aml110impact

8/3/2019 AML110Impact

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AML 110

Impact



1. Smooth impact of two unconstrained

rigid bodies

Linear momentum conservation for individual bodies (4 eqns)Body 1 :

Body 2 :

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv

bbt t

vvvv1

'

11

'

1==

bbt t

vvvv2

'

22

'

2==




rigid bodies

Linear momentum conservation for the system ( 1 eqn)

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv

2211

'

22

'

11vmvmvmvmrrrr

+=+

nnnn

vmvmvmvm2111

'

21

'

11+=+⇒




rigid bodies

Moment of momentum conservation for each body ( 6 eqns)Body 1:

Body 2:

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv

111111 1

'

1

'

C OC C C OC C vmr H vmr H r

r

r

r

×+=×+

222222 2

'

2

'

C OC C C OC C vmr H vmr H rr

r

rr

r

×+=×+




rigid bodies

Empirical Equation using coefficient of restitution( 1 eqn)

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv

)nnnn

A B A Bvvevv −−=−

''




rigid bodies

Summary

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv

( )nnnn A B A B

vvevv −−=−''

bbt t vvvv 1

'

11

'

1 ==bbt t vvvv 2

'

22

'

2 ==

nnnn

vmvmvmvm2111

'

21

'

11+=+

111111 1

'

1

'


r

r

r

×+=×+

222222 2

'

2

'

C OC C C OC C vmr H vmr H rr

r

rr

r

×+=×+




rigid bodies

If Body 2 is massive

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv

( )nnnn A B A B

vvevv −−=−'

bbt t vvvv 1

'

11

'

1 ==bbt t vvvv 2

'

22

'

2 ==

nn

vv2

'

2=

111111 1

'

1

'


r

r

r

×+=×+

2

'

2ω ω

rr

=




rigid bodies (2D)


Body 2 :

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω

rrrr

vv

6 Unknowns :''''

2121,,, ω ω vv

rr

t t

vv1

'

1=

t t

vv2

'

2=

1ω

2ω




rigid bodies (2D)

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω

rrrr

vv

6 Unknowns :''''

2121,,, ω ω vv

rr

1ω

2ω

Linear momentum conservation for the system ( 1 eqn)2211

'

22

'

11vmvmvmvmrrrr

+=+




rigid bodies (2D)

Moment of momentum conservation for each body ( 2 eqns)Body 1:

Body 2:

11

1

11

1

1133

'

1

'

133 C OC

C

C OC

C vmr I vmr I rrrr

×+=×+ ω ω

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω

rrrr

vv

6 Unknowns :''''

2121,,, ω ω vv

rr

1ω

2ω

22

2

22

2

2233

'

2

'

233 C OC

C

C OC

C vmr I vmr I rrrr

×+=×+ ω ω




rigid bodies (2D)

Empirical Equation using coefficient of restitution( 1 eqn)

Further,

en

et

.C2

.C1

A

BO

Given : 2121,,, ω ω vv

rr

6 Unknowns :

)nnnn A B A B

vvevv −−=−''

''''

2121,,, ω ω vv

rr

1ω

2ω

222111r vvr vv

nnnn B Aω ω −=−=

r1 r2




rigid bodies (2D)

Summary

en

et

.C2

.C1

A

BO


rr

6 Unknowns :''''

2121,,, ω ω vv

rr

1ω

2ω

r1 r2

t t

vv1

'

1=

t t

vv2

'

2=

2211

'

22

'

11vmvmvmvmrrrr

+=+

11

1

11

1

1133

'

1

'

133 C OC

C

C OC

C vmr I vmr I rrrr

×+=×+ ω ω

22

2

22

2

2233

'

2

'

233 C OC

C

C OC

C vmr I vmr I rrrr

×+=×+ ω ω

)nnnn


''




rigid bodies (2D)


en

et

.C2

.C1

A

BO


rr

6 Unknowns :''''

2121,,, ω ω vv

rr

1ω

2ω

r1 r2

t t

vv1

'

1=

t t

vv2

'

2=

nn

vv2

'

2=

11

1

11

1

1133

'

1

'

133 C OC

C

C OC

C vmr I vmr I rrrr

×+=×+ ω ω

2

'

2ω ω =

)nnnn


'



3. Smooth central impact of two

unconstrained rigid bodies


Body 2 :

en

et

.C2.C1A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv

bbt t

vvvv1

'

11

'

1==

bbt t

vvvv2

'

22

'

2==

Central impact : C1 , C2 lie on line of impact

v1

v2





Linear momentum conservation for system (1 eqn)

en

et

.C2.C1A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv


v1

v2

2211

'

22

'

11vmvmvmvmrrrr

+=+

nnnn

vmvmvmvm2111

'

21

'

11+=+⇒





Moment of momentum conservation for each body (6 eqns)Body 1

Body 2

en

et

.C2.C1A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv


v1

v2

1

'

1ω ω

rr

=

2

'

2ω ω

rr

=





Empirical equation (1 eqn)

en

et

.C2.C1A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv


v1

v2

)nnnn A B A B

vvevv −−=−''

)nnnn

vvevv12

'

1

'

2−−=−⇒





Summary

en

et

.C2.C1A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv


v1

v2

bbt t vvvv 1

'

11

'

1 == bbt t vvvv 2

'

22

'

2 ==

nnnn

vmvmvmvm2111

'

21

'

11+=+

1

'

1

ω ω

rr

=

2

'

2

ω ω

rr

=

)nnnn

vvevv12

'

1

'

2−−=−






en

et

.C2.C1A

BO

Given : 2121,,, ω ω

rrrr

vv

Unknowns :''''

2121,,, ω ω

rrrr

vv


v1

v2

bbt t vvvv 1

'

11

'

1 == bbt t vvvv 2

'

22

'

2 ==

nn

vv2

'

2=

1

'

1

ω ω

rr

=

2

'

2

ω ω

rr

=

)nnnn

vvevv12

'

12−−=−

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