PERIODIC MOTIONS AND NONLINEAR DYNAMICS OF A WHEELSET MODEL

1999 ◽  
Vol 09 (10) ◽  
pp. 1983-1994 ◽  
Author(s):  
CORNELIA FRANKE
Keyword(s):  
Nonlinear Dynamics ◽  
Periodic Solutions ◽  
Special Class ◽  
Multibody System ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Hopf Bifurcations ◽  
Algebraic Equations ◽  
Periodic Motions

Periodic motions and nonlinear dynamics of a wheelset model are investigated numerically. The equations of motion of this multibody system belong to a special class of differential-algebraic equations (DAEs). In contrast to previous investigations of wheelset models the equations are treated directly as DAEs and are not reduced by simplifications to an explicit ODE. Further, it is shown how basic tools for the analysis of Hopf bifurcations and stability of periodic solutions can be transferred to this class of DAEs.

Rotorcraft Trim Analysis Using a General-Purpose Multibody Code

2008 ◽  
Author(s):  
L. Federico ◽  
A. Russo
Keyword(s):  
Aerodynamic Force ◽  
Dimensional Space ◽  
Equations Of Motion ◽  
Original System ◽  
Differential Algebraic Equations ◽  
General Purpose ◽  
Integration Scheme ◽  
Algebraic Equations ◽  
Periodic Motions ◽  
Wide Range

Rotorcraft dynamics represents a major analytical challenge to aeronautical industry and research centres. Complexities arising from large rigid motions, body elasticity, aerodynamic loads and control systems have to be taken into account in order to ensure the accuracy of a comprehensive analysis. Architected for the nonlinearities associated with large motion in three-dimensional space, the ADAMS general-purpose multibody code allows to automatically formulate and integrate the equations of motion for a wide range of mechanisms, including rotary wing systems (once provided with an aerodynamic force field description). However, the ADAMS simulation system lacks the capability to calculate periodic motions, as required in the helicopter trim analysis and stability evaluation. The prediction of the trimmed periodic motions of the rotor system implies the numerical solution of differential-algebraic boundary value problem. In this work we present a new approach to perform this task inside the ADAMS numerical environment. Thia approach is based on the perturbation of the minimal set of Ordinary Differential Equations (ODEs), being equivalent to the original system of Differential Algebraic Equations (DAEs) which defines the rotorcraft equation of motion. The transformation of DAEs to ODEs is based on the linearization of the local constraint manifold defined by the algebraic constraint equations, as suggested by Maggi in his work [1–3]. The proposed method is quite general and can be used to drive the ADAMS integration scheme within the periodic motion analysis of mechanical systems. The algotithm is adopted to simulate the wind tunnel trim test of a ECD BO105 machscaled model (EU HeliNOVI project [4]). Comparisons between numerical and experimental results are provided.

Analytical Modelling of the Dynamics of the Four-Bar Mechanism: A Comparison of Some Methods

2018 ◽  
Author(s):  
J. P. Meijaard ◽  
V. van der Wijk
Keyword(s):  
Virtual Work ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Force Balance ◽  
Dynamic Force ◽  
Algebraic Equations ◽  
Principle Of Virtual Work ◽  
Principal Vector ◽  
Reaction Forces ◽  
Principal Points

Some thoughts about different ways of formulating the equations of motion of a four-bar mechanism are communicated. Four analytic methods to derive the equations of motion are compared. In the first method, Lagrange’s equations in the traditional form are used, and in a second method, the principle of virtual work is used, which leads to equivalent equations. In the third method, the loop is opened, principal points and a principal vector linkage are introduced, and the equations are formulated in terms of these principal vectors, which leads, with the introduced reaction forces, to a system of differential-algebraic equations. In the fourth method, equivalent masses are introduced, which leads to a simpler system of principal points and principal vectors. By considering the links as pseudorigid bodies that can have a uniform planar dilatation, a compact form of the equations of motion is obtained. The conditions for dynamic force balance become almost trivial. Also the equations for the resulting reaction moment are considered for all four methods.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

scholarly journals Periodic solutions of differential-algebraic equations

2020 ◽  
Vol 25 (4) ◽  
pp. 1383-1395
Author(s):  
Yingjie Bi ◽  
◽  
Siyu Liu ◽  
Yong Li ◽  
◽  
...  
Keyword(s):  
Periodic Solutions ◽  
Differential Algebraic Equations ◽  
Algebraic Equations

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

Automating the Derivation of the Equations of Motion of a Multibody Dynamic System With Uncertainty Using Polynomial Chaos Theory and Variational Work

2019 ◽  
Vol 15 (1) ◽  
Author(s):  
Paul S. Ryan ◽  
Sarah C. Baxter ◽  
Philip A. Voglewede
Keyword(s):  
Chaos Theory ◽  
Polynomial Chaos ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Multibody Dynamic ◽  
Mass Spring ◽  
Polynomial Chaos Theory ◽  
Multibody Dynamic System ◽  
Crank Mechanism

Abstract Understanding how variation impacts a multibody dynamic (MBD) system's response is important to ensure the robustness of a system. However, how the variation propagates into the MBD system is complicated because MBD systems are typically governed by a system of large differential algebraic equations. This paper presents a novel process, variational work, along with the polynomial chaos multibody dynamics (PCMBoD) automation process for utilizing polynomial chaos theory (PCT) in the analysis of uncertainties in an MBD system. Variational work allows the complexity of the traditional PCT approach to be reduced. With variational work and the constrained Lagrangian formulation, the equations of motion of an MBD PCT system can be constructed using the PCMBoD automated process. To demonstrate the PCMBoD process, two examples, a mass-spring-damper and a two link slider–crank mechanism, are shown.

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