10 plus - multi dof - modal analysis

8/3/2019 10 Plus - Multi DOF - Modal Analysis

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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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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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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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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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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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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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Step 8: Return to the physical coordinate

Which is the same as Example 4.1.7

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10 plus - multi dof - modal analysis

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