FEM Equations of Motion for Elastic Systems

The paper presents a novel spectral approach, accompanied by an asymptotic model and numerical simulations for slender elastic systems such as long bridges or tall buildings. The focus is on asymptotic approximations of solutions by Bloch waves, which may propagate in a infinite periodic waveguide. Although the notion of passive mass dampers is conventional in the engineering literature, it is not obvious that an infinite waveguide problem is adequate for analysis of long but finite slender elastic systems. The formal mathematical treatment of a Bloch wave would reduce to a spectral analysis of equations of motion on an elementary cell of a periodic structure, with Bloch–Floquet quasi-periodicity conditions imposed on the boundary of the cell. Frequencies of some classes of standing waves can be estimated analytically. One of the applications discussed in the paper is the “dancing bridge” across the river Volga in Volgograd.

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Analytical formulation of three-dimensional dynamic homogenization for periodic elastic systems

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2011.0698 ◽

2012 ◽

Vol 468 (2142) ◽

pp. 1629-1651 ◽

Cited By ~ 60

Author(s):

A. N. Norris ◽

A. L. Shuvalov ◽

A. A. Kutsenko

Keyword(s):

Effective Medium Theory ◽

Equations Of Motion ◽

Three Dimensional ◽

Expansion Method ◽

Bloch Wave ◽

Dimensional System ◽

Medium Theory ◽

Material Parameters ◽

Elastic Systems ◽

Arbitrary Frequency

Homogenization of the equations of motion for a three-dimensional periodic elastic system is considered. Expressions are obtained for the fully dynamic effective material parameters governing the spatially averaged fields by using the plane wave expansion method. The effective equations are of Willis form with coupling between momentum and stress and tensorial inertia. The formulation demonstrates that the Willis equations of elastodynamics are closed under homogenization. The effective material parameters are obtained for arbitrary frequency and wavenumber combinations, including but not restricted to Bloch wave branches for wave propagation in the periodic medium. Numerical examples for a one-dimensional system illustrate the frequency dependence of the parameters on Bloch wave branches and provide a comparison with an alternative dynamic effective medium theory, which also reduces to Willis form but with different effective moduli.

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Energy of Accelerations Used to Obtain the Motion Equations of a Three- Dimensional Finite Element

Symmetry ◽

10.3390/sym12020321 ◽

2020 ◽

Vol 12 (2) ◽

pp. 321 ◽

Cited By ~ 7

Author(s):

Sorin Vlase ◽

Iuliu Negrean ◽

Marin Marin ◽

Maria Luminița Scutaru

Keyword(s):

Finite Element ◽

Equations Of Motion ◽

Three Dimensional ◽

The Finite Element Method ◽

Method Of Calculation ◽

Elastic Systems ◽

Dimensional Finite Element ◽

Three Dimensional Finite Element ◽

Multi Body ◽

Elastic Elements

When analyzing the dynamic behavior of multi-body elastic systems, a commonly used method is the finite element method conjunctively with Lagrange’s equations. The central problem when approaching such a system is determining the equations of motion for a single finite element. The paper presents an alternative method of calculation theses using the Gibbs–Appell (GA) formulation, which requires a smaller number of calculations and, as a result, is easier to apply in practice. For this purpose, the energy of the accelerations for one single finite element is calculated, which will be used then in the GA equations. This method can have advantages in applying to the study of multi-body systems with elastic elements and in the case of robots and manipulators that have in their composition some elastic elements. The number of differentiation required when using the Gibbs–Appell method is smaller than if the Lagrange method is used which leads to a smaller number of operations to obtain the equations of motion.

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Maggi’s Equations Used in the Finite Element Analysis of the Multibody Systems with Elastic Elements

Mathematics ◽

10.3390/math8030399 ◽

2020 ◽

Vol 8 (3) ◽

pp. 399 ◽

Cited By ~ 2

Author(s):

Sorin Vlase ◽

Marin Marin ◽

Maria Luminița Scutaru

Keyword(s):

Finite Element ◽

Multibody Systems ◽

Equations Of Motion ◽

The Finite Element Method ◽

Element Analysis ◽

Main Method ◽

The Finite Element Analysis ◽

Elastic Systems ◽

Lagrange’S Multipliers ◽

Elastic Elements

The main method used to determine the equations of motion of a multibody system (MBS) with elastic elements is the method of Lagrange’s multipliers. The assembly of equations for the whole system represents an important step in the elastodynamic analysis of such a system. This paper presents a new method of approaching this stage, by applying Maggi’s equations. In this way, the links that exist between the finite elements and the connections that exist between different bodies of the MBS system are conveniently taken into account, each body having a distinct velocity and acceleration field. Although Maggi’s equations have been used, sporadically, in some applications so far, we are not aware that they have been used in the study of elastic systems using the finite element method. Finally, an algorithm is presented that uses the Maggi formalism to obtain the equations of motion for an MBS system.

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The use of splitting of the equations of motion in the numerical solution of problems of the dynamics of elastic systems

Journal of Applied Mathematics and Mechanics ◽

10.1016/j.jappmathmech.2012.05.016 ◽

2012 ◽

Vol 76 (2) ◽

pp. 250-252

Author(s):

S.V. Morgunov

Keyword(s):

Numerical Solution ◽

Equations Of Motion ◽

Elastic Systems

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A Discrete Quasi-Coordinate Formulation for the Dynamics of Elastic Bodies

Journal of Applied Mechanics ◽

10.1115/1.2189873 ◽

2006 ◽

Vol 74 (2) ◽

pp. 231-239 ◽

Cited By ~ 6

Author(s):

G. M. T. D’Eleuterio ◽

T. D. Barfoot

Keyword(s):

Equations Of Motion ◽

Computational Cost ◽

Dynamical Equations ◽

Generalized Coordinates ◽

Order Form ◽

Elastic Bodies ◽

Alternative Approach ◽

Elastic Systems ◽

Computationally Intensive ◽

Derivatives Of

The discretized equations of motion for elastic systems are typically displayed in second-order form. That is, the elastic displacements are represented by a set of discretized (generalized) coordinates, such as those used in a finite-element method, and the elastic rates are simply taken to be the time-derivatives of these displacements. Unfortunately, this approach leads to unpleasant and computationally intensive inertial terms when rigid rotations of a body must be taken into account, as is so often the case in multibody dynamics. An alternative approach, presented here, assumes the elastic rates to be discretized independently of the elastic displacements. The resulting dynamical equations of motion are simplified in form, and the computational cost is correspondingly lessened. However, a slightly more complex kinematical relation between the rate coordinates and the displacement coordinates is required. This tack leads to what may be described as a discrete quasi-coordinate formulation.

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Application of Variational Equation of Motion to the Nonlinear Vibration Analysis of Homogeneous and Layered Plates and Shells

Journal of Applied Mechanics ◽

10.1115/1.3630109 ◽

1963 ◽

Vol 30 (1) ◽

pp. 79-86 ◽

Cited By ~ 18

Author(s):

Yi-Yuan Yu

Keyword(s):

Nonlinear Vibration ◽

Nonlinear Vibrations ◽

Variational Equation ◽

Equations Of Motion ◽

Equation Of Motion ◽

Free Vibrations ◽

Layered Plates ◽

Variational Approximations ◽

Elastic Systems ◽

Plates And Shells

An integrated procedure is presented for applying the variational equation of motion to the approximate analysis of nonlinear vibrations of homogeneous and layered plates and shells involving large deflections. The procedure consists of a sequence of variational approximations. The first of these involves an approximation in the thickness direction and yields a system of equations of motion and boundary conditions for the plate or shell. Subsequent variational approximations with respect to the remaining space coordinates and time, wherever needed, lead to a solution to the nonlinear vibration problem. The procedure is illustrated by a study of the nonlinear free vibrations of homogeneous and sandwich cylindrical shells, and it appears to be applicable to still many other homogeneous and composite elastic systems.

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Energy Considerations in Systems With Varying Stiffness

Journal of Applied Mechanics ◽

10.1115/1.1574060 ◽

2003 ◽

Vol 70 (4) ◽

pp. 465-469 ◽

Cited By ~ 2

Author(s):

J. R. Barber ◽

K. Grosh ◽

S. Oh

Keyword(s):

Energy Conservation ◽

Total Energy ◽

Dynamic Behavior ◽

Contact Force ◽

Elastic System ◽

Equations Of Motion ◽

Local Deformation ◽

Elastic Systems ◽

External Forces ◽

Work Done

If the stiffness of an elastic system changes with time, a conventional Newtonian statement of the equations of motion will generally lead to solutions that violate the fundamental mechanics principle that the work done by the external forces be equal to the increase in total energy of the system. Timoshenko’s discussion of the problem of a vehicle driven across an elastic bridge is generalized to show that energy conservation can be restored only if the local deformation of the components is taken into account in determining the direction of the contact force. This result has important consequences for the interaction of elastic systems in general, including, for example, the dynamic behavior of meshing gears.

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The motion of a lunar orbiter

Symposium - International Astronomical Union ◽

10.1017/s0074180900105686 ◽

1966 ◽

Vol 25 ◽

pp. 373

Author(s):

Y. Kozai

Keyword(s):

Degrees Of Freedom ◽

Dimensional Space ◽

Artificial Satellite ◽

Equations Of Motion ◽

Triaxial Ellipsoid ◽

The Moon ◽

Secular Motion ◽

The Earth ◽

Close Satellite ◽

The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

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On the motion of comet P/d’Arrest

International Astronomical Union Colloquium ◽

10.1017/s025292110003387x ◽

1974 ◽

Vol 22 ◽

pp. 145-148

Author(s):

W. J. Klepczynski

Keyword(s):

Equations Of Motion ◽

Variational Equations ◽

Partial Derivatives

AbstractThe differences between numerically approximated partial derivatives and partial derivatives obtained by integrating the variational equations are computed for Comet P/d’Arrest. The effect of errors in the IAU adopted system of masses, normally used in the integration of the equations of motion of comets of this type, is investigated. It is concluded that the resulting effects are negligible when compared with the observed discrepancies in the motion of this comet.

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