Vibration of 2DOF Systems matix approach voice.ppt

Vibration of systems refers to the oscillations or repetitive motions of a system's components or structures due to external or internal forces. It is a fundamental concept in various fields, including:

1. Mechanical Engineering: Vibration analysis is crucial for designing and optimizing systems like engines, gearboxes, and mechanical structures.
2. Civil Engineering: Vibration studies help ensure the safety and durability of buildings, bridges, and other infrastructure under various loads.
3. Electrical Engineering: Vibration is essential in understanding the behavior of electrical systems, like motors and generators.
4. Aerospace Engineering: Vibration analysis is critical for ensuring the stability and performance of aircraft and spacecraft.
5. Physics: Vibration is a fundamental aspect of quantum mechanics and classical mechanics, describing the behavior of particles and systems.

Some common types of vibration include:

1. Free vibration
2. Forced vibration
3. Damped vibration
4. Resonance
5. Harmonic vibration

Understanding vibration is crucial for designing, optimizing, and maintaining various systems to ensure their reliability, efficiency, and safety.

1. ME 440
Intermediate Vibrations
Th, April 9, 2009
Chapter 5: Vibration of 2DOF Systems, Section 5.5
© Dan Negrut, 2009
ME440, UW-Madison
Quote of the Day:
A pessimist sees the difficulty in every opportunity; an
optimist sees the opportunity in every difficulty.
-Winston Churchill

2. Before we get started…
 Last Time:
 Exam
 Today:
 Matrix-Vector approach to 2 DOF problem
 EOM decoupling
 Principal coordinates
 HW Assigned (due April 16)
 5.28:
1.Derive EOM, natural frequencies, and modal matrix
2.Using information obtained at 1, evaluate the following quantities and in case they are zero explain why
 5.38 (use MATLAB to solve the eigenvalue problem in the Matrix-Vector approach):
1.Derive EOM and use MATLAB to get the modal matrix and natural frequencies
2.Find Principal Coordinates (PCs)
3.Indicate the IVP satisfied by the PCs (that is, the differential equations and the Initial Conditions)
2

3. [Cntd]
Review of Matrix Algebra
How do you solve an eigenvector problem?
Solve first for eigenvalues, then solve for eigenvectors
This starts to sound familiar with the vibration problem: we first solve for
the natural frequencies (the eigenvalues) and then found the modal vectors
(the eigenvectors)
Keep in mind this parallel
3
Going back to the original question, if you want to find nontrivial {u}, it
better be that the following determinant is zero (see two lectures ago
to understand why):

4. [Cntd]
Review of Matrix Algebra
Properties of eigenvectors/eigenvalues, PART 1
Assume matrix [A] is symmetric, and you have just found two
eigenvector/eigenvalue pairs: ({u}
(1), 
(1))and ({u}
(2), 
(2))
If 
(1)and 
(2)are different, then the two eigenvectors are orthogonal, in
other words, their dot product is zero:
4
Proof:

5. [Cntd]
Review of Matrix Algebra
Properties of eigenvectors/eigenvalues, PART 2
This result is slightly artificial now, but it’ll become clear why we need it in a little while
First, recall that an arbitrary matrix [B] is positive definite if for any nonzero
vector {u}, the value {u}
T
[B] {u}>0
The result of interest is as follows
Assume there are two symmetric positive definite matrices [M] and [K] that are used to
compute [A] as follows
5
Then, the eigenvalues of [A] are positive
Specifically, if [A]{u}={u}, then > 0
Proof:

6. [Cntd]
Review of Matrix Algebra
Properties of eigenvectors/eigenvalues, PART 3
Assume again that there are two symmetric positive definite matrices [M]
and [K] that are used to compute [A] as follows
6
Moreover, assume that for matrix [A] you identified two eigenvector/eigenvalue
pairs ({u}
(1), 
(1)) and ({u}
(2), 
(2)), and that 
(1)and 
(2)are different
Then the two eigenvectors are both M and K orthogonal
In other words:

7. [Cntd]
Review of Matrix Algebra
7
Proof:

8. [Back to Normal Program. New Topic]
The Matrix-Vector Approach to the 2DOF Vibration Problem
Motivation:
Pose the problem in a matrix-vector form since this is the only sensible
way to deal with systems with a large number of degrees of freedom
Very illuminating to understand how to obtain the equation that provides
the natural frequencies for arbitrarily large systems
More on this in Chapter 6
Approach relies on eigenvalue/eigenvector theory
So how do you go about it?
Do exactly what we’ve done in first lecture of Chapter 5, except that
We’ll use matrix notation this time, and
We’ll prove and then use the following key observation
8

9. [Cntd.]
The Matrix-Vector Approach to the 2DOF Vibration Problem
Just like we did at the beginning of Chapter 5, assume a solution of the form
9
We hope that this trial solution will satisfy the following EOM:
Using our matrix/vector notation, we express the above set of two ODEs as
As far as time derivatives are concerned, we have:

10. Then, the following should hold:
10
Since q(t), a function of time, cannot be zero all the time, necessarily
This is equivalently expressed as
What this is telling me is that if I define the matrix [A]=[M
-1
][K], then
the quantities {u} and are an eigen-(vector/value) pair of matrix [A]
[Cntd.]
The Matrix-Vector Approach to the 2DOF Vibration Problem

11. [Cntd.]
The Matrix-Vector Approach to the 2DOF Vibration Problem
A couple of remarks
Recall that [M] and [K] are symmetric, and that [A]=[M]
-1
[K]
Since the dimension of [A] is 2, it means that we’ll have two pairs of
(eigenvectors/eigenvalues)
We’ll denote them by
11
Then, based on the “artificial” results of several slides back, the eigenvalues
(that is, 
2
n(1)and 
2
n(1)) are positive
Also, we have the [M] and [K] orthogonality result that holds:

12. [Cntd.]
The Matrix-Vector Approach to the 2DOF Vibration Problem
Take a step back before we get carried away
We already started making statements about {u}
(1)and {u}
(2)and talking about 
2
n(1)
and 
2
n(2)without actually computing them
That’s good, actually
But eventually you need to compute this quantities
How do you compute them?
You have [M] and [K]. Then evaluate [A], and solve the associated eigenvalue problem
The eigenvalues 
2
n(1)and 
2
n(2)are obtained as the solution of the following equation:
12
Condition above becomes:
This looks like our old equation that gave us the natural frequencies, so it’s all good…

13. [Cntd.]
The Matrix-Vector Approach to the 2DOF Vibration Problem
Next, you need compute the normalized eigenvectors:
13
Basically, you have to solve the linear systems
As expected, you’ll get

14. [Cntd.]
The Matrix-Vector Approach to the 2DOF Vibration Problem
One final remark…
14
Define the diagonal matrix []as
Because we have
… I can express the above two equalities in a more compact form (this
is where I use []) as

15. [AO]
Example, 2DOF:
m
1=1kg, m
2=2kg
k
1=9N/m
k
2=k
3=18N/m
Find natural frequencies
Find modal vectors {u}
(1), {u}
(2)
Verify that [A][u]=[u][]holds as
indicated on the previous slide
15

17. 17
[New Topic. Very Important Topic]
Decoupling of the Eqs. of Motion
Matrix-vector flavor of the solution:
Are you with me question: for free response, what was the expression of q
1(t) and q
2(t)?

18. [Cntd]
Decoupling of the Eqs. of Motion
Note that since [u] is constant
18
Then, the equation of motion assumes the form,
Multiply by inverse of mass matrix and recall that [A][u]=[u][]to obtain
Since the modal matrix [u] is nonsingular, we have that

19. [Cntd]
Decoupling of the Eqs. of Motion
Rewrite the previous equation to understand the “decoupling” claim:
19
Note that the differential equations for q
1(t) and q
2(t) look like the EOMs
for two uncoupled springs:
Note that the off-diagonal terms are all zero –both dynamics and static
decoupling

20. [Cntd]
Decoupling of the Eqs. of Motion
I still owe you the initial conditions for the ODE on the previous slide
Things are all good on this front
Remember that you have the initial conditions in the {x} generalized coordinates:
20
Because {x}=[u]{q}, the initial conditions in {f} follow immediately:
To conclude, one needs to solve the following IVP:

21. [Concluding Remarks]
Decoupling of the Eqs. of Motion
Another way to look at what we did:
Recall that we said that coupling is most likely a consequence of a bad choice of
generalized coordinates
If they are not ideal, find and work with the ideal generalized coordinates
It turns out that if you solve the right eigenvalue problem you get the modal matrix
[u] and the natural frequencies 
2
n(1)and 
2
n(2)
The ideal set of generalized coordinates is then obtained as {q} =[u]
-1
{x}
What you do is a change of variables, so to speak…
The new coordinates {q} are called principal coordinates
The new set of differential equations that provide the time evolution of {f} is very
civilized (full decoupling…)
21