Geometric degree of nonconservativity: Set of solutions for the linear case and extension to the differentiable non-linear case

• Published in: Applied mathematical modelling 2016-06, Vol.40 (11-12), p.5930-5941
• Main Authors: Lerbet, J., Challamel, N., Nicot, F., Darve, F.
• Format: Article
• Language: English
• Subjects: Darboux’s theorem; Degrees of freedom; Differential forms; Engineering Sciences; Exterior forms; Geometric degree of nonconservativity; Kinematics; Lagrangian planes; Mathematical analysis; Mathematical models; Mechanics; News; Nonlinearity; Stiffness matrices; Stiffness matrix; Symplectic geometry
• ISSN: 0307-904X; 1872-8480
• DOI: 10.1016/j.apm.2016.01.030

•We consider constrained non-conservative systems.•We compute their Geometric Degree of Non-conservativity.•Lagrangian planes of appropriate symplectic space give the solution in a linear context.•Darboux’s theorem is used to extend the result to a nonlinear framework.•Effectiveness of the methods is questioned through a four d.o.f. example. This paper deals with nonconservative mechanical systems as those subjected to nonconservative positional forces and leading to non-symmetric tangential stiffness matrices. In a previous work, the geometric degree of nonconservativity of such systems, defined as the minimal number ℓ of kinematic constraints necessary to convert the initial system into a conservative one is found to be, in the linear framework, the half of the rank of the skew-symmetric part of the stiffness matrix. In the present paper, news results are reached. First, a more efficient solution of the initial linear problem is proposed. Second, always in the linear framework, the issue of describing the set of all corresponding kinematic constraints is given and reduced to the one of finding all the Lagrangian planes of a symplectic space. Third, the extension to the local non-linear case is solved. A four degree of freedom system exhibiting a maximal geometric degree of nonconservativity (s=2) is used to illustrate our results. The issue of the global non-linear problem is not tackled. Throughout the paper, the issue of the effectiviness of the solution is systematically addressed.