Chapter 15

結構動力分析Structural Dynamic Analysis

2/41

Contents15.1 何謂動力分析？

What Are Dynamic Analyses? 15.2 解題方法

Solution Methods 15.3 質量與阻尼

Mass and Damping 15.4 實例：圓柱形子彈的撞擊模擬

Example: Copper Cylinder Impacting on a Rigid Wall 15.5 動態負載

Dynamic Loads 15.6 初始條件

Initial Conditions 15.7 積分時間間隔

Integration Time Steps 15.8 練習題：火箭的飛行

Exercise: Rocket Flight

3/41

第 15.1 節

何謂動力分析？What Are Dynamic Analyses?

4/41

Dynamic Effects

• Inertia force• Damping force• Elastic Force• External force• Dynamic Effects

FKDDCDM

FKD

5/41

15.1.1 Transient Dynamic Analysis

FKDDCDM

6/41

15.1.2 Modal Analysis (1/3)

0KDDCDM

7/41

15.1.2 Modal Analysis (2/3)

0KDDCDM

0KDDM

21 ud ff

2eR

8/41

15.1.2 Modal Analysis (3/3)

• Avoid resonance

• Exploit resonance

• Assess structural stiffness

• Structural modal degrees of freedom

• Further dynamic analyses

• etc.

9/41

15.1.3 Harmonic Response Analysis

tsinFKDDCDM

10/41

第 15.2 節

解題方法Solution Methods

11/41

Solution Methods

Solution Methods for Equation of Motion

Direct Integration Mode Superposition

Implicit Explicit

ReduceFull Full Reduce

12/41

15.2.1 Direct Integration

• Implicit method (ANSYS)

• Explicit method (LS-DYNA)

13/41

15.2.2 Implicit vs. Explicit Methods

ttttttt DDDfD ,,...,

ttttttt DDDfD ,,..., 2

Implicit method

Explicit method

14/41

15.2.3 Mode Superposition Method

nnCCCC MMMMD ...332211

15/41

15.2.4 Reduced Method

FKD

s

m

s

m

sssm

msmm

F

F

D

D

KK

KK

FDK

m

msmssss DKFKD 1

smssmsmm KKKKK 1

sssmsm FKKFF 1

where

16/41

15.2.5 Methods for Nonlinear Dynamic Analysis

• For nonlinear analysis, the only methods applicable is DIRECT INTEGRATION method.

• Reduced method can not be used for nonlinear analysis.

• Either implicit or explicit methods can be used.

17/41

第 15.3 節

質量與阻尼Mass and Damping

18/41

15.3.1 Consistent vs. Lumped Mass Matrices

x

x

x

x

x

x

xxxx

xxxx

xx

xxxx

xxxx

xx

ROTZ

UY

UX

ROTZ

UY

UX

j

j

j

i

i

i

00000

00000

00000

00000

00000

00000

00

00

0000

00

00

0000

Consistent mass matrix

Lumped mass matrix

19/41

15.3.2 Damping

• Damping effects is the total of all energy dissipation mechanisms– Hysteresis (solid damping)– Viscous damping– Dry-friction (Coulomb damping)

20/41

15.3.3 Idealization of Structural Damping

• Structural dampings are usually small (2%-7%).

• Equivalent viscous damping is assumed in ANSYS, i.e.,

DCF D

21/41

15.3.4 How ANSYS Forms Damping Matrix?

• Alpha damping• Beta damping• Material dependent beta damping• Element damping matrices• Frequency-dependent damping matrix

CCKKMC

em N

kk

N

jjj

mjc

11

2

22/41

第 15.4 節

實例 : 圓柱型子彈的撞擊模擬Copper Cylinder Impacting on a Rigid Wall

23/41

15.4.1 Problem Description

x

y

L D

Initial Velocity Vo

24/41

15.4.2 Modeling Consideration

• Material: bilinear plastic model.

• VISCO106 (2D viscoplastic solid) is used.

• Use axisymmetric model.

25/41

15.4.3 ANSYS Procedure (1/4)01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

21

FINISH

/CLEAR

/TITLE, UNITS: SI

/PREP7

  ET, 1, VISCO106,,, 1

  MP,   EX, 1, 117E9

  MP, NUXY, 1,  0.35

  MP, DENS, 1,  8930

  TB, BISO, 1

  TBDATA,, 400E6, 100E6

  TBPLOT, BISO, 1

  RECTNG, 0, 0.0032, 0, 0.0324

  LESIZE, 1,,, 4

  LESIZE, 2,,, 20

  MSHAPE, 0, 2D

  MSHKEY, 1

  AMESH, ALL

  FINISH

26/41

15.4.3 ANSYS Procedure (2/4)23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

/SOLU

  ANTYPE, TRANS

  TRNOPT, FULL

  NLGEOM, ON

  IC, ALL, UY, 0, -227

  NSEL, S, LOC, X, 0

  D, ALL, UX, 0

  NSEL, S, LOC, Y, 0

  D, ALL, UY, 0

  NSEL, ALL

  /PBC, U,, ON

  EPLOT

  TIME, 80E-6

  DELTIM, 0.4E-6

  KBC, 1

  OUTRES, ALL, 4

  SOLVE

  FINISH

27/41

15.4.3 ANSYS Procedure (3/4)

44

45

46

47

48

49

50

51

52

53

54

55

56

57

/POST26

  TOPNODE = NODE(0,0.0324,0)

  NSOL, 2, TOPNODE, U, Y, DISP

  DERIV, 3, 2, 1,, VELO

  /GRID, 1

  /AXLAB, X, TIME s

  /AXLAB, Y, DISPLACEMENT m

  PLVAR, 2

  /AXLAB, Y, VELOCITY m/s

  PLVAR, 3

  FINISH

28/41

15.4.3 ANSYS Procedure (4/4)59

60

61

62

63

64

65

/POST1

  SET, LAST

  PLDISP, 2

  PLNSOL, EPTO, EQV

  ANTIME, 30

29/41

第 15.5 節

動態負載Dynamic Loads

30/41

Dynamic Loads: An Example01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

/SOLU

  ...

  F, ...        ! 22.5 at the nodes

  TIME, 0.5     ! Ending time

  DELTIM, ...   ! Integration step

  KBC, 0        ! Ramped loading

  AUTOTS, ON    ! Option

  OUTRES, ...   ! Option

  SOLVE         ! Load step 1

  F, ...        ! 10 at the nodes

  TIME, 1       ! Ending time

  SOLVE         ! Load step 2

  FDELE, ...    ! Zero the force

  TIME, 1.5     ! Ending time

  KBC, 1        ! Stepped loading

  SOLVE         ! Load step 3

0 0.5 1.0 1.5Time (s)

Force (N)22.5

10

31/41

第 15.6 節

初始條件Initial Conditions

32/41

15.6.1 Example: An Stationary Plate Subjected to an Impulse

Load

• This is the default initial condition. No input is needed.

33/41

15.6.2 Example: Initial Velocity on a Golf Club Head

• This simple initial condition can be specified by using IC command.

  NSEL, ALL

  IC, ALL, UY, 0, V0

34/41

15.6.3 Example: Plucking a Cantilever Beam

01

02

03

04

05

06

07

08

09

10

11

12

13

14

/SOLU

  ANTYPE, TRANS

  ...

  TIMINT, OFF  ! Transient effects off

  TIME, 0.001  ! Small time interval

  D, ...       ! Apply displacement at desired nodes

  KBC, 1       ! Stepped loads

  NSUBST, 2    ! To avoid non-zero velocity

  SOLVE

  TIMINT, ON   ! Transient effects on

  TIME, ...    ! Actual time at end of load

  DDELE, ...   ! Delete the applied displacement

  SOLVE

35/41

15.6.4 Example: Dropping an Object from Rest

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

/SOLU

  TIMINT, OFF     ! Transient effects off

  TIME, 0.001     ! Small time interval

  NSEL, ...       ! Select all nodes on the object

  D, ALL, ALL, 0  ! Temporarily fix them

  NSEL, ALL

  ACEL, ...       ! Apply acceleration

  KBC, 1          ! Stepped loads

  NSUBST, 2       ! To avoid non-zero velocity

  SOLVE           ! Load step 1

  TIMINT, ON      ! Transient effects on

  TIME, ...       ! Actual time at end of load

  NSEL, ...       ! Select all nodes on the object

  DDELE, ALL, ALL ! Release them

  NSEL, ALL

  SOLVE           ! Load step 2

36/41

第 15.7 節

積分時間間隔Integration time Steps

37/41

15.7.1 Response Frequency

Response

Time

Minimumresponse

time

20

pt

38/41

15.7.2 Abrupt Changes in Loading

0 0.5 1.0 1.5Time (s)

Force (N)

22.5

10

39/41

15.7.3 Contact Frequency

30

Tt

40/41

15.7.4 Wave Propagation

c

xt

3

41/41

15.8 Exercise: Rocket Flight

y

140 in.

Thrust

Time

100 lb

1 sec.1

2

3