Dynamics of Large Scale Mechanical Models Using Multilevel Substructuring

2006 ◽  
Vol 2 (1) ◽  
pp. 40-51 ◽  
Author(s):  
C. Papalukopoulos ◽  
S. Natsiavas
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Equations Of Motion ◽  
Numerical Results ◽  
Ground Vehicles ◽  
Response Characteristics ◽  
Sparsity Pattern ◽  
Mechanical Models ◽  
Set Up

An appropriate substructuring methodology is applied in order to study the dynamic response of very large scale mechanical systems. The emphasis is put on enabling a systematic study of dynamical systems with nonlinear characteristics, but the method is equally applicable to systems possessing linear properties. The accuracy and effectiveness of the methodology are illustrated by numerical results obtained for example vehicle models, having a total number of degrees of freedom lying in the order of a million or even bigger. First, the equations of motion of each component are set up by applying the finite element method. The order of the resulting models is so high that the classical substructuring methodologies become numerically ineffective or practically impossible to apply. However, the method developed overcomes these difficulties by imposing a further, multilevel substructuring of each component, based on the sparsity pattern of the stiffness matrix. In this way, the number of the equations of motion of the complete system is substantially reduced. Consequently, the numerical results presented demonstrate that besides the direct computational savings, this reduction in the dimensions enables the application of numerical codes, which capture response characteristics of dynamical systems sufficiently accurate up to a prespecified level of forcing frequencies. The study concludes by investigating biodynamic response of passenger-seat subsystem models coupled with complex mechanical models of ground vehicles resulting from deterministic or random road excitation.

Pointwise Optimal Control of Dynamical Systems Described by Constrained Coordinates

1999 ◽  
Vol 121 (4) ◽  
pp. 594-598 ◽  
Author(s):  
V. Radisavljevic ◽  
H. Baruh
Keyword(s):  
Optimal Control ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Control Law ◽  
Algebraic Equations ◽  
The Stability ◽  
The Mathematical Model

A feedback control law is developed for dynamical systems described by constrained generalized coordinates. For certain complex dynamical systems, it is more desirable to develop the mathematical model using more general coordinates then degrees of freedom which leads to differential-algebraic equations of motion. Research in the last few decades has led to several advances in the treatment and in obtaining the solution of differential-algebraic equations. We take advantage of these advances and introduce the differential-algebraic equations and dependent generalized coordinate formulation to control. A tracking feedback control law is designed based on a pointwise-optimal formulation. The stability of pointwise optimal control law is examined.

Periodic Steady State Response of Complex Vehicle Models Using Multi-Level Substructuring

2007 ◽  
Author(s):  
I. Stavrakis ◽  
C. Theodosiou ◽  
S. Natsiavas
Keyword(s):  
Nonlinear Models ◽  
Equations Of Motion ◽  
Response Spectra ◽  
Computationally Efficient ◽  
Sparsity Pattern ◽  
Strongly Nonlinear ◽  
Road Excitation ◽  
Multi Level ◽  
Set Up ◽  
Coordinate Reduction

A systematic methodology is presented for investigating long term ride dynamics of large order vehicle models in a computationally efficient way. First, the equations of motion for each of the main structural components of the vehicle are set up by applying the finite element method. As a consequence of the geometric complexity of these components, the number of the resulting equations is so high that the classical coordinate reduction methodologies become numerically ineffective to apply. In addition, the composite model possesses strongly nonlinear characteristics. However, the method applied overcomes some of these difficulties by imposing a multi-level substructuring procedure, based on the sparsity pattern of the stiffness matrix. In this way, the number of the equations of motion of the complete system is substantially reduced. Subsequently, this allows the application of appropriate numerical methodologies for predicting response spectra of the nonlinear models to periodic road excitation. Results obtained by direct integration of the equations of motion are also presented. Where possible, the accuracy and validity of the applied methodology is verified by comparison with results obtained for the original models.

A Method for Solving Large-Scale Multiloop Constrained Dynamical Systems Using Structural Decomposition

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Tao Xiong ◽  
Jianwan Ding ◽  
Yizhong Wu ◽  
Liping Chen ◽  
Wenjie Hou
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
The State ◽  
State Variables ◽  
Structural Decomposition ◽  
Multiple Degrees Of Freedom ◽  
Constraint Equations ◽  
Constrained Dynamical Systems ◽  
Algebraic Constraint

A structural decomposition method based on symbol operation for solving differential algebraic equations (DAEs) is developed. Constrained dynamical systems are represented in terms of DAEs. State-space methods are universal for solving DAEs in general forms, but for complex systems with multiple degrees-of-freedom, these methods will become difficult and time consuming because they involve detecting Jacobian singularities and reselecting the state variables. Therefore, we adopted a strategy of dividing and conquering. A large-scale system with multiple degrees-of-freedom can be divided into several subsystems based on the topology. Next, the problem of selecting all of the state variables from the whole system can be transformed into selecting one or several from each subsystem successively. At the same time, Jacobian singularities can also be easily detected in each subsystem. To decompose the original dynamical system completely, as the algebraic constraint equations are underdetermined, we proposed a principle of minimum variable reference degree to achieve the bipartite matching. Subsequently, the subsystems are determined by aggregating the strongly connected components in the algebraic constraint equations. After that determination, the free variables remain; therefore, a merging algorithm is proposed to allocate these variables into each subsystem optimally. Several examples are given to show that the proposed method is not only easy to implement but also efficient.

scholarly journals A new dynamical systems perspective on atmospheric predictability: Eastern Mediterranean weather regimes as a case study

2019 ◽  
Vol 5 (6) ◽  
pp. eaau0936 ◽  
Author(s):  
Assaf Hochman ◽  
Pinhas Alpert ◽  
Tzvi Harpaz ◽  
Hadas Saaroni ◽  
Gabriele Messori
Keyword(s):  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Climate Model ◽  
Eastern Mediterranean ◽  
Cmip5 Models ◽  
Local Dimension ◽  
Geographical Regions ◽  
Recent Developments ◽  
New Perspective

The atmosphere is a chaotic system displaying recurrent large-scale configurations. Recent developments in dynamical systems theory allow us to describe these configurations in terms of the local dimension—a proxy for the active number of degrees of freedom—and persistence in phase space, which can be interpreted as persistence in time. These properties provide information on the intrinsic predictability of an atmospheric state. Here, this technique is applied to atmospheric configurations in the eastern Mediterranean, grouped into synoptic classifications (SCs). It is shown that local dimension and persistence, derived from reanalysis and CMIP5 models’ daily sea-level pressure fields, can serve as an extremely informative qualitative method for evaluating the predictability of the different SCs. These metrics, combined with the SC transitional probability approach, may be a valuable complement to operational weather forecasts and effective tools for climate model evaluation. This new perspective can be extended to other geographical regions.

scholarly journals Modular Multibody Formulation for Simulating Off-Road Tracked Vehicles

2014 ◽  
Vol 1 (2) ◽  
pp. 77 ◽  
Author(s):  
Mohamed A Omar
Keyword(s):  
Degrees Of Freedom ◽  
Large Scale ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Dynamic Equations ◽  
Solution Stability ◽  
Simulation Code ◽  
Tracked Vehicles ◽  
Main System ◽  
6 Degrees Of Freedom

This paper presents a formulation and procedure for incorporating the multibody dynamics analysis capability of tracked vehicles in large-scale multibody system.  The proposed self-contained modular approach could be interfaced to any exiting multibody simulation code without need to alter the existing solver architecture.  Each track is modeled as a super-component that can be treated separate from the main system.  The super-component can be efficiently used in parallel processing environment to reduce the simulation time.  In the super-component, each track-link is modeled as separate body with full 6 degrees of freedom (DoF).  To improve the solution stability and efficiency, the joints between track links are modeled as complaint connection.  The spatial algebra operator is used to express the motion quantities and develop the link’s nonlinear kinematic and dynamic equations of motion.  The super-component interacts with the main system through contact forces between the track links and the driving sprocket, the support rollers and the idlers using self-contained force modules.  Also, the super-component models the interaction with the terrain through force module that is flexible to include different track-soil models, different terrain geometries, and different soil properties.  The interaction forces are expressed in the Cartesian system, applied to the link’s equation of motion and the corresponding bodies in the main system.  For sake of completeness, this paper presents dynamic equations of motion of the links as well as the main system formulated using joint coordinates approach.

Vibration Control of Plates by Plate-Type Dynamic Vibration Absorbers

1995 ◽  
Vol 117 (3A) ◽  
pp. 332-338 ◽  
Author(s):  
T. Aida ◽  
K. Kawazoe ◽  
S. Toda
Keyword(s):  
Degrees Of Freedom ◽  
Large Scale ◽  
Equations Of Motion ◽  
Harmonic Excitation ◽  
Vibration Absorber ◽  
Dynamic Vibration Absorber ◽  
Two Degrees Of Freedom ◽  
Tuning Method ◽  
Dynamic Vibration ◽  
Plate Type

In this paper, a new plate-type dynamic vibration absorber is presented for controlling the several predominant modes of vibration of plate (mainplate) under harmonic excitation, which consists of a plate (dynamic absorbing plate) under the same boundary condition as the main plate and with uniformly distributed connecting springs and dampers between the main and dynamic absorbing plates. Equations of motion of the system in the modal coordinates of the main plate become equal to those of the two-degrees-of-freedom system with two masses and three springs. Formulas for optimum design of the plate-type dynamic vibration absorber are presented using the optimum tuning method of a dynamic absorber in two-degrees-of-freedom system, obtained by the Den Hartog method. Moreover, for practical problems regarding large-scale plates, an approximate tuning method of the plate-type dynamic absorbers with several sets of concentrated connecting springs and dampers is also presented. The numerical calculations demonstrate the effectiveness of the plate-type dynamic absorbers.

scholarly journals VII.—On the Adelphic Integral of the Differential Equations of Dynamics

1918 ◽  
Vol 37 ◽  
pp. 95-116 ◽  
Author(s):  
E. T. Whittaker
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Two Degrees Of Freedom ◽  
Image Position ◽  
Kinetic And Potential Energies

§ 1. Ordinary and singular periodic solutions of a dynamical system. — The present paper is concerned with the motion of dynamical systems which possess an integral of energy. To fix ideas, we shall suppose that the system has two degrees of freedom, so that the equations of motion in generalised co-ordinates may be written in Hamilton's formwhere (q1q2) are the generalised co-ordinates, (p1, p2) are the generalised momenta, and where H is a function of (q1, q2, p1, p2) which represents the sum of the kinetic and potential energies.

Some Applications of Automatic Differentiation to Rigid, Flexible, and Constrained Multibody Dynamics

2005 ◽  
Author(s):  
D. Todd Griffith ◽  
James D. Turner ◽  
John L. Junkins
Keyword(s):  
Dynamical Systems ◽  
Multibody Dynamics ◽  
Multibody Systems ◽  
Automatic Differentiation ◽  
Equations Of Motion ◽  
Numerical Results ◽  
Flexible Multibody Systems ◽  
New Approach ◽  
Constrained Dynamical Systems ◽  
Constrained Multibody Dynamics

In this paper, we discuss several applications of automatic differentiation to multibody dynamics. The scope of this paper covers the rigid, flexible, and constrained dynamical systems. Particular emphasis is placed on the development of methods for automating the generation of equations of motion and the simulation of response using automatic differentiation. We also present a new approach for generating exact dynamical representations of flexible multibody systems in a numerical sense using automatic differentiation. Numerical results will be presented to detail the efficiency of the proposed methods.

scholarly journals Technical note: Characterising and comparing different palaeoclimates with dynamical systems theory

2021 ◽  
Vol 17 (1) ◽  
pp. 545-563
Author(s):  
Gabriele Messori ◽  
Davide Faranda
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Degrees Of Freedom ◽  
Large Scale ◽  
Climate Science ◽  
Dynamical Systems Theory ◽  
Technical Note ◽  
Sea Level Pressure ◽  
High Resolution Data ◽  
Systems Framework

Abstract. Numerical climate simulations produce vast amounts of high-resolution data. This poses new challenges to the palaeoclimate community – and indeed to the broader climate community – in how to efficiently process and interpret model output. The palaeoclimate community also faces the additional challenge of having to characterise and compare a much broader range of climates than encountered in other subfields of climate science. Here we propose an analysis framework, grounded in dynamical systems theory, which may contribute to overcoming these challenges. The framework enables the characterisation of the dynamics of a given climate through a small number of metrics. These may be applied to individual climate variables or to several variables at once, and they can diagnose properties such as persistence, active number of degrees of freedom and coupling. Crucially, the metrics provide information on instantaneous states of the chosen variable(s). To illustrate the framework's applicability, we analyse three numerical simulations of mid-Holocene climates over North Africa under different boundary conditions. We find that the three simulations produce climate systems with different dynamical properties, such as persistence of the spatial precipitation patterns and coupling between precipitation and large-scale sea level pressure patterns, which are reflected in the dynamical systems metrics. We conclude that the dynamical systems framework holds significant potential for analysing palaeoclimate simulations. At the same time, an appraisal of the framework's limitations suggests that it should be viewed as a complement to more conventional analyses, rather than as a wholesale substitute.

The motion of a lunar orbiter

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.

Sign in / Sign up

Export Citation Format

Share Document