{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:38:43Z","timestamp":1760402323455,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T00:00:00Z","timestamp":1578009600000},"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>This paper presents a new approach to the advanced dynamics of mechanical systems. It is known that in the movements corresponding to some mechanical systems (e.g., robots), accelerations of higher order are developed. Higher-order accelerations are an integral part of higher-order acceleration energies. Unlike other research papers devoted to these advanced notions, the main purpose of the paper is to present, in a matrix form, the defining expressions for the acceleration energies of a higher order. Following the differential principle in generalized form (a generalization of the Lagrange\u2013D\u2019Alembert principle), the equations of the dynamics of fast-moving systems include, instead of kinetic energies, the acceleration energies of higher-order. To establish the equations which characterize both the energies of accelerations and the advanced dynamics, the following input parameters are considered: matrix exponentials and higher-order differential matrices. An application of a 5 d.o.f robot structure is presented in the final part of the paper. This is used to illustrate the validity of the presented mathematical formulations.<\/jats:p>","DOI":"10.3390\/sym12010095","type":"journal-article","created":{"date-parts":[[2020,1,3]],"date-time":"2020-01-03T11:55:07Z","timestamp":1578052507000},"page":"95","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":29,"title":["A New Approach in Analytical Dynamics of Mechanical Systems"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0018-5183","authenticated-orcid":false,"given":"Iuliu","family":"Negrean","sequence":"first","affiliation":[{"name":"Department of Mechanical Systems Engineering, Faculty of Machine Building, Technical University of Cluj\u2013Napoca, 400641 Cluj\u2013Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4017-9063","authenticated-orcid":false,"given":"Adina-Veronica","family":"Cri\u0219an","sequence":"additional","affiliation":[{"name":"Department of Mechanical Systems Engineering, Faculty of Machine Building, Technical University of Cluj\u2013Napoca, 400641 Cluj\u2013Napoca, Romania"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8679-2579","authenticated-orcid":false,"given":"Sorin","family":"Vlase","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Faculty of Mechanical Engineering, Transilvania University of Bra\u0219ov, 500036 Bra\u0219ov, Romania"}]}],"member":"1968","published-online":{"date-parts":[[2020,1,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"065008","DOI":"10.1088\/0143-0807\/37\/6\/065008","article-title":"Beyond velocity and acceleration: Jerk, snap and higher derivatives","volume":"37","author":"Eager","year":"2016","journal-title":"Eur. J. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2603","DOI":"10.1088\/0264-9381\/21\/11\/006","article-title":"Jerk, Snap and the Cosmological Equation of State","volume":"21","author":"Visser","year":"2004","journal-title":"Class. Quantum. Grav."},{"key":"ref_3","unstructured":"Negrean, I., and Negrean, D.C. (2002, January 23\u201325). The Acceleration Energy to Robot Dynamics. Proceedings of the A&QT-R International Conference on Automation, Quality and Testing, Robotics, Cluj-Napoca, Romania."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Negrean, I., and Cri\u0219an, A.-V. (2019). Synthesis on the Acceleration Energies in the Advanced Mechanics of the Multibody Systems. Symmetry, 11.","DOI":"10.3390\/sym11091077"},{"key":"ref_5","unstructured":"Thompson, P. (2011). Snap, Crackle, and Pop, Systems Technology."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.compstruc.2015.02.030","article-title":"An unconditionally stable implicit time integration algorithm: Modified quartic B-spline method","volume":"153","author":"Shojaee","year":"2015","journal-title":"Comput. Struct."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1253","DOI":"10.2514\/3.20976","article-title":"Active Vibration Absorber for the CSI Evolutionary Model\u2013Design and Experimental Results","volume":"15","author":"Bruner","year":"1992","journal-title":"J. Guid. Control Dyn."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1007\/BF01385735","article-title":"On acceleration methods for coupled nonlinear elliptic systems","volume":"60","author":"Kerkhoven","year":"1991","journal-title":"Numer. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2829","DOI":"10.1016\/j.na.2006.09.043","article-title":"Infinite-dimensional second order ordinary differential equations via (TM)-M-2","volume":"67","author":"Aghasi","year":"2007","journal-title":"Nonlinear Anal. Theor."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3010","DOI":"10.1093\/mnras\/stu1080","article-title":"Power requirements for cosmic ray propagation models involving re-acceleration and comment on second-order Fermi acceleration theory","volume":"442","author":"Thornbury","year":"2012","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"103512","DOI":"10.1103\/PhysRevD.86.103512","article-title":"Generalized modified gravity with the second-order acceleration equation","volume":"86","author":"Gao","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1409","DOI":"10.1029\/92JA02191","article-title":"The Ponderomotive Force of Standing Alfven Waves in a Dipolar Magnetoshere","volume":"98","author":"Allan","year":"1993","journal-title":"J. Geophys. Res. Space Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/BF00982498","article-title":"Cosmic-Ray Diffusion at Energies of 1-MEV to 10(5) GEV","volume":"216","author":"Hartquist","year":"1994","journal-title":"Astrophys. Space Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1109\/7.366291","article-title":"2nd-Order Acceleration Models for an MMAE Target Tracker","volume":"31","author":"Wheaton","year":"1995","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1007\/s10103-017-2202-1","article-title":"Calculus migration characterization during Ho:YAG laser lithotripsy by high-speed camera using suspended pendulum method","volume":"32","author":"Zhang","year":"2017","journal-title":"Laser Med. Sci."},{"key":"ref_16","first-page":"125","article-title":"An-Harmonic Analysis and the Arterial Pulse","volume":"30","author":"Voltairas","year":"2009","journal-title":"Adv. Top. Scatt. Biomed. Eng."},{"key":"ref_17","unstructured":"Appell, P. (1899). Sur Une Forme G\u00e9n\u00e9rale des Equations de la Dynamique, Gauthier-Villars. [1st ed.]."},{"key":"ref_18","unstructured":"Appell, P. (1903). Trait\u00e9 de M\u00e9canique Rationnelle, Garnier Fr\u00e8res. [1st ed.]."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3340","DOI":"10.1175\/JAS4015.1","article-title":"Equatorial jets in decaying shallow-water turbulence on a rotating sphere","volume":"64","author":"Kitamura","year":"2007","journal-title":"J. Atmos. Sci."},{"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, 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. Struc., 6631650.","DOI":"10.1080\/15397734.2019.1665542"},{"key":"ref_22","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_23","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 drivings 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_24","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 Dynam."},{"key":"ref_25","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. Contr. Appl. Met."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"904","DOI":"10.1177\/1077546316654854","article-title":"Oblique Impact of Multi-Flexible-Link Systems","volume":"24","author":"Shafei","year":"2018","journal-title":"J. Vib. Control"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Bachau, O.A. (2011). Flexible Multibody Dynamics, Springer. [1st ed.].","DOI":"10.1007\/978-94-007-0335-3"},{"key":"ref_28","unstructured":"Fu, K.S., Gonzalez, R.C., and Lee, C.G. (1987). Control, Sensing, Vision and Intelligence, McGraw-Hill Book Co.. International Edition."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Marin, M., Vlase, S., Ellahi, R., and Bhatti, M. (2019). On the Partition of Energies for the Backward in Time Problem of Thermoelastic Materials with a Dipolar Structure. Symmetry, 11.","DOI":"10.3390\/sym11070863"},{"key":"ref_30","unstructured":"Negrean, I., and Negrean, D.C. (2001, January 10\u201312). Matrix Exponentials to Robot Kinematics. Proceedings of the 17th International Conference on CAD\/CAM, Robotics and Factories of the Future, CARS&FOF, Durban, South Africa."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1109\/9.280779","article-title":"Computational Aspects of the Product-of-Exponentials Formula for Robot Kinematics","volume":"39","author":"Park","year":"1994","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Jazar, R.N. (2010). Theory of Applied Robotics: Kinematics, Dynamics, and Control, Springer. [2nd ed.].","DOI":"10.1007\/978-1-4419-1750-8"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Schiehlen, W., and Eberhard, P. (2014). Applied Dynamics, Springer. [1st ed.].","DOI":"10.1007\/978-3-319-07335-4"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Gattringer, H., and Gerstmayr, J. (2013). Multibody System Dynamics, Robotics and Control, Springer. [1st ed.].","DOI":"10.1007\/978-3-7091-1289-2"},{"key":"ref_35","first-page":"503","article-title":"Advanced Equations in Analytical Dynamics of Systems","volume":"60","author":"Negrean","year":"2017","journal-title":"Acta Tech. Napoc. Ser. Appl. Math. Mech. Eng."},{"key":"ref_36","unstructured":"Pars, L.A. (2007). A Treatise on Analytical Dynamics, Heinemann."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Cassel, K. (2013). Variational Methods with Applications in Science and Engineering, Cambridge University Press.","DOI":"10.1017\/CBO9781139136860"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/95\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:28:43Z","timestamp":1760362123000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/1\/95"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1,3]]},"references-count":37,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2020,1]]}},"alternative-id":["sym12010095"],"URL":"https:\/\/doi.org\/10.3390\/sym12010095","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,1,3]]}}}