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by: Lee Brakus

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# Computational Methods in Mechanical Engineering MAE 340

Marketplace > Old Dominion University > Aerospace Engineering > MAE 340 > Computational Methods in Mechanical Engineering
Lee Brakus
ODU
GPA 3.71

Shizhi Qian

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COURSE
PROF.
Shizhi Qian
TYPE
Class Notes
PAGES
10
WORDS
KARMA
25 ?

## Popular in Aerospace Engineering

This 10 page Class Notes was uploaded by Lee Brakus on Monday September 28, 2015. The Class Notes belongs to MAE 340 at Old Dominion University taught by Shizhi Qian in Fall. Since its upload, it has received 35 views. For similar materials see /class/215349/mae-340-old-dominion-university in Aerospace Engineering at Old Dominion University.

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Date Created: 09/28/15
Chapter 1 Mathematical Modeling Numerical Methods and I r0blem sowing Chapter Objectives Learning how mathematical models can be formulated on the basis of scientific principles to simulate the behavior of a simple physical system Understanding how numerical methods afford a means to generalize solutions in a manner that can be implemented on a digital computer Understanding tne different types of conservation laws tnat lie beneath the models used in the various engineering disciplines and appreciating the difference between steady state and dynamic solutions of these models Learning about the different types of numerical methods we will cover in ths book A Simple Mathematical Model A mathematical model can be broadly defined as a formulation or equation that expresses the essential features of a physical system or process in mathematical terms Models can be represented by a functional relationshi between de endent variables independent variables parameters and forcing functions Model Function Dependent f independent parameters forcing variable variables functions Dependent variable a characteristic that usually reflects the behavior or state of the system lndependent variables dimensions such as time and space along which the system s behavior is being determined Parameters constants reflective of the system s properties or composition Forcing functions external influences acting upon the system Model Function Example Assuming a bungee jumper is in mid flivht an anal tical model for the um er s velocity accounting for drag is vt J tanhtj Dependent variable velocity v Independent variables time t Parameters mass m drag coefficient cd Forcing function gravitational acceleration g Upward force due to air resistance Downward fo rce d ue to gravity Model Results Using a computer or a calculator the model can be used to generate a graphical represemauun on me system rUI example the graph below represents the velocity of a 681 kg jumper assuming a drag coefficient of 025 kgm 50 Terminal velocity Numerical Modeling Some system models will be given as implicit functions or as differential equations these can be solved either using analytical methods or numerical methods Example the bungee jumper velocity equation from before is the analytical solution to the differential equation dv Cd 2 where the change in velocity is determined by the gravitational forces acting on the jumper versus the drag force Numerical Methods To solve the problem using a numerical method note that the time rate of change of velocity can be approximated as Uhill Q N Q vtl1 vtl Av dt At t t i1 l ulti True slope dvdt Approximate slope vtvtl A lil ti Numerical Results As shown in later chapters the efficiency and accuracy of numerical methods will depend upon how the method is applied Applying tne prevrous metnoo In 2 s Intervals yields Terminal velocity Approximate 40 numerical solution 1 ms Exact analytical solution 20 Bases for Numerical Models Conservation laws provide the foundation for many model functions Different fields of engineering and science apply these laws to different paradigms within the field Among these laws are Conservation OT mass Conservation of momentum Conservation u change Conservation of energy

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