ME340-AL2 Homework #8 Solution
Prob. 1
Computation of homogeneous solution
The characteristic equation is
2
2
2
n
n
2
0
4
2
n
2
2
n
4
2
n
2
n
1
n
n
As we have only one root for the characteristic equation, the homogenous solution is given in the
following
ME340-AL2 Homework #5 Solution
Prob. 1
(i) First way of solution
The equilibrium states of the given system can be calculated by letting x1
2
2
2x
x1 3 x2
0
0 as
2
2
3 2
2
x2 e
1 0
x2
x1e
To get the linearized system close to the equilibrium position ( x1
ME 340 Fall 2014
Homework # 9
Due Monday, November 3
1. Compute the Laplace transforms of the following functions:
~ 2
2. Solve the following problem using Laplace transform (where () is the Heaviside function):
4 = 2 5 (),
(0 +) = 1
3. Solve the followi
ME 340 Fall 2014
Homework # 13 (The End)
Due Wednesday, December 10
1. (50 points). Consider the following system that is in equilibrium when 1 = 2 = 0. We want to compute its forced
response using modal analysis. Assume initial conditions 1 (0) = 1 , 1 (
ME340-AL2 Homework #2 Solution
Prob. 1
As long as the small disk is in contact with the big stationary disk, the trajectory of the center of the small
disk will form an arc with the radius of R1+R2. The kinetic energy accounting for this translational mot
ME 340 Fall 2014
Homework # 8
Due Monday, October 27
1. Derive the general solution based on the convolution integral of the following critically damped mechanical oscillator:
+ 2 + 2 = (), (0) = 0 , (0) = 0 , = 1
Follow the methodology that we developed
ME 340 Fall 2014
Homework # 11
Due Monday, November 17
1. We reconsider the mathematical model of a car moving towards a bump with a constant velocity V that we studied
in problem 5 of HW8, but now we assume that the bump is defined as follows (in meters)
ME 340 Fall 2014
Homework#7
DueMonday,October20
1.Weconsidertheweaklyforcedresponseoftheundampedharmonicoscillator:
0.01
, 0
0,
0
0
Compute the analytical response of this system for (i)
1, and (ii)
1. What happens to the response in the
versus foreach
ME340-AL2 Homework #6 Solution
Prob. 1
(i)
x x 5y 1
y 2y x 5
, x(0)
x0 , y(0)
y0
At t=0,
x(0)
x0 5 y0 1
y (0)
2 y0
x0 5
Taking the derivative of the first equation with the time t leads to
x x 5y
0
y 2y x 5
x x 5
2y x 5
0
x x 10 y 5 x 25 0
x x 2 x x 1
5 x
Wm M/ W a mu/wme aim I?! rm/<7 MW}; ak/m?
Y) )(2(f)
k m K 21< -. .
E r ' "*k*"m*
WE/05m ' QmXszJrkpszmro
We rm +v SfMoé/ 515. WW of He {yam
Mae +0 +48 wiry/m 72/0113. [37 «cf/mg? ljwt =9 7'78
7712/1:
th _,)/A_
mi. +ll<)(./<Aé=8" Q1 6.3. we assume 5mm
2m£1
ME 340 AL2: Dynamics of Mechanical Systems
Fall 2014
Instructor:
Alexander F. Vakakis, avakakis@illinois.edu
Office hours: Fridays 12noon-1pm (218 MEB), and Thursdays 12noon-1pm (3003 MEL)
The 12noon-1pm weekly meeting in 218 MEB will be used by the instr
In physics and other sciences, a nonlinear system, in contrast to a linear system, is
a system which does not satisfy the superposition principle meaning that the
output of a nonlinear system is not directly proportional to the input.
In mechanics, the de
ME340-AL2 Homework #7 Solution
Prob. 1
(i) 1
x(t ) x(t ) 0.01sin t , x(0) 0, x(0) 0
The characteristic equation is
2
1 0
j
The homogenous solution is
xh C1 cos t C2 sin t
The particular solution is assumed as
xp
a sin t b cos t
xp
a cos t b sin t
xp
2
a
s
ME340-AL2 Homework #9 Solution
Prob. 1
(i) From the graph on the left, f(t) can be written as
t
f (t )
2(t
0
t
2
2
t
3
3)
0
t
3
By applying the integral for Laplace transformation, we have
F (s)
Lcfw_ f (t )
2
0
2
0
3
te st dt 2
t 3 e st dt
2
3
3
te st dt
ME 340 Fall 2014
Homework # 5
Due Monday, October 6
1. Compute all equilibrium states of the following dynamical systems. For each problem, if the system is nonlinear
linearize it close to each equilibrium state and write the resulting linearized system i
ME 340 Fall 2014
Homework # 5
Due Monday, October 6
1. Compute all equilibrium states of the following dynamical systems. For each problem, if the system is nonlinear
linearize it close to each equilibrium state and write the resulting linearized system i
ME 340 Fall 2014
Homework # 13 (The End)
Due Wednesday, December 10
1. (50 points). Consider the following system that is in equilibrium when 1 = 2 = 0. We want to compute its forced
response using modal analysis. Assume initial conditions 1 (0) = 1 , 1 (
ME 340 Fall 2014
Homework#3
DueMonday,September22
1.Foreachofthefollowingsystemsderivetheequation(s)ofmotionusing(a)Newtonssecondlaw,and(b)Lagranges
equations.Ineachcasedefinethestatevariablesandfindthedegreesoffreedom.
(i)Equilibriumpositionwhen
0.
(ii)
ME 340 Fall 2014
Homework#1
DueMonday,September8,
1.Forthesystemshownbelowweshowthedisplacementsofthemassesfromthepositionofequilibrium:
(i)Derivetheequationsofmotion.Characterizethesystemintermsofthethreeclassificationswediscussedinclass.
(ii)Howmanydeg