Journal of Engineering for Industry","volume":"94","author":"Erdman","year":"1972","journal-title":"ASME Trans."},{"key":"ref_2","unstructured":"Bagci, C. (1983, January 15\u201320). Elastodynamic Response of Mechanical Systems using Matrix Exponential Mode Uncoupling and Incremental Forcing Techniques with Finite Element Method. Proceedings of the Sixth Word Congress on Theory of Machines and Mechanisms, New Delhi, India."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/0094-114X(76)90026-4","article-title":"Finite Element Vibrational Analysis of Planar Mechanisms","volume":"11","author":"Bahgat","year":"1976","journal-title":"Mech. Mach. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1016\/0094-114X(81)90014-8","article-title":"Finite Element Analysis of High-Speed Flexible Mechanisms","volume":"16","author":"Cleghorn","year":"1981","journal-title":"Mech. Mach. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1016\/0045-7949(86)90251-8","article-title":"Nonlinear finite element modelling of the dynamics of unrestrained flexible structures","volume":"23","author":"Christensen","year":"1986","journal-title":"Comput. Struct"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"603","DOI":"10.1016\/0094-114X(78)90028-9","article-title":"Finite element approach to mathematical modeling of high-speed elastic linkages","volume":"13","author":"Midha","year":"1978","journal-title":"Mech. Mach. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/0094-114X(80)90004-X","article-title":"Steady-state response of mechanism with elastic links by finite element methods","volume":"15","author":"Nath","year":"1980","journal-title":"Mech. Mach. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Vlase, S., Marin, M., \u00d6chsner, A., and Scutaru, M.L. (2019). Motion equation for a flexible one-dimensional element used in the dynamical analysis of a multibody system. Contin. Mech. Thermodyn.","DOI":"10.1007\/s00161-018-0722-y"},{"key":"ref_9","first-page":"676","article-title":"Dynamical Response of a Multibody System with Flexible Elements with a General Three-Dimensional Motion","volume":"57","author":"Vlase","year":"2012","journal-title":"Rom. J. Phys."},{"key":"ref_10","first-page":"476","article-title":"Finite Element Analysis of a Two-Dimensional Linear Elastic Systems with a Plane \"rigid Motion","volume":"59","author":"Vlase","year":"2014","journal-title":"Rom. J. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1016\/j.compstruc.2007.01.028","article-title":"Dynamic responses of flexible-link mechanisms with passive\/active damping treatment","volume":"86","author":"Galucio","year":"2008","journal-title":"Comput. Struct."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1016\/j.jsv.2008.05.014","article-title":"Dynamic analysis of flexible linkage mechanisms under uniform temperature change","volume":"319","author":"Hou","year":"2009","journal-title":"J. Sound Vib."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6860","DOI":"10.1016\/j.cma.2005.08.009","article-title":"Composite materials in flexible multibody systems","volume":"195","author":"Neto","year":"2006","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"849","DOI":"10.1016\/j.mechmachtheory.2004.12.007","article-title":"Dynamic finite-element analysis of a planar high-speed, high-precision parallel manipulator with flexible links","volume":"40","author":"Piras","year":"2005","journal-title":"Mech. Mach. Theory"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1006\/jsvi.2001.3614","article-title":"The Modeling and Vibration Control of Beams with Active Constrained Layer Damping","volume":"245","author":"Shi","year":"2001","journal-title":"J. Sound Vib."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1265","DOI":"10.1016\/S0045-7949(01)00019-0","article-title":"Dynamic responses of flexible linkage mechanisms with viscoelastic constrained layer damping treatment","volume":"79","author":"Zhang","year":"2001","journal-title":"Comput. Struct."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.1016\/j.jsv.2006.08.002","article-title":"Simultaneous optimal structure and control design of flexible linkage mechanism for noise attenuation","volume":"299","author":"Zhang","year":"2007","journal-title":"J. Sound Vib."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"49","DOI":"10.2307\/2369196","article-title":"On the fundamental formulae of dynamics","volume":"2","author":"Gibbs","year":"1879","journal-title":"Am. J. Math."},{"key":"ref_19","first-page":"459","article-title":"Sur une forme g\u00e9n\u00e9rale des equations de la dynamique","volume":"129","author":"Appell","year":"1899","journal-title":"C.R. Acad. Sci. Paris"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Mirtaheri, S.M., and Zohoor, H. (2018). The Explicit Gibbs-Appell Equations of Motion for Rigid-Body Constrained Mechanical System. Book Series: RSI International Conference on Robotics and Mechatronics ICRoM, IEEE.","DOI":"10.1109\/ICRoM.2018.8657637"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Mehrjooee, O., Dehkordi, S.F., and Korayem, M.H. (2019). Dynamic modeling and extended bifurcation analysis of flexible-link manipulator. Mech. Based Des. Struct. Mach., 1\u201324.","DOI":"10.1080\/15397734.2019.1665542"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1016\/j.apm.2018.08.035","article-title":"Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs-Appell formulation","volume":"65","author":"Korayem","year":"2019","journal-title":"Appl. Math. Model."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/s11044-015-9496-1","article-title":"A systematic method for the hybrid dynamic modeling of open kinematic chains confined in a closed environment","volume":"38","author":"Shafei","year":"2017","journal-title":"Multibody Syst. Dyn."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Amini, S., Dehkordi, S.F., and Fahraji, S.H. (2017, January 25\u201327). Motion equation derivation and tip-over evaluations for K mobile manipulators with the consideration of motors mass by the use of Gibbs-Appell formulation. Proceedings of the 5th RSI International Conference on Robotics and Mechatronics (IcRoM), Tehran, Iran.","DOI":"10.1109\/ICRoM.2017.8466214"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2041","DOI":"10.1007\/s11071-017-3569-z","article-title":"Derivation of dynamic equation of viscoelastic manipulator with revolute-prismatic joint using recursive Gibbs-Appell formulation","volume":"89","author":"Korayem","year":"2017","journal-title":"Nonlinear Dyn."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1002\/oca.2302","article-title":"Theoretical and experimental study of dynamic load-carrying capacity for flexible robotic arms in point-to-point motion","volume":"38","author":"Shafei","year":"2017","journal-title":"Optim. Control Appl. Methods"},{"key":"ref_27","unstructured":"Negrean, I., Kacso, K., Schonstein, C., and Duca, A. (2012). Mechanics, Theory and Applications, UTPRESS."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Negrean, I., and Cri\u0219an, A.-D. (2019). Synthesis on the Acceleration Energies in the Advanced Mechanics of the Multibody Systems. Symmetry, 11.","DOI":"10.3390\/sym11091077"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Negrean, I., Cri\u0219an, A.-D., and Vlase, S. (2020). A New Approach in Analytical Dynamics of Mechanical Systems. Symmetry, 12.","DOI":"10.3390\/sym12010095"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0093-6413(87)90005-X","article-title":"Elimination of Lagrangian Multipliers","volume":"14","author":"Vlase","year":"1987","journal-title":"Mech. Res. Commun."},{"key":"ref_31","unstructured":"Massonet, C., Deprez, G., Maquoi, R., Muller, R., and Fonder, G. (1972). Calcul des Structures sur Ordinateur, EYROLLES."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3821083","DOI":"10.1155\/2016\/3821083","article-title":"Dynamical Analysis of the Mechanical System with Two Degrees of Freedom Applied to the Transmission of the Wind Turbine","volume":"2016","author":"Scutaru","year":"2016","journal-title":"Math. Probl. Eng."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/321\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:01:07Z","timestamp":1760173267000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/2\/321"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2,23]]},"references-count":32,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2020,2]]}},"alternative-id":["sym12020321"],"URL":"https:\/\/doi.org\/10.3390\/sym12020321","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,2,23]]}}}