10 Plus - Multi DOF - Modal Analysis
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Transcript of 10 Plus - Multi DOF - Modal Analysis
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8/3/2019 10 Plus - Multi DOF - Modal Analysis
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Week 10 plus
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Part 3. Mainly based on the text book by D.J. Inman
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3. Modal Analysis
We know that there are three ways to solve an undampedvibration problem in matrix form:
Which one should we use?
For 2-DOF systems
calculation can be done by hand
the most straightforward way is to use approach (i)
For problems with more than 2-DOF
use a computational code to avoid mistakes and insure accuracy
The most efficient way is to use approach (iii)
(i) 2Mu Ku (ii) 2u M1Ku (iii) 2v M
12KM
12v
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Physical coordinates are not always the easiest to work in
Eigenvectors provide a convenient transformation to modal
coordinates
Modal coordinates are linear combination of physical coordinates
Say we have physical coordinates xand want to transform to
some other coordinates u
u1 x
1 3x
2
u2 x
1 3x
2
u
1
u2
1 3
1 3
x
1
x2
Modal Analysis
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Week 10 plus
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Review: Eigenvalue Problem
We have the equation of a vibration problem in the physicalcoordinate:
Where x is a vector, Mand Kare matrices. The initial conditions
are and .
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Review: Eigenvalue Problem
(4.55)
Now we have a symmetric, real matrix
Guaranteesreal eigenvalues and distinct, mutually orthogonal
eigenvectors
Mode shapes are solutions to in physicalcoordinates. Eigenvectors v are characteristics of matrices.
The two are related by a simple transformation
but they are the synonymous.
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Decoupling of Equation of Motion
Back to the symmetric eigenvalue problem:
Make the 2nd coordinate transformation and multiply by PT
The matrix of eigenvectors Pcan be used to decouple the equations
of motion.
(4.59)
Now we have decoupled the EOM, i.e., we have nindependent 2ndorder systems in modal coordinates r(t)
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Decoupling of Equation of Motion
Writing out equation (4.59) we get:
Also transform the initial conditions:
(4.62)
(4.60)
(4.65)
(4.63)
(4.64)
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Week 10 plus
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Decoupling of Equation of Motion
This transformation takes the problem from coupled equations in
the physical coordinates into decoupled equations in the modal
coordinates.
x M1
2 Pr
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Decoupling of Equation of Motion
The modal transformation transforms our 2-DOF system into twoSDOF systems
This allows us to solve the two decoupled SDOF systems
independently.
Then we can recombine using the inverse transformation to obtainthe solution in terms of the physical coordinates.
The free response is calculated for each mode independently using
the formulas for SDOF:
Or (4.66 & 4.67)
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Returning to Physical Coordinate
Once the solution in modal coordinates is determined (ri) then theresponse in Physical Coordinates is computed by:
With n DOFs these transformations are:
x M1
2 Pr
12
1 1
where
nxn
(t) S (t)
n nn n
S M P
nxnn n
x r
(where n = 2 in the previous slides)
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Steps in Solving via Modal Analysis
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Example: using modal analysis
Related to Examples 4.3.1 in the text book of D.J. Inman.
Calculate the solution of the 2-DOF system of Example 4.1.5.
Follow the given steps (slide 11). From examples 4.1.5, 4.2.1, 4.2.3 and
4.2.4 we have calculated:
These are steps1 4 in slide 11
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Example: using modal analysis
Step 5: Calculate matrix S and its inverse
Check that SS-1 = I. Also note that STMS = I
See that the value between the square bracket of matrix Sis similar to
the mode shapes found in Example 4.1.6
W k 10 l
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Step 6: Calculate the modal initial conditions
So inserting the above to Equation 4.66 & 4.67 (step 7), we get solutions:
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Example: using modal analysis
W k 10 l
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Step 8: Return to the physical coordinate
Which is the same as Example 4.1.7
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Example: using modal analysis