Quasi-velocities definition in Lagrangian multibody dynamics

Based on Lagrangian mechanics, use of velocity constraints as a special set of quasi-velocities helps derive explicit equations of motion. The equations are applicable to holonomic and nonholonomic constrained multibody systems. It is proved that in proposed quasi-spaces, the Lagrange multipliers are eliminated from equations of motion; however, it is possible to compute these multipliers once the equations of motion have been solved. The novelty of this research is employing block matrix inversion to find the analytical relations between the parameters of quasi-velocities and equations of motion. In other words, this research identifies arbitrary submatrices and their effects on equations of motion. Also, the present study aimed to provide appropriate criteria to select arbitrary parameters to avoid singularity, reduce constraints violations, and improve computational efficiency. In order to illustrate the advantage of this approach, the simulation results of a 3-link snake-like robot with nonholonomic constraints and a four-bar mechanism with holonomic constraints are presented. The effectiveness of the proposed approach is demonstrated by comparing the constraints violation at the position and velocity levels, conservation of the total energy, and computational efficiency with those obtained via the traditional methods.

Dynamic Modeling of Planar Multi-Link Flexible Manipulators

A closed-form dynamic model of the planar multi-link flexible manipulator is presented. The assumed modes method is used with the Lagrangian formulation to obtain the dynamic equations of motion. Explicit equations of motion are derived for a three-link case assuming two modes of vibration for each link. The eigenvalue problem associated with the mass boundary conditions, which changes with the robot configuration and payload, is discussed. The time-domain simulation results and frequency-domain analysis of the dynamic model are presented to show the validity of the theoretical derivation.

The motion of point vortices on closed surfaces

We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is non-zero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the Laplace–Beltrami operator must vanish). On variable Gaussian curvature surfaces, this results in self-induced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid. We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as two vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where seven vortices are marginally unstable (Thomson 1883 A treatise on the motion of vortex rings , pp. 94–108; Dritschel 1985 J. Fluid Mech. 157 , 95–134 ( doi:10.1017/S0022112088003088 )), and the sphere, where four vortices may be unstable if sufficiently close to the equator (Polvani & Dritschel 1993 J. Fluid Mech. 255 , 35–64 ( doi:10.1017/S0022112093002381 )).

Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics

We present the new, general, explicit form of the equations of motion for constrained mechanical systems applicable to systems with singular mass matrices. The systems may have holonomic and/or non-holonomic constraints, which may or may not satisfy D'Alembert's principle at each instant of time. The equation provides new insights into the behaviour of constrained motion and opens up new ways of modelling complex multi-body systems. Examples are provided and applications of the equation to such systems are illustrated.

Mechanical Systems With Nonideal Constraints: Explicit Equations Without the Use of Generalized Inverses

In this paper we obtain the explicit equations of motion for mechanical systems under nonideal constraints without the use of generalized inverses. The new set of equations is shown to be equivalent to that obtained using generalized inverses. Examples demonstrating the use of the general equations are provided.

What is the General Form of the Explicit Equations of Motion for Constrained Mechanical Systems?

This paper presents the general form of the explicit equations of motion for mechanical systems. The systems may have holonomic and/or nonholonomic constraints, and the constraint forces may or may not satisfy D’Alembert’s principle at each instant of time. The explicit equations lead to new fundamental principles of analytical mechanics.