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

© 1999 Yijun Liu, University of Cincinnati

157

Chapter 7. Structural Vibration and Dynamics

•

Natural frequencies and modes

•

Frequency response (F(t)=F

o

sin

ωt)

•

Transient response (F(t) arbitrary)

I. Basic Equations

A. Single DOF System

From Newton’s law of motion (ma = F), we have

u

c

u

k

f(t)

u

m

&

&&

−

−

=

,

i.e.

f(t)

u

k

u

c

u

m

=

+

+

&

&&

, (1)

where u is the displacement,

dt

du

u

/

=

&

and

.

/

2

2

dt

u

d

u

=

&

&

F(t)

m

m

f=f(t)

k

c

f(t)

u

c

ku

&












force

-

)

(

damping

-

stiffness

-

mass

-

t

f

c

k

m

x, u

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

© 1999 Yijun Liu, University of Cincinnati

158

Free Vibration:

f(t) = 0 and no damping (c = 0)

Eq. (1) becomes

0

=

+

u

k

u

m &

&

.

(2)

(meaning: inertia force + stiffness force = 0)

Assume:

t)

(

U

u(t)

ω

sin

=

,

where

ω

is the frequency of oscillation, U the amplitude.

Eq. (2) yields

0

sin

sin

2

=

+

−

t)

ù

(

U

k

t)

ù

(

m

ù

U

i.e.,

[

]

0

2

=

+

−

U

k

m

ω

.

For nontrivial solutions for U, we must have

[

]

0

2

=

+

−

k

m

ω

,

which yields

m

k

=

ω

.

(3)

This is the circular natural frequency of the single DOF
system (rad/s). The cyclic frequency (1/s = Hz) is

π

ω

2

=

f

,

(4)

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

© 1999 Yijun Liu, University of Cincinnati

159

With non-zero damping c, where

m

k

m

c

c

c

2

2

0

=

=

<

<

ω

(c

c

= critical damping) (5)

we have the damped natural frequency:

2

1

ξ

ω

ω

−

=

d

,

(6)

where

c

c

c

=

ξ

(damping ratio).

For structural damping:

15

.

0

0

<

≤

ξ

(usually 1~5%)

ω

ω

≈

d

.

(7)

Thus, we can ignore damping in normal mode analysis.

u

t

U

U

T = 1 / f

U n d a m p e d F r e e V i b r a t i o n

u = U s i n w t

u

t

Damped Free Vibration

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

© 1999 Yijun Liu, University of Cincinnati

160

B. Multiple DOF System

Equation of Motion

Equation of motion for the whole structure is

)

(t

f

Ku

u

C

u

M

=

+

+

&

&

&

,

(8)

in which:

u



nodal displacement vector,

M



mass matrix,

C



damping matrix,

K



stiffness matrix,

f



forcing vector.

Physical meaning of Eq. (8):

Inertia forces + Damping forces + Elastic forces

= Applied forces

Mass Matrices

Lumped mass matrix (1-D bar element):

1

ρ,A,L 2

u

1

u

2

Element mass matrix is found to be

4

4 3

4

4 2

1

matrix

diagonal

2

0

0

2





















=

AL

AL

ρ

ρ

m

2

1

AL

m

ρ

=

2

2

AL

m

ρ

=

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

© 1999 Yijun Liu, University of Cincinnati

161

In general, we have the consistent mass matrix given by

dV

V

T

∫

=

N

N

m

ρ

(9)

where N is the same shape function matrix as used for the
displacement field.

This is obtained by considering the kinetic energy:

( )

( ) ( )

u

N

N

u

u

N

u

N

u

m

u

m

&

43

42

1

&

&

&

&

&

&

&

&

∫

∫

∫

∫

=

=

=

=

=

Κ

V

T

T

V

T

V

T

V

T

dV

dV

dV

u

u

dV

u

mv

ρ

ρ

ρ

ρ

2

1

2

1

2

1

2

1

)

2

1

(cf.

2

1

2

2

Bar Element (linear shape function):

[

]

3

/

1

6

/

1

6

/

1

3

/

1

1

1

2

1

u

u

AL

ALd

V

&

&

&

&













=

−











 −

=

∫

ρ

ξ

ξ

ξ

ξ

ξ

ρ

m

(10)

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

© 1999 Yijun Liu, University of Cincinnati

162

Element mass matrices:

⇒

local coordinates

⇒

to global coordinates

⇒

assembly of the global structure mass matrix M.

Simple Beam Element:

4

22

3

13

22

156

13

54

3

13

4

22

13

54

22

156

420

2

2

1

1

2

2

2

2

θ

θ

ρ

ρ

&

&

&

&

&

&

&

&

v

v

L

L

L

L

L

L

L

L

L

L

L

L

AL

dV

T

























−

−

−

−

−

−

=

=

∫

V

N

N

m

(11)

Units in dynamic analysis (make sure they are consistent):

Choice I

Choice II

t (time)

L (length)

m (mass)

a (accel.)

f (force)

ρ (density)

s

m

kg

m/s

2

N

kg/m

3

s

mm

Mg

mm/s

2

N

Mg/mm

3

1

1

θ

v

2

2

θ

v

ρ, A, L