scholarly journals Speed Oscillations of a Vehicle Rolling on a Wavy Road

2021 ◽  
Vol 11 (21) ◽  
pp. 10431
Author(s):  
Walter V. Wedig
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Travel Speed ◽  
Differential Algebraic Equations ◽  
Polar Coordinates ◽  
Algebraic Equations ◽  
Traffic Lights ◽  
Analytical Approximations ◽  
Vertical Vibrations ◽  
Nonlinear Equations Of Motion

Every driver knows that his car is slowing down or accelerating when driving up or down, respectively. The same happens on uneven roads with plastic wave deformations, e.g., in front of traffic lights or on nonpaved desert roads. This paper investigates the resulting travel speed oscillations of a quarter car model rolling in contact on a sinusoidal and stochastic road surface. The nonlinear equations of motion of the vehicle road system leads to ill-conditioned differential–algebraic equations. They are solved introducing polar coordinates into the sinusoidal road model. Numerical simulations show the Sommerfeld effect, in which the vehicle becomes stuck before the resonance speed, exhibiting limit cycles of oscillating acceleration and speed, which bifurcate from one-periodic limit cycle to one that is double periodic. Analytical approximations are derived by means of nonlinear Fourier expansions. Extensions to more realistic road models by means of noise perturbation show limit flows as bundles of nonperiodic trajectories with periodic side limits. Vehicles with higher degrees of freedom become stuck before the first speed resonance, as well as in between further resonance speeds with strong vertical vibrations and longitudinal speed oscillations. They need more power supply in order to overcome the resonance peak. For small damping, the speeds after resonance are unstable. They migrate to lower or supercritical speeds of operation. Stability in mean is investigated.

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.

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.

Ride Dynamics of a Flexible Multibody Model of an Urban Bus

2007 ◽  
Author(s):  
G. Georgiou ◽  
A. Badarlis ◽  
S. Natsiavas
Keyword(s):  
Degrees Of Freedom ◽  
Ride Comfort ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Accurate Information ◽  
Algebraic Equations ◽  
Multibody Model ◽  
Road Excitation ◽  
Urban Bus ◽  
Stiffness And Damping

Dynamic response of a large order mechanical model of an urban bus is investigated. The emphasis is first put on developing a quite complete model, which can be utilized in order to extract sufficiently reliable and accurate information related to its dynamics in a fast way. Since some of the components of the bus undergo large rigid body rotation, in addition to motion resulting from their deformability, a multibody dynamics framework is adopted. This implies that the resulting equations of motion appear in the form of a strongly nonlinear set of differential-algebraic equations, which are difficult to handle even numerically. In fact, the modeling becomes more involved because all the significant nonlinearities appearing in the interconnections of the structural components and especially in the front and rear suspension subsystems of the bus are taken into account. In order to alleviate some of these complexities, the number of degrees of freedom of each component, associated with its deformability, is reduced drastically by applying an appropriate coordinate condensation methodology. Finally, this model is employed and numerical results are obtained for motions resulting from typical road excitation. In particular, selected response quantities related to ride comfort are examined for characteristic combinations of the bus suspension stiffness and damping parameters.

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.

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.

APPLICATION OF THE FINITE STRIP METHOD TO MODELING OF VIBRATIONS OF COLLECTING ELECTRODES

2013 ◽  
Vol 13 (07) ◽  
pp. 1340001 ◽  
Author(s):  
IWONA ADAMIEC-WÓJCIK ◽  
ANDRZEJ NOWAK ◽  
STANISŁAW WOJCIECH
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Geometrical Parameters ◽  
Experimental Measurements ◽  
Newmark Method ◽  
Algebraic Equations ◽  
Finite Strip ◽  
Strip Method ◽  
Finite Strip Method ◽  
Large Length

The paper presents an application of the finite strip method to modeling of vibrations of the collecting electrodes, which are shells with large length (up to 16 m), width of 0.5 m and thickness of 0.002 m. The models and computer programs have been worked out and validated. Comparison of results obtained from numerical simulations and experimental measurements are presented and discussed. The equations of motion have been solved using methods for solution of sparse algebraic equations and Newmark method. The strip method has proved to be numerically effective. The programs enable us to carry out calculations for a system with several hundred thousands of degrees of freedom with time of analysis requiring thousand integration steps during less than 90 min on a PC computer. High numerical efficiency enables the geometrical parameters of the collecting electrodes to be selected in order to ensure large accelerations caused by a beater to be spread evenly over the surface of the electrodes. Conclusions concerning the influence of length of the collecting electrodes on the normal and tangentz accelerations are formulated.

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.

Application of Scaling to Multibody Dynamics Simulations

2015 ◽  
Author(s):  
William Prescott
Keyword(s):  
Equations Of Motion ◽  
Industrial Applications ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Virtual Lab ◽  
Long Run ◽  
Multibody Dynamic ◽  
Dynamics Simulations ◽  
Stability Issues

This paper will examine the importance of applying scaling to the equations of motion for multibody dynamic systems when applied to industrial applications. If a Cartesian formulation is used to formulate the equations of motion of a multibody dynamic system the resulting equations are a set of differential algebraic equations (DAEs). The algebraic components of the DAEs arise from appending the joint equations used to model revolute, cylindrical, translational and other joints to the Newton-Euler dynamic equations of motion. Stability issues can arise in an ill-conditioned Jacobian matrix of the integration method this will result in poor convergence of the implicit integrator’s Newton method. The repeated failures of the Newton’s method will require a small step size and therefore simulations that require long run times to complete. Recent advances in rescaling the equations of motion have been proposed to address this problem. This paper will see if these methods or a variant addresses not only stability concerns, but also efficiency. The scaling techniques are applied to the Gear-Gupta-Leimkuhler (GGL) formulation for multibody problems by embedding them into the commercial multibody code (MBS) Virtual. Lab Motion and then use them to solve an industrial sized automotive example to see if performance is improved.

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