The goal of this study is to present a method for deriving equations of motion capable of modeling the controlled motion of an open loop multibody structure comprised of an arbitrary number of rigid bodies and slender beams. The procedure presented here for deriving equations of motion for flexible multibody systems is carried out by means of the Principle of Virtual Work (often referred to in the dynamics literature as d'Alembert's Principle). We first consider the motion of a general flexible body relative to the inertial space, and then derive specific formulas for both rigid bodies and slender beams. Next, we make a small motions assumption, with the end result being equations for a Rayleigh beam, which include terms which account for the axial motion, due to bending, of points on the beam central axis. This process includes a novel application of the exponential form of an orthogonal matrix, which is ideally suited for truncation. Then, the generalized coordinates and quasi-velocities used in the mathematical model, including those needed in the spatial discretization process of the beam equations are discussed. Furthermore, we develop a new set of recursive relations used to compute the inertial motion of a body in terms of the generalized coordinates and quasi-velocities. This research was motivated by the desire to model the controlled motion of a flexible space robot, and consequently, we use the multibody dynamics equations to simulate the motion of such a structure, providing a demonstration of the computer program. For this particular example we make use of a new sequence of shape functions, first used by Meirovitch and Stemple to model a two dimensional building frame subjected to earthquake excitations. / Ph. D.

The coupling and interactions between ship motion and inner-tank sloshing are investigated by a potential-viscous hybrid method in time domain. For the time domain simulation of vessel motion, the hydrodynamic coefficients and wave forces are obtained by a potential-theory-based 3D diffraction/radiation panel program in frequency domain. Then, the corresponding simulations of motions in time domain are carried out using the convolution-integral method. The liquid sloshing in a tank is simulated in time domain by a Navier-Stokes solver. A finite difference method with SURF scheme, assuming a singlevalued free surface profile, is applied for the direct simulation of liquid sloshing. The computed sloshing forces and moments are then applied as external excitations to the ship motion. The calculated ship motion is in turn inputted as the excitation for liquid sloshing, which is repeated for the ensuing time steps. For comparison, linear inner-fluid motion was calculated using a 3D panel program and it is coupled with the vessel motion program in the frequency domain. The developed computer programs are applied to a barge-type FPSO hull equipped with two partially filled tanks. The time domain simulation results show reasonably good agreement when compared with MARIN's experimental results. The frequency domain results qualitatively reproduce the trend of coupling effects but the peaks are usually over-predicted. It is seen that the coupling effects on roll motions appreciably change with filling level. The most pronounced coupling effects on roll motions are the shift or split of peak frequencies. The pitch motions are much less influenced by the inner-fluid motion compared to roll motions. A developed program is also applied to a more realistic offloading configuration where a LNG-carrier is moored with a floating terminal in a side-by-side configuration. First, a hydrodynamic interaction problem between two bodies is solved successfully in frequency and time domain. A realistic mooring system, including fender, hawser, and simplified mooring system, is also developed to calculate the nonlinear behavior of two bodies in time domain simulation. Then, the LNG-carrier and sloshing problem are coupled in frequency and time domain, similar to the method in the MARIN-FPSO case. Sloshing effect on LNG-carrier motion is investigated with respect to different tank filling levels including various conditions such as gap distance between two bodies, selection of dolphin mooring system, and different cases of environmental conditions using wave, wind, and current.

This Master of Science Thesis focuses on the modeling of an automotive system. Several of the main automotive systems are combined to represent a full vehicle model. One system is a road plane model with degrees of freedom in yaw and lateral acceleration. The model at first includes a two dimensional representation of a steering system and then later expands the steering model to three dimensions. Also included is a five degree of freedom, two-dimensional multibody model in order to model the response of the chassis/suspension system due to an applied step steer input. The tire system incorporates the Magic Formula tire model. Furthermore, a graphic user interface is developed to facilitate setting up the initial conditions and inputs to the full vehicle model, and to ease the use of the simulation.

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The objective of the work carried out in this thesis was to develop analytical and computational tools to model and investigate musculoskeletal human joints. It was recognised that the FEA was used by many researchers in modelling human musculoskeletal motion, loading and stresses. However the continuum mechanics played only a minor role in determining the articular joint motion, and its value was questionable. This is firstly due to the computational cost and secondly due to its impracticality for this application. On the other hand, there isn’t any suitable software for precise articular joint motion analysis to deal with the local joint stresses or non standard joints. The main requirement in orthopaedics field is to develop a modeller software (and its associated theories) to model anatomic joint as it is, without any simplification with respect to joint surface morphology and material properties of surrounding tissues. So that the proposed modeller can be used for evaluating and diagnosing different joint abnormalities but furthermore form the basis for performing implant insertion and analysis of the artificial joints. The work which is presented in this thesis is a new frame work and has been developed for human anatomic joint analysis which describes the joint in terms of its surface geometry and surrounding musculoskeletal tissues. In achieving such a framework several contributions were made to the 6DOF linear and nonlinear joint modelling, the mathematical definition of joint stiffness, tissue path finding and wrapping and the contact with collision analysis. In 6DOF linear joint modelling, the contribution is the development of joint stiffness and damping matrices. This modelling approach is suitable for the linear range of tissue stiffness and damping properties. This is the first of its kind and it gives a firm analytical basis for investigating joints with surrounding tissue and the cartilage. The 6DOF nonlinear joint modelling is a new scheme which is described for modelling the motion of multi bodies joined by non-linear stiffness and contact elements. The proposed method requires no matrix assembly for the stiffness and damping elements or mass elements. The novelty in the nonlinear modelling, relates to the overall algorithmic approach and handling local non-linearity by procedural means. The mathematical definition of joint stiffness is also a new proposal which is based on the mathematical definition of stiffness between two bodies. Based on the joint stiffness matrix properties, number of joint stiffness invariants was obtained analytically such as the centre of stiffness, the principal translational stiffnesses, and the principal rotational stiffnesses. In corresponding to these principal stiffnesses, their principal axes have been also obtained. Altogether, a joint is assessed by six principal axes and six principal stiffnesses and its centre of stiffness. These formulations are new and show that a joint can be described in terms of inherent stiffness properties. It is expected that these will be better in characterising a joint in comparison to laxity based characterisation. The development of tissue path finding and wrapping algorithms are also introduced as new approaches. The musculoskeletal tissue wrapping involves calculating the shortest distance between two points on a meshed surface. A new heuristic algorithm was proposed. The heuristic is based on minimising the accumulative divergence from the straight line between two points on the surface and the direction of travel on the surface (i.e. bone). In contact and collision based development, the novel algorithm has been proposed that detects possible colliding points on the motion trajectory by redefining the distance as a two dimensional measure along the velocity approach vector and perpendicular to this vector. The perpendicular distance determines if there are potentially colliding points, and the distance along the velocity determines how close they are. The closest pair among the potentially colliding points gives the “time to collision”. The algorithm can eliminate the “fly pass” situation where very close points may not collide because of the direction of their relative velocity. All these developed algorithms and modelling theories, have been encompassed in the developed prototype software in order to simulate the anatomic joint articulations through modelling formulations developed. The software platform provides a capability for analysing joints as 6DOF joints based on anatomic joint surfaces. The software is highly interactive and driven by well structured database, designed to be highly flexible for the future developments. Particularly, two case studies are carried out in this thesis in order to generate results relating to all the proposed elements of the study. The results obtained from the case studies show good agreement with previously published results or model based results obtained from Lifemod software, whenever comparison was possible. In some cases the comparison was not possible because there were no equivalent results; the results were supported by other indicators. The modelling based results were also supported by experiments performed in the Brunel Orthopaedic Research and Learning Centre.

Indiana University-Purdue University Indianapolis (IUPUI) / In this thesis, an explicit time integration code which integrates multibody dynamics (MBD) and the discrete element method (DEM) is validated using three previously published steady-state physical experiments for non-cohesive sand-type material, namely: shear-cell for measuring shear stress versus normal stress; penetroplate pressure-sinkage test; and wheel drawbar pull-torque-slip test. The test results are used to calibrate the material properties of the DEM soft soil model and validate the coupled MBD-DEM code. All three tests are important because each test measures specific mechanical characteristics of the soil under various loading conditions. Shear strength of the soil as a function of normal load help to understand shearing of the soil under a vehicle wheel contact patch causing loss of traction. Penetroplate pressure-sinkage test is used to calibrate and validate friction and shear strength characteristics of the soil. Finally the rigid wheel-soil interaction test is used to predict drawbar pull force and wheel torque vs. slip percentage and normal stress for a rigid wheel. Wheel-Soil interaction test is important because it plays the role of ultimate validation of the soil model tuned in the previous two experiments and also shows how the soil model behaves in vehicle mobility applications. All the aforementioned tests were modeled in the multibody dynamics software using rigid bodies and various joints and actuators. The sand-type material is modeled using discrete cubical particles. A penalty technique is used to impose normal contact constraints (including particle-particle and particle-wall contact). An asperity-based friction model is used to model friction. A Cartesian Eulerian grid contact search algorithm is used to allow fast contact detection between particles. A recursive bounding box contact search algorithm enabled fast contact detection between the particles and polygonal body surfaces (such as walls, penetrometer, and wheel). The governing equations of motion are solved along with contact constraint equations using a time-accurate explicit solution procedure. The results show very good agreement between the simulation and the experimental measurements. The model is then demonstrated in a full-scale application of high-speed off-road vehicle mobility on the sand-type soil.

<p>In this thesis, an explicit time integration code which integrates multibody dynamics (MBD) and the discrete element method (DEM) is validated using three previously published steady-state physical experiments for non-cohesive sand-type material, namely: shear-cell for measuring shear stress versus normal stress; penetroplate pressure-sinkage test; and wheel drawbar pull-torque-slip test. The test results are used to calibrate the material properties of the DEM soft soil model and validate the coupled MBD-DEM code. All three tests are important because each test measures specific mechanical characteristics of the soil under various loading conditions. Shear strength of the soil as a function of normal load help to understand shearing of the soil under a vehicle wheel contact patch causing loss of traction. Penetroplate pressure-sinkage test is used to calibrate and validate friction and shear strength characteristics of the soil. Finally the rigid wheel-soil interaction test is used to predict drawbar pull force and wheel torque vs. slip percentage and normal stress for a rigid wheel. Wheel-Soil interaction test is important because it plays the role of ultimate validation of the soil model tuned in the previous two experiments and also shows how the soil model behaves in vehicle mobility applications.</p> <p><br></p> <p>All the aforementioned tests were modeled in the multibody dynamics software using rigid bodies and various joints and actuators. The sand-type material is modeled using discrete cubical particles. A penalty technique is used to impose normal contact constraints (including particle-particle and particle-wall contact). An asperity-based friction model is used to model friction. A Cartesian Eulerian grid contact search algorithm is used to allow fast contact detection between particles. A recursive bounding box contact search algorithm enabled fast contact detection between the particles and polygonal body surfaces (such as walls, penetrometer, and wheel). The governing equations of motion are solved along with contact constraint equations using a time-accurate explicit solution procedure. The results show very good agreement between the simulation and the experimental measurements. The model is then demonstrated in a full-scale application of high-speed off-road vehicle mobility on the sand-type soil.</p>

While some attempts have been made to model the behaviour of the grand piano action (the mechanism that translates a key press into a hammer striking a string), most researchers have reduced the system to a simple model with little relation to the components of a real action. While such models are useful for certain applications, they are not appropriate as design tools for piano makers, since the model parameters have little physical meaning and must be calibrated from the behaviour of a real action. A new model for a piano action is proposed in this thesis. The model treats each of the five main action components (key, whippen, jack, repetition lever, and hammer) as a rigid body. The action model also incorporates a contact model to determine the normal and friction forces at 13 locations between each of the contacting bodies. All parameters in the model are directly measured from the physical properties of individual action components, allowing the model to be used as a prototyping tool for actions that have not yet been built. To test whether the model can accurately predict the behaviour of a piano action, an experimental apparatus was built. Based around a keyboard from a Boston grand piano, the apparatus uses an electric motor to actuate the key, a load cell to measure applied force, and optical encoders and a high speed video camera to measure the positions of the bodies. The apparatus was found to produce highly repeatable, reliable measurements of the action. The behaviour of the action model was compared to the measurements from the experimental apparatus for several types of key blows from a pianist. A qualitative comparison showed that the model could very accurately reproduce the behaviour of a real action for high force blows. When the forces were lower, the behaviour of the action model was still reasonable, but some discrepancy from the experimental results could be seen. In order to reduce the discrepancy, it was recommended that certain improvements could be made to the action model. Rigid bodies, most importantly the key and hammer, should be replaced with flexible bodies. The normal contact model should be modified to account for the speed-independent behaviour of felt compression. Felt bushings that are modelled as perfect revolute joints should instead be modelled as flexible contact surfaces.

This thesis presents the theoretical derivations and the implementation of a contact dynamics modelling system based on compliant contact models. The system was designed to be used as a general-purpose modelling tool to support the task planning process space-based robot manipulator systems. This operational context imposes additional requirements on the contact dynamics modelling system beyond the usual ones of fidelity and accuracy. The system must not only be able to generate accurate and reliable simulation results, but it must do it in a reasonably short period of time, such that an operations engineer can investigate multiple scenarios within a few hours. The system is easy to interface with existing simulation facilities. All physical parameters of the contact model can be identified experimentally or can be obtained by other means through analysis or theoretical derivations based on the material properties. Similarly, the numerical parameters can be selected automatically or by using heuristic rules that give an indication of the range of values that would ensure that the simulations results are qualitatively correct. The contact dynamics modelling system is comprised of two contact models. On one hand, a point contact model is proposed to tackle simulations involving bodies with non-conformal surfaces. Since it is based on Hertz theory, the contacting surfaces must be smooth and without discontinuity, i.e., no corners or sharp edges. The point contact model includes normal damping and tangential friction and assumes the contact surface is very small, such that the contact force is assumed to be acting through a point. An expression to set the normal damping as a function of the effective coefficient of restitution is given. A new seven-parameter friction model is introduced. The friction model is based on a bristle friction model, and is adapted to the context of 3-dimensional frictional impact modelling with introduction of load-dependent bristle stiffness and damping terms, and with the expression of the bristle deformation in vectorial form. The model features a dwell-time stiction force dependency and is shown to be able to reproduce the dynamic nature of the friction phenomenon. A second contact model based on the Winkler elastic foundation model is then proposed to deal with a more general class of geometries. This so-called volumetric contact model is suitable for a broad range of contact geometries, as long as the contact surface can be approximated as being flat. A method to deal with objects where this latter approximation is not reasonable is also presented. The effect of the contact pressure distribution across the contact surface is accounted for in the form of the rolling resistance torque and spinning friction torque. It is shown that the contact forces and moments can be expressed in terms of the volumetric properties of the volume of interference between the two bodies, defined as the volume spanned by the intersection of the two undeformed geometries of the colliding bodies. The properties of interest are: the volume of the volume of interference, the position of its centroid, and its inertia tensor taken about the centroid. The analysis also introduces a new way of defining the contact normal; it is shown that the contact normal must correspond to one of the eigenvectors of the inertia tensor. The investigation also examines how the Coulomb friction is affected by the relative motion of the objects. The concept of average surface velocity is introduced. It accounts for both the relative translational and angular motions of the contacting surfaces. The average surface velocity is then used to find dimensionless factors that relate friction force and spinning torque caused by the Coulomb friction. These latter factors are labelled the Contensou factors. Also, the radius of gyration of the moment of inertia of the volume of interference about the contact normal was shown to correlate the spinning Coulomb friction torque to the translational Coulomb friction force. A volumetric version of the seven-parameter bristle friction model is then presented. The friction model includes both the tangential friction force and spinning friction torque. The Contensou factors are used to control the behaviour of the Coulomb friction. For both contact models, the equations are derived from first principles, and the behaviour of each contact model characteristic was studied and simulated. When available, the simulation results were compared with benchmark results from the literature. Experiments were performed to validate the point contact model using a six degrees-of-freedom manipulator holding a half-spherical payload, and coming into contact with a flat plate. Good correspondence between the simulated and experimental results was obtained.

While some attempts have been made to model the behaviour of the grand piano action (the mechanism that translates a key press into a hammer striking a string), most researchers have reduced the system to a simple model with little relation to the components of a real action. While such models are useful for certain applications, they are not appropriate as design tools for piano makers, since the model parameters have little physical meaning and must be calibrated from the behaviour of a real action. A new model for a piano action is proposed in this thesis. The model treats each of the five main action components (key, whippen, jack, repetition lever, and hammer) as a rigid body. The action model also incorporates a contact model to determine the normal and friction forces at 13 locations between each of the contacting bodies. All parameters in the model are directly measured from the physical properties of individual action components, allowing the model to be used as a prototyping tool for actions that have not yet been built. To test whether the model can accurately predict the behaviour of a piano action, an experimental apparatus was built. Based around a keyboard from a Boston grand piano, the apparatus uses an electric motor to actuate the key, a load cell to measure applied force, and optical encoders and a high speed video camera to measure the positions of the bodies. The apparatus was found to produce highly repeatable, reliable measurements of the action. The behaviour of the action model was compared to the measurements from the experimental apparatus for several types of key blows from a pianist. A qualitative comparison showed that the model could very accurately reproduce the behaviour of a real action for high force blows. When the forces were lower, the behaviour of the action model was still reasonable, but some discrepancy from the experimental results could be seen. In order to reduce the discrepancy, it was recommended that certain improvements could be made to the action model. Rigid bodies, most importantly the key and hammer, should be replaced with flexible bodies. The normal contact model should be modified to account for the speed-independent behaviour of felt compression. Felt bushings that are modelled as perfect revolute joints should instead be modelled as flexible contact surfaces.