HW 3 Solutions
EA3 Spring 2015
Table of common point deductions
Error
Wrong sign (in equation)
Wrong sign convention (arrow pointing in wrong direction
in FBD, etc)
Missing/wrong unit (when units provided in problem
statement)
Matlab graph missing label
M

The Vector-Valued Convolution Equation
In the previous section, we assumed that x is a scalar-valued quantity, which naturally leads to the
question of how to obtain solutions for differential equations of the form x = Ax + Bu where x is
a vector. To thin

15
The Diffusion Equation with Accumulation
Last time we talked about the diffusion equation in equillibriumwhich means that the concentration does not change with timeand without accumulation. Today we talk about the case
where there is accumulation. Acc

14
The Diffusion Equation with No Accumulation
Last time we discussed Ficks law and how ux is related to the derivative of concentration with
respect to the position x. We also discussed that this linear constitutive law holds because the uid
is very cont

13
Chemical Diffusion and Ficks Law
For the last 12 lectures we have been learning about mechanical systems and how to model mechanical systems that have linear constitutive laws relating mechanical variables, like position x
and velocity v, to forces. To

10
Imaginary Numbers Continued
Last time we talked about imaginary numbers and using imaginary numbers as a way of encoding
oscillation into the exponential solution of an ordinary differential equation. But most systems
do not just oscillate foreverthey

11
Vector and Matrix Representations
So far, we have only worked with very, very simple systems. But you can imagine that if we have
lots of springs, dampers, and masses all connected togetherlike the slinky I used in the class
lecture where I connected s

12
Vector Solutions to Ordinary Differential Equations
Last time we learned that vector and matrix representations can make your life easier, specically
when you implement Euler integration for a complicated system involving many states. But how
do we use

Why We Use Exponential Solutions
A very reasonable question that comes up is what is so special about ert ? Why not use a different
number than e, like 2.5 or or something else? And why an exponential? Why not arctan(rt) or
sin(rt)? There are a couple of

8
Newtons Laws with Several Masses
Last time we learned that Newtons laws with mass lead to second-order ODEs that we convert
into rst-order ODEs with multiple equations. What if we want to go backwards from a rst-order
ODE with multiple equations to a se

Superposition in Vector Linear ODEs with Complex Eigenvalues
On the previous page, and in the lecture video, I talk about superposition of solutions to ODEs
when the eigenvalues are real-valued, but I dont say anything about what happens when the eigenval

16
Time-Varying Diffusion
Last time we talked about diffusion processes with accumulation. But everything we have talked
about up until this point is dependent on the idea that the concentration is in equillibrium. However,
most of our experience with dif

17
The Convolution Equation
So far we have talked about mechanical systems and chemical systems, and we have learned that
linear, constant-coefcient differential equations describe these systems in many settings. In both
cases, and soon in electrical syst

19
Kirchhoffs Laws
Last time we introduced electrical systemsvoltage, charge, current, and the constitutive laws that
relate them. What principles do we use to generate mathematical models of electrical systems? In
the case of mechanical system, the princ

18
Modeling Electrical Components
Near the beginning of this course we talked about how we build models of mechanical systems
from models of mechanical components. The same thing is true of electrical systems.
Today we are going to investigate how we mode

System Identication Demonstration
Everything is the Same: Week 3
Introduction
In this demo we will show you how to identify and compute parameters for a given system. This particular
demo will consist of nding the spring constant of a simple spring mass s

MATLAB Tutorial Solutions
MATLAB Variables:
A1. 1 and 2
MATLAB Mathematical Functions:
A1. iii
A2. x = atan(3.14)
MATLAB as a Calculator:
A1.
s = 0.5; m = 1; x = 1.2
y = (exp(-(x-m)^2/(2*s^2)/(2.5*s)
A2. iii
Vector Creation via Concatenation of

24
Everything Is The SameAlmost
We are going to end with what I promised you at the beginning of the classthat everything is
the same. Well, at least with all the caveats that the system is engineered and that we can use
constitutive laws and all of the o

S p r in g S im u la t o r N o t e s & C h a n g e s
E v e r y t h in g is t h e S a m e
M
T h
a n d m
to p o f
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is
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th
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s im u la t o r p r
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n c o v e re d , so
o d
rs
so
p
u c e s
c a n
ftw a
le a s e
a
b

23
Interpretation of Mathematical Expressions as Physical Systems
Last time we talked about analogies between physical systems. Today we are going to talk about
using physical systems to understand and reason about mathematical expressions. If I give you

21
Vector and Matrix Representations in Kirchhoffs Laws
Last time we saw that electrical systems have the same basic characteristic behavior as mechanical
systemsthey have exponential solutions and, with an inductor, can experience oscillation. Today
I wa

20
Kirchhoffs Laws with Inductors
Last time we discussed Kirchhoffs laws, but I didnt say anything about how inductors work.
Today we are going to talk about the role that inductors play in circuits and how inductors complete
the analogy between mechanica

22
Mechanical/Electrical Analogies
Last time we talked about using vector and matrix notation to represent electrical circuits. But
what if we want to use one physical system to represent another physical system? How do we
know when a parallel between two

7
Newtons Laws with Mass
Previously, we learned about Newtons laws in Lecture 3, but in the absence of mass. Today we
are going to look at what happens when the mass at a point is nonzero. That is, we are going to
look at
F = ma
where m = 0. Remember that

2
Modeling Components
Last time I mentioned that Analytical Reasoning, Computational Skills, and Physical Knowledge
are all critical components of an engineers judgment. Building those skills starts with understanding how to model components of a physical

Diusion Instructions
As we have discussed in class, diusion of a chemical in a solution is governed
by what is called the diusion equation. The form of the equation for a system
that is not reacting is
d
2
C(x, t) = D 2 C(x, t)
dt
dx
where C is a function

Note: Also correct if state vector is [v3 x1] instead of
[x1 v3] as long as elements are in the consistent
order. For this problem, you can switch rows 1 and 2.
3
Note, matrix form is still correct if the row order is
changed consistently on both sides of

% EA3 HW4 Problem 5-2: w/o the 100kg mass
% Given values
k = 2000;
b = 1200;
X_spring(1) = -0.02;
% Time interval of dt = 0.01
dt = 0:0.01:8;
% Force values: change between F = 0 and the actual values for F
% depending on the question
for ii = 1:length(dt