Adina Veronica Crisan

In multibody systems (MBS), such as robot structures, classical modeling is often based on simplified assumptions concerning mass geometry. This paper introduces a formal theoretical model to overcome these limitations by introducing the concept of mass dis-tribution, which describes the continuous nature of mass properties within kinetic as-semblies. Furthermore, the research integrates higher-order acceleration energies into the dynamic formulation – a topic less explored in conventional approaches. By applying the principles of analytical dynamics, particularly a generalized form of D'Alem-bert-Lagrange principle, a comprehensive model based on higher-order acceleration en-ergies is developed. Matrix exponentials and higher-order differential operators are ap-plied to determine the dynamic equations. Generalized forces are also analyzed as es-sential dynamical parameters, directly related to generalized variables and characterized by mass properties, including mass centers, inertial tensors, and pseudo-inertial tensors. The dynamic behavior of the system is described by using matrix-based expressions for defining kinetic and acceleration energies, and their time derivatives. The paper proposes a unified, matrix-based theoretical framework for modeling advanced dynamics in MBS, emphasizing the role of mass distribution and higher-order acceleration energies. This formulation facilitates a deeper understanding of inertial properties and dynamic inter-actions in complex mechanical systems such as robots.

In multibody systems (MBS), such as robot structures, classical modeling is often based on simplified assumptions concerning mass geometry. This paper introduces a formal theoretical model to overcome these limitations by in-troducing the concept of mass distribution, which describes the continuous nature of mass properties within kinetic assemblies. Furthermore, the research integrates high-er-order acceleration energies into the dynamic formulation – a topic less explored in conventional approaches. By applying the principles of analytical dynamics, particularly a generalized form of D'Alembert-Lagrange principle, a comprehensive model based on higher-order acceleration energies is developed. Matrix exponentials and higher-order differential operators are applied to determine the dynamic equations. Generalized forces are also analyzed as essential dynamical parameters, directly related to general-ized variables and characterized by mass properties, including mass centers, inertial tensors, and pseudo-inertial tensors. The dynamic behavior of the system is described by using matrix-based expressions for defining kinetic and acceleration energies, and their time derivatives. The paper proposes a unified, matrix-based theoretical framework for modeling ad-vanced dynamics in MBS, emphasizing the role of mass distribution and higher-order acceleration energies. This formulation facilitates a deeper understanding of inertial properties and dynamic interactions in complex mechanical systems such as robots.

The dynamic study of current and rapid movements of rigid and multibody mechanical systems, according to differential principles from dynamics, is based on advanced concepts from analytical mechanics: kinetic energy, higher-order acceleration energies, and their absolute time derivatives. In advanced dynamics, the study will extend to higher-order acceleration energies. This paper, reflecting the authors’ research, presents new and revised formulations in advanced kinematics and dynamics, with a focus on acceleration energies of the higher order. Explicit and matrix representations of the defining expressions for higher-order acceleration energies, relevant to the current and rapid movements of rigid bodies and multibody mechanical systems, are presented. These formulations include higher-order absolute time derivatives of advanced concepts, following the specific equations from analytical dynamics. Based on the authors’ findings, acceleration energies play a central, decisive role in formulating higher-order differential equations, which describe both rapid and transient motion behavior in rigid and multibody systems.

We construct the conformable actuated pendulum model in the conformable Lagrangian formalism. We solve the equations of motion in the absence of force and in the case of a specific force resulting from torques which generalizes a well known mechanical model. Our study shows that the conformable model captures essential information about the physical system encoded in the parameters which depend on the conformability factor α. This dependence can describe internal variations such as viscous friction, transmission or environmental effects. We solve the equations of motion analytically for α=1/2 and by Frobenius’ method for α≠1/2.

The dynamic study of current and rapid movements of rigid and multibody mechanical systems, according to differential principles from dynamics, is based on advanced concepts from analytical mechanics: kinetic energy, higher-order acceleration energies and their absolute time derivatives. In advanced dynamics, the study will extend to higher-order acceleration energies. This paper, reflecting the authors’ research, presents new and revised formulations in advanced kinematics and dynamics, with a focus on acceleration energies of higher order. Explicit and matrix representations of the defining expressions for higher-order acceleration energies, relevant to the current and rapid movements of rigid bodies and multibody mechanical systems, will be presented. These formulations include higher-order absolute time derivatives of advanced concepts, following the specific equations from analytical dynamics. Based on authors’ findings, acceleration energies play a central, decisive role in formulating higher-order differential equations, which describe both rapid and transient motion behavior in rigid and multibody systems.

The D’Alembert–Lagrange principle is a fundamental concept in analytical mechanics that simplifies the analysis of multi-degree-of-freedom mechanical systems, facilitates the dynamic response prediction of structures under various loads, and enhances the control algorithms in robotics. It is essential for solving complex problems in engineering and robotics. This theoretical study aims to highlight the advantages of using acceleration energy to obtain the differential equations of motion and the generalized driving forces, compared to the classical approach based on the Lagrange equations of the second kind. It was considered a mechanical structure with two degrees of freedom (DOF), namely, a planar robot consisting of two homogeneous rods connected by rotational joints. Both the classical Lagrange approach and the acceleration energy model were applied. It was noticed that while both approaches yielded the same results, using acceleration energy requires only a single differentiation operation, whereas the classical approach involves three such operations to achieve the same results. Thus, applying the acceleration energy method involves fewer mathematical steps and simplifies the calculations. This demonstrates the efficiency and effectiveness of using acceleration energy in dynamic system analysis. By incorporating acceleration energy into the model, enhanced robustness and accuracy in predicting system behavior are achieved.

Multibody mechanical systems (i.e., serial, and parallel robots) have a wide range of applications in the industrial field. In technological processes, these systems perform mechanical movements, in which the active forces have a certain time variation law and, hence, induce higher-order accelerations in the mechanical system, which become central functions in acceleration energies. The advanced dynamics study of multibody systems, often characterized by symmetry, is conducted by applying the differential and variational principles. Lagrange–Euler equations and their time derivatives are commonly used. Here, the central function is the kinetic energy and its higher-order time derivatives. Additionally, the generalization of Gibbs–Appell equations, where the central function is represented by the first and higher-order acceleration energy, can be applied. This paper aims to establish a relation between the kinetic energy and acceleration energy for different material systems. This purpose is achieved by applying the absolute second-order time derivative on the expressions of kinetic energy, corresponding to different material systems. Following this differential calculation and by applying some constraints, the relationship between kinetic energy and acceleration energy is obtained. For validating the relation between kinetic energy and acceleration energy of the first, second and third order, an application is presented.

We construct a class of knot solutions of the time-dependent gravitoelectromagnetic (GEM) equations in vacuum in the linearized gravity approximation by analogy with the Rañada–Hopf fields. For these solutions, the dual metric tensors of the bi-metric geometry of the gravitational vacuum with knot perturbations are given and the geodesic equation as a function of two complex parameters of the time-dependent GEM knots are calculated. By taking stationary potentials, which formally amount to particularizing to time-independent GEM equations, we obtain a set of stationary fields subjected to constraints from the time-dependent GEM knots. Finally, the Landau–Lifshitz pseudo-tensor and a scalar invariant of the static fields are computed.