Lecture Notes:  Introduction to Finite Element Method                       Chapter 7.  Structural Vibration and Dynamics

© 1999 Yijun Liu, University of Cincinnati

167

III. Damping

Two commonly used models for viscous damping.

A.

 

 Proportional Damping (Rayleigh Damping)

K

M

C

β

α

+

=

                             

(17)

where the constants 

α

 & 

β

 are found from

,

2

2

,

2

2

2

2

2

1

1

1

ω

β

αω

ξ

ω

β

αω

ξ

+

=

+

=

with 

2

1

2

1

 

&

 

 

,

 

,

ξ

ξ

ω

ω

 (damping ratio) being selected.

B. Modal Damping

Incorporate the viscous damping in modal equations.

Dam

p

ing

 r

a

ti

o

Lecture Notes:  Introduction to Finite Element Method                       Chapter 7.  Structural Vibration and Dynamics

© 1999 Yijun Liu, University of Cincinnati

168

IV. Modal Equations

•

 

Use the normal modes (modal matrix) to transform the
coupled system of dynamic equations to uncoupled
system of equations.

We have

[

]

n

1,2,...,

     

,

2

=

=

−

i

i

i

0

u

M

K

ω

                  (18)

where the normal mode 

i

u  satisfies:







=

=

,

0

,

0

j

T

i

j

T

i

u

M

u

u

K

u

 for 

i 

≠

 j,

and







=

=

,

,

1

2

i

i

T

i

i

T

i

ω

u

K

u

u

M

u

for i = 1, 2, …, n.

Form the modal matrix:

        

[

]

n

n

n

u

u

u

Ö

 

 

2

1

)

(

L

=

×

               

(19)

Can verify that

.

,

matrix)

 

Spectral

(

0

0

0

0

0

0

 

2

n

2

2

2

1

I

M

Ö

Ö

Ù

K

Ö

Ö

=

























=

=

T

T

ω

ω

ω

L

O

M

M

L

  

(20)

Lecture Notes:  Introduction to Finite Element Method                       Chapter 7.  Structural Vibration and Dynamics

© 1999 Yijun Liu, University of Cincinnati

169

Transformation for the displacement vector,

z

u

u

u

u

Φ

=

+

+

+

=

n

n

z

z

z

L

2

2

1

1

,     (21)

where

                























=

)

(

)

(

)

(

2

1

t

z

t

z

t

z

n

M

z

are called principal coordinates.

Substitute (21) into the dynamic equation:

Pre-multiply by 

Φ

T

, and apply (20):

),

( t

p

z

z

C

z

=

Ω

+

+

&

&&

φ

                            

(22)

where 

Ω

+

=

β

α

φ

I

C

     (proportional damping),

   

)

( t

T

f

p

Φ

=

.

Using Modal Damping

























=

n

n

ω

ξ

ω

ξ

ω

ξ

φ

2

0

2

0

0

0

2

2

2

1

1

L

M

O

M

L

C

. (23)

).

( t

f

z

K

z

C

z

M

=

Φ

+

Φ

+

Φ

&

&

&

Lecture Notes:  Introduction to Finite Element Method                       Chapter 7.  Structural Vibration and Dynamics

© 1999 Yijun Liu, University of Cincinnati

170

Equation (22) becomes,

),

(

2

2

t

p

z

z

z

i

i

i

i

i

i

i

=

+

+

ω

ω

ξ

&

&&

 i = 1,2,…,n. (24)

Equations in (22) or (24) are called modal equations.
These are uncoupled, second-order differential equations,
which are much easier to solve than the original dynamic
equation (coupled system).

To recover u from z, apply transformation (21) again, once
z is obtained from (24).

Notes:

•

 

Only the first few modes may be needed in constructing
the modal matrix 

Φ

 (i.e., 

Φ

 could be an n

×

m rectangular

matrix with m<n).  Thus, significant reduction in the
size of the system can be achieved.

•

 

Modal equations are best suited for problems in which
higher modes are not important (i.e., structural
vibrations, but not shock loading).

Lecture Notes:  Introduction to Finite Element Method                       Chapter 7.  Structural Vibration and Dynamics

© 1999 Yijun Liu, University of Cincinnati

171

V. Frequency Response Analysis
     (Harmonic Response Analysis)

3

2

1

&

&&

loading

 

Harmonic

sin t

ω

F

Ku

u

C

u

M

=

+

+

                    (25)

Modal method:  Apply the modal equations,

 

,

sin

2

2

t

p

z

z

z

i

i

i

i

i

i

i

ω

ω

ω

ξ

=

+

+

&

&&

   i=1,2,…,m.      (26)

These are 1-D equations. Solutions are

                                   

),

sin(

)

2

(

)

1

(

)

(

2

2

2

2

i

i

i

i

i

i

i

t

p

t

z

θ

ω

η

ξ

η

ω

−

+

−

=

     (27)

                                   where













=

=

=

−

=

ratio

damping

 

,

2

,

angle

 

phase

 

,

1

2

arctan

i

2

i

i

c

i

i

i

i

i

i

i

m

c

c

c

ω

ξ

ω

ω

η

η

η

ξ

θ

   Recover u from (21).

Direct Method:   Solve Eq. (25) directly, that is, calculate

the inverse. With

t

i

e

ω

u

u

=

(complex notation), Eq. (25)

becomes

[

]

.

2

F

u

M

C

K

=

−

+

ω

ω

i

This equation is expensive to solve and matrix is ill-
conditioned if 

ω

 is close to any 

ω

i

.

z

i

ω

/

ω

i