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

© 1999 Yijun Liu, University of Cincinnati

172

VI. Transient Response Analysis

(Dynamic Response/Time-History Analysis)

•

Structure response to arbitrary, time-dependent loading.

f(t)

t

u(t)

t

Compute responses by integrating through time:

t

0

t

1

t

2

t

n

t

n+1

u

1

u

2

u

n

u

n+1

t

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

© 1999 Yijun Liu, University of Cincinnati

173

Equation of motion at instance

n

t , n = 0, 1, 2, 3,

⋅⋅⋅

:

.

n

n

n

n

f

Ku

u

C

u

M

=

+

+

&

&&

Time increment:

∆

t=t

n+1

-t

n

, n=0, 1, 2, 3,

⋅⋅⋅

.

There are two categories of methods for transient analysis.

A. Direct Methods (Direct Integration Methods)

•

Central Difference Method

Approximate using finite difference:

)

2

(

)

(

1

),

(

2

1

1

1

2

1

1

−

+

−

+

+

−

∆

=

−

∆

=

n

n

n

n

n

n

n

t

t

u

u

u

u

u

u

u

&

&

&

Dynamic equation becomes,

,

)

(

2

1

)

2

(

)

(

1

1

1

1

1

2

n

n

n

n

n

n

n

t

t

f

Ku

u

u

C

u

u

u

M

=

+









−

∆

+













+

−

∆

−

+

−

+

which yields,

)

(

1

t

n

F

Au

=

+

where

( )

( )

( )

























∆

−

∆

−













∆

−

−

=

∆

+

∆

=

−

.

2

1

1

2

)

(

,

2

1

1

1

2

2

2

n

n

n

t

t

t

t

t

t

u

C

M

u

M

K

f

F

C

M

A

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

© 1999 Yijun Liu, University of Cincinnati

174

u

n+1

is calculated from u

n

& u

n-1

, and solution is

marching from

,

,

1

,

,

1

,

0

L

L

+

n

n

t

t

t

t

until convergent.

This method is unstable if

∆

t is too large.

•

Newmark Method:

Use approximations:

[

]

[

]

,

)

1

(

)

(

,

2

)

2

1

(

2

)

(

1

1

1

1

2

1

+

+

+

+

+

+

−

∆

+

≈

=

→

+

−

∆

+

∆

+

≈

n

n

n

n

n

n

n

n

n

n

t

t

t

u

u

u

u

u

u

u

u

u

u

&&

&&

&

&

L

&&

&&

&&

&

γ

γ

β

β

where

β

&

γ

are chosen constants. These lead to

)

(

1

t

n

F

Au

=

+

where

).

,

,

,

,

,

,

,

,

(

)

(

,

)

(

1

1

2

n

n

n

n

t

f

t

t

t

u

u

u

M

C

f

F

M

C

K

A

&&

&

∆

=

∆

+

∆

+

=

+

β

γ

β

β

γ

This method is unconditionally stable if

4

1

,

2

1

.,

.

e

.

2

1

2

=

=

≥

≥

β

γ

γ

β

g

which gives the constant average acceleration method.

Direct methods can be expensive! (the need to
compute A

-1

, often repeatedly for each time step).

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

© 1999 Yijun Liu, University of Cincinnati

175

B. Modal Method

First, do the transformation of the dynamic equations using
the modal matrix before the time marching:

),

(

2

,

)

(

1

t

p

z

z

z

t

z

i

i

i

i

i

i

i

m

i

i

i

=

+

+

Φ

=

=

∑

=

ω

ω

ξ

&

&&

z

u

u

i = 1,2,

⋅⋅⋅

, m.

Then, solve the uncoupled equations using an integration

method. Can use, e.g., 10%, of the total modes (m= n/10).

•

Uncoupled system,

•

Fewer equations,

•

No inverse of matrices,

•

More efficient for large problems.

Comparisons of the Methods

Direct Methods

Modal Method

•

Small model

•

More accurate (with small

∆

t)

•

Single loading

•

Shock loading

•

…

•

Large model

•

Higher modes ignored

•

Multiple loading

•

Periodic loading

•

…

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

© 1999 Yijun Liu, University of Cincinnati

176

Cautions in Dynamic Analysis

•

Symmetry: It should not be used in the dynamic analysis
(normal modes, etc.) because symmetric structures can
have antisymmetric modes.

•

Mechanism, rigid body motion means

ω

= 0. Can use

this to check FEA models to see if they are properly
connected and/or supported.

•

Input for FEA: loading F(t) or F(

ω

) can be very

complicated in real applications and often needs to be
filtered first before used as input for FEA.

Examples

Impact, drop test, etc.