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# ME 440 Intermediate Vibrations - PowerPoint PPT Presentation

ME 440 Intermediate Vibrations. Th, April 16, 2009 Chapter 6: Multi-degree of Freedom (MDOF) Systems. Quote of the Day: Space may be the final frontier But it's made in a Hollywood basement - Red Hot Chili Peppers (Californication). © Dan Negrut, 2009 ME440, UW-Madison.

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### ME 440Intermediate Vibrations

Th, April 16, 2009

Chapter 6: Multi-degree of Freedom (MDOF) Systems

Quote of the Day:

Space may be the final frontierBut it's made in a Hollywood basement- Red Hot Chili Peppers (Californication)

• Last Time:

• Example, Matrix-Vector approach, talked about MATLAB use

• Flexibility Matrix

• Influence Coefficient: stiffness, damping, flexibility

• Today:

• Examples

• Modal Analysis

• Response of forced undamped and underdamped MDOF systems

• Free Vibration

• Forced Vibration

• HW Assigned: 6.22 and 6.54

• Next Tuesday, April 21, I will be out of town

• Professor Engelstad will cover for me

• There will be a demo of vibration modes, draws on Chapter 5

• Class meets in the usual room in ME building

2

[AO, from text] Example, Computing the Stiffness Matrix

• Derive EOMs

3

[AO]Example

• Determine the natural frequencies and modal matrix

• Determine the time evolution for the following ICs:

4

[New Topic. Short Detour]Semidefinite Systems

• Under certain circumstances the flexibility coefficients are infinite

• Consider the following example:

• If a unit force F1=1 is applied, there is nothing to prevent this system from moving to the right for ever…

• Therefore, a11= a21= a31=1

• In this situation, fall back on the stiffness matrix approach

• This situation will correspond to the case when at least one of the eigenvalues is zero there is a zero natural frequency there is a rigid body motion embedded in the system

• We’ll elaborate on this later…

5

[Matrix Algebra Detour]The Diagonalization Property

• Recall the K and M orthogonality of the vibration modes (eigenvectors):

• This means two things:

• [u]T[m][u] is diagonal

• [u]T[k][u] is diagonal

6

[Matrix Algebra Detour]The Diagonalization Property

• To conclude

• One says that the modal matrix [u] is diagonalizing both [m] and [k]

• Nomenclature:

• The diagonal matrices obtained after diagonalization are called [M] and [K]

• My bad: before, I was pretty loose with the notation and used the notation [M] and [K] when I should have not used it

7

[New Topic]Modal Analysis

• Finding the time evolution of a MDOF system is simple because it relies on a powerful method: modal analysis

• At cornerstone of modal analysis are two properties of the modal matrix [u]:

• Modal matrix is nonsingular

• Modal matrix simultaneously diagonalizes the [m] and [k] matrices

• Recall the equation of motion, in matrix-vector form:

• Also recall that modal matrix [u] is obtained based on the eigenvectors (modal vectors) associated with the matrix [m]-1[k]

8

[Cntd]Modal Analysis

• Important consequence of nonsingularity of the modal matrix [u]:

• Since [u] is nonsingular, any vector {x} you give me, I can find a vector {q} such that [u]{q}={x}

• How do I get {q}?

• Recall that the modal matrix is nonsingular and also constant. So I just solve a linear system [u]{q}={x} with {x} being the RHS. I can find {q} immediately (kind of, solving a linear system is not quite trivial…)

• Another key observation: since {x} is a function of time (changes in time), so will {q}:

• The nice thing is that [u] is *constant*. Therefore,

9

[Cntd]Modal Analysis [Context: Forced Undamped Response]

• Let’s look now at the forced vibration response, the undamped case:

Generalized force, definition:

10

[Cntd]Modal Analysis [in the Context of Undamped Response]

• What’s on the previous slide is what you needed for the third bullet of problem 5.38

• This modal analysis thing is nothing more than just a decoupling of the equations of motion

• Please keep in mind how you choose the initial conditions:

11

m1=1kg, m2=2kg

k1=9N/m

k2=k3=18N/m

[AO]Example: Forced Undamped Response

• Find response of the system

12

MDOF Forced Response of Underdamped Systems

• General form of EOMs for MDOF m-c-k type linear system:

• The novelty here is the presence of [c] in the above equation…

• Two basic approaches to solve this problem

• Go after the simultaneous solution of coupled differential equations of motion

• You might have to fall back on a numerical approximation method (Euler Method, Runge-Kutta, etc.) if expression of {f(t)} is not civilized (harmonic functions)

• Use modal-analysis method

• This involves the solution of a set of *uncoupled* differential equations

• How many of them? – As many DOFs you have

• Each equation is basically like the EOM of a single degree of freedom system and thus is easily solved as such

13

[Cntd]MDOF Forced Response of Underdamped Systems

• The dirty trick (when hard to solve, simplify the heck out of the problem):

• Introduce the concept of proportional damping

• Specifically, assume that the damping matrix is proportional to the stiffness matrix

• Why? – One of the reasons: so that we can solve this problem analytically

• Since  is a scalar, it means that

• Therefore,

14

[Cntd]MDOF Forced Response of Underdamped Systems

• To conclude, if modal damping is present, we have

• If you explicitly state the entries in the matrices above you get

• Equivalently,

• Note that in all matrices the off-diagonal elements are zero

• We accomplished full decoupling, solve now p one DOF problems…

15

[Cntd]MDOF Forced Response of Underdamped Systems

• The EOMs are decoupled

• Each generalized principal coordinate satisfies a 2nd order diff eq of the form

• Let’s do one more trick (not as dirty as the first one)

• Recall that  is given to you (attribute of the system you work with)

• For each generalized coordinate i, define a damping ratio i like this:

• The equation of motion for the ith principal coordinate becomes

• Above, I used the notation:

16

[End]MDOF Forced Response of Underdamped Systems

• Quick remarks:

• Recall that you have to start with the correct set of initial conditions

• Also, note the relationship between i and :

• The following damping coefficient ci corresponds to this damping ratio:

• At the end of the day, the entire argument here builds on one thing: the ability to come up with the coefficient  such that

• Figuring out how to pick  (modal damping) is an art rather than a science

• Recall that after all, we have a National Academy of Sciences and another National Academy of Engineering. The members of the latter get to pick …

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