Equations Of Motion For System Research Articles

Quantization of Holonomic Systems Using WKB Approximation

The Lagrange multipliers for holonomic systems are introduced as generalized coordinates, then, the system is enlarged to be singular system. The Hamilton-Jacobi function is obtained. This function is used to determine the solution of the equations of motion for holonomic systems and to quantize these systems using the WKB approximation. Two examples are considered to demonstrate the application of our formalism. The solution of the two examples are found to be in exact agreement with the Euler-Lagrange equations.

Direct Linearization of Continuous and Hybrid Dynamical Systems

Nonlinear equations of motion are often linearized, especially for stability analysis and control design applications. Traditionally, the full nonlinear equations are formed and then linearized about the desired equilibrium configuration using methods such as Taylor series expansions. However, it has been shown that the quadratic form of the Lagrangian function can be used to directly linearize the equations of motion for discrete dynamical systems. This procedure is extended to directly generate linearized equations of motion for both continuous and hybrid dynamical systems. The results presented require only velocity-level kinematics to form the Lagrangian and find equilibrium configuration(s) for the system. A set of selected partial derivatives of the Lagrangian are then computed and used to directly construct the linearized equations of motion about the equilibrium configuration of interest, without first generating the entire nonlinear equations of motion. Given an equilibrium configuration of interest, the directly constructed linearized equations of motion allow one to bypass first forming the full nonlinear governing equations for the system. Examples are presented to illustrate the method for both continuous and hybrid systems.

Generating human-like soccer primitives from human data

Recently, interest in analysis and generation of human and human-like motion has increased in various areas. In robotics, in order to operate a humanoid robot, it is necessary to generate motions that have strictly dynamic consistency. Furthermore, human-like motion for robots will bring advantages such as energy optimization. This paper presents a mechanism to generate two human-like motions, walking and kicking, for a biped robot using a simple model based on observation and analysis of human motion. Our ultimate goal is to establish a design principle of a controller in order to achieve natural human-like motions. The approach presented here rests on the principle that in most biological motor learning scenarios some form of optimization with respect to a physical criterion is taking place. In a similar way, the equations of motion for the humanoid robot systems are formulated in such a way that the resulting optimization problems can be solved reliably and efficiently. The simulation results show that faster and more accurate searching can be achieved to generate an efficient human-like gait. Comparison is made with methods that do not include observation of human gait. The gait has been successfully used to control Robo-Erectus, a soccer-playing humanoid robot, which is one of the foremost leading soccer-playing humanoid robots in the RoboCup Humanoid League.

From least action in electrodynamics to magnetomechanical energy—a review

The equations of motion for electromechanical systems are traced back to the fundamental Lagrangian of particles and electromagnetic fields, via the Darwin Lagrangian. When dissipative forces can be neglected the systems are conservative and one can study them in a Hamiltonian formalism. The central concepts of generalized capacitance and inductance coefficients are introduced and explained. The problem of gauge independence of self-inductance is considered. Our main interest is in magnetomechanics, i.e. the study of systems where there is exchange between mechanical and magnetic energy. This throws light on the concept of magnetic energy, which according to the literature has confusing and peculiar properties. We apply the theory to a few simple examples: the extension of a circular current loop, the force between parallel wires, interacting circular current loops and the rail gun. These show that the Hamiltonian, phase space, form of magnetic energy has the usual property that an equilibrium configuration corresponds to an energy minimum.

An Argument Against Augmenting the Lagrangean for Nonholonomic Systems

Although it is known that correct dynamical equations of motion for a nonholonomic system cannot be obtained from a Lagrangean that has been augmented with a sum of the nonholonomic constraint equations weighted with multipliers, previous publications suggest otherwise. One published example that was proposed in support of augmentation purportedly demonstrates that an accepted method fails to produce correct equations of motion whereas augmentation leads to correct equations. This present paper shows that, in fact, the opposite is true. The correct equations, previously discounted on the basis of a flawed application of the Newton–Euler method, are verified by using Kane’s method together with a new approach for determining the directions of constraint forces.

Dynamic response of a rotating multi-span shaft with general boundary conditions subjected to a moving load

This paper investigates the dynamic response of a rotating multi-span shaft with general boundary conditions subjected to an axially moving load. The system equations of motion are derived based on the global assumed mode method and a modified transformation matrix is proposed to deal with the multi-span geometric constraints. The numerical results are solved by the Runge–Kutta method. To ensure the validity of the method, the convergence test for dynamic response and the comparison of natural frequencies obtained by both the present method and finite element method are demonstrated. Numerical simulations are performed to study the effects of the moving speed of the load, the number of spans, and the rotational speed of the shaft on the deflections under the moving load. In addition, the effects on the occurring position of maximum deflection are presented. The numerical results show that the maximum deflection in the direction of the load may occur at any span and near the midpoint of one span. However, the maximum deflection induced by the gyroscopic effect in the direction perpendicular to the load always occurs at the first span. Moreover, the occurring position of maximum deflection is independent of the rotational speed.

Special Lie Symmetry and Hojman Conserved Quantity of Appell Equations for a Holonomic System

Special Lie symmetry and Hojman conserved quantity of Appell equations for a holonomic system are studied. Appell equations and differential equations of motion for holonomic mechanic systems are established. Under special Lie infinitesimal transformations in which the time is invariable, the determining equation of the special Lie symmetry and the expressions of Hojman conserved quantity for Appell equations of holonomic systems are presented. Finally, an example is given to illustrate the application of the results.

부분구조물의 단순모델화를 이용한 시스템의 동적해석

Complex systems are made of many subsystems, those are developed and manufactured by many part companies. Even though the information for a part is necessary to analyze the performance of the other part, it is not so easy to get the information for that part from other companies due to many reasons like security or compatibilities. If the modal parameters of a system between the connecting points are available, we can reconstruct a reduced model for that system in a physical coordinate not in a generalized coordinate. The assemble of the equations of motion for the main system and the reduced equations of motion for the connected system can give a response of the main system considering the effects of connected systems. The results show that the proposed method can give the response of a system accurately. The rule for the selection of modes is to use the fundamental modes whose natural frequencies are low.

Seismic History Analysis of Asymmetric Buildings with Soil–Structure Interaction

An efficient approximate method that uses the multi-degrees-of-freedom (MDOF) modal equations of motion for the seismic response history analysis of asymmetric elastic buildings with soil–structure interaction (SSI) is presented in this paper. The systems considered are two-way asymmetric shear buildings resting on the surface of an elastic half-space, which are excited by two-directional seismic ground motions. The SSI forces were simulated using frequency-independent soil springs and dashpots. First, the MDOF modal equations of motion for the SSI systems were derived. The modal response histories were obtained by solving the MDOF modal equations of motion using the step-by-step integration method. Subsequently, the seismic response histories of the whole SSI system were determined from the arithmetic summation of the modal response histories. The MDOF modal equations of motion retain the property of nonproportional damping of the original SSI system. The proposed method has the advantages of the convent...

A generalization of the Helmholtz conditions for the existence of a first‐order Lagrangian using nonholonomic velocities

AbstractThe paper presents a generalization of the Helmholtz conditions for the existence of a first‐order kinetic potential (Lagrangian) in that cases where the motion equations of mechanical, electrical, or electromechanical systems are given in terms of nonholonomic velocities. It is assumed that the transformation between holonomic and nonholonomic velocities is known. It is shown how the generalized Helmholtz conditions for the simultaneous existence of a first‐order Lagrangian and a first‐order dissipation function follow from the generalized Helmholtz conditions for systems with holonomic velocities. Moreover, the classical Helmholtz conditions in case of nondissipative systems can be directly generalized to motion equations which include nonholonomic velocities. In both cases the Boltzmann‐Hamel equations are used. The approaches are demonstrated by two examples: a heavy gyroscope and a 3‐dimensional rotating electrical machine.

Dynamic Response of a Rotating Multi-Span Shaft with Elastic Bearings Subjected to a Moving Load

This paper studies the dynamic response of a rotating multi-span shaft with elastic bearings subjected to an axially moving load. The moving load is motivated by the cutting forces in machining process. The multi-span shaft with elastic bearings models a system that a rotating shaft is mounted by the tailstock and chuck and the steady rests are installed between the ends of shaft. The tailstock and steady rest are modeled as the translational springs and the chuck is assumed as the combination of the translational and rotational springs. The system equations of motion are derived based on the global assumed mode method. The numerical results are obtained to examine the effects of the moving speed of the load, the rotational speed of the shaft, the number of spans, and the stiffnesses of the translational and rotational springs. The numerical results show that the deflection of shaft under the moving load significantly decreases due to the intermediate supports. Moreover, the dynamic behaviors of shaft will be similar when the stiffness of translational spring exceeds the critical stiffness value.

Variational principles for spin systems and the Kirchhoff rod

We obtain the affine Euler-Poincaré equations by standard Lagrangian reduction and deduce the associated Clebsch-constrained variational principle. These results are illustrated in deriving the equations of motion for continuum spin systems and Kirchhoff's rod, where they provide a unified geometric interpretation.

NONLINEAR RESPONSES OF A STRING-BEAM COUPLED SYSTEM WITH FOUR-DEGREES-OF-FREEDOM AND MULTIPLE PARAMETRIC EXCITATIONS

The nonlinear dynamic responses of a string-beam coupled system subjected to harmonic external and parametric excitations are studied in this work in the case of 1:2 internal resonance between the modes of the beam and string. First, the nonlinear governing equations of motion for the string-beam coupled system are established. Then, the Galerkin's method is used to simplify the nonlinear governing equations to a set of ordinary differential equations with four-degrees-of-freedom. Utilizing the method of multiple scales, the eight-dimensional averaged equation is obtained. The case of 1:2 internal resonance between the modes of the beam and string — principal parametric resonance-1/2 subharmonic resonance for the beam and primary resonance for the string — is considered. Finally, nonlinear dynamic characteristics of the string-beam coupled system are studied through a numerical method based on the averaged equation. The phase portrait, Poincare map and power spectrum are plotted to demonstrate that the periodic and chaotic motions exist in the string-beam coupled system under certain conditions.

Improved equations for eccentricity generation in hierarchical triple systems

In a series of papers, we developed a technique for estimating the inner eccentricity in hierarchical triple systems, with the inner orbit being initially circular. However, for certain combinations of the masses and the orbital elements, the secular part of the solution failed. In the present paper, we derive a new solution for the secular part of the inner eccentricity, which corrects the previous weakness. The derivation applies to hierarchical triple systems with coplanar and initially circular orbits. The new formula is tested numerically by integrating the full equations of motion for systems with mass ratios from 10 3 to 10 3 . We also present more numerical results for short term eccentricity evolution, in order to get a better picture of the behaviour of the inner eccentricity.

Newtonian-based methodologies in multi-body dynamics

Three commonly used methods of formulating the equations of motion for multi-body systems are reviewed in this paper. Two of these methods are Newtonian-based; i.e. the entire formulations can be derived simply based on the knowledge of Newton's laws of motion. These two methodologies result in formulations that contain many algebraic and differential equations suitable for computational procedures. The number of equations in these formulations increases as the number of moving bodies and the number of kinematic joints increase. The third reviewed methodology provides a systematic process to transform the large number of equations from any of the other two formulations to a smaller set, possibly to as many as the number of degrees of freedom in the system. This process preserves the simplicity of the original Newtonian-based formulation and, at the same time, provides significant improvement in the computational efficiency and numerical stability in any dynamic analyses.

Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods

This paper employs a new finite element formulation for dynamics analysis of a viscoelastic flexible multibody system. The viscoelastic constitutive equation used to describe the behavior of the system is a three-parameter fractional derivative model. Based on continuum mechanics, the three-parameter fractional derivative model is modified and the proposed new fractional derivative model can reduce to the widely used elastic constitutive model, which meets the continuum mechanics law strictly for pure elastic materials. The system equations of motion are derived based on the absolute nodal coordinate formulation (ANCF) and the principle of virtual work, which can relax the small deformation assumption in the traditional finite element implementation. In order to implement the viscoelastic model into the absolute nodal coordinate, the Grunwald definition of the fractional derivative is employed. Based on a comparison of the HHT-I3 method and the Newmark method, the HHT-I3 method is used to solve the equations of motion. Another particularity of the proposed method based on the ANCF method lies in the storage of displacement history only during the integration process, reducing the numerical computation considerably. Numerical examples are presented in order to analyze the effects of the truncation number of the Grunwald series (fading memory phenomena) and the value of several fractional model parameters and solution convergence aspects.

DYNAMIC NON-LINEAR ANALYSIS OF A CRACKED BLADE

A cracked blade non‐linear dynamic analysis was performed, taking into account contact interaction between crack sides. The contact‐induced non‐linear problem was solved by using the harmonic balance method. Accuracy and computational efficiency are demonstrated by comparing the results with the time integration of the system's motion equation Problem size reduction was performed using system fixed‐interface method. They suppose that a crack forms an interface between two sub‐structures and use a relative degree of freedom to describe the motion of crack sides. The influence of centrifugal forces was investigated in order to understand the necessity of problem non‐linear formulation depending on crack location and frequency of rotation. Santrauka Mentelės su įtrūkimu netiesinė dinamikos analizė atlikta įvertinant įtrūkimo kraštų kontaktinę sąveiką. Netiesinis uždavinys sprendžiamas harmoninės analizės metodu. Tokio metodo rezultatų tinkamumas ir efektyvumas demonstruojamas lyginant su sistemų dinaminių lygčių skaitinio sprendimo rezultatais. Sprendžiamo uždavinio eilė mažinama kraštų fiksavimo metodas. Šiuo atveju buvo laikoma, kad plyšys suformuoja ribą tarp dviejų konstrukcijos dalių ir plyšio ribų kitimo aprašymui naudojamos santykinės koordinatės. Buvo ištirta išcentrinių jėgų įtaka netiesiniam formulavimui, priklausomai nuo plyšio padėties ir kampinio greičio.

On the theory of motion of nonholonomic systems. The reducing-multiplier theorem

This classical paper by S.A. Chaplygin presents a part of his research in non-holonomic mechanics. In this paper, Chaplygin suggests a general method for integration of the equations of motion for non-holonomic systems, which he himself called the “reducing-multiplier method”. The method is illustrated on two concrete problems from non-holonomic mechanics. This paper produced a considerable effect on the further development of the Russian non-holonomic community. With the help of Chaplygin’s reducing-multiplier theory the equations for quite a number of non-holonomic systems were solved (such systems are known as Chaplygin systems). First published about a hundred years ago, this work has not lost its scientific significance and is hoped to be estimated at its true worth by the English-speaking world.

Structural response of a ship in severe seas considering global hydroelastic vibrations

In this paper, a consistent structural analysis procedure to estimate the global and local load effects considering symmetric and anti-symmetric hydroelastic vibrations in high waves is proposed. The procedure consists of motion analysis followed by a structural analysis. First, a system equation of motion by modal approach is set-up and solved in the time domain. In the load evaluations, not only linear but also nonlinear forces such as quadratic component in Bernoulli's theory and body nonlinearity due to the fluctuation of wetted surface, which might have high priority from the viewpoint of structural analysis, are considered by using three-dimensional potential theory. The accuracy of the structural analysis results depends upon the consistency between the models employed in each analysis as well as the accuracy of the analysis at each stage itself. In order to maintain the consistency, the mass terms and the modal forces are evaluated based on the FE model of the whole ship in which all the mass properties are included, and also on modal analysis results on the same model. Then, the inertia forces and pressure distributions at selected time steps are statically applied on to the FE model considering dynamic amplification effects arising from the flexible modes. The procedure is applied to a large container ship. Local and global effects are evaluated. Fluctuating component in torsional mode, or springing in torsion is found.

Dynamic stability of shear-flexible beck’s columns based on Engesser’s and Haringx’s buckling theories

Based on Engesser’s and Haringx’s buckling theories for shear-flexible columns, dynamic stability behaviors of damped Beck’s columns subjected to sub-tangential follower forces are intensively investigated using fifth-order Hermitian beam elements. For this purpose, dimensionless equations of motion for the non-conservative systems are firstly derived based on the buckling approaches of Engesser and Haringx. FE procedures using the shear-flexible Hermitian interpolation functions are next presented by obtaining the mass matrix, the internal and external damping matrices, the elastic and the geometric stiffness matrices, and the load correction stiffness matrices due to sub-tangential follower forces, respectively. Evaluation schemes for flutter and divergence loads of non-conservative systems are described and the static buckling loads and natural frequencies of the shear-flexible beam-columns are compared through numerical examples using the two buckling theories. Finally, the influences of sub-tangentiality, shear deformation and Rayleigh damping on the dynamic stability of the non-conservative system are investigated and compared based on two buckling approaches.