An object-oriented approach to multibody systems analysis

Methods and identities for computation of kinematic and kinetic derivatives required for a broad spectrum of multibody system analyses are presented. Analyses such as implicit numerical integration of the differential–algebraic equations of multibody dynamics, dynamic sensitivity analysis, and workspace analysis are shown to require computation of three derivatives of algebraic constraint functions and first derivatives of inertia and force expressions. Computationally efficient derivative calculation methods and associated identities are presented for Cartesian generalized coordinates, with Euler parameters for orientation. Results presented enable practical and efficient computation of all derivatives required in multibody mechanical system analysis.

The double pendulum, a simple system that exhibits complex dynamics, is used to demonstrate a method based on Hamilton’ s weak principle (HWP) for assembling and solving the maximum coordinate set equations for a multibody system. Results calculated using HWP are compared to a reference solution obtained by integrating the ordinary differential equations (ODE) in minimum coordinate set form. The HWP solution is shown to be comparable to the ODE solution in accuracy and computational efe ciency. These results suggest an alternative architecture for multibody system analysis, which implements the maximum coordinate set form of the system equations, and a simple, efe cient, and robust solution algorithm.

Different finite‐element‐based strategies used to represent the components' flexibility in multibody systems lead to various sets of co‐ordinates. For systems in which the bodies only experience small elastic deformations it is common to use mode component synthesis to reduce the number of generalized elastic co‐ordinates and, consequently, the equations of motion are written in terms of modal co‐ordinates. However, when the system components experience non‐linear deformations the use of reduction methods is not possible, in general, and the finite element nodal co‐ordinates are the generalized co‐ordinates used. Furthermore, depending on the type of finite elements used to represent each flexible body, the nodal co‐ordinates may include all node rotations and translations or only some of each. Regardless of the type of generalized co‐ordinates adopted it is required that kinematic joints are defined. The complete set of joints available in a general‐purpose multibody code must include, for each particular type of joint, restrictions involving only rigid bodies, or only flexible bodies, or flexible and rigid bodies. Therefore, the effort put into the development and implementation of any joint is at least three times as much as the initial work done in the implementation of joints with rigid bodies only. The concept of virtual bodies provides a general framework to develop general kinematic joints for flexible multibody systems with minimal effort, regardless of the flexible co‐ordinates used. Initially, only a rigid constraint between the flexible and a massless rigid body is developed. Then, any kinematic joint that involves a flexible body is set with the massless rigid body instead, using the regular joint library of the multibody code. The major drawback is that for each kinematic joint involving a flexible body it is required to use six more co‐ordinates per virtual body and six more kinematic constraints. It is shown in this work that for small elastic deformations, for which the mode component synthesis is applied, the use of sparse matrix solvers can compensate for the computational overhead of involving more co‐ordinates and kinematic constraints in the system, due to the use of virtual bodies. For non‐linear deformations, where the generalized co‐ordinates are the global positions of the finite‐element nodes, the use of the virtual body concept does not require an increase in the number of system co‐ordinates or kinematic constraints. By introducing the rigid joint between the flexible body nodal co‐ordinates and the virtual body, with the use of Lagrange multipliers, and then solving the equations explicitly for these multipliers the resulting equations of motion for the subsystem have the same degrees of freedom as the original flexible body alone. The difference is that degrees of freedom associated to the virtual body are used as co‐ordinates of the subsystem instead of the nodal co‐ordinates of the nodes of the flexible body attached to the virtual body. Copyright ©2003 John Wiley & Sons, Ltd.

Dynamic recursive decoupling method for closed-loop flexible mechanical systems

Generally speaking, the flexible manufacturing systems can be classified into two main groups: open-loop and closed-loop systems. In this investigation, a recursive formulation is developed for the dynamic analysis of open-loop flexible manufacturing systems. The nonlinear generalized Newton-Euler equations are developed for rigid and deformable bodies that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars, vectors and matrices that depend on the spatial coordinates as well as the assumed displacement fields, and these time invariant quantities represent the dynamic coupling between the rigid body motion and elastic deformation. The method to solve equations of motion for open-loop systems consisting of interconnected rigid and deformable bodies is presented in this paper. This method applies recursive method with the Newton-Euler method for deformable bodies to obtain a large, loosely coupled system equations of motion. The solution techniques used to solve for the system equations of motion can be more efficiently implemented in the modern computer systems. The algorithms presented in this paper are demonstrated by using cylindrical joints that can be easily extended to revolute, slider and rigid joints. The recursive formulation developed in this investigation is illustrated by a practical numerical example.

The aim of this work is to implement a new nondestructive testing (NDT) method based on the metal magnetic memory (MMM) effect into comprehensive diagnostic approach for power-line connectors, which is of great practical importance. Essence of this method lies in detection of stress concentration zones (SCZs) in material of the considered element during its operation (before micro-cracks and fracture occurs). If to compare with the traditional NDT methods, which indicate SCZs after the failure (post factum), this method provides major progress in the field of NDT. In this paper an approach of connection of numerical structural simulation (static and dynamic) with MMM-based testing method is also proposed. SCZs in power-line mechanical connector were evaluated and dynamic analysis of a complete power line, which included the analyzed connector, was carried out by means of MSC.ADAMS. Uniqueness of this analysis consist in integration of FEM analysis, MSC.ADAMS multibody system analysis (for testing of working conditions in advance - during virtual prototyping) and verification using MMM method. This hybrid approach enables to accomplish diagnostic analysis of selected construction elements and provides the possibility to take into consideration actual working conditions (e.g. vibrations) of the element. Proposed diagnostics method can substantially increase operational safety of electrical connectors and lines.

For efficient dynamic simulation of mechanical and aerospace systems, the use of different sets of coordinate types may be necessary. The components of multibody systems can be rigid, flexible or very flexible and can be subject to contact forces. Examples of challenging problems encountered when multibody systems are considered are crashworthiness, problems of cables used in rescue operations and heavy load handling, belt drives, leaf spring system design, tire deformations, large deformation of high-speed rotors, stability problems, and contact problems. Most large displacement problems in structural mechanics are being solved using incremental solution procedures. On the other hand, general purpose flexible multibody computer tools and methodologies in existence today are not, in general, capable of systematically and efficiently solving applications that include, in addition to rigid bodies and bodies that undergo small deformations, bodies that experience very large deformations. The objective of this paper is to discuss the development of new computational algorithms, based on non-incremental solution procedures, for the computer simulation of multibody systems with flexible components. These new algorithms that do not require special measures to satisfy the principles of work and energy and lead to optimum sparse matrix structure can be used as the basis for developing a new generation of flexible multibody computer programs. The proposed non-incremental algorithms integrate three different formulations and three different sets of generalized coordinates for modeling rigid bodies, flexible bodies with small deformations, and very flexible bodies that undergo large deformations. The implementation of a general contact model in multibody algorithms is also presented as an example of mechanical systems with non-generalized coordinates. The kinematic equations that describe the contact between two surfaces of two bodies in the multi body system are formulated in terms of the system-generalized coordinates and the non-generalized surface parameters. Each contact surface is defined using two independent parameters that completely determine the tangent and normal vectors at an arbitrary point on the body surface. In the contact model presented in this study, the points of contact are determined on line during the dynamic simulation by solving the non-linear differential and algebraic equations of the constrained multibody system. The augmented form of the equations of motion expressed in terms of the generalized coordinates and non-generalized surface parameters is presented in this paper.

Abstract. Current space missions require predicting the spacecraft dynamics with considerable reliability. Among the various components of a spacecraft, subsystems like payload, structures, and power depend heavily on the dynamic behavior of the satellite during its operational life. Therefore, to ensure that the results obtained through numerical simulations correspond to the actual behavior, an accurate dynamical model must be developed. In this context, an implementation of Kane’s method is presented to derive the dynamical equations of a spacecraft composed of both rigid and flexible bodies connected via joints in tree topology. Starting from the kinematics of two generic interconnected bodies, a systematic approach is derived and the recursive structure of the equations is investigated. The Kane’s formulation allows a relatively simple derivation of the equation of motion while obtaining the minimum set of differential equations, which implies lower computational time. On the other hand, this formulation excludes reaction forces and torques from the dynamical equations. Nevertheless, in this work a strategy to compute them a posteriori without further numerical integrations is presented. Flexibility is introduced through the standard modal decomposition technique, so that modal shapes obtained by FEA software can be directly utilized to characterize the elastic motion of the flexible bodies. A spacecraft composed of a rigid bus and several flexible appendages is modeled and numerical simulations point out that this systematic method is very effective for this illustrative example.

<div class="htmlview paragraph">In this paper methods for the nonlinear dynamic analysis of multibody vehicle systems are presented. A total Lagrangian finite element formulation for deformable bodies of the vehicle systems that undergo large translational and rotational displacements is developed. A set of nonlinear dynamic differential equations are developed in a closed form for each finite element in terms of the reference as well as the nodal elstic coordinates and their time derivatives. These equations are expressed in terms of a set of element invariants that depend on the assumed displacement field. The invariants of the deformable body that undergoes large reference translational and rotational displacements can be obtained by assembling the invariants of its elements using a standard finite element procedure. The nonlinear dynamic equations of the deformable body can then be obtained by assembling the nonlinear element equations. Both the consistent and lumped mass finite element formulations are discussed. The dynamic formulations presented in this paper are exemplified using multibody vehicle systems that consist of interconnected rigid and deformable bodies. Numerical results are obtained using the general purpose computer program <b>DAMS</b> (Dynamic Analysis of Multibody Systems).</div>

This chapter provides a brief and very selective survey of some software useful for solving problems in analytical dynamics. I have chosen to include only software intended for academic and industrial use that has been validated for modeling accuracy. Consequently, I have omitted the many dynamics engines used for video games and entertainment purposes. General Purpose Mathematical Software A number of powerful tools for mathematical analysis exist. Two of the most popular are Mathematica by Wolfram Research Inc. and MATLAB by The MathWorks Inc. They provide capabilities for both symbolic manipulation of expressions (i.e., computer algebra systems) and numerical computation and evaluation of expressions. While the basic Mathematica and MATLAB applications do not provide high-level programming abstractions specific to multibody system modeling and analysis, they are extremely useful for the low-level symbolic and numerical processing required in solving problems in analytical dynamics (see Figure 10.1 for a screenshot of Mathematica). This has a benefit in the learning process, as they require the student to understand the central concepts of analytical dynamics and to set up the solution procedure in a detailed manner, while alleviating much of the tedium of the algebraic and numerical work. Packages and Extensions for Multibody Dynamics Wolfram Research and MathWorks both provide domain-specific extensions for analyzing multibody systems. This gives the user access to high-level abstractions for modeling, simulating, and analyzing multibody systems. These high-level abstractions can be accessed through a programming language or through a diagrammatic interface supporting both acausal and causal modeling. Acausal modeling involves component-based schematic representations of the physical system, whereas causal modeling involves block diagram representations of the underlying mathematical description of the system. Simscape Multibody (formerly SimMechanics) by The MathWorks Inc. allows users to model multibody systems using a library of blocks associated with bodies, joints, constraints, spatial transformations, force elements, and sensors (see Figure 10.2 for a screenshot of Simscape Multibody). Entire CAD assemblies can be imported along with associated geometry, inertial properties, joint parameters, and constraints. Simscape Multibody then formulates and integrates the equations of motion for the multibody system. Animations are generated during simulation to allow visualization of the motion of the system.

This paper describes a novel, system-level design methodology based on a genetic algorithm (GA) using freeform geometries for microelectromechanical systems (MEMS) devices. A MEMS accelerometer comprising a freeform mechanical motion preamplifier embedded in a closed-loop control system is presented to demonstrate the effectiveness of the design approach. The optimization process improves the main figure-of-merit (FOM) by 482%. Measurements show that the displacement of the MEMS accelerometer in the closed-loop system is decreased by 86% with a 4.85 V feedback voltage for 1 g acceleration at 100 Hz compared with an open-loop system.

Ionic wind has attracted increasing interest in promising applications of heat transfer, flow control of aircraft, solid-state propulsion, and plasma-assisted combustion. To enhance the stability of the ionic wind speed and the ability to withstand external disturbance, we developed a closed-loop control system of the ionic wind speed. The system mainly includes an air differential pressure sensor, microcontrollers, a homemade nanosecond pulsed power supply, and a surface dielectric barrier discharge (SDBD), which generates the ionic wind. When the applied voltage increases from 10.3 to 16.1 kV at a pulse frequency of 10 kHz, the ionic wind speed increases from 0 to 2.92 m/s with an average power consumption of less than 2.5 W. On the other hand, at a fixed voltage of 15.3 kV, the ionic wind speed increases almost linearly from 0.56 to 2.61 m/s with the increase of the pulse frequency from 2 to 10 kHz. Two feedback-loop control strategies based on the relationship among the pulse voltage, pulse frequency, and the ionic wind speed are designed based on the proportion-integral control of the repetition frequency and voltage amplitude. The comparison between the open-loop system and the closed-loop system of the ionic wind speed shows that the closed-loop system works with excellent stability and a solid ability to withstand external disturbance.

We present a Lagrangian–Lagrangian approach for the simulation of fully resolved Fluid Solid/Structure Interaction (FSI) problems. In the proposed approach, the method of Smoothed Particle Hydrodynamics (SPH) is used to simulate the fluid dynamics in a Lagrangian framework. The solid phase is a general multibody dynamics system composed of a collection of interacting rigid and deformable objects. While the motion of arbitrarily shaped rigid objects is approached in a classical 3D rigid body dynamics framework, the Absolute Nodal Coordinate Formulation (ANCF) is used to model the deformable components, thus enabling the investigation of compliant elements that experience large deformations with entangling and self-contact. The dynamics of the two phases, fluid and solid, are coupled with the help of Lagrangian markers, referred to as Boundary Condition Enforcing (BCE) markers which are used to impose no-slip and impenetrability conditions. Such BCE markers are associated both with the solid suspended particles and with any confining boundary walls and are distributed in a narrow layer on and below the surface of solid objects. The ensuing fluid–solid interaction forces are mapped into generalized forces on the rigid and flexible bodies and subsequently used to update the dynamics of the solid objects according to rigid body motion or ANCF method. The robustness and performance of the simulation algorithm is demonstrated through several numerical simulation studies.

In this paper, we discuss the motion of a tethered system. In general, a tether is a cable or wire rope, and a tethered system consists of a tether and attached equipment. A tethered subsatellite in space is an example of this system. We consider the tethered system consisting of a very flexible body (the tether) and rigid bodies at one end as our analytical model. A flexible body in planer motion is described using the Absolute Nodal Coordinate Formulation. Using this method, the motion of a flexible body with large deformation, rotation and translation can be expressed with the accuracy of rigid body motion. The combination of flexible body motion and rigid body motion is performed and the interaction between them is discussed. We also performed experiments to investigate the fundamental motion of the tethered system and to evaluate the validity of the numerical formulation. The first experiments were conducted using a steel tether and rubber tether in gravity space. We also conducted experiment of the motion of the tethered system with a rigid body in microgravity space. The numerical solutions using the proposed methods for the modeling and formulation for the tethered system are in good agreement with the experimental results.

Many identification strategies for closed loop system structure assume that an open loop system is closed by a feedback mechanism, which contains a known and linear controller. This assumption means these identification strategies are feasible under some prior knowledge of feedback controller. To relax this assumption for some complex systems with unknown controller or nonlinear controller, a new stealth identification strategy is proposed to tackle the identification problem for closed loop system with unknown controller or nonlinear controller. Stealth identification modifies closed loop system, so that the new prediction error and inverse covariance matrix are all independent of the unknown controller or nonlinear controller. This independence simplifies the problem of estimating parameter vector and designing optimal input. The Robbins-Monro algorithm is applied to identify the unknown parameter vector in closed loop system with unknown controller, and the nonlinear controller is replaced by its equivalent linear time invariant controller. Some consideration about the linear approximation to nonlinear system are studied, and two linear approximation forms are constructed to approximate the nonlinear system. The effectiveness of our proposed stealth identification strategy is demonstrated through simulation example.