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Institut für Technische Mechanik####### Teilinstitut Dynamik / Mechatronik❙❑❘■P❚▼❊❍❘❑Ö❘P❊❘❉❨◆❆▼■❑Pr♦❢✳❉r✳✲■♥❣✳❲✳❙❡❡♠❛♥♥✕❏✉❧✐✷✵✶✾✕■♥❤❛❧ts✈❡r③❡✐❝❤♥✐s✸✳✹✳✸ ▲❛❣r❛♥❣❡s❝❤❡●❧❡✐❝❤✉♥❣❡♥❜ ❡✐♥✐❝❤t❤♦❧♦♥♦♠❡♥❇✐♥❞✉♥❣❡♥ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✻✸ ✸✳✹✳✹ ▲❛❣r❛♥❣❡s❝❤❡●❧❡✐❝❤✉♥❣❡♥❜ ❡✐❑rä❢t❡♥♠✐t❡✐♥❡♠P♦t❡♥t✐❛❧ ✳✳✳✳✳✳✳✳✳✳✳✳✳✳ ✻✺ ✸✳...

Description

Institut für Technische Mechanik Teilinstitut Dynamik / Mechatronik

R3

x, y, z

r, ϕ, ϑ

~v ~v

q

~v

A

q

A

q

q

B

~ p

O q1 q2 q3

B

P C

A

• ~ p

q1 q2 q3

• ~ p

~ p (q1 )

• ~ p

~ p (q1 , q2 )

• ~ p

~ p (q1 , q2 , q3 )

C

B A D

R3 ~a1 ~a2 ~a3

A

~a1 ~a2 ~a3 R3 ~v

R3

~v = v1~a1 + v2~a2 + v3~a3 ~ai v1 v2 v3 vi~ai

~v

q

A

q ~ai

~a1 ~a2 ~a3 ~ p = α1~a1 + α2~a2 + α3~a3

~v

~a1 ~a2 ~a3 A ~ p

vi

~b1 ~b2 ~b3 b~1 b~2 b~3

~ai

~v

A B

B

~ p = β1b~1 + β2 b~2 + β3b~3 R

~ p

~ p = R(c 1b~2 + s1 b~3 ) c i = cos qi si = sin qi β1 = 0 β2 = Rc 1 β3 = Rs1 . ~ai

~ p

~ai

α1

=~ p · ~a1

α2

=~ p · ~a2

= R(c 1b~2 · ~a1 + s1~b3 · ~a1 ), = R(c 1b~2 · ~a2 + s1~b3 · ~a2 ),

α3

=~ p · ~a3

= R(c 1b~2 · ~a3 + s1~b3 · ~a3 )

~b2 · ~a1 = 0, ~b2 · ~a3 = 0, ~b3 · ~a2 = 0,

α1

=

Rs1 c 2 ,

α2

=

Rc 1 ,

α3

=

Rs1 s2 .

~v

~b2 · ~a2 = 1, ~b3 · ~a1 = cos q2 = c 2 , ~b3 · ~a3 = sin q2 = s2 ,

n

q1 , . . . , qn

A

~v = v1~a1 + v2~a2 + v3~a3

A

B

n 3

A

X ∂vi ∂~v := · a~i , ∂qr ∂qr

r = 1, . . . , n

i=1

~v

A

~v

t A

3 3 X X ~v vi := · a~i = v˙i · a~i t t i=1

~v

t

i=1

A

~ p A

∂~ p ∂q1

=

∂α1 ∂α2 ∂α3 · ~a3 · ~a1 + · ~a2 + ∂q1 ∂q1 ∂q1 R · (c 1 c 2~a1 − s1~a2 + c 1 s2~a3 ),

=

R · (−s1 s2~a1 + s1 c 2~a3 ),

=

~0,

=

A

∂~ p ∂q2 A

∂~ p ∂q3 B

∂~ p ∂q1

=

∂β1 ~ ∂β2 ~ ∂β3 ~ · b1 + · b2 + · b3 ∂q1 ∂q1 ∂q1 R · (−s1 b~2 + c 1 b~3 ),

=

~0,

=

~0.

=

B

∂~ p ∂q2 B

∂~ p ∂q3

qi A

~ p t B ~ p t

=

R · [(c 1 c 2 q˙1 − s1 s2 q˙2 ) · ~a1 − s1 q˙1~a2 + (c 1 s2 q˙1 + s1 c 2 q˙2 ) · ~a3 ]

=

R · [−s1 q˙1 b~2 + c 1 q˙1 b~3 ]

A

b~i

d~ p dt

~a1

s2~b1 + c 2b~3 , ~b2 , −c 2~b1 + s2b~3

=

~a2

=

~a3

=

A

∂~ p ∂q1

=

R · [(c 1 c 2 s2 − c 1 s2 c 2 ) · ~b1 − s1b~2 + (c 1 c 2 c 2 + c 1 s2 s2 ) ·b~3 ]

=

R · [−s1~b2 + c 1~b3 ],

=

−Rs1~b1

=

~0,

=

R · [−s1 q˙2~b1 − s1 q˙1~b2 + c 1 q˙1b~3 ].

A

∂~ p ∂q2 A ∂~ p ∂q3 A

~ p t

~v1

~vN

N N X ∂~vi ∂ X ~vi = ∂qr ∂qr i=1 i=1

q1 r = 1, . . . , n.

s

~v

∂ (s · ~v) ∂qr

∂~v ∂s +s· ∂qr ∂qr

= ~v ·

∂ (~v · w) ~ = ∂qr

w ~·

∂w ~ ∂~v + ~v · ∂qr ∂qr

s

=

s(q1 , . . . , qn ),

~ p

= =

~ p(q1 , . . . , qn ), ~ p·~ p

~ p2

w ~

q1 , . . . , qn r = 1, . . . , n

r = 1, . . . , n.

a = s~ p·~ p

∂ (s · ~ p·~ p) ∂qr

∂~ p ∂~ p ∂s ·~ p+s·~ p· ·~ p·~ p+s· ∂qr ∂qr ∂qr ∂s ∂~ p ·~ p·~ p + 2s · ~ p· ∂qr ∂qr

= =

~ p2 ~ p·

∂~ p =0 ∂qr ∂~ p ∂qr

~ p

B

∂ ∂qs B

t

A

∂~v ∂qr

A

~v t





A

6=

∂ ∂qr A

6=

t

qn

B

∂~v ∂qs

B

~v t





,

r = 1, . . . , n

~v ~v

A

A

~ai

B B

~bi ~v ~v = t~a1 ~a1 = s2b~1 + c 2~b3

  ~v = t s2~b1 + c 2~b3 . A

B

t

A

~v t B

A

t

B

t

~v t

=

~a1

=

s2b~1 + c 2~b3 ,



=

q˙2 · (c 2~b1 − s2~b3 )

=

−q˙2~a3 ,

~v t

=

s2~b1 + c 2~b3 + tq˙2 (c 2~b1 − s2b~3 )

=

~a1 − tq˙2~a3 ,

=

− (q˙2 + t¨ q2 ) ~a3

B

~v t



A

~v t



A

6=

t

B

~v t



.

q1 , . . . , qn A

~v

n A ~v X A ∂~v ∂~v q˙r + = t ∂t ∂q r r=1 A

~ v t

∂~v ∂ ~v , = ∂qr t t ∂qr

q˙r = ~v

r = 1, . . . , n

t

qr . t

t



∂~v ∂qr



=

∂ ∂q1

=

∂ ∂qr

=

∂ ∂qr

=

∂ ∂qr

 ∂~v ∂qr         ∂~v ∂~v ∂~v ∂~v ∂ ∂ ∂ q˙n + q˙2 + · · · + q˙1 + ∂qr ∂t ∂qr ∂qn ∂qr ∂q2 ∂q1   ∂~v ∂~v ∂~v ∂~v · q˙n + · q˙2 + · · · + · q˙1 + ∂qn ∂q2 ∂q1 ∂t   ~v . t 

∂~v ∂qr



q˙1 +

∂ ∂q2



∂~v ∂qr



q˙2 + · · · +

∂ ∂qn



∂~v ∂qr



q˙n +

~r ~r =

∂ ∂t



B

E

a1~e1 + a2~e2 + a3~e3 , A1~b1 + A2~b2 + A3~b3

~r =

r∼E

=

r∼B

=



 a1  a2  = (a1 , a2 , a3 ) , a3   A1  A2  = (A1 , A2 , A3 ) . A3

~a

~b

a

∼

b∼

a b =b a , ~a · ~b = ∼ ∼ ∼ ∼ = (a1 , a2 , a3 ) a ∼ 

0 ˜a =  a3 ∼ −a2

 −a3 a2 0 −a1  a1 0

~c = ~a × ~b a˜∼ c∼ = ∼ b.

~a ˙ = (a˙ 1 , a˙ 2 , a˙ 3 ) . a ∼

a b ∼ ∼ t

˙ b + a b˙ =a ∼ ∼ ∼ ∼

b∼ = (b1 , b2 , b3 )

~c = ~a × ~b ˜a˙ b + a ˜ ˙b. c˙ = ∼ ∼ ∼∼ ∼

~k1

~i1 ~i2 ~i3

~k2 ~k3 ~k1 ~k2

= =

m11~i1 + m12~i2 + m13~i3 , m21~i1 + m22~i2 + m23~i3 ,

~k3

=

m31~i1 + m32~i2 + m33~i3 , mij

~ij

i

mij = ~ki · ~ij . ~ki K

~ij 

m11 I  m21 = m ∼ m31

m12 m22 m32

mij 

m13 m23  m33

   ~k1 ~i1  ~  K I ~  m . i2  k2  = ∼ ~k3 ~i3 

I

K

m ∼



m ¯ 11 ¯ 21 = m m ¯ 31

m ¯ 12 m ¯ 22 m ¯ 32

~km

~il

 m ¯ 13 m ¯ 23  , m ¯ 33

m ¯ lm = ~il · ~km

I

K I ) . m = (K m ∼ ∼

I

I −1 K ) m = (K m ∼ ∼

P ∼ · Q| = det(P ) · det(Q ) = |P | · |Q| |P ∼ ∼ ∼ ∼

∼

∼

Q ∼

K I K ( m ) , I∼ = I m ∼ ∼

K 2 )) 1 = (det(I m ∼

K ) = 1. det(I m ∼

B

E

B

~e3 = ~b3

E

ϕ

ϕ



  ~b1 cos ϕ ~ =  − sin ϕ  b2  ~b3 0

B



cos ϕ =  − sin ϕ m ∼ 0 E

det(B mE ) ∼

B

E E · (B m ) m ∼ ∼

= =

m ∼ r T r∼j = δij , ∼i

r∼j m ∼

j

m ∼

sin ϕ cos ϕ 0

sin ϕ cos ϕ 0

  ~e1 0 0   ~e2  . 1 ~e3

 0 0  1

,

B

E

( m ) ∼



cos ϕ =  sin ϕ 0

E cos2 ϕ + sin2 ϕ = 1 = det((B m ) ), ∼   1 0 0  0 1 0 . 0 0 1

m ∼

− sin ϕ cos ϕ 0

 0 0 . 1

~e3

S

S

m ∼

I

T

S m ∼

K

T m ∼

T

 cos α sin α 0 =  − sin α cos α 0  , 0 0 1   1 0 0 sin β  , =  0 cos β 0 − sin β cos β   cos γ sin γ 0 =  − sin γ cos γ 0  . 0 0 1 

I   ~k1  ~  K T T SS I  m m  k2  = ∼ m ∼ ∼ ~k3 

K

I

m ∼

1, 2′ , 3′′ S

  ~i1 ~i2  = K m I  ∼ ~i3

K

 ~i1 ~i2  . ~i3



 sin β sin γ cos γ sin β  . cos β

T

α

cos α cos γ − cos β sin α sin γ cos γ sin α + cos α cos β sin γ =  − cos β cos γ sin α − cos α sin γ cos α cos β cos γ − sin α sin γ sin α sin β − cos α sin β

β

I





~s1  ~s2  ~s3

=



 ~t1  ~t2  = ~t3   ~k1  ~  =  k2  ~k3

K

• • • • • • • • • •

I m ∼



K  1  0 0

0 cos α − sin α



cos β  0 sin β



 ~  i1 0 sin α   ~i2  , ~i3 cos α

  0 − sin β ~s1   ~s2  , 1 0 ~s3 0 cos β

cos γ  − sin γ 0

cos β cos γ =  − sin γ cos β − sin β

sin γ cos γ 0

  ~t1 0 0   ~t2  . ~t3 1

cos γ sin α sin β + cos α sin γ − cos α cos γ − sin α sin β sin γ − cos β sin α

 − cos α cos γ sin β + sin α sin γ cos γ sin α + cos α sin β sin γ  . cos α cos β

• •

~v E

B

θ ~v

v = (x, y, z)T v∼E = ∼B



 x  y = z 





E

mB ∼

E

E

−1

E

 x mB  y  ∼ z 

     x x x  T  y  = EmB  y = y  ∼ z z z



B − I∼ m ∼



B − λI∼ m ∼

   x 0  y  =  0 . z 0





   0 x  y  =  0 , z 0



x, y, z

    B =0   Em − λI ∼ ∼ λ

λ1 = 1

λ2,3 = e±jθ θ x, y, z λ1

v∼ ~v

θ B

A

~r

~rneu = ~r + ∆~r

B

~rneu

~r

E

~r ∆~r p~1

θ p~1

~v

p~2

~v

~r

|~ p1 | = a sin θ

a = |~r| sin α = |~v × ~r|, ~v

p~1 ~i1 = ~v × ~r |~v × ~r|

p~1 = a sin θ · ~i1 = a sin θ

~v × ~r = sin θ ~v × ~r |~v × ~r|

p~2 θ |~ p2 | = a(1 − cos θ) = 2a sin2 . 2 p~2

p~1

~v

~i2 = ~v × ~i1 = ~v × (~v × ~r) a

p~2 = 2a sin2

θ θ ~v × (~v × ~r) = 2[~v × (~v × ~r)] sin2 . 2 2 a ~rneu = ~r + ∆~r

~rneu = ~r + ∆~r = ~r + ~p1 + p~2 = ~r + sin θ(~v × ~r) + 2[~v × (~v × ~r)] sin2

θ . 2

B r, (~v × ~r)B = v∼ ˜∼



0 −v3 0 v∼ ˜ =  v3 −v2 v1 v˜∼

r + sin θ r∼neu,E = ∼B

r∼neu,E

 = I + sin θ ∼

r

=

∼neu,E

E

E

v˜ r + 2 sin2

∼∼B

v˜∼ +

2 2˜ v∼

sin2

B m r ∼ ∼B

B =∼ I + sin θ m ∼

2

v˜∼ + 2˜ v∼ sin2

θ 2

θ 2



v2 −v1  , 0

v˜ v˜ r

∼ ∼∼B

 θ r , 2 ∼B



 r1 r =  r2  , ∼ r3

~v

sin θ = 2sin

E

θ θ cos 2 2

B v∼ sin m =∼ I + 2˜ ∼

Θ0

θ = cos , 2

Θ1

=

Θ2

=

Θ3

=

θ 2



I∼ cos

θ θ + v˜ sin 2 2 ∼



.

θ v1 sin , 2 θ v2 sin , 2 θ v3 sin 2

θ∼ = [Θ1 , Θ2 , Θ3 ]T

E

B ˜) m =∼ I + 2 ˜θ∼(I∼Θ0 + θ∼ ∼

˜θ

=

∼

=

3 X

Θ2k



0 −Θ3  Θ3 0 −Θ2 Θ1



0 −v3  v3 0 −v2 v1

  0 Θ2  −Θ1  =  v3 sin 2θ 0 −v2 sin θ2  v2 θ θ −v1  sin = v˜ · sin ∼ 2 2 0

−v3 sin θ2 0 v1 sin θ2

=

cos2

θ θ θ θ + v12 sin2 + v22 sin2 + v23 sin2 2 2 2 2

=

cos2

θ θ + ( v12 + v22 + v32 ) sin2 | {z } 2 2

k=0

v2 sin 2θ

 −v1 sin θ2  0

=1,

=

3 X

1.

Θ2k = 1.

k=0

E



1 − 2Θ22 − 2Θ32



2(Θ1 Θ2 − Θ0 Θ3 ) 2(Θ1 Θ3 + Θ0 Θ2 )



B = 2(Θ2 Θ3 − Θ0 Θ1 )  m   2(Θ1 Θ2 + Θ0 Θ3 ) 1 − 2Θ12 − 2Θ32 ∼ 2(Θ1 Θ3 − Θ0 Θ2 ) 2(Θ2 Θ3 + Θ0 Θ1 ) 1 − 2Θ12 − 2Θ22



 2(Θ20 + Θ12) − 1 2(Θ1 Θ2 − Θ0 Θ3 ) 2(Θ1 Θ3 + Θ0 Θ2 )   E B =  2(Θ1 Θ2 + Θ0 Θ3 ) 2(Θ20 + Θ22 ) − 1 2(Θ2 Θ3 − Θ0 Θ1 )  . m ∼ 2(Θ1 Θ3 − Θ0 Θ2 ) 2(Θ2 Θ3 + Θ0 Θ1 ) 2(Θ20 + Θ23 ) − 1 θ

~e3 T

v∼ = (0, 0, 1)

θ θ Θ0 = cos , Θ1 = 0, Θ2 = 0, Θ3 = sin . 2 2 E

E

m ∼

  2Θ20 − 1 −2Θ0 Θ3 0   2 2Θ 0 − 1 0  =  2Θ0 Θ3 0 0 2(Θ20 + Θ23 ) − 1   2 cos2 θ2 − 1 −2 sin θ2 cos 2θ 0   =  2 sin θ2 cos θ2 0 2 cos2 θ2 − 1  0 0 2(cos2 θ2 + sin2 2θ ) − 1

B

θ −1 2 θ θ 2 cos sin 2 2 2 cos2

E

m ∼

B

B m ∼

=

cos θ,

=

sin θ



cos θ =  sin θ 0

 0 0 . 1

− sin θ cos θ 0

~ω A

~v = t

B

~v + t

A

~ω B × ~v A

B A

~ω

B

A

=

A

! ~b2 ~ · b3 ~b1 + t ~b1 , ~b2 , b~3

A

~ω B := ~b1

A

! ~b3 ~ · b1 ~b2 + t

A

! ~b1 ~ · b2 ~b3 t A

      ~˙b2 · ~b3 + ~b2 ~b˙ 3 · ~b1 + ~b3 ~b˙ 1 · ~b2 .

() t

= (˙ )

~ω B

~b1 × ~b1 = 0~

~b1 A

    ˙ ˙ ~ω B × ~b1 = ~b2 × ~b1 ~b3 ·b~1 + ~b3 × ~b1 ~b1 · ~b2 . ~b1 ~b2

~b3

~b2 × ~b1 = −~b3 , ~b3 × ~b1 = ~b2

A

    ˙ ˙ ~ω B × ~b1 = − ~b3 · ~b1 ~b3 + ~b1 · ~b2 b~2 . ~b1 · ~b1 = 1

~b3 · ~b1 = 0

~˙b1 · ~b1 = 0

˙ ~˙b3 · ~b1 = −~b3 · ~ b1 ,

A

      ˙ ˙ ˙ ~ω B × ~b1 = ~b1 · ~b1 ~b1 + ~b1 · ~b2 ~b2 + ~b1 · ~b3 ~b3 . A

A

~ω B × ~b2

A

~ω B × ~b3

~˙b1

~b 1 t

~˙b1 · ~b1 ~˙b1 · ~b2

B

˙ = ~b2 , ˙ = ~b3

E

α

B

~ ωB ~bi

~e3

~e3 ~ei ~b1 ~b2 ~b3

=

cos α ~e1 + sin α ~e2 ,

=

− sin α ~e1 + cos α ~e2 ,

=

~e3 .

E

~˙b1 · b~3

~ bi ~˙b1 ~˙b2 ~˙b3

E

=

E

() t

−α˙ sin α ~e1 + α˙ cos α ~e2 ,

=

−α˙ cos α ~e1 − α˙ sin α ~e2 ,

=

~0.

~ ωB

( ˙) =

E

0 · ~b1 + 0 · b~2 + (−α˙ sin α ~e1 + α˙ cos α ~e2 ) · (− sin α ~e1 + cos α ~e2 ) ~b3 2 α(sin ˙ α + cos2 α) ~b3 = α˙ ~b3 .

= =

A ~b

B

ϕ(t) A

~ω B = ϕ˙ ~b β~

A

~ β = t

A

B

~ω B × β~

A B ~bi

~ai ~b1 ~b2 ~b3

A

=

cos θ~a1 + sin θ~a2 ,

=

− sin θ~a1 + cos θ~a2 ,

=

~a3 .

~ ωB

=

=

 i −θ˙ cos θ~a1 − θ˙ sin θ~a2 · ~a3 ~b1 + 0 · ~b2 h  i + −θ˙ sin θ~a1 + θ˙ cos θ~a2 · (− sin θ~a1 + cos θ~a2 ) ~b3 h

θ˙~b3 .

~ β = −L1~b1 − L2~b3 A

A

β~ t 2~ β

t2

~b1 , ~b2 , ~b3

~a1 , ~a2 , ~a3 ~a3 , ~b3

L1 , L2

  θ˙~b3 × −L1~b1 − L2~b3 = −L1 θ˙~b2 , ! ! A ~ A ~ A B A ~ β β β = +A ~ ωB × t t t t t   −L1 ¨θ~b2 − L1 θ˙2~b3 × ~b2 = L1 θ˙2~b1 − θ¨~b2 .

= = =

β~

~ai ~ai

~ai ~bi

~v   = v ∼ A

  A B m v ∼ ∼

B

β~

~v   v˙ = ∼ A

  A B ˙ m v + ∼ ∼

A

  B v˙ . m ∼ ∼ B

B

A

  A B v ˙ m ∼ B ∼   A ˙B m v ∼ ∼

~ v t B

A

B

B

B

A

~ v t

:

~ω B × ~v :





A

A

B m ∼

A

mB ∼

A   

~ v t

 ~ω B × ~v 

−1

−1

A



A

mB ∼

−1

  ˙  ˙ = ∼ v∼B v m ∼ B T    A ˙ B A B m = m v ∼B ∼ ∼

A

B

B

  ˜ω · v ∼ ∼

˜ω = ∼



B

A



B m ∼

T

0 ω ˜∼ =  ω3 −ω2

A

˙B m ∼

 −ω3 ω2 0 −ω 1  ω1 0

ω1 ω2

ω3

A

  ˙B v m ∼ ∼

B

      

B

~k

B

A

A B B

A

B

A

A

~ω B = ωk~ ~k

B

ω = ϑ˙

ϑ

LA ϑ

LB B

A A

A B C D

B

B A

~ ω

B

= q˙1~b,

B

~ ω

D

A C

= q˙2~c , A

C C

D

B ~ ω

D

A

D ~ = −q˙3d.

B

C

C

B

D

C

~v A

A

B

~v = t

~v + t

A

A

B

~ω B × ~v

~ω B

B

A

K ~2 + I3 ω3~k3 ~ = I1 ω1~k1 + I2 ω2k H...