Mecanica de Solidos BUAP

7/24/2019 Mecanica de Solidos BUAP

1/9

dtpdF

=

LABEvC2EvC1MQEXfinalNOTA 1.01.01.01.06.0 ++++=


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

=


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= + +


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


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


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 )()( =


7/9

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

==

.


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 +=


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=

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