Numerical integration scheme using singular perturbation method

Some multi degree-of-freedom dynamical systems exhibit a response that contain fast and slow variables. An example of such systems is a multibody system with rigid and deformable bodies. Standard numerical integration of the resultant equations of motion must adjust the time step according to the frequency of the fastest variable. As a result, the computation time is sacrificed. The singular perturbation method is an analysis technique to deal with the interaction of slow and fast variables. In this study, a numerical integration scheme using the singular perturbation method is discussed, its absolute stability condition is derived, and its order of accuracy is investigated. Copyright © 2013 by ASME

CFD investigation of a complete floating offshore wind turbine

This chapter presents numerical computations for floating offshore wind turbines for a machine of 10-MW rated power. The rotors were computed using the Helicopter Multi-Block flow solver of the University of Glasgow that solves the Navier-Stokes equations in integral form using the arbitrary Lagrangian-Eulerian formulation for time-dependent domains with moving boundaries. Hydrodynamic loads on the support platform were computed using the Smoothed Particle Hydrodynamics method. This method is mesh-free, and represents the fluid by a set of discrete particles. The motion of the floating offshore wind turbine is computed using a Multi-Body Dynamic Model of rigid bodies and frictionless joints. Mooring cables are modelled as a set of springs and dampers. All solvers were validated separately before coupling, and the loosely coupled algorithm used is described in detail alongside the obtained results

Methods to solve the equations of motion

"Series title: Springerbriefs in applied sciences and technology, ISSN 2191-530X"This chapter presents several methods to solve the equations of motion of spatial multibody systems. In particular, the standard approach, the Baumgarte method, the penalty method and the augmented Lagrangian formulation are revised here. In this process, a general procedure for dynamic analysis of multibody systems based on the standard Lagrange multipliers method is described. Moreover, the implications in terms of the resolution of the equations of motion, accuracy and efficiency are also discussed in this chapter.(undefined

Equations of motion for constrained systems

"Series title: Springerbriefs in applied sciences and technology, ISSN 2191-530X"In this chapter, the formulation of motion’s equations of multi-rigid body systems is described. The generalized coordinates are the centroidal Cartesian coordinates, being the system configuration restrained by constraint equations. The present formulation uses the Newton-Euler’s equations of motion, which are augmented with the constraint equations that lead to a system of differential algebraic equations. This formulation is straightforward in terms of assembling the equations of motion and providing all reaction forces.(undefined

Correction of the initial conditions

"Series title: Springerbriefs in applied sciences and technology, ISSN 2191-530X"This chapter presents a general approach to deal with the correction of the initial conditions at the position and velocity levels. This procedure is of paramount importance to avoid constraints violation during the numerical resolution of the equations of motion. The material presented here closely follows the standard methodologies available in the literature. Thus, in this chapter, a simple and efficient approach to correct the initial conditions at the position and velocity levels is revised.(undefined

Numerical integration scheme using singular perturbation method

Some multi degree-of-freedom dynamical systems exhibit a response that contain fast and slow variables. An example of such systems is a multibody system with rigid and deformable bodies. Standard numerical integration of the resultant equations of motion must adjust the time step according to the frequency of the fastest variable. As a result, the computation time is sacrificed. The singular perturbation method is an analysis technique to deal with the interaction of slow and fast variables. In this study, a numerical integration scheme using the singular perturbation method is discussed, its absolute stability condition is derived, and its order of accuracy is investigated. Copyright © 2013 by ASME

Local Linearization Method in the integration of multibody equations

Computational efficiency of solving the dynamics of highly oscillatory systems is an important issue due to the requirement of small step size of explicit numerical integration algorithms. A system is considered to have an oscillatory solution if it contains a fast solution that varies regularly about a slow solution. This paper investigates the use of the so-called Local Linearization Method (LLM) in the integration of multibody equations of motion that exhibit oscillatory behavior. The LLM is an exponential method that is based on the piecewise linear approximation of the equations through a firstorder Taylor expansion at each time step, where the solution at the next time step is determined by the analytic solution of the approximated linear system. In this paper the LLM is applied to simple examples. The results show that the LLM can improve computational efficiency, without jeopardizing the accuracy, when the multibody system is highly oscillatory

Angular velocity and acceleration

In this chapter, a complete characterization of the angular velocity and angular acceleration for rigid bodies in spatial multibody systems are presented. For both cases, local and global formulations are described taking into account the advantages of using Euler parameters. In this process, the transformation between global and local components of the angular velocity and time derivative of the Euler parameters are analyzed and discussed in this chapter.(undefined

Definition of multibody system

This chapter presents a general view of multibody system concept and definition by describing the main features associated with spatial systems. The mechanical components, which can be modeled as rigid or flexible, are constrained by kinematic pair of different types. Additionally, the bodies can be actuated upon by force elements and external forces due to interaction with environment. This chapter also presents some examples of application of multibody systems that can include automotive vehicles, mechanisms, robots and biomechanical systems.(undefined