Rose-Hulman Institute of TechnologyMechanical Engineering

Vibrations

Todays Objectives:

Students will be able to:

a) Find the homogeneous or steady state solution using analytical modal analysis

Analytical Modal Analysis

Note: Bring your laptop to class tomorrow!

Rose-Hulman Institute of TechnologyMechanical Engineering

Vibrations

Overview (from Pete Avitabile)

Rose-Hulman Institute of TechnologyMechanical Engineering

Vibrations

Analytical Modal Analysis(MDOF free and forced vibration)

EOM (1)

Solve the eigenvalue problem

For eigenvalues:

For eigenvectors:

Mass normalize the eigenvectors:

Form the modal matrix:

[ ]{ } [ ]{ } [ ]{ } { }FxKxCxM =++ &&&

Rose-Hulman Institute of TechnologyMechanical Engineering

Vibrations

Modal analysis (cont.)

Apply a coordinate transformation

Into (1) gives:

Premultiply by []T gives:

Gives:

Notes: we get n uncoupled differential equations! qi(t) is called a principle coordinate of, if [] is mass normalized, a

normal coordinate

[ ][ ]{ } [ ][ ]{ } { }FqKqM =+ &&[ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ] { }FqKqM TTT =+ &&

Rose-Hulman Institute of TechnologyMechanical Engineering

Vibrations

Modal analysis (cont.)

Then uncoupled equations are:

We know how to solve this!!

How do we fine Ai and Bi?

iiii Qqq =+ 2&&

( ) ( ) ( )tqtqtqPH iii

+=Particular solution(depends on RHS)

Homogeneous solution=Aicosit + Bisinit

Rose-Hulman Institute of TechnologyMechanical Engineering

Vibrations

Initial conditions for normal coordinates

We need

Recall:

But if we mass normalize the modes we have

So

So we get

One we find {q(t)} we can find {x(t)} by transforming back

( ) ( )0 and 0 q q &

( ){ } = 0q( ){ } = 0q&

( ){ } [ ] ( ){ }tqtx =

Rose-Hulman Institute of TechnologyMechanical Engineering

Vibrations

Summary of Analytical Modal Analysis

1. Find EOM2. Solve the eigenvalue problem to find frequencies and modes

Mass normalize the modes

3. Apply coordinate transformation (basically any motion can be considered as a superposition of the normal modes) Obtain the decoupled equations of motion:

Find the generalized force: Qi(t) = []T{F(t)} Find the initial conditions for {q(t)}:

4. Solve the decoupled equations and apply ICs to find qi(t)5. Transform back to find {x(t)}

[ ]{ } [ ]{ } { }FxKxM =+&&[ ]{ } [ ]{ }XKXM =2 [ ] [ ] and

[ ] [ ][ ] [ ] [ ] [ ][ ] [ ] so == KIM TT( ){ } [ ] ( ){ }tqtx =

( ){ } ( ){ }0 and 0 qq &iiii Qqq =+ 2&&

( ){ } [ ] ( ){ }tqtx =

Rose-Hulman Institute of TechnologyMechanical Engineering

Vibrations

Example

The 3-DOF system shown is found to have the natural frequencies

0.4

0.3

0.1

23

22

21

===

and the mass normalized modal matrix:

[ ]

=

4083.05.02886.04083.005774.0

4083.05.02886.0 and mass matrix [ ]

=

200020002

M

a) Using modal analysis determine the time response of each mass if the system is given the initial conditions

0)0()0()0()0( 3211 ==== xxxx &&& and 1)0()0( 32 == xx b) What is the stiffness matrix for this problem?

x1 x2 x3