@G ezne"*l- R Loh*g*lr

V

'llrrne de+i vah ve

.fi17' Yv-lo v tn6

of a vephnrtA '

'{', *.d

Ydahv e Jo

111, A

x'lz:

Concil*lohs dQrle /aa\lT)ryZ

/ dA \@)xlL

& rrvctFrYe.G.

-tJ/\

Fw lt-r's P*rPose' a Ynov* t'?

\gosr h on veclrlr f ri' r4erem ca

R = Irar,s lotron wlot^]Y ,

ut z om6u)'av \'.e'1" *

pa-dr'ctr- P t^ruld cu

A u.dL ; Cohsr"lere-o\ '

a.bbilaail7,

=O

(nx A

Yxlewd t'h'ese cohsr d eral-ior.u Jo t n cLucle Etn,u

,d VLu[onr

"fe-\orfean cA-

A Nttch rS hot rn€cesSo Y i)'^f {,*r-d

'referer,"e'

0ur

Cah

ruz-

?

12p orclrrl to rr{ex-er,c-

nr\z^xt + VJ +-z'tL

\iheru- L, 3 , T" orle un't} ve&u'c $- vt\onrer*' vaz'

/41\ +i + ht + 2 tL"* )*a''

No,"t 1;1/& do-r1Vahv e ? r ur t \h respu; {o hv*-o' *"/1-

\tc XY Z rreJe-'rer'"c-o- r

l" e\ =( :xi *yi +2?*) +('-a* )*r,

Lxt+U3+='t)

&d*

I ar\Lr/

: tcxl , 4= @*'i / # = urxl'c

"Jf

/el \xyz= W )"t

- lar \@ )ry,

fdl) |W )1-=

/z/),

d,y

dj

Lox 1zi* 3i *&)Ir

t^:X f

{ Ccrnh"m*-t1i

Rele}-i.tnsrup &haer"^ vb'lp-erhl.u- "&

a p&Lbi cl-r. +.^. cLfferertt rd"re\^cen '

J- Velo csh'el t P*1".1' P ?relohvs +" H^{ X)Z

0lArX wLJT' nre-terrenac-t c^^-tr- ) xrt^ Per'ttrel*

V /4\ V*'r z-) @)r'

t dz\tll-l

L* )n'7,

,+"= R+f

Vry, =

vry ,= [ + U),r-* dxr

vry, g* V*J, .oxf

te\ ) fdrtlM lrr, @ l*r^

)@

7

c

R-,

Ac"e-le rah o n

df+e-ernf

Q*tr,

o{ a pa'r}rc,t'- {otQor\enrer" eea .

= Vry>+ e+ tBxf o

x

=ffit,ril^yL W)r*= l+1r"",)l*,: W)=,t4 -rw

X\Z'/a

"ffuxil+ R+lz dJ' ;r-

,} E) XP.:__*@

I \ry\,e- {or, )

Iawv4\ .. 1

\arl /xy

."x k:l' (d) lxv

Vxy,

0xy,

uIa@

tlerren\t * ^t t^s.r'{'

Vrvf = Qxy,/X'fz

: W'v"RQ,Y,

mL-/

N ouc

/4r)W bv'

@ SujDshfuh \r^. o5o,,ne, Tel*'ka/\^a '"^ q O

Qv, t 6x'*U- +v

x?) + cc x I

' ouDx Vx'r= {-R+ hxf +oy-feJ

t$XL\iDXI)

o./o $u-[otv vel o *% ? 'xtz- x-e]'o!'f,ve {o XY z-

o,/nflu-\-r^v auld-ci""*-lt t^,, q +r^"- a4L ''se-\enr*'t-l^Cr3 =-o

tJ 2-

(al'*\tqa,r )ry-

[df\ + .axYl-M/Y'

oyl;- =

ndoJl^l* \D Y \z-

Z@Xv*y) = Co^t i o;i s acee)era')too vethx

A -{- tt}V'l.tz

tnx Vru-

SECTION 15'8 ACCELERATION OFA PARTICLEFOR DIFFERENTREFERENCES 757

A stationary truck is carrying a cockpit for a worker who repairs overhead fix-\rres.Ar\{r*trrs\xr\s\sNrr\rrB\.\13\,\se\useDrsrs\$a\tsrgu\tsslee\r% of '\.raillsec-*rfi q = 2 radlseg rerative to the truck. ar* aa is rotating

1 .anSular speed ar, of .2.radlsec with dr, = .8 rad/sec2 relative to DA. Cockpit

c is rotating re\ative to AB so as to a\ways keep {ne man upnght. whatare {neve\ocity and acce\emtion vectors of the man re\atile to the ground it c[ = 4s.and B = 3ff atthe instant of interest? Talce DA = 13 m.

Because of the ro\,ation of {ie cockpit C re\ative to arm AB to keepthe man vertical, clearly, each particle in that body including the man hasLhe same motion as point. B of arm AB. Therefore, we shall concentrate ourattention on thrs potnt

Fix xyz tcr arm DA.F\xX]Zto truck.

This situation is shown in Fis. 15.3i

Example 15.11

Figure 15.31. ry1 fixed to DA; XyZ fixed to truck.

{. Motion of B relati.ve to "ry2

p = 3(cos Bi - sin Fil = Z.SOi - 1.5j m

't.tbr

at=.2radlseca), = .8 rad./sec2

@z= .1 radlsecdz=.2radlsec2

Figure 15.30. Truck with moving cockpit

0r = .2 rad/sec

X di= 8radlsec2

0: = 1 rad/secd: = 2 rad/sec2

:-

SECTION 15'8 ACCELERATION OFA PARTICLEFOR DIFFERENTREFERENCES 757

A stationary truck is carrying a cockpit for a worker who repairs overhead fix-\rres.Ar\{r*trrs\xr\s\sNrr\rrB\.\13\,\se\useDrsrs\$a\tsrgu\tsslee\r% of '\.raillsec-*rfi q = 2 radlseg rerative to the truck. ar* aa is rotating

1 .anSular speed ar, of .2.radlsec with dr, = .8 rad/sec2 relative to DA. Cockpit

c is rotating re\ative to AB so as to a\ways keep {ne man upnght. whatare {neve\ocity and acce\emtion vectors of the man re\atile to the ground it c[ = 4s.and B = 3ff atthe instant of interest? Talce DA = 13 m.

Because of the ro\,ation of {ie cockpit C re\ative to arm AB to keepthe man vertical, clearly, each particle in that body including the man hasLhe same motion as point. B of arm AB. Therefore, we shall concentrate ourattention on thrs potnt

Fix xyz tcr arm DA.F\xX]Zto truck.

This situation is shown in Fis. 15.3i

Example 15.11

Figure 15.31. ry1 fixed to DA; XyZ fixed to truck.

{. Motion of B relati.ve to "ry2

p = 3(cos Bi - sin Fil = Z.SOi - 1.5j m

't.tbr

at=.2radlseca), = .8 rad./sec2

@z= .1 radlsecdz=.2radlsec2

Figure 15.30. Truck with moving cockpit

0r = .2 rad/sec

X di= 8radlsec2

0: = 1 rad/secd: = 2 rad/sec2

:-

764 cHAprER r5 KrNEr.,rAncs oF RIGrD BoDTES: RELATTvE MorIoN

Fxample 15.12 (Continued)

Aiso,

&xyz = axy- + i? + 2a x Vr. +<il x p + (l) x (to x p)

= -l0k - s00j + 2(5i - 10ft) x (-20k) + (-50i) x ft+ (5i - 10ft) x [(5' - i0ft) x r]

&xyz = -100i = 30Oj '' 35k ftlsecz

Example 15.13

In Example 15.12, the wheel accelerates at the instant under discussionwith r)t, = 5 rad/sec2, and the platform accelerates with rb, = l0 rad/sec2(see Fig. 15.32). Find the velocity and acceleration of the valve gate A.

If we review th€ contents of parts A and B of Example 15.12, it willbe ciear that only Ii and dr are affected by the fact that dt, = l0 rad/sec2and a, = 5 rad/sec2. In this regard, consider ror. It is no longer of con-stant value and cannot be considered as fixed in the platform. However wecan express tnzas o?i' at all times, wherein i' ts fixed in the platform as

shown in Fi-e. 15.32. Thus. we can say for o:

*or

Therefore, :. .et=a)2t'+a{'+an1= 5i' + 5(ro, x i') * 10k

= 5i' + 5(-10fr) x i' - l\kAt the instant of interest, i' = i. Hence,

-' ;5Oj: " -. 1A k. rad t sec2

Hence, we use the above<ir in part B of Example 15.12 to compute Yr,and a*rr. The computation of 1i is straightforward and so we can computear* accordingly. We leave the details to the reader.

l}

:

I

i,1

);i

dlil

,

angulprobltexamline it <

of theence k

r Exe

I TotI opeo

I lorn.

I auorr

I axrs s

I to sinI rusrj eanh

pilot s

as sho

Ipm aabout ,

cons|aofaccethe thn

3

C-

B

Er"::itions c:::t rno\e ile

.-.L a

: 9.E1 m.:

:

CHAPTER 15 KINEMATICS OFRIGID BODIES: RELATIVE MOTION

Example 1 5.14 (Continued)

In Fig. 15.34 the arm of the centrifuge rotates relative to the groundat an angular velocity of atr. The cockpit meanwhile rotates relative to thearm with angular speed alr. Finally, the seat rotates relative to the cockpitat an angular speed arri Eor constant a2, we see that, because of bearingsin the arm, the vector or,,is "fixed" in the arm. Also, for constant al,because of bearings in the Qockpit, the vector ro, is "fixed" in the cockpit.Before we examine the acceleration of the pilot's head, note that at theinstant of interest:

T-3',1

CT

@r = A2 = l0 rpm = 1.048 rad/sec

dr = 5rpm2 =.00873rad/sec2ro, = 5 rpm = .524 rad/sec

We shall do this problem using three different kinds of movingreferences xyz.

ANALYSIS I

'Fix.ryz,,to.arm.

Ftx XYZ, to ground.

Note in Fig. 15.35 that ryzand the arm ro which it is fixed are shown dark.Note also that the axes r_y-z and XYZ are parallel to each other at the instantof inierest.

Exi

, A.llo

Note i:of icrr-

I'i,:

i:

Cleil.,rh. --r|u ! la r*

a-

B. ]Iorr,.r

\ote !:.:io XIZ :.:

i

I

I

i

Figure 15.34. Centrifuge listing r,r's

7@ cHAprER t5 KINEMATICS oF RtcID BoDIES: RELATIvE MorIoN

Example 15.14 (Continued)Example 15.14 (Continuecl)

-

We can now substitute into the following equation:

axyz = aq, + h+ 2o X Vo, +'bx P+@ x ((,) xP)

Therefore,

axYZ = 3'64i + 50'5j - 4J2k ftlsecz

r | ,[i.A,+2+5o.52+4.122 - l57gglaxrzl= -----EZ :*a&-*L.

A. Motion of particle relative to ryz

P = 3kftNote that p is fixed to the seat, which has an angular velocity of o, rela-tive to the cockpit and thus relative to ryz. Hence,

Vr, = @3 x p = .524i x 3k = -1.572i ft/sec

/dro,) (dp\'"' = [7i ),,* o +

",3 x

lffi )r.

But co, is constant as seen from the cockpit and thus from.ryz. Hence,

Qr, = 0 x P + (o3 x V*y, = .524i x (-1.572i)

= -.824k ftlsecz

ANALYSN II

Figure 15.36. Centrifuge with rya fixed to cockpit.

766 cHAprER 15 KINEMATICs oF RIcID BoDIES: RELATTvE MorIoN

Example 15.14 (Continued)

A. Motion of particle relative to xyz

P = 3kft

Since the particle is fixed to the seat and is thus fixed in xyz, we can say:

vrr, = oorr, = o

B. Motion of xyz relative to XYZ. Again, the origin of ryz has identically

the same motion as in the previous analyses. Thus, we have the same

results as before for R and its derivatives.

R = -40,/ ftIt = 41.9i ft/sec

fr = $.9j + .349i ftlsec2o = @l +ro2 +or = 1.048ft +1.048j +'524itir=<lrr+d2+63

Note that rb, is given. Also, to, is fixed in the arm, which rotates withangular speed or, relative to XYZ. Finally. to, is fixed in the cockpit, which

has an angular velocity Gr2 + (o r relative to XYZ. Thus,

<b = (r)z +G)l x to, +(to, +ror) x ., V= .00873k + (1.048& x 1.0487) + (1.048t + 1.0487) x (.524i)

= -1.098i + .549j - 54Ak

We now go to the basic equation, 15.24.

Txyz = o.o, ! R + 2ut x Vo. + to x p + co x (to x p)

Substituting, we get

axyz= 3.64i + 50'5j - 4.12kft/secz

lorol = 1.5?8s

E

AX)A

i

i

I

I

IEFigthe

In the final example of this series, we have a case where it is adv

geous to use cllindrical coordinates in parts of the problem and then later

convert to rectangular coordinates.

Ar the.3 rad/rhorizon

Two spl*ith the

timeEB= m

;haageinstarr

----=lll-

D

768 cHAprER 15 KINEMATICS oF RIGID BoDIES: RELATIvE MorIoN

Exarnple 1 5.1 5 (Continued)

We will first consider the motion of the center of mass of the rotat-

ing spheres which clearly must be G. We now proceed to get the accelera-

tion of G relative to XYZ (see Fig. 15.39).

Sx:rYr'O*c vesscl dr{.Fir XI-Zo the gurd.

A. Motion of G relative to xyz (using cylindrical coordinates). Use Figs.

15.38 and 15.39.

p = .25er = .25j m

= iG, + r6e, + 2e- = -3er + (.25)(.3)e, +.5e.

' -3j - .015i + .5k = -.0'l5i -3i +.5Ir m/s

= (7 - r07)<, + (rg + 2f 0)er+ -te,

=l-2 -(.25)(.3)2]e, + [(.2s)(.1) + (2X-3X.3))e, + .2e

= -2.023er - 1.7e, + .2e- = l.7i - 2.023j + .2k mlsz

B. Motion of xye relative to XYZ

R = 3i + .67 rn/s

ii = 2i + 3J - .5k mlsz<o = .3k rad/s

to = 0 rad/s2

We may now express arrrfor point G' Thus

axyz =4.n,- + il * 2o x Y',. *6 x p+ Gr x (<o x p)

= (1.]i - 2.023j + .2k) + (2i + 3i - .5k) +2(.3k)

x (-.075i -3j +.5/<) + 0 x p + (.3k) x (.3k x .25,r)

= 5.5i + .9095j - .3ft m/sl

Now we apply Newton's law to the mass center G at the instant of inter-

est. Denoting the force from the rod AB onto G as Fooo, we get

&.o, - 2mglc = 5.5i + .9095j - .3k

.'. &.oo = (2)(lx9.8l)k + 5.5i + .909si - .3k

Fnop = 5'5i.-+ 'g0g5i + 19'32* N' t'.

This is our desired result.

15.91

of 3 n

withoangulr

r:id /se

the ce

vnz

Figure 15.39. Reference xyz fixed tothe vessel.

{rFa

ryz

5

t -i.92.

':-:eJ ,-

-t _

---,... j

- :-::l:

-i ;-t. -