Problem 1 Solution
Part (a)
Multiply by test function w(x) and integrate over domain
( ()
) + () =
()
) + ()
0
0
0
Integrate first term by parts and pick up boundary conditions terms
0
()
) + () )
Problem 1 (70 points)
Given is a tapered elastic bar of length L, cross sectional area A(x) (which is a function of position x),
Youngs Modulus E. The bar experiences a body force b(x). The left end o
STANDARD TOLERANCES
AND
FITS
ME 240
INTRODUCTION TO DESIGN AND
MANUFACTURING
Northwestern University
STANDARD TOLERANCES
Northwestern University
Tolerancing
(Recap)
Tolerance is the total amount a di
GEOMETRIC DIMENSIONING
AND TOLERANCING GD&T
Part 1
ME 240
INTRODUCTION TO DESIGN AND
MANUFACTURING
Northwestern University
WHAT IS GD&T?
&
BASIC DEFINITIONS
Northwestern University
Three Categories of
TOLERANCES
ME 240
INTRODUCTION TO DESIGN AND
MANUFACTURING
Northwestern University
Tolerances
Definition:
Tolerance is defined as the total amount by which a specific
dimension is permitted to vary. T
THE DESIGN PROCESS
ME 240
INTRODUCTION TO DESIGN AND
MANUFACTURING
Northwestern University
Engineering Design
The design engineers main task is to apply his scientific
and pragmatic knowledge to the s
GEOMETRIC DIMENSIONING
AND TOLERANCING GD&T
Part 2
ME 240
INTRODUCTION TO DESIGN AND
MANUFACTURING
Northwestern University
Specifying Geometric Characteristics
SYMBOLS:
Northwestern University
TOLERAN
DESIGN PROCESS
AND
ENGINEERING DRAWING
ME 240
INTRODUCTION TO DESIGN AND
MANUFACTURING
Northwestern University
Steps in the Design Process
Conceptual Design
Preliminary Design
Detail Design
Northweste
INTRODUCTION
ME 240
INTRODUCTION TO DESIGN AND
MANUFACTURING
Northwestern University
ME 240 Course Objectives
Students should be able to:
1. Design discrete mechanical parts with all necessary details
About Bond graphs - The System Modeling World
Page 1 of 25
About Bond Graphs
1. Introduction
2. Power variables
3. Standard elements
4. Power directions
5. Bond numbers
6. Causality
7. System equation
Dynamic variables and modeling
When you make a physical diagram (picture, circuit diagram,
etc) be sure to show each element distinct from all the others.
This can be easily overlooked for an effect l
ME 390 INTRODUCTION TO DYNAMIC SYSTEMS
Homework #1
1. Write a differential equation for x(t) using this block diagram:
x(t)
-1
A dt
C
B
+
d
dt
+
1
+
u(t)
You can assume that u(t) is a kno
Reduction rules
1 of 3
file:/C:/z%20Northwestern%20Data/C90/c90%20website%20content/Readings/BGreduction.html
Reduction rules
You can (and should) reduce a bond graph as far as possible using the redu
1. Consider the circuit below.
C
Vs
L
R
a.
b.
c.
d.
Develop a bond graph for the circuit. Reduce and annotate.
Identify state variables.
Write junction relations and constitutive relations in prop
1. Sketch, reduce and annotate bond graph models of the following systems. Gravity
acts downward in all cases and there is no friction.
2. Sketch, reduce and annotate a bond graph model of the followi
Junctions
M.A. Peshkin
Common effort / common flow
A system consisting of two elements shares both an effort, and a flow. (Shares magnitude, but not necessarily sign)
A system consisting of three ele
Ugly
1 of 2
file:/C:/z%20Northwestern%20Data/C90/c90%20website%20content/Readings/BGugly.html
Ugly Method
First rule of Ugly Method: don't use Ugly Method. It is much easier and quicker and more relia
Arbitrary Causality and State Equation Loops
When assigning causality to your bond graph, if you find that you can arbitrarily assign
the causality of your resistive elements, you will hit a loop when
MANE 4240 & CIVL 4240
Introduction to Finite Elements
Prof. Suvranu De
Higher order elements
Reading assignment:
Lecture notes
Summary:
Properties of shape functions
Higher order elements in 1D
Hig
Chapter 1
Tension and Compression in Bars
1
1 Tension and Compression in
Bars
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Stress.
Strain.
Constitutive Law .
Single Bar under Tension or Compression.
Statically Det
Computer Homework #1
Description:
Starting with the attached Matlab code, develop a program for onedimensional elasticity. The parts that need to be added are the
computational of strain and stress
% CEE-ME_327 computer homework #1 %
clear;
%= Preprocessing =%
nel=5; % number of elements
totlength=10; % total length of the domain
b = 10; % body force [force/length]
youngs=1; % elastic modulus
ar
ME-CEE 327 Finite Element Methods in Mechanics Fall 2016
Instructors:
Professor Wing Kam Liu with Dr. Mark Fleming of Caulfield Engineering as Guest Lecturer
Days and Times:
Tu, Th 12:30pm-1:50pm, Tec
1D Elasticity (Boundary Value Problem (BVP) or Strong Form)
Given a bar of cross-sectional area A(x), with thermal expansion coefficient (x), temperature
distribution t(x), Youngs modulus E, acted on
[Ref: Jacob Fish and Ted Belytschko, The first Course in Finite Element, John Wiley & Son, 2007]
HEAT CONDUCTION:
1. FOURIER LAW (LIKE HOOKES LAW FOR STRESS ANAYSIS)
dT
d TEMPERATURE
g=
HEAT FLUX =
)
Section 4: Implementation of Finite Element
Analysis Other Elements
1. Quadrilateral Elements
2. Higher Order Triangular Elements
3. Isoparametric Elements
Implementation of FEA:
Other Elements
-1-
Se
FORMULATION OF 2 AND 3 DIMENSIONAL BVPs
Laplacian Equations, e.g. Heat Conduction, Incompressible Potential Flow
Preliminary
NSD # of space dimensions
open set, domain (not including boundary)
boundar
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cfw_Design Specificat