Search Results for integral calculus

This article is a comprehensive collection of formulas, tables, and theorems associated with integral calculus. It also summarizes the applications of single, double, and triple integrations in various fields.

This article begins with information on fundamental Laplace transformation rules. Some general theorems concerning operations on transforms are provided. The article also discusses the applications of Laplace transforms. It concludes with a table that lists the values of Laplace transforms.

This article describes the models of physical phenomena involving solution of differential equations such as ordinary or partial differential equations. It reviews the basic concepts of differential calculus and tabulates the expansion of functions into power series. A table of linear partial differential equations is also presented.

This article presents an overview of the use of mechanism analysis (kinematics and dynamics) and simulation. It provides indications of the directions in which mechanism simulation is growing and how it is integrated in the evolving computer aided design and computer aided engineering (CAD/CAE) fields. Mechanism simulation is best used as part of a concurrent CAD/CAE approach to design. The article discusses the state, evolution, and direction of application for these techniques in a variety of fields.

Computational fluid dynamics (CFD) is one of the tools available for understanding and predicting the performance of thermal-fluids systems. This article qualitatively describes the basic principles of CFD. The numerical methods, such as geometry description and discretization, used to solve the CFD equations are discussed. The article also demonstrates the application of CFD to a few casting problems.

This article discusses tools that are used for the systematic optimization of engineering designs. It focuses on the practical application of optimization technology in a computer-aided engineering environment. The article presents numerical optimization algorithms and provides some background on how these algorithms make decisions when searching for the optimal design. It also provides information on structural optimization, topology optimization, materials processing optimization, multidisciplinary optimization, and global optimization.

Computational fluid dynamics (CFD) is a computationally intensive three-dimensional simulation of thermal fluids systems where non-linear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction to the CFD techniques. It introduces some common techniques for discretizing the fluid-flow equations and methods for solving the discrete equations. These include finite-difference methods, finite-element methods, spectral methods, and computational particle methods. The article describes the approaches for grid generation with complex geometries. It discusses the four-step procedures used in the CFD process for engineering design: geometry acquisition, grid generation and problem specification, flow solution, and post-processing and synthesis. The article also provides information on the engineering applications of the CFD. It concludes with a discussion on issues and directions for engineering CFD.

Casting offers a great amount of component design flexibility. This article discusses six casting design parameters that drive the geometry of casting design from a process standpoint. It provides information on the design of junctions and addresses considerations of secondary operations in design. The article describes the factors that control casting tolerances and presents specific tips for designing castings with uniform wall thickness, unequal sections, thin sections, economical coring, functional packaging, and core design. The article provides a framework for analyzing all manners of manufacturing as possible conversion candidates for casting. It concludes with a discussion on different metalcasting design projects.

This article introduces the fundamental concepts of chemical thermodynamics and chemical kinetics in describing presolidification phenomena. For metallurgical systems, the most important thermodynamic variables are enthalpy and Gibbs free energy. A qualitative demonstration of the interrelationship between phase diagrams and thermodynamics is presented. The article discusses processes that generally limit the rates of chemical processes. These include nucleation of the product phase and interphase mass transport. The article provides a discussion on the dissolution of alloy with melting point lower than bath temperature and dissolution of alloy that is solid at bath temperatures.

This article describes concepts and tools that can be used by the failure analyst to understand and address deformation, cracking, or fracture after a stress-related failure has occurred. Issues related to the determination and use of stress are detailed. Stress is defined, and a procedure to deal with stress by determining maximum values through stress transformation is described. The article provides the stress analysis equations of typical component geometries and discusses some of the implications of the stress analysis relative to failure in components. It focuses on linear elastic fracture mechanics analysis, with some mention of elastic-plastic fracture mechanics analysis. The article describes the probabilistic aspects of fatigue and fracture. Information on crack-growth simulation of the material is also provided.

This article describes the underlying fundamentals, applications, the relevance and necessity of performing proper stress analysis in conducting a failure analysis. It presents an introduction to the stress analysis of bodies containing crack-like imperfections and the topic of fracture mechanics. The fracture mechanics approach is an important part of stress analysis at the tips of sharp cracks or discontinuities. The article reviews fracture mechanics concepts, including linear elastic fracture mechanics, elastic-plastic fracture mechanics, and subcritical fracture mechanics. It also provides information on the applications of fracture mechanics in failure analysis.

Computational fluid dynamics (CFD) is reserved for computationally intensive three-dimensional simulations of thermal fluids systems where nonlinear momentum transport plays an important role. This article presents the governing equations of fluid dynamics and an introduction to the CFD techniques for their solution. It introduces discretization techniques that are used by finite-difference, finite-volume, finite-element, spectral, and some particle methods. Associated concepts of numerical stability and accuracy are also reviewed. The article describes two approaches for grid generation with complex geometries: the use of unstructured grids and the use of special differencing methods on structured grids. The article describes the four-step procedures of the CFD process: geometry acquisition, grid generation and problem specification, flow solution, and post-processing and synthesis. It provides information on the applications of the engineering CFD. Issues and directions for the engineering CFD are also described.

This article discusses the fundamental aspects of phase-field microstructure modeling. It describes the evolution of microstructure modeling, including nucleation, growth, and coarsening. The article reviews two approaches used in the modeling nucleation of microstructure: the Langevin force approach and explicit nucleation algorithm. Calculation of activation energy and critical nucleus configuration is discussed. The article presents the deterministic phase-field kinetic equations for modeling growth and coarsening of microstructure. It also describes the material-specific model inputs, chemical free energy and kinetic coefficients, for phase-field microstructure modeling. The article provides four examples that illustrate some aspects of phase-field modeling.

Quenchant agitation can be obtained by circulating quenchant in a quench tank through pumps and impellers. The selection of the agitation method depends on the tank design, type and volume of the quenchant, part design, and the severity of quench required. This article describes flow measurement methods, temperature control, materials handling, and filtration processes during the agitation process. The maintenance of quenching installations is also discussed.

This article describes dental alloy compositions and its properties. It discusses the safety and efficacy considerations of dental alloy devices. The article defines and compares interstitial fluid and oral fluid environments. Artificial solutions developed for the testing and evaluation of dental materials are summarized. The article examines the effects of restoration contact on electrochemical parameters and reviews the concentration cells developed by dental alloy-environment electrochemical reactions. The composition and characterization of biofilms, corrosion products, and other debris that deposit on dental material surfaces are discussed. The article evaluates the types of alloys available for dental applications, including direct filling alloys, crown and bridge alloys, partial denture alloys, porcelain fused to metal alloys, wrought wire alloys, soldering alloys, and implant alloys. The effects of composition and microstructure on the corrosion of each alloy group are also discussed. The article concludes with information on the tarnishing and corrosion behavior of these alloys.

This article discusses some of the statistical aspects of design from an engineer's perspective. It reviews the commonly used statistical terms and distributions for providing some guidance on the practical engineering applications of these distributions. The article describes the basic statistical procedures that can be used to address variability and uncertainty in an engineering analysis. It contains a table that lists the relevant statistics standards published by the American Society of Testing and Materials.

Agitation is one of the most critical areas of quench system design. This article provides an overview of the impact of agitation on quench uniformity, followed by a general discussion of the selection and use of various types of agitators, including recirculation pumps, jet mixers, forced air (sparging), and impellers. A brief overview of heat-exchanger types and their selection criteria is also provided, along with simplified calculations for approximating heat-exchange requirements. The methods of selecting a quenchant are provided. Recommendations for system maintenance are also described. Much effort is placed on the proper design of the furnace for temperature and atmosphere uniformity, proper temperature control, and exact carbon potential. However, the design of the quench tank can have a drastic effect on the overall system performance, with proper design ensuring proper mechanical properties (hardness, strength, and fracture toughness) as well as distortion control.