1 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Review MDOF Theory

Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell

[ K ]n

[ M ]n [ M ]a[ K ]a [ E ]a

[ ω ]2

Structural Dynamic Modeling Techniques & Modal Analysis Methods

m

k

1

1 c1

mp

3

3

c3

m

k

2

2 c2

f3

k 3

p 2 f2f1p 1

MODE 1 MODE 2 MODE 3

2 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Multiple Degree of Freedom Systems

• Referred to as a Multiple Degree of Freedom • An NDOF system has ‘N’ independent degrees of freedom to describe the system

• There is one natural frequency for every DOF in the system description

Systems with more than one DOF:

3 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Multiple Degree of Freedom Systems

• Each natural frequency has a displacement configuration referred to as a ‘normal mode’

• Mathematical quantities referred to as ‘eigenvalues’ and ‘eigenvectors’ are used to describe the system characteristics

• While the resulting motion appears more complicated, the system set of equations can always be decomposed into a set of equivalent SDOF systems for each mode of the system.

Properties of a MDOF system:

4 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Multiple Degree of Freedom Systems

• lumped mass• stiffness proportional to displacement

• damping proportional to velocity

• linear time invariant• 2nd order differential equations

Assumptions

m

m

k

k

c

c

1

1

2

2

1

2

f 1

f 2

x1

x2

5 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Multiple Degree of Freedom Systems

Free body diagram

f 1

f 2

(k 2 x1x2 )- (c 2 x1x2 )-

x1

x2

k 1x1 c1x1

m2

m1

6 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Multiple Degree of Freedom Systems

Newton’s Second Law

Rearrange terms

( ) ( ) ( )( ) ( ) ( )122122222

1221112211111

xxkxxctfxmxxkxkxxcxctfxm

−−−−=−+−−+−=

&&&&

&&&&&

( ) ( ) ( )( )tfxkxkxcxcxm

tfxkxkkxcxccxm

22212221222

1221212212111

=+−+−=−++−++

&&&&

&&&&

f 1

f 2

(k 2 x1x2 )- (c 2 x1x2 )-

x1

x2

k 1x1 c1x1

m2

m1

7 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

MDOF Equations of Motion

Equation of Motion for 2 DOF

can be written in compact matrix form as

( )

( )

=

−

−++

−

−++

)t(f)t(f

xx

kkkkkxx

ccccc

xx

mm

2

1

2

1

22

221

2

1

22

221

2

1

2

1

&

&

&&

&&

[ ] [ ] [ ] FxKxCxM =++ &&&

Matrices and Linear Algebra are important !!!

8 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

MDOF Equations of Motion

This coupled set of equations can be uncoupled by performing an eigensolution to obtain ‘eigenpairs’for each mode of the system, that is

‘eigenvalues’ and ‘eigenvectors’

or

‘frequencies’ (poles) and ‘mode shapes’

9 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

MDOF Equations of Motion

Equation of Motion[ ] [ ] [ ] )t(FxKxCxM =++ &&&

Eigensolution

Frequencies (eigenvalues) and Mode Shapes (eigenvectors)

[ ] [ ][ ] 0xMK =λ−

[ ] [ ]L2122

21

2 uuUand\\

\=

ω

ω=

Ω

10 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

MDOF - Orthogonality of Eigenvectors

Normal modes are ‘orthogonal’ with respect to the system mass and stiffness matrices.The eigen problem can be written as

For the ith mode of the system, the eigenproblemcan be written as

[ ][ ] [ ][ ][ ]2UMUK Ω=

[ ] [ ] iii uMuK λ=

11 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

MDOF - Orthogonality of Eigenvectors

Premultiply that equation by the transpose of a different vector uj

and write a second equation for uj and premultiplyit by the transpose of vector ui

[ ] [ ] iTjii

Tj uMuuKu λ=

[ ] [ ] jTijj

Ti uMuuKu λ=

12 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

MDOF - Orthogonality of Eigenvectors

Subtracting the previous two equations, yields

Since the eigenvalues for each mode are different,

When i=j, then the values are not zero

( ) [ ] 0uMu jT

iji =λ−λ

[ ] [ ]

jikuKumuMu

iiiT

i

iiiT

i =

==

[ ] ji0uMu jT

i ≠=

13 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Matrix and Modal Space Transformation

Define the modal matrix as the collection of modal vectors for each mode organized in column fashion in the modal matrix

This modal matrix is then used to define the modal transformation equation with a new coordinate with ‘p’ as the principal coordinate

[ ] [ ]L321 uuuU =

[ ] [ ]

==

M

L 2

1

21 pp

uupUx

14 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Space Transformation

Substitute the modal transformation into the equation of motion

In order to put the equations in normal form, this equation must be premultiplied by the transpose of the projection operator to give

[ ] [ ][ ] [ ] [ ][ ] [ ] [ ][ ] [ ] FUpUKUpUCUpUMU TTTT =++ &&&

[ ][ ] [ ][ ] [ ][ ] FpUKpUCpUM =++ &&&

15 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Space Transformation

The first term of the modal acceleration can be expanded as

[ ] [ ][ ]

[ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] ( )

=

OMMM

L

L

L

3T

3T

31T

3

3T

22T

21T

2

3T

12T

11T

1

T

uMu2uMuuMuuMuuMuuMuuMuuMuuMu

UMU

16 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Space Transformation

Recall from orthogonality that

[ ] ji0uMu jT

i ≠=

so that

[ ] [ ][ ]

[ ] ( ) [ ] ( )

[ ] ( )

=

OMMM

L

L

L

3T

3

2T

2

1T

1

T

uMu000uMu000uMu

UMU

17 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Mass, Modal Damping, Modal Stiffness

The mass becomes

The stiffness becomes

The damping becomes (under special conditions)

[ ] [ ][ ]

=

OMMM

L

L

L

33

22

11

T

m000m000m

UMU

[ ] [ ][ ]

=

OMMM

L

L

L

33

22

11

T

k000k000k

UKU

[ ] [ ][ ]

=

OMMM

L

L

L

33

22

11

T

c000c000c

UCU

18 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Unit Modal Mass Scaling

The mass orthogonality results in an arbitrary mass called the modal mass

It may be desirable to scale the mode shapes so that the modal mass is 1.0The mode shape must be scaled by the inverse of the square root of the modal mass

[ ] iiiT

i muMu =

[ ] iiiT

i muMu =

19 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Unit Modal Mass Scaling

Each mode can be conveniently scaled as follows

[ ] [ ]

=

OMMM

L

L

L

33

22

11

n

m100

0m10

00m1

UU

20 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Scaled Unit Modal Mass, Damping, and Stiffness

Modal Mass

Modal Damping

Modal Stiffness

[ ] [ ][ ]

=

OMMM

L

L

L

100010001

UMU nTn

[ ] [ ][ ]

ωω

ω

=

OMMM

L

L

L

23

22

21

nTn

000000

UKU

[ ] [ ][ ]

ωζωζ

ωζ

=

OMMM

L

L

L

33

22

11

nTn200020002

UCU

21 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Space Transformation

The physical set of highly coupled equations are transformed to modal space through the modal transformation equation to yield a set of uncoupled equations

Modal equations (uncoupled)

=

+

+

MMM

&

&

M

&&

&&

FuFu

pp

\k

kpp

\c

cpp

\m

mT

2

T1

2

1

2

1

2

1

2

1

2

1

2

1

22 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Space Transformation

Diagonal Matrices -Modal Mass Modal Damping Modal Stiffness

Highly coupled system

transformed into simple system

[ ] FUp\

K\

p\

C\

p\

M\

T=

+

+

&&&

m

k

1

1 c1

mp

3

3

c3

m

k

2

2 c2

f3

k 3

p 2 f2f1p 1

MODE 1 MODE 2 MODE 3

23 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Space Transformation

It is very clear to see that these modal space equations result in a set of independent SDOF systemsThe modal transformation equation uncouples the highly coupled set of equationsThe modal transformation appropriates the force to each modal oscillator in modal spaceThe modal transformation equation combines the response of all the independent SDOF systems to identify the total physical response

24 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Space Response Analysis

Since the MDOF system is reduced to equivalent SDOF systems with appropriate force, the response of each SDOF system can be determined using SDOF approaches discussed thus far.The total response due to each of the SDOF systems can be determined using the modal transformation equation

25 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Space Transformation

[ M ]p + [ C ]p + [ K ]p = [U] F(t)T

m

k

1

1 c1

f1p1

MODE 1

m

k

2

2 c2

p2 f2

MODE 2

mp

3

3

c3

f3

k3

MODE 3

MODAL

PHYSICAL MODEL[M]x + [C]x + [K]x = F(t)

x = [U]p = [

SPACE

x = [U]p = ++ u p 33u p 22u p 11

u 3u 2u 1 ]p

.. .

.. .

+

+

Σ=

26 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Numerical Response - Mode Superposition

Using the modal space formulation, a simple integration on the SDOF system can be performed

=

+

+

MMM

&

&

M

&&

&&

FuFu

pp

\k

kpp

\c

cpp

\m

mT

2

T1

2

1

2

1

2

1

2

1

2

1

2

1

m

k

1

1 c 1

f 1 p 1

MODE 1

m

k

2

2 c 2

p 2 f 2

MODE 2

m

p

3

3

c 3

f 3

k 3

MODE 3

x =

x =

x =

u p 1 1

u p 2 2

u p 3 3

+

x = [U]p = + + u p 3 3

u p 2 2

u p 1 1

+ Σ =

27 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Modal Space - Modal Matrices

[ ] [ ][ ]

=

\M

\UMU 111

T1

[ ] [ ][ ]

=

\C

\UCU 111

T1

[ ] [ ][ ]

=

\K

\UKU 111

T1Modal Stiffness

Modal Damping

Modal Mass

The mode shapes are linearly independent and orthogonal w.r.t the mass and stiffness matrices

TRUE !!!

???????

TRUE !!!

28 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Proportional Damping

[ ] [ ] [ ][ ][ ]

β+α=β+α

\KM

\UKMU 1

T1

The damping matrix is only uncoupled for a special case where the damping is assumed to be proportional to the mass and/or stiffness matrices

Many times proportional damping is assumed since we do not know what the actual damping isThis assumption began back when computational power was limited and matrix size was of critical concern. But even today we still struggle with the damping matrix !!!

29 Dr. Peter AvitabileModal Analysis & Controls Laboratory22.515 – Review MDOF Theory

Non-Proportional Damping

If the damping is not proportional then a state space solution is required (beyond our scope here)

[ ] [ ][ ] [ ]

[ ] [ ][ ] [ ]

=

−

−

F0

xx

K00M

xx

CMM0

1

1

11

1 &

&

&&

[ ] [ ] QYAYB 11 =−&

[ ] [ ] [ ][ ] [ ]

=

11

11 CM

M0B [ ] [ ] [ ]

[ ] [ ]

−

=1

11 K0

0MA

[ ] [ ][ ] 0YBA 11 =λ− [ ] 11 pY Φ=

The equation of motion can be recast as

The eigensolution and modal transformation is then