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ME 440 Intermediate Vibrations

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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)

© Dan Negrut, 2009ME440, UW-Madison

- 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

- Looking ahead, please keep in mind:
- 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

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

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- Derive EOMs

3

- Determine the natural frequencies and modal matrix
- Determine the time evolution for the following ICs:

4

- 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

- 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

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- 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

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- 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

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

- 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]

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- 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,

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

Generalized force, definition:

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- 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:

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m1=1kg, m2=2kg

k1=9N/m

k2=k3=18N/m

- Find response of the system

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- 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

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

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

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- 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,

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- 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…

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- 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:

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- 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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