MAE 340: Homework 4
Name and Section:
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, marg
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
Homework 14
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
Assume all in
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)
b) 3(t) = u(t) ; the two outputs are y1 (t) = x(t), and
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
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
1. A low-pass lter, with bandwidth 0 100 and act
MAE 340
HW21: Modeling of Systems
Find state-space models for each of the following systems:
1.
Masses m1 and m2 are connected to the ground through
the springs and dampers as shown. Each is also connected to a exible attachment to the pulley with inertia
MAE 340
HW20: Modeling of 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 the pulley
MAE 340
Example: Boundary Conditions in the Homogeneous Solution
2015 D. Joseph Mook
We have seen that the general form of the homogeneous solution is
xh (t) = A1 e1 t + A2 e2 t + + An en t
where the s are found from the Characteristic Equation.
The As ma
MAE 340
HW26: Motor Design
NOTE: Use the posted solution from HW25 as your model!
The system shows an electrically-powered rack-and-pinion steering system in a car. The input to this system is the
voltage source, which we assume to be proportional to the
MAE 340
HW25: Modeling Transducers
The system shows an electrically-powered rack-and-pinion system. The input is the voltage source. The electric
system drives the output shaft of the EM transducer, which ends at the pinion gear. The shaft has inertia J a
MAE 340: Homework 8
Name and Section:
For each of the following characteristic equations,
a) Draw the root locus plot for 0 K
b) Find the range of K (if any) that results in a stable system
c) Find the range of K (if any) that results in a non-oscillator
MAE 340
2015 D. Joseph Mook
Frequency Domain
From our knowledge of particular solutions, we know that:
If input = Asin(t)
then particular solution = Bsin(t + )
The response to a sinusoidal input is a sinusoidal output with the same frequency
Magnitude m
MAE 340: Homework 10
Name and Section:
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
b) Find the range of K that
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
Modeling of Ideal Transducers
Transducers represent physical devices that couple different types of subsystems together.
Transducers are called ideal if they do not consume any energy or power.
Since ideal transducers do not consume any energy or
MAE 340
Laplace Transforms
2013 D. Joseph Mook
Laplace transform:
Z
X(s) L(x(t)
x(t)est ds
0
Use capital letter for Laplace domain, lower case letter for time domain
Inverse Laplace transform:
x(t) = L1 (X(s)
In engineering practice, its rare to actual
MAE 340
Frequency Domain
2016 D. Joseph Mook
From our knowledge of particular solutions, we know that:
If input = Asin(t)
then particular solution = Bsin(t + )
The response to a sinusoidal input is a sinusoidal output with the same frequency
Magnitude m
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 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 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 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: Homework 6
Routh Hurwitz with axis-shifting
2015 D Joseph Mook
For each of the following homogeneous LTI ODEs,
a) use Routh-Hurwitz to determinethe values of K (if any) for which the system is stable
b) use Routh-Hurwitz along with axis-shifting
MAE 340: Homework 8
Name and Section:
For each of the following characteristic equations,
a) Draw 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 range of K (if any) that r
MAE 340: Homework 5
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 are not
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
Modeling of Physical Systems
The general process for creating LTI ODE models for physical systems follows a common approach that is essentially
the same regardless of what type of system is being modeled:
1. Use standard elements.
(a) Each element