Kinematic constraint equations

This chapter presents a general methodology for the formulation of the kinematic constraint equations at position, velocity and acceleration levels. Also a brief characterization of the different type of constraints is offered, namely the holonomic and nonholonomic constraints. The kinematic constraints described here are formulated using generalized coordinates. The chapter ends with a general approach to deal with the kinematic analysis of multibody systems.

Basic constraints between two vectors

This chapter deals with the characterization of the basic constraints between two vectors. This issue plays a crucial role in the formulation of constraint equations for mechanical joints. In particular, relations between two parallel and two perpendicular vectors are derived. Moreover, formulation for a vector that connects two generic points is presented. The material described here is developed under the framework of multibody systems formulation for spatial systems.

Kinematic joints constraints

"Series title: Springerbriefs in applied sciences and technology, ISSN 2191-530X"

Force elements and reaction forces

"Series title: Springerbriefs in applied sciences and technology, ISSN 2191-530X"

Methods to solve the equations of motion

"Series title: Springerbriefs in applied sciences and technology, ISSN 2191-530X"

Rakenteellisen jouston kuvaus reaaliaikasimuloinnissa

Työn tavoitteena oli tuottaa rakenteellisen jouston huomioiva monikappaledynmiikan simulointiohjelma Matlab-ympäristöön. Rakenteellinen jousto huomioitiin kelluvan koordinaatiston menetelmällä ja joustavuutta kuvaavat muodot ratkaistiin elementtimenetelmällä. Tehdyn ohjelman avulla voidaan koostaa joustavista kappaleista koostuvia avaruusmekanismeja ja tutkia niiden dynaamista käyttäytymistä. Simulointitulosta verrattiin kaupallisen ohjelmiston tuottamaan tulokseen. Työssä havaittiin, että kelluvan koordinaatiston menetelmä on käyttökelpoinen reaaliaikaiseen simulointiin. Työssä toteutetun ohjelman tulokset vastasivat kaupallisen simulointiohjelman tuloksia.

Liikkuvan kivimurskaimen syötinosan rakenneanalyysi osarakennetekniikan avulla

Työn tarkoituksena oli selvittää, miten osarakennetekniikkaa voidaan soveltaa siirrettävän kivimurskaimen syötinosan simuloinnissa. Tätä tutkittiin luomalla kahdella eri ohjelmistolla simulaatiomalli syötinosasta ja mallintamalla syötinosan runko joustavaksi kappaleeksi osarakennetekniikan avulla. Luotujen simulointimallien tarkkuutta selvitettiin vertaamalla niistä saatuja rungon jännityksiä tutkittavan rakenteen rungosta mitattuihin jännityksiin. Työn tarkoituksena oli myös tutkia, miten hyvin simulaatiomallit soveltuvat käytettäväksi syötinosan tuotekehityksessä. Tässä työssä käytettiin syötinosan simulaatiomallin luomiseen ANSYS-ohjelmistoa ja ADAMS-ohjelmistoa. Simulaatiomalleihin lisättiin tutkittavasta järjestelmästä mitattu ohjaussignaali sekä syötinosan jousien arvot. Järjestelmän rakenneominaisuudet saatiin suoraan valmistajan luovuttamista tiedoista. ADAMS-ohjelmistolla mallinnetussa simulaatiomallissa runko mallinnettiin joustavaksi ANSYS-ohjelmistossa, josta se siirrettiin ADAMS-ohjelmistoon. Saaduista tuloksista kävi ilmi, että osarakennetekniikkaa voidaan hyödyntää syötinosan joustavan rungon simuloinnissa. Tutkittavasta järjestelmästä mitatuissa jännityksissä ja simulaatiomalleista saaduissa jännityksissä oli eroja, mutta jännityshistorian muodot ja suuruusluokat vastasivat pääosin toisiaan. Tulosten parantamiseksi tulee selvittää lisää alkuarvoja tutkittavasta järjestelmästä ja varmistua nyt saatujen jousiparametrien oikeellisuudesta.

Development of Generic Simulation Model for Mobile Working Machines

This master’s thesis has been done for Drive! –project in which a new electric motor solution for mobile working machines is developed. Generic simulation model will be used as marketing and development tool. It can be used to model a wide variety of different vehicles with and without electric motor and to show customer the difference between traditionally build vehicles and those with new electric motor solution. Customers can also use simulation model to research different solutions for their own vehicles. At the start of the project it was decided that MeVEA software would be used as main simulation program and Simulink will only be used to simulate the operation of electrical components. Development of the generic model started with the research of these two software applications, simulation models which are made with them and how these simulation models can be build faster. Best results were used for building of generic simulation model. Finished generic model can be used to produce new tractor models for real-time simulations in short notice. All information about model is collected to one datasheet which can be easily filled by the user. After datasheet is filled a script will automatically build new simulation model in seconds. At the moment generic model is capable of building simulation models for wide variety of different tractors but it can be easily altered for other vehicle types too which would also benefit greatly from electric drive solution. Those could be for example wheel loaders and harvesters.

Análise do exercício de agachamento utilizando o método de Kane

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

O Método de Kane aplicado a um modelo de corpo humano para obtenção da pressão intradiscal na coluna lombar

Nos últimos anos tem surgido vários métodos para se estudar sistemas multicorpos. A vantagem destes métodos é a obtenção das equações do movimento de forma mais fácil e precisa. Robôs, manipuladores, mecanismos, satélites, máquinas em geral e sistemas biomcânicos, pelas suas características, podem ser vantajosamente tratados por formalismo multicorpo, mas precisamente, o método de Kane a um modelo do corpo humano, com a finalidade de se determinar a pressão intradiscal entre as vértebras L4/L5 da coluna lombar, em algumas atividades normais de vida diária, como: sentar, abaixar, pegar e elevar cargas, realizar movimentos de flexão e extensão. Pretendeu-se, também obter os esforços internos nas diversas articulações deste modelo. A pressão intradiscal representa um parâmetro importante para caracterizar a sobrecarga na coluna, podendo ter relação com a degeneração do disco intervertebral, o que produz muita dor em indivíduos. Inicialmente, o modelo foi proposto e o método de Kane foi aplicado para se obter as equações do movimento. Para resolvê-las, foi necessário realiza a filmagem de um sujeito realizando as atividades citadas, a fim de se obter as equações das trajetórias dos diversos segmentos do corpo humano. Após a simulação, os parâmetros desejados foram obtidos e comparados com valores experimentais citados na literatura.

Application of virtual certification techniques to vehicle design and track maintenance

Virtual certification partially substitutes by computer simulations the experimental techniques required for rail vehicle certification. In this paper, several works were these techniques were used in the vehicle design and track maintenance processes are presented. Dynamic simulation of multibody systems was used to virtually apply the EN14363 standard to certify the dynamic behaviour of vehicles. The works described are: assessment of a freight bogie design adapted to meter-gauge, assessment of a railway track layout for a subway network, freight bogie design with higher speed and axle load, and processing of the data acquired by a track recording vehicle for track maintenance.

Methodologies for Designing and Developing New Concepts in Vertical Transportation

Escalator and moving walkway are multibody systems with a design of more than a century. Developed methodology allows studying and improving any subsystem of both systems. In addition, new concepts can be developed and tested without the necessity and cost of a real construction. CITEF (Railway Technologies Research Centre) has been modelling escalators for more than four years. Several complex and innovative models has been developed to characterize static, kinematic and dynamic escalator behaviour. The high number of mechanical elements that are part of escalators complicate modelling task. In this way, methodologies and tools have been developed in order to automate these task and saving computational and time costs. Developed methodologies have been validated with the results of comparing real measurements and simulated outputs from a dynamic model.

Tratamiento consistente de restricciones e impacto en dinámica de sistemas multicuerpo

Esta tesis está enmarcada en el estudio de diferentes procedimientos numéricos para resolver la dinámica de un sistema multicuerpo sometido a restricciones e impacto, que puede estar compuesto por sólidos rígidos y deformables conectados entre sí por diversos tipos de uniones. Dentro de los métodos numéricos analizados se presta un especial interés a los métodos consistentes, los cuales tienen por objetivo que la energía calculada en cada paso de tiempo, para un sistema mecánico, tenga una evolución coherente con el comportamiento teórico de la energía. En otras palabras, un método consistente mantiene constante la energía total en un problema conservativo, y en presencia de fuerzas disipativas proporciona un decremento positivo de la energía total. En esta línea se desarrolla un algoritmo numérico consistente con la energía total para resolver las ecuaciones de la dinámica de un sistema multicuerpo. Como parte de este algoritmo se formulan energéticamente consistentes las restricciones y el contacto empleando multiplicadores de Lagrange, penalización y Lagrange aumentado. Se propone también un método para el contacto con sólidos rígidos representados mediante superficies implícitas, basado en una restricción regularizada que se adaptada adecuadamente para el cumplimiento exacto de la restricción de contacto y para ser consistente con la conservación de la energía total. En este contexto se estudian dos enfoques: uno para el contacto elástico puro (sin deformación) formulado con penalización y Lagrange aumentado; y otro basado en un modelo constitutivo para el contacto con penetración. En el segundo enfoque se usa un potencial de penalización que, en ausencia de componentes disipativas, restaura la energía almacenada en el contacto y disipa energía de forma consistente con el modelo continuo cuando las componentes de amortiguamiento y fricción son consideradas. This thesis focuses on the study of several numerical procedures used to solve the dynamics of a multibody system subjected to constraints and impact. The system may be composed by rigid and deformable bodies connected by different types of joints. Within this framework, special attention is paid to consistent methods, which preserve the theoretical behavior of the energy at each time step. In other words, a consistent method keeps the total energy constant in a conservative problem, and provides a positive decrease in the total energy when dissipative forces are present. A numerical algorithm has been developed for solving the dynamical equations of multibody systems, which is energetically consistent. Energetic consistency in contacts and constraints is formulated using Lagrange multipliers, penalty and augmented Lagrange methods. A contact methodology is proposed for rigid bodies with a boundary represented by implicit surfaces. The method is based on a suitable regularized constraint formulation, adapted both to fulfill exactly the contact constraint, and to be consistent with the conservation of the total energy. In this context two different approaches are studied: the first applied to pure elastic contact (without deformation), formulated with penalty and augmented Lagrange; and a second one based on a constitutive model for contact with penetration. In this second approach, a penalty potential is used in the constitutive model, that restores the energy stored in the contact when no dissipative effects are present. On the other hand, the energy is dissipated consistently with the continuous model when friction and damping are considered.

Simulación en tiempo real de vehículos industriales con modelos multicuerpo de gran complejidad

El objetivo de esta Tesis ha sido la consecución de simulaciones en tiempo real de vehículos industriales modelizados como sistemas multicuerpo complejos formados por sólidos rígidos. Para el desarrollo de un programa de simulación deben considerarse cuatro aspectos fundamentales: la modelización del sistema multicuerpo (tipos de coordenadas, pares ideales o impuestos mediante fuerzas), la formulación a utilizar para plantear las ecuaciones diferenciales del movimiento (coordenadas dependientes o independientes, métodos globales o topológicos, forma de imponer las ecuaciones de restricción), el método de integración numérica para resolver estas ecuaciones en el tiempo (integradores explícitos o implícitos) y finalmente los detalles de la implementación realizada (lenguaje de programación, librerías matemáticas, técnicas de paralelización). Estas cuatro etapas están interrelacionadas entre sí y todas han formado parte de este trabajo. Desde la generación de modelos de una furgoneta y de camión con semirremolque, el uso de tres formulaciones dinámicas diferentes, la integración de las ecuaciones diferenciales del movimiento mediante métodos explícitos e implícitos, hasta el uso de funciones BLAS, de técnicas de matrices sparse y la introducción de paralelización para utilizar los distintos núcleos del procesador. El trabajo presentado en esta Tesis ha sido organizado en 8 capítulos, dedicándose el primero de ellos a la Introducción. En el Capítulo 2 se presentan dos formulaciones semirrecursivas diferentes, de las cuales la primera está basada en una doble transformación de velocidades, obteniéndose las ecuaciones diferenciales del movimiento en función de las aceleraciones relativas independientes. La integración numérica de estas ecuaciones se ha realizado con el método de Runge-Kutta explícito de cuarto orden. La segunda formulación está basada en coordenadas relativas dependientes, imponiendo las restricciones por medio de penalizadores en posición y corrigiendo las velocidades y aceleraciones mediante métodos de proyección. En este segundo caso la integración de las ecuaciones del movimiento se ha llevado a cabo mediante el integrador implícito HHT (Hilber, Hughes and Taylor), perteneciente a la familia de integradores estructurales de Newmark. En el Capítulo 3 se introduce la tercera formulación utilizada en esta Tesis. En este caso las uniones entre los sólidos del sistema se ha realizado mediante uniones flexibles, lo que obliga a imponer los pares por medio de fuerzas. Este tipo de uniones impide trabajar con coordenadas relativas, por lo que la posición del sistema y el planteamiento de las ecuaciones del movimiento se ha realizado utilizando coordenadas Cartesianas y parámetros de Euler. En esta formulación global se introducen las restricciones mediante fuerzas (con un planteamiento similar al de los penalizadores) y la estabilización del proceso de integración numérica se realiza también mediante proyecciones de velocidades y aceleraciones. En el Capítulo 4 se presenta una revisión de las principales herramientas y estrategias utilizadas para aumentar la eficiencia de las implementaciones de los distintos algoritmos. En primer lugar se incluye una serie de consideraciones básicas para aumentar la eficiencia numérica de las implementaciones. A continuación se mencionan las principales características de los analizadores de códigos utilizados y también las librerías matemáticas utilizadas para resolver los problemas de álgebra lineal tanto con matrices densas como sparse. Por último se desarrolla con un cierto detalle el tema de la paralelización en los actuales procesadores de varios núcleos, describiendo para ello el patrón empleado y las características más importantes de las dos herramientas propuestas, OpenMP y las TBB de Intel. Hay que señalar que las características de los sistemas multicuerpo problemas de pequeño tamaño, frecuente uso de la recursividad, y repetición intensiva en el tiempo de los cálculos con fuerte dependencia de los resultados anteriores dificultan extraordinariamente el uso de técnicas de paralelización frente a otras áreas de la mecánica computacional, tales como por ejemplo el cálculo por elementos finitos. Basándose en los conceptos mencionados en el Capítulo 4, el Capítulo 5 está dividido en tres secciones, una para cada formulación propuesta en esta Tesis. En cada una de estas secciones se describen los detalles de cómo se han realizado las distintas implementaciones propuestas para cada algoritmo y qué herramientas se han utilizado para ello. En la primera sección se muestra el uso de librerías numéricas para matrices densas y sparse en la formulación topológica semirrecursiva basada en la doble transformación de velocidades. En la segunda se describe la utilización de paralelización mediante OpenMP y TBB en la formulación semirrecursiva con penalizadores y proyecciones. Por último, se describe el uso de técnicas de matrices sparse y paralelización en la formulación global con uniones flexibles y parámetros de Euler. El Capítulo 6 describe los resultados alcanzados mediante las formulaciones e implementaciones descritas previamente. Este capítulo comienza con una descripción de la modelización y topología de los dos vehículos estudiados. El primer modelo es un vehículo de dos ejes del tipo chasis-cabina o furgoneta, perteneciente a la gama de vehículos de carga medianos. El segundo es un vehículo de cinco ejes que responde al modelo de un camión o cabina con semirremolque, perteneciente a la categoría de vehículos industriales pesados. En este capítulo además se realiza un estudio comparativo entre las simulaciones de estos vehículos con cada una de las formulaciones utilizadas y se presentan de modo cuantitativo los efectos de las mejoras alcanzadas con las distintas estrategias propuestas en esta Tesis. Con objeto de extraer conclusiones más fácilmente y para evaluar de un modo más objetivo las mejoras introducidas en la Tesis, todos los resultados de este capítulo se han obtenido con el mismo computador, que era el top de la gama Intel Xeon en 2007, pero que hoy día está ya algo obsoleto. Por último los Capítulos 7 y 8 están dedicados a las conclusiones finales y las futuras líneas de investigación que pueden derivar del trabajo realizado en esta Tesis. Los objetivos de realizar simulaciones en tiempo real de vehículos industriales de gran complejidad han sido alcanzados con varias de las formulaciones e implementaciones desarrolladas. ABSTRACT The objective of this Dissertation has been the achievement of real time simulations of industrial vehicles modeled as complex multibody systems made up by rigid bodies. For the development of a simulation program, four main aspects must be considered: the modeling of the multibody system (types of coordinates, ideal joints or imposed by means of forces), the formulation to be used to set the differential equations of motion (dependent or independent coordinates, global or topological methods, ways to impose constraints equations), the method of numerical integration to solve these equations in time (explicit or implicit integrators) and the details of the implementation carried out (programming language, mathematical libraries, parallelization techniques). These four stages are interrelated and all of them are part of this work. They involve the generation of models for a van and a semitrailer truck, the use of three different dynamic formulations, the integration of differential equations of motion through explicit and implicit methods, the use of BLAS functions and sparse matrix techniques, and the introduction of parallelization to use the different processor cores. The work presented in this Dissertation has been structured in eight chapters, the first of them being the Introduction. In Chapter 2, two different semi-recursive formulations are shown, of which the first one is based on a double velocity transformation, thus getting the differential equations of motion as a function of the independent relative accelerations. The numerical integration of these equations has been made with the Runge-Kutta explicit method of fourth order. The second formulation is based on dependent relative coordinates, imposing the constraints by means of position penalty coefficients and correcting the velocities and accelerations by projection methods. In this second case, the integration of the motion equations has been carried out by means of the HHT implicit integrator (Hilber, Hughes and Taylor), which belongs to the Newmark structural integrators family. In Chapter 3, the third formulation used in this Dissertation is presented. In this case, the joints between the bodies of the system have been considered as flexible joints, with forces used to impose the joint conditions. This kind of union hinders to work with relative coordinates, so the position of the system bodies and the setting of the equations of motion have been carried out using Cartesian coordinates and Euler parameters. In this global formulation, constraints are introduced through forces (with a similar approach to the penalty coefficients) are presented. The stabilization of the numerical integration is carried out also by velocity and accelerations projections. In Chapter 4, a revision of the main computer tools and strategies used to increase the efficiency of the implementations of the algorithms is presented. First of all, some basic considerations to increase the numerical efficiency of the implementations are included. Then the main characteristics of the code’ analyzers used and also the mathematical libraries used to solve linear algebra problems (both with dense and sparse matrices) are mentioned. Finally, the topic of parallelization in current multicore processors is developed thoroughly. For that, the pattern used and the most important characteristics of the tools proposed, OpenMP and Intel TBB, are described. It needs to be highlighted that the characteristics of multibody systems small size problems, frequent recursion use and intensive repetition along the time of the calculation with high dependencies of the previous results complicate extraordinarily the use of parallelization techniques against other computational mechanics areas, as the finite elements computation. Based on the concepts mentioned in Chapter 4, Chapter 5 is divided into three sections, one for each formulation proposed in this Dissertation. In each one of these sections, the details of how these different proposed implementations have been made for each algorithm and which tools have been used are described. In the first section, it is shown the use of numerical libraries for dense and sparse matrices in the semirecursive topological formulation based in the double velocity transformation. In the second one, the use of parallelization by means OpenMP and TBB is depicted in the semi-recursive formulation with penalization and projections. Lastly, the use of sparse matrices and parallelization techniques is described in the global formulation with flexible joints and Euler parameters. Chapter 6 depicts the achieved results through the formulations and implementations previously described. This chapter starts with a description of the modeling and topology of the two vehicles studied. The first model is a two-axle chassis-cabin or van like vehicle, which belongs to the range of medium charge vehicles. The second one is a five-axle vehicle belonging to the truck or cabin semi-trailer model, belonging to the heavy industrial vehicles category. In this chapter, a comparative study is done between the simulations of these vehicles with each one of the formulations used and the improvements achieved are presented in a quantitative way with the different strategies proposed in this Dissertation. With the aim of deducing the conclusions more easily and to evaluate in a more objective way the improvements introduced in the Dissertation, all the results of this chapter have been obtained with the same computer, which was the top one among the Intel Xeon range in 2007, but which is rather obsolete today. Finally, Chapters 7 and 8 are dedicated to the final conclusions and the future research projects that can be derived from the work presented in this Dissertation. The objectives of doing real time simulations in high complex industrial vehicles have been achieved with the formulations and implementations developed.

Nonlinear models for lateral dynamics of railway vehicles on bridges subject to wind action

The study of lateral dynamics of running trains on bridges is of importance mainly for the safety of the traffic, and may be relevant for laterally compliant bridges. These studies require threedimensional coupled vehicle-bridge models, wheree consideration of wheel to rail contact is a key aspect. Furthermore, an adequate evaluation of safety of rail traffic requires nonlinear models. A nonlinear coupled model is proposed here for vehicle-structure vertical and lateral dynamics. Vehicles are considered as fully three-dimensional multibody systems including gyroscopic terms and large rotation effects. The bridge structure is modeled by means of finite elements which may be of beam, shell or continuum type and may include geometric or material nonlinearities. The track geometry includes distributed track alignment irregularities. Both subsystems (bridge and vehicles) are described with coordinates in absolute reference frames, as opposed to alternative approaches which describe the multibody system with coordinates relative to the base bridge motion. The wheelrail contact employed is a semi-Hertzian model based on realistic wheel-rail profiles. It allows a detailed geometrical description of the contact patch under each wheel including multiple-point contact, flange contact and uplift. Normal and tangential stresses in each contact are integrated at each time-step to obtain the resultant contact forces. The models have been implemented within an existing finite element analysis software with multibody capabilities, Abaqus (Simulia Ltd., 2010). Further details of the model are presented in Antolín et al. (2012). Representative applications are presented for railway vehicles under lateral wind action on laterally compliant viaducts, showing the relevance of the nonlinear wheel-rail contact model as well as the interaction between bridge and vehicle.