Velocity as a function of state in general-relativistic mechanics

It has been shown that realization of the fact that, in the mechanics of general relativity (GR), the velocity is a function of state, along with the refined metric of Fock’s first-approximation metric and Euler’s hydrodynamic formula, makes it possible to solve the unambiguity problem for the relativistic equations of rotational motion in GR mechanics.     

Discrete hill's equations

In the work, based on the formulas of discrete mechanics in a rotating frame, a discretization of classical Hill's equations of the moon motion is worked out. It is proved that the proposed discretization conserves Jacobi's constant of motion. For computational purposes an algorithm to the solution of obtained discrete Hill's equations is given.     

Structural aspects of stability in non-linear systems Part I

A stability analysis of non-linear systems, represented by a set of first-order differential equations, is performed by examination of the structural properties of these equations. These properties are mapped into a vector field which enables the transient motion to be expressed as a resultant of gradient and rotational components. It is shown how this approach can usefully supplement other well-known techniques.     

Calculation of frequency responses of a flexible string in a rotational motion

A system of quasi-linear partial differential equations describing the dynamic behavior of a flexible string in a rotational motion is obtained. To define frequency responses of a string the obtained system is linearized and then transformed to a two-point boundary-value problem. To solve the latter shooting methods are used. The paper develops an efficient original numerical algorithm based on the application of Taylor calculus and allowing for reduction of computational expenses, characteristic for traditional methods of a numerical solution of an initial-value problem, arising when the shooting methods are realized. The given example of the calculation of string frequency responses in a rotational motion illustrates the serviceability of the suggested procedure on the whole and its efficiency.     

Inhomogeneous universe in f(T) theory

We obtain the equations of motions of the f(T) theory considering the Lemaître-Tolman-Bondi’s metric for a set of diagonal and non-diagonal tetrads. In the case of diagonal tetrads, the equations of motion of the f(T) theory impose a constant torsion or the same equations as in general relativity (GR), while in the case of a non-diagonal set, the equations are quite different from that obtained in GR. We show a simple example of a universe dominated by matter for the two cases. The comparison of the masses in the non-diagonal case shows a sort of increase with respect to the diagonal case. We also find two examples for the non-diagonal case. The first one concerns a Schwarzschild-type black hole solution, which presents a temperature higher than that of Schwarzschild, and a black hole in a dust-dominated universe.     

Nonlinear finite element modeling of the dynamics of unrestrained flexible structures

A finite element modeling and solution technique capable of determining the time response of unrestrained flexible structures which are undergoing large elastic deformations coupled with gross nonsteady translational and rotational motions with respect to an inertial reference frame has been developed. The governing equations of motion are derived using momentum conservation principles and the principle of virtual work. The finite element approximation is applied to the equations of motion and the resulting set of nonlinear second order matrix differential equations is solved timewise by a direct numerical integration scheme based on the trapezoidal rule and Newton-Raphson type iterations. The solution technique is tested on some simple structures consisting of long, slender, uniform beams attached to a rigid mass.     

A Lagrange–Eulerian formulation of an axially moving beam based on the absolute nodal coordinate formulation

In the scope of this paper, a finite-element formulation for an axially moving beam is presented. The beam element is based on the absolute nodal coordinate formulation, where position and slope vectors are used as degrees of freedom instead of rotational parameters. The equations of motion for an axially moving beam are derived from generalized Lagrange equations in a Lagrange–Eulerian sense. This procedure yields equations which can be implemented as a straightforward augmentation to the standard equations of motion for a Bernoulli–Euler beam. Moreover, a contact model for frictional contact between an axially moving strip and rotating rolls is presented. To show the efficiency of the method, simulations of a belt drive are presented.     

关于非完整系统的Ценов方程和Mac Millan方程

Ценов方程和Mac Millan方程是非完整力学史上较晚出现的运动方程.这些方程的建立不仅有历史意义,而且有理论和应用的优势.本文对Ценов方程和Mac Millan方程的形成提供一些史料,并提出一些看法.     

Divergence of a rotating shaft with an intermediate support and conservative axial loads

The dynamic behavior of a rotating shaft with an intermediate support and conservative axial loads is analyzed using Euler beam theory and the assumed mode method. The equations of motion are then transformed to the standard form of eigenvalue problems for determining the critical dimensionless rotational speeds and the critical axial loads corresponding to the divergence-type instability of the shaft. Results of numerical simulations are presented for various combinations of support locations, axial loads and rotational speeds.     

Numerical integration of the time-dependent equations of motion for taylor vortex flow

A method is presented for the finite difference solution of the equations of fluid motion. The complete Navier-Stokes equations are expressed in terms of tangential velocity, vorticity and stream function. The transformed equations are solved using an alternating direction implicit scheme. The classical problem of hydrodynamic stability of the rotational Couette flow is solved in two dimensions. Comparison with other numerical and experimental works shows that the method reported here is computationally stable, even when used with coarse grids and relatively large time increments.     

Modeling and control of two constrained manipulators

The coordinated control of two manipulators in the presence of environment constraints is studied in this paper. Such a control method is needed in applications in which the two manipulators grasp a common object whose motion is constrained by environments. The two manipulators are not only constrained with each other, but also constrained by the environment in their workspace. It is realized that the motion and constraint equations obtained directly from mechanics are not suitable for the control purpose. A set of equivalent equations are derived, which are in the standard form of the nonlinear system representation with clear state equations and output equations. A nonlinear feedback is found which exactly linearizes and decouples the dynamic nonlinear system of the two constrained manipulators. The coordinated controller design is then carried out based on the linearized system by using linear system theory.     

Control of a Parametrically Excited Flexible Beam Undergoing Rotation and Impacts

The linear control of a parametrically excited impacting flexible link in rotational motion is considered. The equation of motion for such a system contains time-periodic coefficients. To suppress the vibrations resulting after impact with an external rigid body, a linear controller is designed via Lyapunov–Floquet transformation. In this approach, the equations of motion with time-periodic coefficients are transformed into a time-invariant form suitable for the application of standard time-invariant controller design techniques. The momentum balance method and an empirical coefficient of restitution is used to model the collision between the two bodies.     

Numerical integration of the equations of motion for rigid polyatomics: The matrix method

A new scheme for numerical integration of motion for classical systems composed of rigid polyatomic molecules is proposed. The scheme is based on a matrix representation of the rotational degrees of freedom. The equations of motion are integrated within the Verlet framework in velocity form. It is shown that, contrary to previous methods, in the approach introduced the rigidity of molecules can be conserved automatically without any additional transformations. A comparison of various techniques with respect to numerical stability is made.     

Energy and angular momentum densities in a Gödel-type universe in teleparallel geometry

The main scope of this research consists in evaluating the energy-momentum (gravitational field plus matter) and gravitational angular momentum densities in the universe with global rotation, considering the Gödel—Obukhov metric. To do that, we use the Hamiltonian formalism of the Teleparallel equivalent of general relativity (TEGR), which is justified for presenting covariant expressions for the quantities under consideration. We have found that the total energy density calculated by the TEGR method is in agreement with the results reported by other authors in the literature using pseudotensors. The result found for the angular momentum density depends on the rotational parameter, as expected. We also show explicitly the equivalence among the field equations of the TEGR and the Einstein equations (GR), considering a perfect fluid and the Gödel—Obukhov metric.     

Finite element analysis of maneuvering spacecraft truss structures

A finite element modeling and solution technique capable of determining the time response of flexible spacecraft truss structures undergoing large angle slew maneuvers has been developed. The elastic deformations of the structure are coupled with large nonsteady translational and rotational motions with respect to an inertial reference frame. The governing equations of motion of the system are derived using momentum conservation principles and the principle of virtual work. The finite element approximation is applied to the equations of motion and the resulting set of nonlinear second order matrix differential equations is solved timewise by an iterative direct numerical integration scheme based on the trapezoidal rule. The solution technique is tested on both planar and three-dimensional maneuvering spacecraft truss structures.     

Dynamics of shells of revolution under axisymmetric load involving shear deformation

A general solution procedure, based on the linear theory, is presented for arbitrary shells of revolution subjected to arbitrary axisymmetric dynamic loads. The equations of motion admit shear deformation and rotational inertia. The numerical solution is obtained by Houbolt's method and by finite differences.     

In-plane vibrations of rotating sectors and rings with and without radial supports by finite element method

The governing differential equations of motion for a ring or sector are modified to include the effects of rotational speed. The finite element equations are systematically and logically derived from these equations using the Galerkin method. A quintic polynomial, satisfying the compatibility of derivatives up to second order, has been used for the ring finite element. The efficiency of the formation has been illustrated by the numerical results presented. An 180°-sector on unsymmetric supports has also been analysed.     

Vibration analysis of rotating rods based on the nonlocal elasticity theory and coupled displacement field

Free longitudinal vibration analysis of a rotating rod based on the Eringen’s nonlocal elasticity is studied in this paper. Rod is supposed to rotate around a fixed axis with a constant angular velocity. To capture the effect of the rotational motion into analysis of the continuous system, a linear proportional relation is introduced between axial and angular velocities. For the first time the mentioned relation is presented based on the internal motions of the infinitesimal element. This novelty makes the rotational displacement as a dependent function of axial displacement playing a significant role through the analysis. Variational approach is adopted to derive the equations of motion for clamped–clamped and clamped-free boundary conditions. For verification of the results obtained from the Galerkin approach, comparison with technical literature is reported. Finally current results illustrate the dependency of the dynamic-vibration analysis of the presented system on the nonlocality and the rotational velocity parameter. This dependency shows the decrement of the frequency with increment in both the angular velocity and the nonlocal parameter. As a result, the mentioned parameters are key factors in the design and analysis of such systems.

     

Rigid body dynamics with a scalable body,quaternions and perfect constraints

In this paper, we present a formulation of the quaternion constraint for rigid body rotations in the form of a standard perfect bilateral mechanical constraint, for which the associated Lagrangian multiplier has the meaning of a constraint force. First, the equations of motion of a scalable body are derived. A scalable body has three translational, three rotational, and one uniform scaling degree of freedom. As generalized coordinates, an unconstrained quaternion and a displacement vector are used. To the scalable body, a perfect bilateral constraint is added, restricting the quaternion to unit length and making the body rigid. This way a quaternion based differential algebraic equation (DAE) formulation for the dynamics of a rigid body is obtained, where the 7×7 mass matrix is regular and the unit length restriction of the quaternion is enforced by a mechanical constraint. Finally, the equations of motion in the form of a DAE are linked to the Newton–Euler equations of motion of a rigid body. The rigid body DAE formulation is useful for the construction of (energy) consistent integrators.