Motion and Stress Analysis by Vector Mechanics
Edward C. Ting
Professor Emeritus of Applied MechanicsPurdue University, West Lafayette, IN
National Central University, ChungLi, Taiwan
a computer framework
for the study of a multi-component structural system with
• component motion • component interactions: connection, contact, collision, penetration• geometrical changes: deformation, displacement, fragmentation, collapse
• stress distribution• behavior and material property changes
a physics approach of mechanics
motion analysis and VFIFE
* vector mechanics ---- particle mechanics * discrete description * intrinsic finite element ---- physical structural element
example: a rod in plane motion
Newton’s law
1. displacement is a motion
1,2,3
, 1,2,3
2. ,
3. 0,
j j j rj jr
rj jr r j
m
x F S
S S
m1
m2
m3
m1
m3
F3
S32F2
F1S21
S12
S23
y
x
x2
m2
analytical mechanics:
For motion analysis, assume
1. rigid body, 2. functional description
ˆc
c jj
jj
x
m
I M
x x e
x F
e
1
xxc
x
01 1
cm
I M
x x
x F
1 1
1 1
1 1
ˆ
( sin 0)
x
I M
I mgd
x x e
pendulum problem: hinged at end 1
motion analysis1. general formulation:
2. complete formulation:
e
1
x
xc
1xd1
x1
1
1 1
21 1
ˆ
ˆ( , )
ˆ ˆ( )
dx
x ds
component
ds f x dx
x dF dS
x x e dS e
e
1ˆ, ( )s sdu dxA AE
stress analysis
assume: 1. deformable body, 2. Hooke’s law
e
1
1ˆdx
1ˆdx
1ˆf dx s ds
sx
1x
X1
dF
1. An approximation
→ separate motion analysis and stress analysis
► continuous bodies: motion--rigid body; stress--deformable body
► variables: motion--displacement; stress--deformation
► governing equations: motion--translation and rotation; stress--equilibrium
2. Described by continuous functions
→ discretization
computation based on analytical mechanics
1,2,3,j j j rj j
r
m
x F S
12 21 1
23 32 2
f
f
S S e
S S e
1 2 1 1
2 3 2 2
u
u
x x
x x
1 11
2 22
( )
( )
f AE u
f AE u
01 1x x
1 2e e
1. Newton’s law
2. behavior model
3. kinematics
4. Hooke’s law
5. pendulum: constraint conditions hinged end: straight rod:
vector mechanics
1
1
2
3
f2e2
-f2e2
f1e1
-f1e1
2
3
e1
e2
l1
l 2
properties:
1. structure: a set of particles
2. always a dynamic process
3. always deformable
advantages:
1. suitable for computation
2. a general and systematic formulation
3. explicit constraint conditions
development needs:
1. describe structural geometry: intrinsic finite element
2. kinematics: fictitious reversed motion
3. continuity requirements
4. mechanics requirements
5. material model: standard tests
elements:
plane rod, plane frame, plane solid, space rod, space frame, 3d membrane, 3d solid, 3d plate shell
V-5 research group:
e. c. ting, c. y. wang, t. y. wu, r. z. wang, c. j. chuang
motion analysis procedure: a simple rod structure
A
B
B
C0t
t
0x
xu
0P
P
x
y
discrete model: mass particles and structural elements
A
B
B
C0t
t
0P
P( )tu
A
B
B
C0t
t( )tu
vector form equation of motion
Am
B
C1f
t
( )tu
P
1f2f
2fB
2
1 22( ) ( ) [ ( ) ( )]
dm t P t t t
dt
uf f
A
B
C
0t
( )tu
at
bt
ct
ft
B
a bt t t
path element
1. element geometry remains unchanged
2. small deformation
discrete path:
kinematics and force calculation
1 material frame: configuration at
2. variable: nodal deformation
3. fictitious reversed motion to define deformation
4. infinitesimal strain and engineering stress
5. nodal forces: use finite element
6. internal forces are in equilibrium
at
reversed motion for nodal deformation
A
A
vB
vB
aB
tB
a
( )
te
ae
u
( )r u
du
l
al
l
f
f( )
d r
a al l
u u u
e
A
A
vB
tB
te
ae
l
al laf
fal
ae
aB
af
f
f
f
a
a
l l
l
ˆˆa
a a
f ff
A A
aE
ˆˆ ( ) aa a a a a a a a
a
l lf f A E A
l
f e e e
ˆtff e
governing equations
21 2
21 2
x x x
y y y
P f fudm
P f fvdt
1 11 11 1 1
1 11
cos
sinx a
a a ay a
f l lE A
f l
2 22 22 2 2
2 22
cos
sinx a
a a ay a
f l lE A
f l
2 2 21 1 1
2 2 22 2 2
( ) ( )
( ) ( )
l b u h v
l b u h v
a bt t t
,a a
a a
u uu ud
v vv vdt
difference equation (symmetrical case)
2
22 , a b
d v h vm P f t t t
dt l
aa a a
a
l lf E A
l
2 2 2( )l b h v , ,a a a
dvt t v v v
dt
2
1 1
( )2 2n
n n n n nn
h vtv P f v v
m l
n a
n a a aa
l lf E A
l
2 2 2( )n nl b h v
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
0
5
10
15
20
25
P*
M otion Analysis
Analytica l so l.
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25
-0 .50
-0.25
0.00
0.25
0.50
0.75
P*
M otion Ana lysis
Analytica l so l.
0 .00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25
-0 .50
-0 .25
0.00
0.25
0.50
0.75
P*
M otion Ana lysis
Analytica l so l.
3* 1
3
PlP
AEh v
h
,
4 node plane solid element
x
y
1u
01
0t
t
02
3u
3au
3u
1au
1u
at
03 04
1a2a
3a
4a
1
2
3
4
estimate the rigid body motion
4
1a
2a
3a
4a
2
3
4
2
3
1
1
1ae
1e
4ae
3ae
2ae
4e
3e
2e1
4
1
1
4 jj
element translation
element rotation
u
fictitious reversed motion
1 ,1,1a
2a
3a
4a
2
3
4
2
3
4
nodal deformation
1( )( )r Ti i η R I x x
cos( ) sin( )
sin( ) cos( )
R
1 ,1,1a
2a
3a
4a
2
3
4
2
3
4
4η
4( )r η
4dη1
1 1
1 1
0
0
( ) , 2,3,4
dxdy
di ix T
di iy i
u u x xi
v v y y
R I
deformation coordinates to define independent variables
ˆˆ x Qx
ˆ ˆ , 1,2,3,4ˆ
di x
di y i
ui
v
Q
1 ,1a
2a
3a
4a
2 2ˆd u η
1e
2
3
4
x
y
2e
x
y
2dη
3v3u
4dη
4v
4u
1 1 2ˆ ˆ ˆ 0u v v
4 4
1 1
ˆ ˆ ˆ ˆ;i i i ii i
x N x y N y
1 2
3 4
1 1;
4 41 1
;4 4
N l s l t N l s l t
N l s l t N l s l t
4 4
1 1
ˆ ˆ ˆ ˆ;i i i ii i
u N u v N v
shape functions:
2 2 3 3 4 4 2 2 3 3 4 4ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ;x N x N x N x y N y N y N y
2 2 3 3 4 4 3 3 4 4ˆ ˆ ˆ ˆ ˆ ˆ ˆ;u N u N u N u v N v N v
*ˆ ˆ n ε Bu
2 3 4
1
JB B B B
*2 3 3 4 5ˆ ˆ ˆ ˆ ˆ ˆ( )Tn u u v u vu
2 2
2
2 2
ˆ ˆ, ,
0
ˆ ˆ, ,
t s s t
s t t s
x N y N
x N x N
B
ˆ ˆ, , 0
ˆ ˆ0 , , , 3,4
ˆ ˆ ˆ ˆ, , , ,
t i s s i t
i s i t t i s
s i t t i s t i s s i t
y N y N
x N x N i
x N x N y N y N
B
ˆ ˆ ˆ
ˆ ˆ ˆ
ˆ ˆ ˆ
x xa x
y ya y
xy xya xy
1 2U U
* * *ˆ ˆ ˆ( ) ( ) ( )a
T T Tn n a a aA
d dA u f u B σ σ
ˆ a σ E ε
*2 3 3 4 4
ˆ ˆ ˆ ˆ ˆˆx x y x yf f f f ff
* * *ˆ ˆ ˆa f f f
* *ˆ ˆˆ ˆ;a a
T Ta a a a a a aA A
d dA d dA f B σ f B σ
ˆ 2 2 2 3 3 3 3 4 4 4 42
ˆ 1 2 3 4
ˆ 1 2 3 4
1ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ0,ˆ
ˆ ˆ ˆ ˆ0,
ˆ ˆ ˆ ˆ0,
z y x y x y x
x x x x x
y y y y y
M f f y f x f y f x f yx
F f f f f
F f f f f
nodal forces
1 ,1a
2a
3a
4a
2
3
4
x
y
3ˆ
axf
3ˆ
ayf
4ˆ
axf
4ˆ
ayf
1ˆaxf
1ˆayf
2ˆ
axf2ˆ
ayf2ˆ
xf
2ˆ
yf
1ˆxf
1ˆ
yf
4ˆ
xf
4ˆ
yf3ˆ
xf
3ˆ
yf
1 ,1a
2a
3a
4a
2
3
4 4xf
4 yf
4ˆ
xf4
ˆyf
2
3
4
1
x
y
x
y
x
y
x
y
4xf
4 yf
4xf
4 yf1u
V
aV
V
ˆˆ
ˆix ixT
iy iy
f f
f f
RQ
stress
x
y
2
3
1
4
2
3
4
1
V
V
x
x
x
x
y
x
y
1u
xSxyS
yS
xS
xyS
yS
ˆ xˆ xy
ˆ y
xxy
y
ˆ ˆˆ ,
ˆ ˆx xy x xy
xy y xy y
S S
S S
σ S
ˆ ˆˆTS Q σQ
x xy
xy y
σ
Tσ RSR
Thank You