Weak formulation and first order form of the equations of motion for a class of constrained mechanical systems

• Published in: International journal of non-linear mechanics Vol. 77; pp. 208 - 222
• Main Authors: Paraskevopoulos, Elias, Natsiavas, Sotirios
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.12.2015
• Subjects: Derivatives; Differential equations; Dynamic response; Dynamical systems; Equations of motion; Hamilton׳s canonical equations; Holonomic and non-holonomic constraints; Integrable and non-integrable variations; Mathematical analysis; Mathematical models; Mechanical systems; Newton׳s law of motion; Strong and weak form of equations of motion; Hamilton׳s canonical equations; Strong and weak form of equations of motion; Integrable and non-integrable variations; Holonomic and non-holonomic constraints; Newton׳s law of motion
• DOI: 10.1016/j.ijnonlinmec.2015.07.007
• ISSN: 0020-7462

Some new theoretical results are presented on modeling the dynamic response of a class of discrete mechanical systems subject to equality motion constraints. Both the development and presentation are facilitated by employing some fundamental concepts of differential geometry. At the beginning, the equations of motion of the corresponding unconstrained system are presented on a configuration manifold with general properties, first in strong and then in a primal weak form, using Newton׳s law of motion as a foundation. Next, the final weak form is obtained by performing a crucial integration by parts step, involving a covariant derivative. This step required the clarification and enhancement of some concepts related to the variations employed in generating the weak form. The second part of this work is devoted to systems involving holonomic and non-holonomic scleronomic constraints. The equations of motion derived in a recent study of the authors are utilized as a basis. The novel characteristic of these equations is that they form a set of second order ordinary differential equations (ODEs) in both the coordinates and the Lagrange multipliers associated to the constraint action. Based on these equations, the corresponding weak form is first obtained, leading eventually to a consistent first order ODE form of the equations of motion. These equations are found to appear in a form resembling the form obtained after application of the classical Hamilton׳s canonical equations. Finally, the new theoretical findings are illustrated by three representative examples. •A three field weak form is presented for constrained mechanical systems.•It is based on a new ODE form of the equations of motion.•The configuration space possesses general geometric properties.•Constraint violation, scaling and coordinate partitioning are avoided.•The equations of motion resemble the classical Hamilton׳s canonical equations.