ME 440
Intermediate Vibrations
Tu, March 31, 2009
Chapter 5: Vibration of 2DOF Systems
© Dan Negrut, 2009
ME440, UW-Madison
Quote of the Lecture:
Opportunity is missed by most people because it is dressed in overalls and it looks like work.
- Thomas Edison
Before we get started…

Last Time:


Started Chapter 5: response of 2DOF systems
Covered:






Concept of singular matrix and the requirement of zero determinant
How to compute the two natural frequencies of the system
How to compute the corresponding vibration modes
The concept of mode ratio
Terminology: stiffness matrix, modal matrix
Today:

HW Assigned (due April 7)



5.5
5.8 (use example in today’s lecture to find critical speed)
Material Covered:

Examples + Using the Initial Conditions (ICs) to compute constants in the
expression of the solution
2
[AO]
Example: Vibration of Two Disks


Two identical disks of centroidal mass moment of inertia are attached to a steel
shaft that is fixed at the right end as shown. Each section of shaft has the same
diameter d, a length l, and a torsional spring constant k
Determine:


The natural frequencies of the system
The modal vectors associated with the free vibration of the 2DOF system
3
[AO]
Example: Vibration of Two Disks
4
a
[AO]
Highway Markers Example
b
m, JJcgcg
k1
k2
u(t)




Car runs over markers that separate lanes on highway
Determine resonant frequencies of the car
Determine vibration modes and mode ratios
What should be the distance between markers if maximum impact should be at 65 mph?
5
Highway Markers Problem
a
b
m, JJcgcg
k2
k1
u(t)

y
k2(y+b-u)
k1(y-a)
6
[Cntd.]
Free Vibration of Undamped Systems

Recall the definition of the modal vectors:

Also, recall the definition of the modal matrix:

This matrix is always nonsingular (more on this later)

Solution was expressed last time as

We are supposed to use ICs (two positions and two velocities) to find:
7
Finding the A’s and ’s

Assume the following ICs are provided:

That means that the following conditions must hold:

Recall that n(1) , n(2) , rn(1) , and rn(2) , are known


Natural frequencies
Mode ratios
8
[Cntd.]
Finding the A’s and ’s

Introduce the following notation:

This notation leads to the following two linear systems:

It’s easy to see that for both these linear systems, the coefficient matrix
has a nonzero determinant


It’s either det([u]), or n(1) n(2) det([u])
Recall that the modal matrix is always nonsingular, so you can always find
the coefficients P1, P2, P3, P4
9
[Cntd.]
Finding the A’s and ’s

Recall:

Once P1, P2, P3, P4 are available, then we can find the A’s and the ’s:
10
Free Response of 2DOF Systems
~ Summary Slide ~

11
Steps for finding the solution
1.
Compute the two natural frequencies of the system
2.
Compute the mode ratios
3.
Use ICs to compute the A’s and ’s :