Due September 26, 2012
ME340 HW#3
ME340 Midterm II
0. Read Sections 2.2, 4.2, Name: in Esfandiari and Lu.
and 4.6
8
1. A mass m pivots on a massless rod with length l about a point which is driven to move vertically, the
8. A mass m pivots the pivot relat
Homework 6 Solution: Time and Frequency Domain
ME 340 Fall 2016
Prof. Amy LaViers
1
Computational Analysis. Consider the catapult system we considered in HW
5 (for this problem, use the solution provided for HW 5 on the course website).
rock
x
cable
f
mot
ME 340 HW#9
Due November 4, 2016
SOLUTIONS
Show all work for full credit!
1. Consider the system shown:
The equations that describe the motion of the three masses are given by
2 + 5 + ( ) = 0
+ ( ) + 2( ) = 0
+ 2( ) + = 0
(a) Find this systems three nat
ME 340 HW#3
Due September 21, 2016
SOLUTIONS
0. Reading Assignment: Read review Notes on Partial Fractions given in the Lectures folder on the course website.
1. Express the following fractions as a sum of their partial fractions
(a)
3+4
2 +3+2
Solution:
ME 340 HW#4
Due September 28, 2016
SOLUTIONS
Show all work for full credit!
1. The free motion of a mass-spring-damper system is governed by the differential equation
() + () + () = 0,
where () denotes the displacement from rest of the mass.
(a) Suppose
ME 340 HW#2
Due September 7, 2016
SOLUTIONS
0. Reading Assignment: Read review notes on partial fractions given in the Lectures folder on the course website.
1. Show that sin + cos can be written as sin( + ) where = 2 + 2 and = tan1 ().
Solution:
1
1
sin
ME 340 HW#6
Due October 17, 2016
SOLUTIONS
Show all work for full credit!
In the following represents convolution operation, () and () represent unit-impulse and unit-step
functions respectively.
1. Compute Laplace transform of the function (), which is g
ME 340 HW#8
Due October 28, 2016
SOLUTIONS
Show all work for full credit!
1. Consider the system of equations below
1 = 71 42
2 = 241 132
()
(a) Let () = [ 1 ]. Rewrite the above system in the form = . Find .
2 ()
()
Solution: Based on our given infor
ME 340 HW#1
Due August 31, 2016
SOLUTIONS
1
30
3
1. Find the real numbers and such that + = ( + )
2
2
coordinates].
Solution: First convert the right-hand side to polar notation:
2
1 2
3
= ( ) + ( ) = 1,
2
2
30
So we evaluate( 3 )
. [Hint: compute the ri
ME 340 HW#11
Due December 2, 2016
SOLUTIONS
1. In simpler times, the world agreed that Pluto was a planet and Charon was its moon. In fact, Pluto and Charon
differ in mass by less than one order of magnitude, and orbit one another as a planar binary syste
Homework 12: Toward Control
ME 340 Fall 2016
Prof. Amy LaViers
Due in class Wednesday, December 7, 2016
1
Linearization. Consider the cart system given below; our goal is to balance the
inverted pendulum upright with the cart.
Figure 1: Inverted pendulum
Homework 3 Solution: Systems in Time, Part II
ME 340 Fall 2016
Prof. Amy LaViers
1
State-space representation. Convert the following three systems into statespace representation. Choose a reasonable input and output (describe your context
for these select
Homework 2 Solution: Systems in Time
ME 340 Fall 2016
Prof. Amy LaViers
1
ODEs. Answer the following questions around ordinary differential equations.
(a) State whether the following ODEs are 1) linear or nonlinear, 2) their order,
and 3) are time-invaria
Reference Slides
ME 340 Dynamical Systems
- all dates and content are subject to change -
Welcome
ME 340 Dynamical Systems
August 22, 2016
MATHEMATICS REVIEW
Complex Numbers
ME 340 Dynamical Systems
August 24, 2016
Eulers Formula
ME 340 Dynamical Systems
ME 340 HW#12
Will not be graded.
SOLUTIONS
1. Let
be the function depicted below
and consider the differential equation
0.
Find a linear differential equation whose solution approximates the deviation variable
is close to 0.
Solution: From the plot of
t
ME340 HW10 Solutions
1(a).
The equations of motions of each mass
For mass m1 :
1 + (k1 + k2 )x1 + b1 x 1
m1 x
k2 x2 = F1 . (1)
For mass m2 :
(m2 x
2 + b2 x 2 + k2 x2 )
k2 x1 = 0 . (2)
b2 x 3
For mass m3 :
b2 x 2 = F3 . (3)
3 + b2 x 3 + k3 x3 )
(m3 x
1(b).
ME 340 HW#5
Due October 5, 2016
SOLUTIONS
Show all work for full credit!
1.
(a) Consider the function () = ( + log ), where 2, , and 1 are positive real
numbers.
i. Show that () > 0 for all > maxcfw_1, + log .
Solution: With the above information, we can
ME 340 HW#7
Due October 21, 2016
SOLUTIONS
Show all work for full credit!
1.
(a) Find the Laplace transform of () = (sin ) ().
Solution: At each step, we will use the table of Laplace transforms, which is attached to the end of this
document.
Let 1 () = s
ME340 HW#7, partial solutions
1. Consider the mechanical suspension shown below.
f (t)
i
xm
0.2 kg
i
0.3 Ns/m
0.1 N/m
Use a free-body diagram and Newtons 2nd law to derive a dierential equation governing x(t) and
determine whether the system is overdamped
ME340 HW#9, partial solutions
1. Consider a convolution with unit impulse response
h(t) = et/2
sin( 2 /4 t)
2 /4
with > 0 and > 2 /4. Show that |h(t)| et/2 /
corresponding convolution is stable.
2 /4. Use this to show that the
Answer: Since > 2 /4, | si
ME340 HW#8, partial solutions
1. Find the natural frequency of the dynamical system shown below.
4
d
X(s)
1
s
+
+
1
s
+
O(s)
c
and express x(t) in terms of the initial values c and d.
Answer: Here,
X(s) =
cs 4d
x(t) = c cos 2t 2d sin 2t
s2 + 4
so the nat
ME340 HW#10, partial solutions
1. Find the equation of motion for a small bead of mass m that is able to translate along the straight line
y = x + in a vertical inertial coordinate system under the inuence of gravity.
y
y = x +
m
g
x
Answer: See solution
Due: April 24, 2015
ME340 HW#12
You may discuss the assignment with other students, e.g., using the course blog, but by signing your assignment, you pledge that all work is your own. You must show all work for full credit!
1. Use the Lagrangian recipe to
ME340 HW#11, partial solutions
1. Use the Newtonian recipe to nd the equation of motion for a slender ring of mass M and radius R
rolling without slipping along an inclined plane that makes an angle with the horizontal, and under
the inuence of a linear s
Due: May 1, 2015
ME340 HW#13
You may discuss the assignment with other students, e.g., using the course blog, but by signing your assignment, you pledge that all work is your own. You must show all work for full credit!
1. Linearize the function f (x) = (