MAE 340: Homework 2
Name and Section:
For each of the following homogeneous LTI ODEs,
a) Find the characteristic equation
b) Find the roots of the characteristic equation
c) Write the general form of the homogeneous solution
Since these are all LTI ODEs,
MAE 340: Homework 5 Solution
Routh Hurwitz
2015 D Joseph Mook
1. For each of the following homogeneous LTI ODEs, use Routh-Hurwitz to determine whether or not the system
is stable. If it is not stable, use Routh-Hurwitz to determine exactly how many roots
MAE 340: Homework 4
Solutions
Homogeneous Solutions of 1st- and 2nd-Order Systems
For each of the following homogeneous LTI ODEs, with accompanying initial conditions, use MATLAB to do the
following:
a) Determine whether the system is stable, marginally s
2015 D Joseph Mook
MAE 340: Quiz 1 Solution
A system is modeled by the homogeneous LTI ODE:
x(t) + 4x(t) + 43.5x(t) = 0
with x(0) = 4, x(0) = 0
a) Find the damping ratio, natural frequency, and settling time for the homogeneous solution
b) Sketch the homo
MAE 340: Homework 6 Solution
2015 D Joseph Mook
Routh Hurwitz with axis-shifting
For each of the following homogeneous LTI ODEs,
a) use Routh-Hurwitz to determine the values of K (if any) for which the system is stable
b) use Routh-Hurwitz along with axis
MAE 340: Homework 9
Name and Section:
For the characteristic equation:
3 + 52 + c + (K 13) = 0
a) Set c = 7, and sketch the root locus plot for 0 K , using the graph provided
b) Find the range of K (if any) that results in a stable system
c) Find the rang
MAE 340: Homework 12
Particular Solutions
1. Find the particular solution for the following systems:
(a)
x(t) + 2x(t) + 10x(t) = f (t)
,
f (t) = 4
Solution: The forcing is a constant, so the form of the particular solution is xp (t) = C. Substituting,
xp
MAE 340: Homework 3
Name and Section:
For each of the following homogeneous LTI ODEs, with accompanying initial conditions, use MATLAB to do the
following:
a) Find the roots of the characteristic equation (function ROOTS)
b) Apply the initial conditions t
MAE 340
Homework 14
Solution
I. Unit Step Response
For each of the given LTI ODEs:
1.
x + 10x + 16x = f (t)
2.
x + 0.8x + 16x = f (t)
3.
x + 8x + 16x = f (t)
Do the following:
(a) Compute the complete solution for a unit step input
f (t) =
0
1
t<0
t0
Assu
LTI ODE General Form:
an
dn1 x(t)
dx(t)
dn x(t)
+ an1
+ . + a1
+ a0 x(t) = f (t)
dtn
dtn1
dt
(1)
The solution is sum of homogeneous and particular:
x(t) = xh (t) + xp (t)
ICBS that
xh (t) = Aet
where
A, = constants, whose values we will determine soon
e =
MAE 376, Fall 2015, Homework 2
Due: Tues. Sept. 29, 2015, in class
Work all problems. Show all work, including any M-les you have written
or adapted. Make sure your work is clear and readable - if the TA cannot
read what youve written, he will not grade i
MAE 340: Homework 1
Name and Section:
1. Determine whether or not each of the following is a linear, time-invariant, ordinary dierential equation (LTI
ODE)
2. If an equation is not an LTI ODE, clearly identify why it is not
3. If an equation is an LTI ODE
MAE 340: Homework 19
Name
In each case below, nd a transfer function that behaves as described, then use MATLAB to construct the Bode
plots of your transfer function to verify that it is indeed as described
The solutions shown are not the only ones possib
MAE 376, Fall 2015, Homework 1
Due: Thurs. Sept. 17, 2015, in class
Work all problems. Show all work, including any M-les you have written
or adapted. Make sure your work is clear and readable - if the TA cannot
read what youve written, he will not grade
MAE 340
HW23: Modeling Mechanical Systems
Find state-space models for each of the following systems:
1.
Mass m1 is connected to the wall through spring k1 and
damper c1 . It slides on the oor without friction. An
inextensible cord connects mass 1, over th
MAE 376, Fall 2015, Homework 3
Due: Tues. Oct. 6, 2015, in class
Work all problems. Show all work, including any M-les you have written or adapted.
Make sure your work is clear and readable - if the TA cannot read what youve written,
he will not grade it.
MAE 340
Ordinary Differential Equations
I Introduction
Real World Mathematical Model
Engineers almost always create mathematical models to represent physical systems, and then, they may perform
a very wide variety of calculations based on these mathematic
Chapter 1
Linear Systems
In this chapter, we on focus engineering systems that can be cast into linear systems of equations. Although a more complete treatment of linear systems is the
subject of courses in linear algebra, we rst review necessary topics i
Useful MATLAB Functions
Name
Description
bode
generate a Bode plot
Page
#
3
conv
multiply two polynomials (convolution)
6
deconv
divide one polynomial by another (deconvolution)
7
for
repeat a set of calculations iteratively (often builds on previous resu
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Fall 2016 SYLLABUS for MAE 340: Dynamic Systems
IMPORTANT Announcement:
Due to crowded classrooms, you MUST attend the section in which you are registered; do not attempt to attend
another section. All graded work will ONLY be accepted in the section in w
Outside Reading Suggestions for MAE 340: Linear Systems
The following books may be useful as additional references on one or more course topics. I have avoided using
edition numbers or dates of publication because most of these have appeared in multiple e
MAE 340: Homework 18
I. Bode Plots of 2nd-Order Systems
A standard 2nd-order system is given by
2
x(t) + 2n x(t) + n x(t) = f (t)
First, nd the three transfer functions for this system, for the outputs y1 = x, y2 = x, and y3 = x. Then use
MATLAB function
MAE 340: HW10
More Root Locus Practice
2015 D Joseph Mook
Plot the root locus plots for each of the systems below, in the order given. This will provide essential extra practice
at creating the plots, and also help to illustrate the eect of dierent roots
MAE 340
HW22 - FL1: Solution
We use element equations, loop equations, and node equations to convert to the form
z(t) = Az(t) + Bu(t) ,
y(t) = Cz(t) + Du(t)
We begin by dening the usual state, input, and output vectors:
PC1
P
y(t) = wasnt specied, so wel
MAE 340: Homework 10 Solution
Given the root locus plot shown, drawn for 0 K :
a) Find the LTI ODE whose characteristic equation generated this plot. Use x(t) for the dependent variable, t for
the independent variable
Solution:
The roots of D() = 1, 1, 4
MAE 340: Homework 13
Conversion to State Space
Find a state space model for each of the following systems. Use a separate input function for each independent
function of time on the RHS of the original equation.
1.
.
x (t) + x(t) + 2x(t) + 10x(t) = f (t)
MAE 340: Homework 16
1. For each of the following systems, nd all transfer functions by using your own algebra, and then nd their
poles and zeros:
a) x(t) + 2x(t) = 3u(t) ; the output is y(t) = 2x(t)
Solution: Take the Laplace transform of the LTI ODE an
MAE 340: Homework 17
For each of the following systems,
a) nd the frequency response function (FRF) as a function of
b) nd the magnitude of the FRF as a function of
c) nd the phase of the FRF as a function of
d) using your results from (2) and (3) abov
CODE PROBLEM 4
clear; clc;
p = [1 9 9];
in = [5 1]';
X = roots(p);
[r, c] = size(X);
lamb1 = X(1,1);
lamb2 = X(2,1);
B = [1 1;
lamb1 lamb2];
A = inv(B) * in;
A1 = A(1,1);
A2 = A(2,1);
TAU = zeros(r,1);
for i = 1:r
TAU(i,1) = -4 * (X(i,1)^(-1);
end
ST = ma
CODE PROBLEM 1
clear; clc;
p = [1 2];
in = [10];
X = roots(p);
A = in;
tau = -(X^(-1);
ST = 4 * tau;
t = [0:0.01:ST];
x = A * exp(-2 * t);
plot(t,x);
grid on
title('Problem 1 : A_1e^cfw_\lambdat')
xlabel('Time (t)')
ylabel('Amplitude')
legend('10e^cfw_-2t