scholarly journals Mathematical Formulation of Soft-Contact Problems for Various Rheological Models of Damper

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Wiesław Grzesikiewicz ◽  
Artur Zbiciak
Keyword(s):  
Mathematical Formulation ◽  
Contact Problems ◽  
Differential Algebraic Equations ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Unilateral Constraints ◽  
Soft Contact ◽  
Rheological Models ◽  
Model Interaction ◽  
Time Histories

The paper deals with analysis of selected soft-contact problems in discrete mechanical systems. Elastic-dissipative rheological schemes representing dampers as well as the notion of unilateral constraints were used in order to model interaction between colliding bodies. The mathematical descriptions of soft-contact problems involving variational inequalities are presented. The main finding of the paper is a method of description of soft-contact phenomenon between rigid object and deformable rheological structure by the system of explicit nonlinear differential-algebraic equations easy for numerical implementation. The results of simulations, that is, time histories of displacements and contact forces as well as hysteretic loops, are presented.

DYNAMICS OF MULTIBODY SYSTEMS WITH UNILATERAL CONSTRAINTS

1999 ◽  
Vol 09 (03) ◽  
pp. 473-478 ◽  
Author(s):  
M. WÖSLE ◽  
F. PFEIFFER
Keyword(s):  
Contact Problem ◽  
Multibody Systems ◽  
Mathematical Formulation ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
Mathematical Form ◽  
Algebraic Equations ◽  
Unilateral Constraints ◽  
Stick Slip ◽  
Dynamics Of Multibody Systems

In couplings of machines and mechanisms, backlash and friction phenomena are always occurring. Whether stick–slip phenomena take place depends on the structure of such couplings. These processes can be modeled as multibody systems with a time-varying topology. Making use of Lagrange multiplier methods with a mathematical formulation of the contact problem is very efficient for large systems with many constraints. In the following, the differential-algebraic equations are transformed into a resolvable mathematical form by means of the contact laws in equation form. Ultimately we get a nonlinear system of equations for the three-dimensional contact problem with dependent constraints. For its solution, the homotopy method will be used and applied to a simple mechanical system.

Dynamics of Multibody Systems Containing Dependent Unilateral Constraints with Friction

1996 ◽  
Vol 2 (2) ◽  
pp. 161-192 ◽  
Author(s):  
M. Wösle ◽  
F. Pfeiffer
Keyword(s):  
Multibody Systems ◽  
Convex Sets ◽  
Equations Of Motion ◽  
Mathematical Formulation ◽  
Differential Algebraic Equations ◽  
Mathematical Form ◽  
Algebraic Equations ◽  
Unilateral Constraints ◽  
Constraint Forces ◽  
Stick Slip

In couplings of machines and mechanisms, backlash and friction phenomena are always occurring. Whether stick-slip phenomena take place depends on the structure of such couplings. These processes can be modeled as multibody systems with a time-varying topology. Making use of Lagrange multiplier methods with a mathematical formulation of the contact problem is very efficient for large systems with many constraints. The differential-algebraic equations of a system are transformed into a resolvable mathematical form by means of contact laws. In the following, rigid multibody systems with dependent constraints and planar friction will be considered. For the evaluation of such problems, an iterative algorithm is presented. This method is based on transformations of the kinematic secondary conditions in the form of inequalities to equations. In mathematical sense, these transformations are projections of the constraint forces on convex sets. Ultimately, we have a solvable nonlinear system of equations consisting of the differential equations of motion, the constraint equations and the projections of the constraint forces.

Adding Non-Smooth Analysis Capabilities to General-Purpose Multibody Dynamics by Co-Simulation

2013 ◽  
Author(s):  
Matteo Fancello ◽  
Pierangelo Masarati ◽  
Marco Morandini
Keyword(s):  
Differential Algebraic Equations ◽  
General Purpose ◽  
Rigid Body Dynamics ◽  
Nonsmooth Dynamics ◽  
Algebraic Equations ◽  
Unilateral Constraints ◽  
Frictional Contacts ◽  
Continuous Contact ◽  
Hard Constraints ◽  
Dynamics Problems

Multi-rigid-body dynamics problems with unilateral constraints, like frictionless and frictional contacts, are characterized by nonsmooth dynamics. The issue of nonsmoothness can be addressed with methods that apply a mathematical regularization, called continuous contact methods; alternatively, hard constraints with complementarity approaches can be proficiently used. This work presents an attempt at integrating consistently modeled unilateral constraints in a general purpose multibody formulation and implementation originally designed to address intrinsically smooth problems. The focus is on the analysis of generally smooth problems, characterized by significant multidisciplinarity, with the need to selectively include nonsmooth events localized in time and in specific components of the model. A co-simulation approach between the smooth Differential-Algebraic Equations solver and the classic Moreau-Jean timestepping approach is devised as an alternative to entirely redesigning a monolithic nonsmooth solver, in order to provide elements subject to frictionless and frictional contact in the general-purpose, free multibody solver MBDyn. The implementation uses components from the INRIA’s Siconos library for the solution of Complementarity Problems. The proposed approach is applied to several problems of increasing complexity to empirically evaluate its properties and versatility. The applicability of the family of second-order accurate, A/L stable multistep integration algorithms used by MBDyn to nonsmooth dynamics is also discussed and assessed.

scholarly journals Two-Mode Resonator and Contact Model for Standing Wave Piezomotor

2001 ◽  
Author(s):  
Brian Andersen ◽  
Mogens Blanke ◽  
Jan Helbo
Keyword(s):  
Standing Wave ◽  
Contact Stiffness ◽  
Ritz Method ◽  
Differential Algebraic Equations ◽  
Contact Forces ◽  
Piezoelectric Motor ◽  
Algebraic Equations ◽  
Bending Mode ◽  
Resonance Frequencies ◽  
Mode Resonator

Abstract The paper presents a model for a standing wave piezoelectric motor with a two bending mode resonator. The resonator is modelled using Hamilton’s principle and the Rayleigh-Ritz method. The contact is modelled using the Lagrange Multiplier method under the assumption of slip and it is shown how to solve the set of differential-algebraic equations. Detailed simulations show resonance frequencies as function of the piezoelement’s position, tip trajectories and contact forces. The paper demonstrates that contact stiffness and stick should be included in such a model to obtain physically realistic results and a method to include stick is suggested.

Planar Multi-body Dynamics of a Tracked Vehicle using Imaginary Wheel Model for Tracks

2017 ◽  
Vol 67 (4) ◽  
pp. 460
Author(s):  
Ilango Mahalingam ◽  
Chandramouli Padmanabhan
Keyword(s):  
Interaction Model ◽  
Differential Algebraic Equations ◽  
Rigid Bodies ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Tracked Vehicle ◽  
The Road ◽  
Wheel Model ◽  
Longitudinal Slip ◽  
Multi Body

<p class="p1">Off-road vehicles achieve their mobility with the help of a track system. A track has large number of rigid bodies with pin joints leading to computational complexity in modelling the dynamic behaviour of the system. In this paper, a new idea is proposed, where the tracks are replaced by a set of imaginary wheels connected to the road wheels using mechanical links. A non-linear wheel terrain interaction model considering longitudinal slip is used to find out the normal and tangential contact forces. A linear trailing arm suspension, where a road arm connecting the road wheel and chassis with a rotational spring and damper system is considered. The differential algebraic equations (DAEs) from the multi-body model are derived in Cartesian coordinates and formulated using augmented formulation. The augmented equations are solved numerically using appropriate stabilisation techniques. The novel proposition is validated using experimental measurements done on a tracked vehicle.</p>

Numerical Computation of Differential-Algebraic Equations for Non-Linear Dynamics of Multibody Systems Involving Contact Forces

1999 ◽  
Vol 123 (2) ◽  
pp. 272-281 ◽  
Author(s):  
B. Fox ◽  
L. S. Jennings ◽  
A. Y. Zomaya
Keyword(s):  
Multibody Systems ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Variational Problems ◽  
Differential Algebraic Equations ◽  
Contact Forces ◽  
Algebraic Equations ◽  
Differential Algebraic Equation ◽  
Lagrange Equations ◽  
Constraint Equations

The well known Euler-Lagrange equations of motion for constrained variational problems are derived using the principle of virtual work. These equations are used in the modelling of multibody systems and result in differential-algebraic equations of high index. Here they concern an N-link pendulum, a heavy aircraft towing truck and a heavy off-highway track vehicle. The differential-algebraic equation is cast as an ordinary differential equation through differentiation of the constraint equations. The resulting system is computed using the integration routine LSODAR, the Euler and fourth order Runge-Kutta methods. The difficulty to integrate this system is revealed to be the result of many highly oscillatory forces of large magnitude acting on many bodies simultaneously. Constraint compliance is analyzed for the three different integration methods and the drift of the constraint equations for the three different systems is shown to be influenced by nonlinear contact forces.

scholarly journals Non-autonomous higher-order Moreau's sweeping process: Well-posedness, stability and Zeno trajectories

2018 ◽  
Vol 29 (5) ◽  
pp. 941-968 ◽  
Author(s):  
BERNARD BROGLIATO
Keyword(s):  
Dynamical Systems ◽  
Differential Algebraic Equations ◽  
Higher Order ◽  
Well Posedness ◽  
Sweeping Process ◽  
Algebraic Equations ◽  
Unilateral Constraints ◽  
Uniqueness Of Solutions ◽  
Autonomous Case ◽  
Moreau’S Sweeping Process

In this article, we study the higher-order Moreau's sweeping process introduced in [1], in the case where an exogenous time-varying functionu(·) is present in both the linear dynamics and in the unilateral constraints. First, we show that the well-posedness results (existence and uniqueness of solutions) obtained in [1] for the autonomous case, extend to the non-autonomous case whenu(·) is smooth and piece-wise analytic, after a suitable state transformation is done. Stability issues are discussed. The complexity of such non-smooth non-autonomous dynamical systems is illustrated in a particular case named the higher-order bouncing ball, where trajectories with accumulations of jumps are exhibited. Examples from mechanics and circuits illustrate some of the results. The link with complementarity dynamical systems and with switching differential-algebraic equations is made.

Markov Jump Linear System Analysis of a Microgrid Operating in Islanded and Grid Tied Modes

2020 ◽  
Author(s):  
Gilles Mpembele ◽  
Jonathan Kimball
Keyword(s):  
Monte Carlo ◽  
Power Systems ◽  
Power System ◽  
System Analysis ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Markov Jump ◽  
Numerical Application ◽  
Conditional Moments ◽  
Markov Jump Linear Systems

<div>The analysis of power system dynamics is usually conducted using traditional models based on the standard nonlinear differential algebraic equations (DAEs). In general, solutions to these equations can be obtained using numerical methods such as the Monte Carlo simulations. The use of methods based on the Stochastic Hybrid System (SHS) framework for power systems subject to stochastic behavior is relatively new. These methods have been successfully applied to power systems subjected to</div><div>stochastic inputs. This study discusses a class of SHSs referred to as Markov Jump Linear Systems (MJLSs), in which the entire dynamic system is jumping between distinct operating points, with different local small-signal dynamics. The numerical application is based on the analysis of the IEEE 37-bus power system switching between grid-tied and standalone operating modes. The Ordinary Differential Equations (ODEs) representing the evolution of the conditional moments are derived and a matrix representation of the system is developed. Results are compared to the averaged Monte Carlo simulation. The MJLS approach was found to have a key advantage of being far less computational expensive.</div>

Sign in / Sign up

Export Citation Format

Share Document