Grundlagen: Mengen, reelle Zahlen, elementare Funktionen, Grenzwerte; Lineare Algebra (wesentlich ergänzt): Vektorräume, lineare Gleichungssysteme, Matrizen, Eigenwerte, analytische Geometrie, Skalarprodukt, Norm; komplexe Zahlen: GAUSSsche Zahlenebene, komplexe Funktionen, Anwendungen in der Technik; Differentialrechnung: Differenzierbarkeit, Ableitungsregeln, Anwendung auf Näherungen und Grenzwerte, NEWTON-Iteration; Integralrechnung: Unbestimmtes, bestimmtes, uneigentliches Integral, Hauptsatz der Differential- und Integralrechnung, Integrationsmethoden, praktische Anwendungen, numerische Integration; Ebene und räumliche Kurven: Parameterdarstellung von Kurven, Kurvengleichung in Polarkoordinaten; Reihen: Konvergenzkriterien, Potenzreihen, FOURIER-Reihen; Funktionen mehrerer Variablen: Partielle und vollständige Differenzierbarkeit, Doppelintegrale, Kurvenintegrale, Flächen im Raum, Umrisse; Differentialgleichungen: Elementare Verfahren für Dgln 1. und 2. Ordnung, lineare Dgln, Dgl-Systeme. Neu enthalten: Lineare Ausgleichsrechnung, Nabla-Kalkül, LAPLACE-Transformation, RUNGE-KUTTA-Verfahren In diesem Lehrbuch werden alle notwendigen Mathematikgrundlagen für Ingenieure und Naturwissenschaftler in einem Band dargestellt. Viele anschauliche Beispiele führen in die Thematik ein und vertiefen das Gelernte anhand von über 300 Grafiken. Mit mehr als 300 Übungsaufgaben mit Lösungen eignet sich das Buch hervorragend zum Selbststudium. Die Erstauflage dieses Buches, 1999 unter dem Titel »Mathematik für Ingenieure« erschienen, entstand aus Vorlesungen, die die beiden Autoren in verschiedenen Fachbereichen der Hochschule München gehalten haben. In der Folge wurden mehrfach Überarbeitungen und Ergänzungen vorgenommen.
Herringbone grooved hydrodynamic bearings are employed in applications that require high precision rotation. One of the most important applications is a hard disk drive (HDD) spindie motor. Over the past decade ball bearings were replaced by hydrodynamic fluid film bearings in order to have fast, quiet, and reliable operation of the spindie motor in a HDD. Reliable operation of such precision equipment depends heavily on the bearing properties.Bearing clearance in a hydrodynamic bearing system of a spindie motor is in the order of micrometers. Having such tight bearing clearances, production tolerances have great influence on the dynamic properties like stiffness and damping. Effects of surface roughness on the dynamic properties of the hydrodynamic journal bearing are not known in detail. In the course of this study investigations by means of numerical simulation have been carried out to find out the effect 01 surface roughness on the orbit of a herringbone grooved hydrodynamic journal bearing.The Reynolds equation is the governing partial differential equation for the pressure distribution in hydrodynamic bearings. It is derived from the Navier-Stokes equations under certain assumptions. Finite element method (FEM) has been employed to solve the Reynolds equation for complex bearing geometries. Analytical solutions available for simple geometries have been utilized to verify the FEM code. Additionally, validation has been carried out by comparison with available experimental studies.A surface roughness generation method is developed which is appropriate lor cylindrical surfaces. Surface roughness has been modeled using appropriate random number generators and fast Fourier transformation. An integration method with constant time step size is used to eliminate interpolation 01 the surface roughness, which may cause undesired disturbanees, during the generation of the time dependent grid. The equation of motion is integrated by coupling the Reynolds solution fully to the lourth order Runge-Kutta method to compute the journal orbit.
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Runge Kutta Fehlberg method (or Fehlberg method) is an algorithm of numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German mathematician Erwin Fehlberg and is based on the class of Runge Kutta methods. The Runge Kutta Fehlberg method uses an O(h4) method together with an O(h5) method that uses all of the points of the O(h4) method, and hence is often referred to as an RKF45 method. Similar schemes with different orders have since been developed. By performing one extra calculation that would be required for an RK5 method, the error in the solution can be estimated and controlled and an appropriate step size can be determined automatically, making this method efficient for ordinary problems of automated numerical integration of ordinary differential equations.
This book deals with the derivation of diagonally implicit Runge-Kutta (DIRK) methods of order four and five which are specially designed for the integration of linear ordinary differential equations (LODEs). The restriction to LODEs with constant coefficients reduces the number of order equations which the coefficients of Runge-Kutta (RK) methods must satisfy. The coefficients of the RK methods are chosen such that the error norm is minimized, this resulted in methods which are almost one order higher than the actual order. The stability polynomials and stability regions of the methods are then obtained using MATHEMATICA package. Codes using C++ programming based on the methods are developed to test sets of problems on linear ordinary differential equations. Numerical results show that the new methods are more efficient than the existing methods.
In this work, similarity and numerical solutions for some problems of unsteady heat transfer over different surfaces in the boundary layer of fluids through a porous medium are discussed. In these problems, we studied the influences of magnetic field, permeability of porous medium, heat generation / absorption, thermal conductivity, variable temperature. The governing equations are first cast into a dimensionless form by a nonsimilar transformation and the resulting equations are solved numerically by using the Runge-Kutta numerical integration, procedure in conjunction with shooting technique. The obtained results are shown in graphics and tabulated representation followed by a quantitative discussion. The main aim of this book which consists of four chapters, is to study similarity solutions of boundary layer of some fluid. In the following, a brief discussion of the chapters is given.
In numerical analysis, the Dormand Prince method is a method for solving ordinary differential equations (Dormand & Prince 1980). The method is a member of the Runge Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth-order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Cash Karp (RKCK).
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In numerical analysis, the Cash Karp method is a method for solving ordinary differential equations (ODEs). The method is a member of the Runge Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Dormand Prince (RKDP).
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In molecular dynamics, orbit, and particle simulations, energy drift is the gradual change in the total energy of a closed system. According to the laws of mechanics, the energy should be a constant of motion and should not change. However, the energy does fluctuate on a short time scale and increase on a very long time scale due to numerical integration artifacts that arise with the use of a finite time step t.Energy drift - usually damping - is substantial for numerical integration schemes that are not symplectic, such as the Runge-Kutta family. Symplectic integrators usually used in molecular dynamics, such as the Verlet integrator family, exhibit increases in energy over very long time scales, though the error remains roughly constant. These integrators do not in fact reproduce the actual Hamiltonian mechanics of the system, instead, they reproduce a closely related "shadow" Hamiltonian whose value they conserve many orders of magnitude more closely. The accuracy of the energy conservation for the true Hamiltonian is dependent on the time step.
The three body problem is one important object of space dynamics, because this problem has many applications on space dynamics objects. Also this problem attract many scientist during last three centuries. The motion of space craft around binary asteroid can be consider as restricted three body problem so that, the treatment this object was done. Where equations of motioncan be solved by using numerical integration methods (Runge Kutta fourth order).