Correction of the Koenig Formula for the Kinetic Energy of a Rotating Solid

An exact solution is obtained for the kinetic energy in the general case of the spatial motion of solids with arbitrary rotation, which differs from the Koenig formula by three additional terms with centrifugal moments of inertia. The description of motion in the Lagrange form and the superposition principle are used, which provides a geometric summation of the velocities and accelerations of the joint motions in the Lagrange form for any particle at any time. The integrand function in the equation for kinetic energy is represented by the sum of the identical velocity components of the joint plane-parallel motions. The moments of inertia in the Koenig formula do not change during movement and can be calculated from the current or initial state of the body. The centrifugal moments change and turn to 0 when rotating relative to the main central axes only for bodies with equal main moments of inertia, for example, for a ball. In other cases, the difference in the main moments of inertia leads to cyclic changes in the kinetic energy with the possible manifestation of precession and nutation, the amplitude of which depends on the angular velocities of rotation of the body. An example of using equations for a robot with one helical and two rotational kinematic pairs is given.

Correction of the Koenig Formula for the Kinetic Energy of a Rotating Solid

An exact solution is obtained for the kinetic energy in the general case of the spatial motion of solids with arbitrary rotation, which differs from the Koenig formula by three additional terms that take into account the change in the centrifugal moments of inertia when the body rotates. The description of motion in the Lagrange form and the superposition principle are used, which provides a geometric summation of the velocities and accelerations of the joint motions in the Lagrange form for any particle at any time. The integrand function in the equation for kinetic energy is represented as the sum of the identical velocity components of the joint plane-parallel motions. In the general case of motion with 6 degrees of freedom, the energy of rotational motion is determined by three axial moments of inertia, as in the Koenig formula, and three additional centrifugal moments, which take into account the rotation of the body. They can be calculated through 6 integral characteristics of the density distribution, determined for the initial position of the body.

Correction of the Koenig Formula for the Kinetic Energy of a Rotating Solid

An exact solution is obtained for the kinetic energy in the general case of the spatial motion of solids with arbitrary rotation, which differs from the Koenig formula by three additional terms that take into account the change in the centrifugal moments of inertia when the body rotates. The description of motion in the Lagrange form and the superposition principle are used, which provides a geometric summation of the velocities and accelerations of the joint motions in the Lagrange form for any particle at any time. The integrand function in the equation for kinetic energy is represented as the sum of the identical velocity components of the joint plane-parallel motions. In the general case of motion with 6 degrees of freedom, the energy of rotational motion is determined by three axial moments of inertia, as in the Koenig formula, and three additional centrifugal moments, which take into account the rotation of the body. They can be calculated through 6 integral characteristics of the density distribution, determined for the initial position of the body.

Variational principles and nonequilibrium thermodynamics

Variational principles play a fundamental role in deriving the evolution equations of physics. They work well in the case of non-dissipative evolution, but for dissipative systems, the variational principles are not unique and not constructive. With the methods of modern nonequilibrium thermodynamics, one can derive evolution equations for dissipative phenomena and, surprisingly, in several cases, one can also reproduce the Euler–Lagrange form and symplectic structure of the evolution equations for non-dissipative processes. In this work, we examine some demonstrative examples and compare thermodynamic and variational techniques. Then, we argue that, instead of searching for variational principles for dissipative systems, there is another viable programme: the second law alone can be an effective tool to construct evolution equations for both dissipative and non-dissipative processes. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.

Robust adaptive controller design for excavator arm

This paper presents a robust adaptive controller that does not depend on the system parameters for an excavator arm. Firstly, the model of the excavator arm is demonstrated in the Euler-Lagrange form considering with overall excavator system. Next, a robust adaptive controller has been constructed from information of state error. In this paper, the stability of overall system is mathematically proven by using Lyapunov stability theory. Also, the proposed controller is model free then the closed loop system is not affected by disturbances and uncertainties. Finally, the simulation is executed in Matlab/Simulink for both presented scheme and the PD controller under some conditions to ensure that the proposed algorithm given the good performances for all cases.

Stability of the Moons Orbits in Solar System in the Restricted Three-Body Problem

We consider the equations of motion of three-body problem in aLagrange form(which means a consideration of relative motions of 3 bodies in regard to each other). Analyzing such a system of equations, we consider in detail the case of moon’s motion of negligible massm3around the 2nd of two giant-bodiesm1,m2(which are rotating around their common centre of masses on Kepler’s trajectories), the mass of which is assumed to be less than the mass of central body. Under assumptions of R3BP, we obtain the equations of motion which describe the relative mutual motion of the centre of mass of 2nd giant-bodym2(planet) and the centre of mass of 3rd body (moon) with additional effective massξ·m2placed in that centre of massξ·m2+m3, whereξis the dimensionless dynamical parameter. They should be rotating around their common centre of masses on Kepler’s elliptic orbits. For negligible effective massξ·m2+m3it gives the equations of motion which should describe aquasi-ellipticorbit of 3rd body (moon) around the 2nd bodym2(planet) for most of the moons of the planets in Solar System.

The Interior Euler-Lagrange Operator in Field Theory

The paper is devoted to the interior Euler-Lagrange operator in field theory, representing an important tool for constructing the variational sequence. We give a new invariant definition of this operator by means of a natural decomposition of spaces of differential forms, appearing in the sequence, which defines its basic properties. Our definition extends the well-known cases of the Euler-Lagrange class (Euler-Lagrange form) and the Helmholtz class (Helmholtz form). This linear operator has the property of a projector, and its kernel consists of contact forms. The result generalizes an analogous theorem valid for variational sequences over 1-dimensional manifolds and completes the known heuristic expressions by explicit characterizations and proofs.

Attitude Coordination Control of Spacecraft Formation

In this paper, we study adaptive attitude synchronization of spacecraft formation with switching topology. By introducing a novel adaptive control architecture and by transforming the attitude dynamics into Euler-Lagrange form, decentralized controllers are developed, which allow for parametric uncertainties and unknown external disturbances. Based upon graph theory and Lyapunov stability theory, rigid mathematical analysis on system stability is provided. A distinctive feature of this work is to address the adaptive attitude synchronization with general directed information flow. It is shown that arbitrary desired constant relative orientations within the group or with respect to any external references can be attained. Simulation results are provided to demonstrate the effectiveness of the obtained results.