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7/24/2019 Mecanica de Solidos BUAP
1/9
dtpdF
=
LABEvC2EvC1MQEXfinalNOTA 1.01.01.01.06.0 ++++=
7/24/2019 Mecanica de Solidos BUAP
2/9
( )( ) ( ), ( ), ( )r t x t y t z t =
, de mdulo
( ), ,r x y z=kzjyixzyxr ),,( ++==
i j
k
222 zyxrr ++==
2 1 2 1 2 1 2 1( , , )r r r x x y y z z = =
2 2 2 2
2 2 2 2, ,
dv d r d x d y d za v r
dt dt dt dt dt
= = = = =
( )r t ( )v t
)(tv
zzyyxx CdttvCdttvCdttvCdttvtr +++=+= )(,)(,)()()(
===
=
dt
dz
dt
dy
dt
dxr
dt
rd
t
rv
t ,,lim 0
( , , )dr dx dy dz=
),0,0()( gta =
)1,0,2()0( ==tv
d
Ta v
Na n
a
( )r t
vdt
vd
0)(
2)(
2
===
dt
vd
dt
vdv
dt
vvd
( )
dt
vdv
dt
dvv
dt
vvd
dt
vdta
)( +===
n
v
a
vd)(tv)(tr
rd
)(tv
=
==
unitariongentevector ta
)()()(
v
vvtvtvtv
NT aandt
vdv
dt
dvvta
+=+=
)(
v v
dt
vdvRCURV
=
C
NR
vna
2
=
CURVCURV R
v
dt
rd
Rdt
d
dt
vd===
1
CURVRd
)( dttv +
ddvvd == CURVR
rdd
v
vd
==
1 =v
CURVRdRdrd CURV==
v
v
v
vtv
==)(
dt
vd
dt
vdtn
)( = dt
dvvaT =
NT
aaa
+=
C
NR
vn
dt
vdvna
2
==
)(tv
dt
vdvRC
=
7/24/2019 Mecanica de Solidos BUAP
3/9
.
pamF
==vmp
=
0=F
1F
2F
F
2m
1N 1kg s=
amF
=
1 2, ...F F
1 2, ...a a
1 2
1 2
...F F
ma a
= = =
( )r t
1 2
C C
0 0 0 0( ) ( )r r t v v t = =
0 1 2 0
0 1 2 0
( , , )
( , , )
r f C C t
fv C C t
t
=
=
0t
F
I
2
1
=
t
t
I Fdt1 2t t
=
I p
0 0, ,r t r r v t =1 2, 0 0,r v
)(tFF
=
2
2
1( ) ( ) ( )t t
d rm F t r t F t dt dt
dt m
= =
ctF
m a F am
= = =
2
1 2
1( )
2
Fr t C C t t
m= + +
0 0 0 0( ) , ( )r t r r t v= =
2
0 0 1 2 0 0 2 0 0
2
0 0 2 0 1 0 0 0 0
1( )
2
1( )
2
F Fr t r C C t t C v t
m m
F Fr t v C t C r v t t
m m
= = + + =
= = + = +
2
0 0 0 0
1( ) ( ) ( )
2
Fr t r v t t t t
m= + +
7/24/2019 Mecanica de Solidos BUAP
4/9
( , )x y
y
F
v
my
x
ctF
m a F mg a gm
= = = = =
vNF,NF DRDDRD ==
v RDF
N RDF
)(tv
t
0v
2
0 0 0 0
1( ) ( ) ( )
2
Fr t r v t t t t
m= + +
1.0=D
1
0 ms3 =v
)( NH llkF =
kxxmlmF === Nllx =
xm
kx =
0 0( ) sin( ( ) )x t A t t = +
00
( )sin
x t =
k
m=
2=T k
EtvtxA
2)()(
2
2
02
0 =+=
m
m0=HF
0=x
0>x
0HF0
7/24/2019 Mecanica de Solidos BUAP
5/9
( )
PY
2
1
( ) ( )= t
P P
t
Y M dt
1 2t t
( ) ( )=
P PY L
( )
PM
PL P = const)(
const)( =OL
const)( =CL
CF
0=F
BT
T
C
C
( ) ( )
( ) 0C C
C
dL dLM
dt dt = =
)(CL
rr
mmGF
GU
2
21=
rr
mmGFGU 2 21=
constL C =)(
a
b
)(CL
32
aT
7/24/2019 Mecanica de Solidos BUAP
6/9
dr
dtvrd =
dtW
t
tii
2
1= vF
ii
=
J1W 1s
= mNJ=
iF
iWiF
rdFW
r
r
ii
2
1:
=
1r
2r
iW
1r
2r
i
F
( )r
1 1 2 2( ) ( ( ), ( ), ( )), ( ) , ( )r x y z P r P r = = =
( )r
dr
1
2
iF
d
d
dzF
d
dyF
d
dxFd
d
rdFrdFW zyxi
r
r
ii 2
1
2
1
2
1:
++===
21
2c
E m v=cEmvW =
= 2
2
1
...... 2121 ++=++== dWdWrdFrdFrdFdW TOTTOT
=
=== 22
2
1
2
1mvddtmv
dt
ddtv
dt
vdmrdamrdFTOT
amFFFF TOT
=++== ...21
cEWW =++ ...21
( )F r
U
)( rU
== z
U
y
U
x
UUF ,,
=
+
+
=++== dUdz
z
Udy
y
Udx
x
UdzFdyFdxFrdFW zyx )(
F
[ ] UUUdUW === )1()2(2
1
21
W U=
( )F r
0cerradacurva = rdF
B
A rdF
ctdzFdyFdxFctrdFU zyx +++=+= )(
( )r
dr
1
2
iF
x y x yU U
dU F dr F dx F dy F F x y
= = = =
yx FFy x
=
xyU
yxU
=
22
ctdxxFxU += )()(
F
rr =
ctdrrFctrdrrFrU +=+= )()()(
rrFrF )()( =
7/24/2019 Mecanica de Solidos BUAP
7/9
cte
dU F dr
F
=
=
ctU F r= +
ctU m g y= +gmFg
=
21( ) ( ) ct
2
U x kx dx k x= = += kxFH
r
krUr
r
kF == )(
2
r
r
mmGFGU 2
21=
rr
qqFC
4
12
21
0=
21
2
1
2
1
21)( ====
NdNrdvNWvNF DDDDRDS
)(consconscons UEWWUE cNOFNOFFc +=+=
+++= rdFrdFrdFrdF n
...21
0
dE
dt=
ctEUEc ==+
...21 ++=== FRamFF TOT
rd
...21 ++= rdFrdRrdam
( ) 0...2 =++ rdamFF n
R
dtrdvR = rd
01 = rdR
F
F
N N
im
)(ext
iF
)(ext
iF
( )122121 // rrrF
=
jm
ijF
jiFim
int
ij
int
ji FF =
iii
j
int
ij
ext
ii pamFFF.
==+=
=====i
i
ext
TOT
i
ext
i
i
iTOT PpFFFF..
dt
PdFF
extext
TOT
==
0=extF
ctP=
=i
iiCM rm
MR
1=
i
imM
CMi i VMpP
== CMext AMPF
==
.
7/24/2019 Mecanica de Solidos BUAP
8/9
( ) ( ) ( )122
122122211 )(11
rrM
mrrmrmM
Mrmrm
MRCM
+=+=+=
12
2121 mMmmmM =+=1r
12 rr
CM
CMR
=+=+=i
iiQQ
i
iQi
i
iiQ
i
iii
i
iiQi
i
iiQi
QvmvMFrvmvvmvamrvmv
dt
Ld
)()()()(
)(
ext
QQ MM )()(
=
== iiiQi
iQiQ vmrLL
)()()(
0)( =
ext
QM
ctL Q =)(
ij
int
ji rF //
QiQi rrr
=)(
CMQ
ext
QQ VMvMdtLd
= )()(
dt
Ld
M Qext
Q
)(
)(
=
CMR
( )122111222112 rdrdFrdFrdFdW intintint
=+=
12 rrrrel
=
( ) ( ) dUrdrUrdrFdWrelrelrelrel
===
2112
ctUEE C =+= +=parejas
int
ij
i
ext
i UUU
)( UEW CCONSNOFext +=
7/24/2019 Mecanica de Solidos BUAP
9/9
Nt
IFmedia 1000=
=
sNvvmpI AD 8.4moduloen),(
==
1m
1( )iniv
1( )fiv
2m2( )iniv
2( )fiv
1m
2m
0extF
ctP=
AD PP
=
2,22
2
,11
2
,22
2
,112
1
2
1
2
1
2
1AADD
vmvmvmvm +=+
1
m
m
n
N
2,1,//
,
//
, == ivv AiDi
////
AD vv =
Av
Dv
Av ,1 Dv ,1
n
=
)(2)(1
)(2)(1
AA
DD
vv
vve
Av ,2 D
v ,2ctP=