ME 563 Fall 2011
Homework Prob. 2.1
A homogeneous wheel having a radius of R and mass of M rolls without slipping on a
horizontal surface. A massless, inextensible cable is attached to the center of mass G of
the wheel and to a block B (having a mass of m
Problem 11.1 - SOLUTION
Consider the undamped single-DOF system shown below, where m = 2 kg and
k1 = 1600 N / m . The applied force, f(t), has an impulse of I = 5 N sec .
x
k1
k2
m
()
ft
x
k2 x 1
k1
a) Derive the EOM for the system in terms of the general
ME 563 - Fall 2004
Midterm Exam
Problem No. 1 20 points
Name
Use the influence coefficient method to determine the flexibility matrix for the threeDOF system shown below.
x3
3k
k
k
2k
x2
x1
2
M E 5 6 3 - F a ll 2 0 0 4
Midterm Exam
Problem No. 2 20 points
Problem 11.1 - SOLUTION
Consider the undamped single-DOF system shown below, where m = 2 kg and
k1 = 1600 N / m . The applied force, f(t), has an impulse of I = 5 N sec .
x
k1
k2
m
()
ft
x
k2 x 1
k1
a) Derive the EOM for the system in terms of the general
Example 2 - usage of the power equation
Find the EOM for the system shown corresponding to the generalized coordinate x,
where the coordinate x is defined such that x = 0 when the spring is unstretched.
Power equation: Example No. 2 Solution
x
m
k
f(t)
c
ME 563 SYLLABUS
MECHANICAL VIBRATIONS
Fall Semester 2010
INSTRUCTOR:
Professor Douglas E. Adams, School of Mechanical Engineering
E-mail: [email protected] (24 hours a day, 7 days a week!)
Office Hrs: MW 3:30-4:20 p.m. ME 361, 765-496-6033
COURSE TEXT:
M
1
Practice with some matrix operations
Use the following vectors a and b as well as the following matrix [A] in doing the following calculations:
1
a= 2
3
4
b= 5
6
123
[A] = 4 5 6
789
Do the following:
1. Determine aT
2. Determine [A]T
3. Compute [A]a
Questions and Answers on Natural Frequencies and
Mode Shapes
Background
The free response of an undamped, discrete system with N DOFs is given by:
N
x ( t) =
f ( j) [ c j cosw j t + s j sinw j t]
(1)
j =1
where w j and f ( j ) (j = 1, 2, , N) are the natu
Example 1 four different single DOF systems with harmonic excitation
Find the steady state response of the systems shown in (a)-(d):
a)
b)
c)
d)
applied harmonic force
support motion relative displacement
support motion absolute displacement
rotating imba
ME 563
Mechanical Vibrations
ME 563
MECHANICAL VIBRATIONS
Fall 2010
Potter
MWF 4:30 p.m.-5:20 p.m.
Instructor: Prof. D. E. Adams
Room: ME 361
Email: [email protected]
Phone: 496-6033
1 -1
Fall 2010
ME 563
Mechanical Vibrations
Fall 2010
1 Introduction to
ME563 Fall 2011
Purdue University
West Lafayette, IN
Homework Set No. 11
Assignment date: Friday, November 11
Due date: Friday, November 18
Please attach this cover sheet to your completed homework assignment.
Name
PUID
Problem 11.1
Problem 11.2
TOTAL
Pro
Example IV.1.7 - beating in forced response
Sketch the steady-state response of the base-excited system shown below (with initial
conditions of x(0) = = 0) for n .
Solution
From before:
x(t) = c cost + s sint + sint
x(0) = 0 = c
= 0 = s +
s=Therefore
x(t
Problem 12.1
SOLUTION
Consider the undamped, four-DOF system shown below.
x1
x2
x3
x4
SYSTEM A
This system has 4x4 symmetric and positive definite mass and stiffness matrices
! M # and ! K # , respectively. x
However, wexare not givenxthe details ofxthe
"
Example IV.1.7 - beating in forced response
Sketch the steady-state response of the base-excited system shown below (with initial
conditions of x(0) = x(0) = 0) for n .
Solution
From before:
y0 n
x(t) = c cosn t + s sinn t +
2
2
n -
2
x(0) = 0 = c
x(0) =
Example 3 - free response of a cantilevered bending beam
Find the general form of the free vibration solution of a cantilevered homogeneous
bending beam having a flexural rigidity EI, cross-sectional area A and mass/volume !.
u(x,t)
x=0
From last example,
MATLAB REVIEW/TUTORIAL
ME 563
C. M. Krousgrill
School of Mechanical Engineering
Table of Contents
Some Matlab Basics . 1
Some Elementary Matlab Commands . 5
Examples of Selected Commands . 10
Matlab N otes:
Some Matlab B asics
Some Matlab Basics
The follo
Example 2 - free response of a simply supported bending beam
Find the general form of the free vibration solution of a simply supported homogeneous
bending beam having a flexural rigidity EI, cross-sectional area A and mass/volume !.
d 4"( x)
# $ 4 "( x)
ME 563
Mechanical Vibrations
Lecture #16
Forced Response
(Step Input, Harmonic Excitation)
1
Free + Forced Response
Because the equations we are solving are linear in nature, we
can simply add the free and forced response components to
obtain the total (g
Example 1 - flexibility matrix for cantilevered beam
Consider a homogeneous, clamped beam having a length of 3L and flexural rigidity of
EI. Find the flexibility matrix for the beam using the transverse deflections at the three
particles as generalized co
ME 563
Mechanical Vibrations
Lecture #17
Frequency Response Functions
1
Sinusoidal Response
When we considered a co-sinusoidal input force,
we obtained the following amplitude/phase information:
xp(t)=Xpcos(t+p )
We can also use Laplace transforms to find
Example 1 - EOM for a tapered shaft
Find the EOM and boundary conditions for the rotational motion of a tapered shaft with
end disk, as shown to the right. The shaft has a circular cross section having a radius of
2R at x = 0 and radius of R at x = L.
G,
ME 563
Mechanical Vibrations
Lecture #18
Multiple Degree of Freedom
Frequency Response Functions
1
Frequency Response
When we considered a single degree of freedom system with
one input force and one output response, the relationship
between the steady st
Example 2 - EOM for a string hanging under its weight
Find the EOM and boundary conditions for the string hanging under its weight. A particle
of mass M is also attached at x = L.