HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE PROCRASTINATORS
HELLO FINE P
Analysis of Microfluidics
Introduction
Microfluidics refers to the control and
operation in the microscopic size and
the testing of complex fluids
technology
in
microelectronics,
MEMS,
biotechnology
and
nanotechnology.
Microfluidics
is
widely applied in b
Todays Agenda:
The Direct Stiness Method
Derive stiness matrix using solid mechanics principles
Focus will be on devising algorithms for assembling
element level matrices
Properties of the stiness matrix
Incorporation of Dirichlet/Displacement boundar
3
Fundamentals of FEA: Strong form,
weighted residual form, weak form and
the principle of minimum total
potential energy
3.1
Introduction
Earlier we studied the direct stiness method which involved applying principles from mechanics of solids to
derive t
AER501
Solutions to exercises1
DRAFT Version: Wednesday 16th September, 2015
19:05
Chapter 3
Problem 3.3.6.1
The residual error corresponding to the assumed approximation u
b(x) = c1 x(1 x2 ) is given by
R(x, c1 ) =
d2 u
b
+ 1000x2 = 6c1 x + 1000x2 .
dx2
7
Finite element analysis of linear
elasticity problems
So far we have considered only one-dimensional (line) elements that form structures such as trusses and frames.
Line elements have properties associated with the other two dimensions in their cross s
Outline
Review of linear elasticity governing equations
General derivation of Ke & f e
3-node constant strain triangle (CST)
4-node rectangular element
Finite element analysis of
linear elasticity problems
AER501
AER501
Finite element analysis of linear e
4
Finite element analysis of rods
We shall look at how the element stiness matrix of rod/bar structures can be evaluated using the weak/minimization
form. Consider an axially loaded rod (with Youngs modulus E and cross-section area A(x) of length 1 cantil
AER501
Topics to be covered
Introduction to structural optimization
Non-gradient optimization methods
Gradient-based optimization methods
Sensitivity analysis
Last 2 sessions course review and
problem solving
Requirements
The design spiral
CONCEPTUA
12
Sensitivity Analysis
12.1
Introduction
Sensitivity analysis is of fundamental importance to design based on computational approaches. It allows the use
of gradient descent methods, reveals when optimal designs have been produced and indicates which var
11
Introduction to gradient-based
numerical optimization
There are several excellent texts on numerical optimization; for example, Nocedal and Wright, Numerical
Optimization, Springer-Verlag, 1999, and C T Kelley, Iterative Methods for Optimization, SIAM,
8
Theoretical and convergence aspects of
finite element methods
8.1
Criteria for monotonic convergence
To ensure monotonic convergence, the elements must be complete and the elements and mesh must be compatible.
In other words
Compatibility + Completeness
H
An Outline of
MSA History
H1
Appendix H: AN OUTLINE OF MSA HISTORY
TABLE OF CONTENTS
Page
H.1.
H.2.
H.3.
H.4.
H.5.
H.6.
H.7.
H.8.
H.9.
H.10.
INTRODUCTION
Background and Terminology
Prolog - Victorian Artifacts: 1858-1930
Act I - Gestation and Birth: 193
6
Finite element analysis of dynamic
problems
6.1
Derivation of weak form
To illustrate how the weak form of time-dependent problems can be constructed, consider the equations governing the axial vibrations of a cantilevered one-dimensional rod structure
Orbital Perturbations
4
1
Orbital Perturbations
4.1
Classification of Perturbations
Consider the two-body problem which addresses the motion of a point mass m1 in the
gravitational field of another point mass m2 (the primary). The two-body problem defines
AER501: Midterm Course Review
1
Finite element analysis
1.1
1.3
In AER501, we have so far dealt with the strong form of the equations governing the static response of rods and beams. The strong
form includes the governing equations AND all the boundary co
5
Finite element analysis of beams and
frame structures
A beam is a bar-like structural component whose primary function is to support transverse loading and carry
it to the supports. A beam resists transverse loading by bending. Under transverse loading,
AER501: Advanced Mechanics of Structures
UTIAS
University of Toronto Institute for Aerospace Studies
http:/utias.utoronto.ca/~pbn
Assignment 1 (7 pts)
Due October 12, 2015
Write a MATLAB code for computing the displacements, stresses and strains
of two-di
1
Gravity-Gradient Stabilization
12
12.1
Gravity-Gradient Stabilization
Reference Frames
At this point, three important reference frames are introduced. The first, F I , is an inertial
frame which is centred at the earth. This frame does not rotate with t
1
Interplanetary Spacecraft
6
6.1
Interplanetary Spacecraft
Introduction
Consider an interplanetary trajectory from earth to mars. Near earth, the spacecraft is
primarily under the influence of the earths gravitational field. For most of the journey, it i
1
Spin Stabilization
9
Spin Stabilization
At this point, we have a general statement for the rotational behaviour of a rigid body,
Eulers equations. In the absence of external torques,
I1 1 + (I3 I2 )2 3 = 0
I2 2 + (I1 I3 )1 3 = 0
I3 3 + (I2 I1 )1 2 = 0
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1
Orbital Dynamics
3
3.1
Orbital Dynamics
The Two-Body Problem
In this section we consider a system of two particles with masses m1 and m2 which experience
only the mutual gravitation acting between them. Although only point masses are covered
by this tre
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Spin Stabilization Revisited
13
1
Spin Stabilization Revisited
13.1
The Thomson Equilibrium
Recap
In the absence of energy dissipation, major and minor axis spins are stable. With energy
dissipation accounted for, minor axis spins are unstable.
Question
W
AER506H1F Spacecraft Dynamics and Control I
Problem Set #1
Due: Wednesday, September 24, 2014
1. Show that C21 r
1 C12 = (C21 r1 ) . Hint: Consider expressing the components of
r s in two different ways where r = F T1 r1 = F T2 r2 and s = F T2 s2 .
~ ~
~
1
The Restricted Three-Body Problem
7
7.1
The Restricted Three-Body Problem
Formulation
Consider three masses that interact gravitationally:
h
m2
r2
~
r m3
r = r3
FI
~ ~
~
1h m1
6
r1
~
O
Assumptions
1. The gravitational effect of m3 on m2 and m1 is