Equations Of Motion For System Research Articles

Methods of reduction for Lagrange systems on time scales with nabla derivatives**Project supported by the National Natural Science Foundation of China (Grant Nos. 11572212 and 11272227) and the Innovation Program for Graduate Student of Jiangsu Province, China (Grant No. KYLX16-0414).

The Routh and Whittaker methods of reduction for Lagrange system on time scales with nabla derivatives are studied. The equations of motion for Lagrange system on time scales are established, and their cyclic integrals and generalized energy integrals are given. The Routh functions and Whittaker functions of Lagrange system are constructed, and the order of differential equations of motion for the system are reduced by using the cyclic integrals or the generalized energy integrals with nabla derivatives. The results show that the reduced Routh equations and Whittaker equations hold the form of Lagrnage equations with nabla derivatives. Finally, two examples are given to illustrate the application of the results.

Hamilton’s principle and Hamilton’s equations with holonomic and non-holonomic constraints

Hamilton’s Principle is written in terms of generalized displacements and momenta. This allows holonomic and non-holonomic constraints to be treated on an equal basis using Lagrange multipliers to derive equations of motion for a dynamical system. Three examples are considered that have been previously considered in the literature by other methods.

Wave propagation behavior of coupled viscoelastic FG-CNTRPC micro plates subjected to electro-magnetic fields surrounded by orthotropic visco-Pasternak foundation

Wave propagation in a coupled visco-elastic micro polyvinylidene fluoride (PVDF) plate reinforced by single-walled carbon nanotubes (SWCNTs) under an electro-magnetic field will be analyzed in this study. Quasi-3D sinusoidal shear deformation theory (SSDT) of plates is presumed in order to simulate plates. The system is surrounded by orthotropic visco-Pasternak foundation. Plates are assumed to have piezoelectric property and CNTs are applied to reinforce plates in different types so micro plates are functionally graded carbon nanotube-reinforced piezoelectric composite (FG-CNTRPC); therefore, electro-magnetic field can be used as a controlling parameter for system. At first, energy equation is written according to local piezoelasticity theory and strain energy, kinetic energy and work of external forces are calculated and motion equation of system will be calculated by applying Hamilton's principle. Finally, motion equation will be solved by using analytical solution in order to obtaining velocity of wave propagation, cut-off and escape frequency of system in different types of wave propagation in system, such as in-phase, out-of-phase and one of plate fixed. After that, the effects of magnetic field, voltage, structure damping and distributions of CNTs on velocity of system and the effects of three type of wave propagation on cut-off and escape frequency of system will be studied. Results show that the effect of out-of-phase type on phase velocity and frequency of system is more than in-phase and one of plate fixed types.

Spacecraft attitude maneuver control using two parallel mounted 3-DOF spherical actuators

A parallel configuration using two 3-degree-of-freedom (3-DOF) spherical electromagnetic momentum exchange actuators is investigated for large angle spacecraft attitude maneuvers. First, the full dynamic equations of motion for the spacecraft system are derived by the Newton-Euler method. To facilitate computation, virtual gimbal coordinate frames are established. Second, a nonlinear control law in terms of quaternions is developed via backstepping method. The proposed control law compensates the coupling torques arising from the spacecraft rotation, and is robust against the external disturbances. Then, the singularity problem is analyzed. To avoid singularities, a modified weighed Moore-Pseudo inverse velocity steering law based on null motion is proposed. The weighted matrices are carefully designed to switch the actuators and redistribute the control torques. The null motion is used to reorient the rotor away from the tilt angle saturation state. Finally, numerical simulations of rest-to-rest maneuvers are performed to validate the effectiveness of the proposed method.

The Hori–Deprit method for averaged motion equations of the planetary problem in elements of the second Poincaré system

We consider an algorithm to construct averaged motion equations for four-planetary systems by means of the Hori–Deprit method. We obtain the generating function of the transformation, change-variable functions and right-hand sides of the equations of motion in elements of the second Poincare system. Analytical computations are implemented by means of the Piranha echeloned Poisson processor. The obtained equations are to be used to investigate the orbital evolution of giant planets of the Solar system and various extrasolar planetary systems.

Robust adaptive antiswing control of underactuated crane systems with two parallel payloads and rail length constraint

The antiswing control and accurate positioning are simultaneously investigated for underactuated crane systems in the presence of two parallel payloads on the trolley and rail length limitation. The equations of motion for the crane system in question are established via the Euler–Lagrange equation. An adaptive control strategy is proposed with the help of system energy function and energy shaping technique. Stability analysis shows that under the designed adaptive controller, the payload swings can be suppressed ultimately and the trolley can be regulated to the destination while not exceeding the pre-specified boundaries. Simulation results are provided to show the satisfactory control performances of the presented control method in terms of working efficiency as well as robustness with respect to external disturbances.

Post-Newtonian celestial mechanics in scalar-tensor cosmology

Applying the recently developed dynamical perturbation formalism on cosmological background to scalar-tensor theory, we provide a solid theoretical basis and a rigorous justification for phenomenological models of orbital dynamics that are currently used to interpret experimental measurements of the time-dependent gravitational constant. We derive the field equations for the scalar-tensor perturbations and study their gauge freedom associated with the cosmological expansion. We find a new gauge eliminating a prohibitive number of gauge modes in the field equations and significantly simplifying post-Newtonian equations of motion for localized astronomical systems in the universe with time-dependent gravitational constant. We identify several new post-Newtonian terms and calculate their effect on secular cosmological evolution of the osculating orbital elements.

The applications of the partial Hamiltonian approach to mechanics and other areas

The partial Hamiltonian systems of the form q̇i=∂H∂pi,ṗi=−∂H∂qi+Γi(t,qi,pi) arise widely in different fields of the applied mathematics. The partial Hamiltonian systems appear for a mechanical system with non-holonomic nonlinear constraints and non-potential generalized forces. In dynamic optimization problems of economic growth theory involving a non-zero discount factor the partial Hamiltonian systems arise and are known as a current value Hamiltonian systems. It is shown that the partial Hamiltonian approach proposed earlier for the current value Hamiltonian systems arising in economic growth theory Naz et al. (2014) [1] is applicable to mechanics and other areas as well. The partial Hamiltonian approach is utilized to construct first integrals and closed form solutions of optimal growth model with environmental asset, equations of motion for a mechanical system with non-potential forces, the force-free Duffing Van der Pol Oscillator and Lotka–Volterra models.

Dynamics of multibody chains in circular orbit: non-integrability of equations of motion

This paper discusses the dynamics of systems of point masses joined by massless rigid rods in the field of a potential force. The general form of equations of motion for such systems is obtained. The dynamics of a linear chain of mass points moving around a central body in an orbit is analysed. The non-integrability of the chain of three masses moving in a circular Kepler orbit around a central body is proven. This was achieved thanks to an analysis of variational equations along two particular solutions and an investigation of their differential Galois groups.

Nonlinear Dynamic Analysis of Telescopic Mechanism for Truss Structure Bridge Inspection Vehicle Under Pedestrian Excitation

Nonlinear dynamic analysis of an axially moving telescopic mechanism for truss structure bridge inspection vehicle under pedestrian excitation is carried out. A biomechanically inspired inverted-pendulum model is utilized to simplify the pedestrian. The nonlinear equations of motion for the beam-pedestrian system are derived using the Hamilton's principle. The equations are transformed into two ordinary differential equations by applying the Galerkin's method at the first two orders. The solutions to the equations are acquired by using the Newmark-β method associated with the Newton-Raphson method. The time-dependent feature of the eigenfunctions for the two beams are taken into consideration in the solutions. Accordingly, the equations of motion for a simplified system, in which the pedestrian is regarded as moving cart, are given. In the numerical examples, dynamic responses of the telescopic mechanism in eight conditions of different beam-telescoping and pedestrian-moving directions are simulated. Comparisons between the vibrations of the beams under pedestrian excitation and corresponding moving cart are carried out to investigate the influence of the pedestrian excitation on the telescopic mechanism. The results show that the displacement of the telescopic mechanism under pedestrian excitation is smaller than that under moving cart especially when the pedestrian approaches the beams end. Additionally, compared with moving cart, the pedestrian excitation can effectively strengthen the vibration when the beam extension is small or when the pedestrian is close to the beams end.

Workspace analysis and verification of cable-driven parallel mechanism for wind tunnel test

Cable-driven parallel mechanism is a special kind of parallel robot in which traditional rigid links are replaced by actuated cables. This provides a new suspension method for wind tunnel test, in which an aircraft model is driven by a number of parallel cables to fulfil 6-DOF motion. The workspace of such a cable robot is limited due to the geometrical and unilateral force constraints, the investigation of which is important for applications requiring large flight space. This paper focuses on the workspace analysis and verification of a redundant constraint 6-DOF cable-driven parallel suspension system. Based on the system motion and dynamic equations, the geometrical interference (either intersection between two cables or between a cable and the aircraft) and cable tension restraint conditions are constructed and analyzed. The hyperplane vector projection strategy is used to solve the aircraft’s orientation and position workspace. Moreover, software ADAMS is used to check the workspace, and experiments are done on the prototype, which adopts a camera to monitor the actual motion space. In addition, the system construction is designed by using a built-in six-component balance to measure the aerodynamic force. The results of simulation and tests show a good consistency, which means that the restraint conditions and workspace solution strategy are valid and can be used to provide guidance for the cable-driven parallel suspension system’s application in wind tunnel tests.

Um estudo sobre grandezas constitutivas e diferença de pressões arbitrárias em uma suspensão particulada com restrição de incompressibilidade

Este trabalho, no contexto de meio poroso com restrição de incompressibilidade, trata de um estudo sobre grandezas constitutivas e diferença de pressões arbitrárias em uma suspensão particulada. São usadas duas soluções exatas de um sistema de equações do movimento da suspensão particulada, com tais soluções inseridas em modelagem matemática do movimento gravitacional de suspensões particuladas em proveta. A modelagem envolve sub-regiões que variam com o tempo e são delimitadas por interfaces em movimento, inclusive pela onda de aceleração. As soluções exatas e os movimentos da onda e das interfaces, juntamente com dados experimentais da literatura de altura versus tempo da interface superior descendente do próprio teste de proveta, são usados para o cálculo de parâmetros envolvidos em equações propostas para as partes constitutivas da força de interação e de cada tensor tensão. Com isso, é determinada a diferença de pressões arbitrárias. São mostrados resultados numéricos do cálculo dos parâmetros envolvidos nas equações do modelo, usando-se como ponto de partida a determinação dos parâmetros básicos velocidade da onda de aceleração, ponto de encontro da onda com a interface superior descendente e ponto de encontro das interfaces.

A Variationally Consistent Approach to Constrained Motion

A variationally consistent approach to constrained rigid-body motion is presented that extends D'Alembert's principle in a way that has a form similar to Kane's equations. The method results in minimal equations of motion for both holonomic and nonholonomic systems without a priori consideration of preferential coordinates.

Model-based control of a third-order nonholonomic system

This paper addresses dynamics modelling and control of mechanical systems subjected to constraints due to the tasks they are to perform. The researched problem refers to the control of a manipulator end-effector motion subjected to a constraint on an acceleration change, that is, jerk. In the presented approach, the acceleration and jerk constraints are incorporated into a system constrained dynamics and then passed to its control dynamics using a unified approach, which is based on an advanced dynamics modelling method, that is, on the generalized programmed motion equations method. This enables one to derive motion equations for systems subjected to high-order constraints. The novelty of this approach consists of the formulation of a unified representation of constraints by differential equations, merging them into system constrained dynamics and then using nonlinear control theory techniques to design controllers for tracking motions along the pre-specified constraints.

Analysis of singularized motion equations of latticed vibroimpact 2D systems in renouncing Newton’s hypothesis

Mathematical models of a 2D latticed structure colliding with a fixed catcher are considered. Consideration of these models is relevant in connection with the development of scientific bases for the design and calculation of machines and structures. The system is analyzed by time–frequency analysis and singularization methods. The calculation techniques of resonant modes are presented and the defining relations are derived.

Formulation of three-dimensional equations of motion for train–slab track–bridge interaction system and its application to random vibration analysis

This study presents the formulation of three-dimensional equations of motion for a train–slab track–bridge interaction system and its application to random vibration analysis using the finite element and pseudo-excitation methods. In this study, a train, slab track, and bridge are regarded as an integrated system, each vehicle is modeled as a four-wheelset mass-spring-damper system with a two-layer suspension system at 23 degrees of freedom, and the rail, slab, girder, and pier are modeled as elastic Bernoulli–Euler beams connected with each other by discrete or continuous spring and damper elements. Three-dimensional equations of motion for the entire system are derived using the energy principle. Dynamic contact forces between moving vehicles and rails are considered as internal forces, and thus, the excitation vectors of load between a wheel and rail, induced by a vehicle's weight and random track irregularities, are easily formulated using the pseudo-excitation method. These equations can be solved by a step-by-step integration method to simultaneously obtain the random dynamic responses of the system. The three-dimensional random vibration characteristics of the system are investigated using an example of a nine-span simply supported beam bridge on which a train consisting of 8 cars travels.

Noether theorem and its inverse for nonlinear dynamical systems with nonstandard Lagrangians

The Noether theorem and its inverse theorem for the nonlinear dynamical systems with nonstandard Lagrangians are studied. In this paper, two kinds of nonstandard Lagrangians, namely exponential Lagrangians and power-law Lagrangians, are discussed. For each case, the Hamilton principle based on the action with nonstandard Lagrangians is established, the differential equations of motion for the dynamical systems with nonstandard Lagrangians are obtained, and two basic formulae for the variation in Hamilton action with nonstandard Lagrangians are derived. The definitions and the criteria of the Noether symmetric transformations and the Noether quasi-symmetric transformations are given. The Noether theorem and its inverse theorem are established, which reveal the intrinsic relation between the symmetry and the conserved quantity for the systems with nonstandard Lagrangians. Two examples are given to illustrate the application of the results.

Model-Based Switching-Crack Identification in a Jeffcott Rotor With an Offset Disk Integrated With an Active Magnetic Bearing

Vibration characteristics of a cracked Jeffcott rotor with an offset disk under the action of an active magnetic bearing (AMB), implemented to improve the radial positioning of the rotor, has been studied. Presence of the AMB suppresses the vibration induced due to the crack and unbalance; identification of the crack could be made by utilizing the vibration signal in conjunction with the controller current of the AMB. A four degrees-of-freedom (DOF) cracked rotor is modeled considering the gyroscopic effect due to the offset disk and a switching crack excitation function (SCEF) to introduce the breathing of crack. The dynamic condensation is applied to eliminate rotational displacements, which would pose practical difficulty in accurate measurement, from the system equations of motion (EOM) to develop an identification algorithm. Frequency domain transformation of the EOM is made with the help of the full spectrum analysis. An algorithm developed with the purpose of crack identification in the form of additive crack stiffness estimates the viscous damping, disk unbalance, and AMB constants as well. The algorithm has been tested for the measurement noise (in the displacement and the current) and bias errors in system parameters, and found to be robust.

エレベーターのロープストリップ推定技術

Elevators in the building are the main vertical traffic system and they transport passengers to their destination floors conveniently. The ordinary elevator system uses the rope traction mechanism which moves the rope and drive pulley simultaneously by the friction force. Therefore, the regular maintenance is required to keep the proper friction force to the drive pulley. In this paper, we propose Rope Slip Observer (RSO) as a support tool for the maintenance. As RSO can estimate the rope slip on the drive pulley induced by friction reduction, it can watch the friction condition of the traction system in real time. Firstly, fundamental equations of motion for the traction elevator system are derived and a simple RSO model is proposed by the equations. Secondly, the proposed RSO model is improved by adding more detailed parameters of the actual elevator system. Finally, the improved RSO model is evaluated by experiment and we can conclude that the estimated rope slip by RSO model shows the same response as the measured one.

Notes on Conservation Laws, Equations of Motion of Matter, and Particle Fields in Lorentzian and Teleparallel de Sitter Space-Time Structures

We discuss the physics of interacting fields and particles living in a de Sitter Lorentzian manifold (dSLM), a submanifold of a 5-dimensional pseudo-Euclidean (5dPE) equipped with a metric tensor inherited from the metric of the 5dPE space. The dSLM is naturally oriented and time oriented and is the arena used to study the energy-momentum conservation law and equations of motion for physical systems living there. Two distinct de Sitter space-time structuresMdSLandMdSTPare introduced given dSLM, the first equipped with the Levi-Civita connection of its metric field and the second with a metric compatible parallel connection. Both connections are used only as mathematical devices. Thus, for example,MdSLis not supposed to be the model of any gravitational field in the General Relativity Theory (GRT). Misconceptions appearing in the literature concerning the motion of free particles in dSLM are clarified. Komar currents are introduced within Clifford bundle formalism permitting the presentation of Einstein equation as a Maxwell like equation and proving that in GRT there are infinitely many conserved currents. We prove that in GRT even when the appropriate Killing vector fields exist it is not possible to define a conserved energy-momentum covector as in special relativistic theories.