Dynamic Modeling of the Dissipative Contact and Friction Forces of a Passive Biped-Walking Robot

This article belongs to the Special Issue Optimization of Motion Planning and Control for Automatic Machines, Robots and Multibody Systems.This work presents and discusses a general approach for the dynamic modeling and analysis of a passive biped walking robot, with a particular focus on the feet-ground contact interaction. The main purpose of this investigation is to address the supporting foot slippage and viscoelastic dissipative contact forces of the biped robot-walking model and to develop its dynamics equations for simple and double support phases. For this investigation, special attention has been given to the detection of the contact/impact between the legs of the biped and the ground. The results have been obtained with multibody system dynamics applying forward dynamics. This study aims at examining and comparing several force models dealing with different approaches in the context of multibody system dynamics. The normal contact forces developed during the dynamic walking of the robot are evaluated using several models: Hertz, Kelvin-Voight, Hunt and Crossley, Lankarani and Nikravesh, and Flores. Thanks to this comparison, it was shown that the normal force that works best for this model is the dissipative Nonlinear Flores Contact Force Model (hysteresis damping parameter - energy dissipation). Likewise, the friction contact/impact problem is solved using the Bengisu equations. The numerical results reveal that the stable periodic solutions are robust. Integrators and resolution methods are also purchased, in order to obtain the most efficient ones for this model.This work was financially supported by the Spanish Government through the MCYT project "RETOS2015: sistema de monitorización integral de conjuntos mecánicos críticos para la mejora del mantenimiento en el transporte-maqstatus

The influence of cracks in rotating shafts

In this paper, the influence of transverse cracks in a rotating shaft is analysed. The paper addresses the two distinct issues of the changes in modal properties and the influence of crack breathing on dynamic response during operation. Moreover, the evolution of the orbit of a cracked rotor near half of the first resonance frequency is investigated. The results provide a possible basis for an on-line monitoring system. In order to conduct this study, the dynamic response of a rotor with a breathing crack is evaluated by using the alternate frequency/time domain approach. It is shown that this method evaluates the nonlinear behaviour of the rotor system rapidly and efficiently by modelling the breathing crack with a truncated Fourier series. The dynamic response obtained by applying this method is compared with that evaluated through numerical integration. The resulting orbit during transient operation is presented and some distinguishing features of a cracked rotor are examined

RKC: An explicit solver for parabolic PDEs

AbstractThe FORTRAN program RKC is intended for the time integration of parabolic partial differential equations discretized by the method of lines. It is based on a family of Runge-Kutta-Chebyshev formulas with a stability bound that is quadratic in the number of stages. Remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size. Moreover, they make it possible to evaluate the explicit formulas in just a few vectors of storage. These characteristics of the program make it especially attractive for problems in several spatial variables. RKC is compared to the BDF solver VODPK on two test problems in three spatial variables

Efficient cardiac simulations using the Runge--Kutta--Chebyshev method

Heart disease is one of the leading causes of death in Canada, claiming thousands of lives each year. Cardiac electrophysiology that studies the electrical activity in the human heart has emerged as an active research field in response to the demand for providing reliable guidance for clinical diagnosis and treatment to heart arrhythmias. Computer simulation of electrophysiological phenomena provides a non-invasive way to study the electrical activity in the human heart and to provide quantitative guidance to clinical applications. With the need to unravel underlying physiological details, mathematical models tend to be large and possess characteristics that are challenging to mitigate. In this thesis, we describe numerical methods for solving widely used mathematical models: the bidomain model and its simplified form, the monodomain model. The bidomain model is a multi-scale cardiac electrophysiology model that includes a set of reaction-diffusion partial differential equations (PDEs) with the reaction term representing cardiac cell models that describes the chemical reactions and flows of ions across the cell membrane of myocardial cells at the micro level and the diffusion term representing current propagation through the heart at the macro level. We use the method of lines (MOL) to obtain numerical solution of this model. The MOL first spatially discretizes the system of PDEs, resulting in a system of ordinary differential equations (ODEs) at each space point, and we obtain fully discrete solutions at each space-time point using time-integration methods for ODEs. In this thesis, we propose innovative numerical methods for the time integration of systems of ODEs based on the Runge--Kutta--Chebyshev (RKC) method. We implement and compare our methods with those used by current research on time integration of ODEs on three problems: time integration of individual cardiac cell models, time integration of the cell model of a monodomain problem, and time integration of spatially discretized tissue equation in a monodomain benchmark problem proposed by S. Niederer et al. in 2011. Numerical methods in cardiac electrophysiology research for solving ODEs include the forward Euler (FE) method, the Rush--Larsen (RL) method, the backward Euler method, and the generalized RL method of first-order. We introduce multistage first-order RKC methods and multistage first-order RL methods that are constructed by replacing the FE method with multistage first-order RKC methods. We implement all the aforementioned methods and test their efficiencies in time integration of 37 cardiac cell models. We find introducing the multistage RKC and RL methods allows larger step sizes to meet prescribed numerical accuracy; the increased time steps sped up time integration of 19 cell models. We replace the FE method with two-stage RKC method in time integration of cell model in a monodomain model. We find the increased time step introduced by applying this method improved the entire solving process by up to a factor of 1.4. We also apply the RKC(2,1) method to time integration of the tissue equation from a monodomain benchmark problem. Results show we have decreased the execution time of this benchmark problem by a factor of two. We note the increase of time step is from stability improvement brought by the numerical method. We finally give a quantitative explanation of stability improvement from introducing multistage RKC and RL methods for solving systems of ODEs considered in this thesis

Rotordynamic Analysis of a Two-Pole Synchronous Motor with Sleeve and Pressure Dam Bearings

A two-pole synchronous motor was recently rewound for the von Karman Gas Dynamics facility at Arnold Engineering Development Complex, Arnold Air Force Base, Tennessee. After installing the rewound rotor, unexpected vibration amplitudes were recorded during motor checkouts. To resolve this issue, an investigation was initiated to investigate the causes of the vibration issues. The investigation discovered that the original design used sleeve bearings rather than pressure dam bearings. A study was formed to determine the effect of changing the pressure dam bearings back to sleeve bearings. Because only one spare bearing shell existed, the bearing with the highest vibration amplitudes was chosen to be switched. A lateral rotordynamic analysis was performed to determine the impact of this switch, prior to performing the bearing swap. The rotordynamic model predicted that the rotor was operating near the second critical speed. Regardless of the bearing change, the second critical speed was not impacted. However, the determination was made to change the bearing to the sleeve bearing due to predicted lower vibration amplitudes. From the motor checkout runs, the model prediction was verified and the sleeve bearing was kept. Vibration amplitudes have been reduced, but issues still remain with the rotor operation near the second critical speed. Further analysis is required to successfully shift the rotor operation away from the second critical speed

Transforming Stiffness and Chaos

Stiff and chaotic differential equations are challenging for time-stepping numerical methods. For explicit methods, the required time step resolution significantly exceeds the resolution associated with the smoothness of the exact solution for specified accuracy. In order to improve efficiency, the question arises whether transformation to asymptotically stable solutions can be performed, for which neighbouring solutions converge towards each other at a controlled rate. Employing the concept of local Lyapunov exponents, it is demonstrated that chaotic differential equations can be successfully transformed to obtain high accuracy, whereas stiff equations cannot. For instance, the accuracy of explicit fourth order Runge-Kutta solution of the Lorenz chaotic equations can be increased by two orders of magnitude. Alternatively, the time step can be significantly extended with retained accuracy.Comment: 27 pages, 11 figure

Efficient Implicit Runge-Kutta Methods for Fast-Responding Ligand-Gated Neuroreceptor Kinetic Models

Neurophysiological models of the brain typically utilize systems of ordinary differential equations to simulate single-cell electrodynamics. To accurately emulate neurological treatments and their physiological effects on neurodegenerative disease, models that incorporate biologically-inspired mechanisms, such as neurotransmitter signalling, are necessary. Additionally, applications that examine populations of neurons, such as multiscale models, can demand solving hundreds of millions of these systems at each simulation time step. Therefore, robust numerical solvers for biologically-inspired neuron models are vital. To address this requirement, we evaluate the numerical accuracy and computational efficiency of three L-stable implicit Runge-Kutta methods when solving kinetic models of the ligand-gated glutamate and gamma-aminobutyric acid (GABA) neurotransmitter receptors. Efficient implementations of each numerical method are discussed, and numerous performance metrics including accuracy, simulation time steps, execution speeds, Jacobian calculations, and LU factorizations are evaluated to identify appropriate strategies for solving these models. Comparisons to popular explicit methods are presented and highlight the advantages of the implicit methods. In addition, we show a machine-code compiled implicit Runge-Kutta method implementation that possesses exceptional accuracy and superior computational efficiency

An improved longitudinal vibration model and dynamic characteristic of sucker rod string

Considering the influence of the nonlinear characteristics of plunger load and the friction of sucker rod string (SRS) on the SRS’s longitudinal vibration, an improved simulation model of SRS’s longitudinal vibration is derived. In the details, based on the flow characteristic of non-Newtonian power law fluid (NNPLF), a velocity model of NNPLF between pump plunger and pump barrel is established. Then the law of the velocity distribution is solved out with Lagrange multiplier method. Therefore, with the law of the velocity distribution of NNPLF, the computing models of nonlinear friction of pump plunger and clearance leakage between pump plunger and barrel are derived. Taking account of the influence of some parameters on the plunger load, such as plunger friction, hydraulic loss of pump and clearance leakage, an improved simulation model of plunger load is derived. The dynamic response is solved out with fourth order Runge-Kutta method. Comparing experiment results with simulated results, good agreement is found, which shows the simulation model is feasible. The influences of the different parameters on pump pressure and pump plunger load are analyzed, such as stroke number, power law exponent, consistency coefficient and gap between plunger and pump barrel. Simulation result indicates that the opening time of standing valve and traveling valve is affected by the parameters, and the maximum and minimum loads of pump plunger are affected by stroke number. In addition, the influence of SRS absorber on SRS’s longitudinal vibration is analyzed