Formulario Dinámica I y II

Vista previa del material en texto

Euro Casanova / Dpto. de Mecánica / Universidad Simón Bolívar. 
Formulario Dinámica I & Dinámica II 
 
Cinemática 
Coordenadas cartesianas kzjyixri ˆˆˆ ++=
r 
kzjyixa
kzjyixv
i
i
ˆˆˆ
ˆˆˆ
&&&&&&
r
&&&
r
++=
++=
 
 
Coordenadas cilíndricas kzrr ri ˆˆ += ε
r 
( ) ( ) kzrrrra
kzrrv
ri
ri
ˆˆ2ˆ
ˆˆˆ
2 &&&&&&&&&
r
&&&
r
+++−=
++=
θ
θ
εθθεθ
εθε
 
 
Coordenadas intrínsecas ( )( )tii Srr rr = 
( )
( ) n
t
tti
tti
e
S
eSa
eSv
ˆˆ
ˆ
2
)(
ρ
&
&&r
&r
+=
=
 
Sistemas no inerciales 
( )
).(2
2
ˆˆ
ˆˆ
ˆˆ
;;;
2 uniplanarmovAVRRaa
AVRRaa
VRvv
K
dt
Kd
J
dt
Jd
I
dt
Id
dt
d
t
RR
dt
RdRrr
RiRiiipi
RiRiiipi
Riipi
i
i
i
ipi
⇔+×+−×+=
+×+××+×+=
+×+=
×=
×=
×=
=+×=+=
rrrrrrrr
rrrrrrrrrr
rrrrr
r
r
r
r
r
r
rr
r
rrr
ωωα
ωωωα
ω
ω
ω
ω
ωα
δ
δω
 
 
 
Dinámica de sistemas de partículas 
 
ii
N
i
ii
N
i
i
N
i
PiPCM
N
i
ii
N
i
i
N
i
ii
CM
N
i
ii
CM
N
i
ii
CM
N
i
i
vmRpRhhvMvmpp
M
am
a
M
vm
v
M
Rm
RmM
rrrrrrrrrr
r
r
r
r
r
r
×=×=====
====
∑∑∑∑∑
∑∑∑
∑
=====
===
=
11111
111
1 
 
).(;
1
2
11
uniplanarmovRmIaMRIM
vMv
td
hdMamRM
td
pdaMamF
N
i
iiPPCMP
E
P
CMP
PE
P
N
i
iii
E
PCM
N
i
ii
E
⇔=×+=
×+=×====
∑
∑∑
=
==
rrrr
rr
r
rrrr
r
rrr
α
 
 
Euro Casanova / Dpto. de Mecánica / Universidad Simón Bolívar. 
( ) ( ) ( )
( )
2
1
2
121
1
2
121
11222121
2
1;; kxU
mghU
rdFW
rdFW
UTUTWW elast
ref
grav
N
i
i
I
i
I
N
i
i
E
i
ncE
IncE
=
=
=
=
+−+=+
∑∫
∑∫
=
→
=
→
→→ r
o
r
r
o
r
 
( ) ).(
2
1
2
1
2
1
22
1
2
1
uniplanarmovRMvIMvT
vmTT
CMPPP
N
i
ii
N
i
i
⇔×++=
== ∑∑
==
rv
o
r ωω
 
 
Dinámica del Cuerpo Rígido 
 
ω
rr
rrrr
rrrr
rr
PXYZP
CMCMCMP
PCMPP
CM HvMRHh
vMRHh
vMp I=
×+=
×+=
= 
).(
;;
uniplanarmov
aMRM
aMRM
aMRM
aMRM
vMv
td
hdM
td
pdaMF
CMCMCM
E
P
PCMP
E
P
CMCMCMXYZCMXYZ
E
P
PCMPXYZPXYZ
E
P
CMP
PE
PCM
E
⇔
×+=
×+=
×+×+=
×+×+=
×+===
rrrr
rrrr
rrrrrr
rrrrrr
rr
r
rrrr
α
α
ωωα
ωωα
I
I
II
II
 
TT
PXYZ
T
ZYXP
CMCM
CMCMCMCM
CMCMCMCMCMCM
PXYZCMPXYZCMCMXYZPXYZ
KJIKJI
YXSym
ZYZX
ZXYXZY
M
′′′==










+
−+
−−+
=+=
′′′
ˆˆˆˆˆˆ;
;
22
22
22
//
LLILI
IIII
 
( ) ( ) ( ) ( )
( )
( ) ).(
2
1
2
1
2
1
2
1
2
1;;
22
2
2
1
2
121112221
uniplanarmovRMvIMvT
RMvMvT
kxU
mghU
rdFWUTUTW
CMPPP
CMPPXYZP
elast
CMref
gravN
i
i
E
i
ncEncE
⇔×++=
×++=
=
=
=+−+= ∑∫
=
→→
rv
o
r
rv
o
rr
o
r
r
o
r
ωω
ωωω I 
 
Ecuaciones de Lagrange 
( )
2
1 1
,,1
)(
21
2
1;;
,,,
xCDrFqQW
q
D
q
U
q
T
q
T
dt
dQ
qqqrr
N
i
N
i
iijjNj
jjjj
nc
j
Nii
&
r
o
r
&&
L
rr
L ===
∂
∂
+
∂
∂
+
∂
∂
−







∂
∂
=
=
∑ ∑
= =
= δδδ
 
 
Teoría de choques 
 
.....
.....
;;
ILcontactoenptoslosdeonaproximacirelVel
ILcontactoenptoslosdeoalejamientrelVel
ehp P
E
P
E
o
orrrr
=∆=∆= MF