Solutions Manual To Accompany
R.W. Clough and J. Penzien
D·YNAMICS OF STRUCTURES .
"."
Second Edition
Francisco Medina University of Puerto Rico, Mayaguez
McGraw-Hili, Inc. New York SI. Louis Son Francisco Auckland Bogota eOlacos Lisbon london Madrid Mexico City Mijon Monlreol New Delhi Sydr,·?y Tokyo TOlOnlo Son Juan Singapore
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Solutions Manual to Accompany Clough/Peiwen
DYNAMICS OF STRUCTURES -
Second Edition
Copyright 0 1995 by McGraw-Hili, Inc. All rights reserved. Printed in the United States of Americo. The contents. or parts thereof. may be reproduced for use with DYNAMICS OF STRUCTURES - Second Edition Joseph Penzien Ray W. Clough provided such reproductions beor copyright notice. but may not be reproduced in any form for any other purpose wifhout permission of the publisher. ISBN 0-07-011395·5 1 2 3 4 5 6 7 8 9 0 BKM BKM 9 0 9 8 7 6 5
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PREFACE
This Solutions Manual contains the solution to all the problems proposed in the text Dynamics of Structures. To solve many of these problems, the student should be very familiar with trigonometric and hyperbolic relationships, integral and differential calculus, matrix algebra, and matrix structural analysis. The student also ~as to have some knowledge of differential equations, and probability and theory of residues (these last two topics for Part IV). The problems were solved from the first edition of the book, systematically and for no particular reason, when I was pursuing graduate studies at Berkeley in the late 70s, without haying in mind that these solutions would later be assembled in a Solutions Manual. Therefore, the way the solutions are carried out may not be optimally didactic.or elegant; however, they present the advantage of showing a step-by-step procedure that a regular student would perform in attempting to solve these problems in a homework or test framework, or simply just for fun. All the problems can be solved by hand, but in a small number of cases it was preferred to present a solution carried out with the help of a computer progra·m or a programmable calculator. Some of the problems may be solved differently (especially in Part IV); however, the answers are correct, and almost all have been independently checked. In order to prepare the Solutions Manual, the existing solutions for the first edition of the book were revised to C!mform to the new edition, and then transcribed into the manual's current format. This procedure was checked carefully. Nevertheless, if the reader notic~s any error, I would be pleased to be advised. Finally, I wish to thank the book's authors for allowing me to have an enjoyable time while studying in Berkeley. The transcription of the solutions was patiently done by Mr. Jose Orozco, who deserves to be given all the credit for it. Francisco Medina Mayagiiez, December 1994
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+ 'NL ]/(.E) [( ~_4\)L20 .2. 1\ .2.1Trt j
Problem 8-3 LiZ
1<:
ret>
z
*
L..
(L t::=:::;~=R=!J=io=:=1'l1=Q=56=l:;e=ss=b=a=r;:::::====1 \}/z •
-<.11;pliCQl flal"
"t=
p(t I
is
I
I
I I.
coso<:::
-mass area.
Problem 8-3 (con'd) =
tr1
r
7T (L)(,c/2)
8
4
I
o
=
7r L
Z
8
r
= 7T L Z
r
LZ
torclZ.<; ac.f:i'j over -t~Q
+
(<:'/02)
"
ICo
5!f <;i ern :
pet- )
5s" -k. -U("-,tl= _k ~ <'(-1:)= _ 3 1c..~U) Sn =- c -0(-L/Z , t) = - C ~(-t) = -!.. c ~(tl 2.
3L/Z
-L/2
31../2
J~lJ
= -111
-i.i(-3.L/Z,
-l)=- r
3
7r(L)(J..JZ).
4
+ ~
C ~U)
~U) =0'
3.1.../2
+ ~L k ~(-n
(co"tinuqd on
-31../2
=
~L pet)
Solh:oi '9 r
P:J~)
7TL<
8
~(-I:)
Probkl71 8- 3 (con'cl)
T1 e
J(amino'1':S princip11l l
m· ~(i:) + co¥. ~(-I:)
+ k'" ~(t)
=:
-n'J~ ==jl..",C;C) [$U(.d"dx-m'"
C'i'
to¥: '-
=fL-rnm [f{):ltJ~ I...
== [ 0 =0
cex.)
+
]<: [PCX) J):. +
(\Od[P(:t>J~d"
j
+
o
Jir:;t
,"'(t)
+
L1·Tf1i
"1'/
oj
QZ,.'
+
5
8-
L i IoJ~·lt
Li (JOt + TT1i"'~) (1t:.J1. L
Q11.E.ICX)
[p'C):)]
"
d;x.
[L EI(:l:)[~'~x)tcix
+ ~i ci ~<:
+
E. Tw.2. f,
t
I
I(t
0
-J .
<-
L.
A!ex)[ Y"'cd .dJ<.
o
f~ru
==
/1.
('(X,t.) Y'(:I:)
dx +
L i F/I:) 5t;,(x)
o
=:
111
[\l'C-3L/2)f +I o [t'C-3{./.2)]'= 1Tl'1el)+51TL'i-1'(_f_)~ 8
1491f L~t I\5Z
or
[ 1. + m .j<
111
1497f L < 1/52.
=
c'" == c
'k... =o
r" U:) ~ m" =
=
If' (31.../ 2 )1 =(S7T L+ r 5
¥
[svc-L/zr"f'
)r
J.<.
= c (1/3)'= C/9 =
1cC Z /3)Z=4k/7
f' (t) $V ( 3 L/Z) =- ,P U)( I )
1497T L 115Z
~
(-31.../:2. ),] [
k"'= k [5l'c L 2
5/2.
0' , c"': ~ , 1<.~ = 41: , F''' (t) = f' (·n '3
1
+
1fL2. &
r.
3L/Z
2J( I
9L ~
I 31.../2-
)Zo
Pro bh11l 8-3 (co,,'d )
'--''''~-I )<.
S \Alp
t) =
= p(.1:) SV(31../Z
I
, . , ( .... Ofltl'i·.(1?o
I
on
r
-+0
J
no1Vlnq . ~
P ({) I
p~JQ )
'-
3'-/2 31../2
8;Z
Problem 8-4 (coh'd)
.
Toec~s art;"5 over t ~e bay-:
F= - [
~
~ ( -!o)
1"11
Jo =- cV
(3L, 1: )
1rAB:::: - l A"
11AB =o_(3 _
p(-!:l
l.: ~A =- 0:
flF
=0
F (3l...)
=0
-
[i
-m
=0
k Z (-0 ]
-
j3 '" - [1
1Tl
4
t-
L"-+
C
0
~ (-/:)
+m
lhnL"-) -r
(3L/Z)"]
ictJ 3L
~U) + k ~(t-) Je3L)
~fO=JD(3L)=O-C
0= -
+
~7nLi(t)-31
p(i) L2.
i<-t) 31.-
Prob1nn 8-4
I"
21.-
L
)I.
31~
L
)I
CD
T oq"1 ma"", = ,." (tmiJoTm
loadl l."j~~
OVal"
arQQ )
~)
I h~>:hT15i b1l!.
;r;;
~~;d unijorm bar
/ool..-M5s=m
cab1~
mossl"ss
1J(X,t)= F
"\1&
1\
VCl<.t I
0<
11
So
~({-)
L' I =m_ o
1<-
~ ~(-l:) 3L
-V(3L,t) =. L/:<
.3- i((1:) t...
I~=.,.." C3L)~= 3t'nL" rz. 4-
A
1----....",.:
(+) =
. d~ 1=1"11o g
lrs
I t- is Q5SlAmeci t~~ coblt on1; tron511lits cotnrre",sio'1 .
• TOrce<. QctinJ
QVil-r
t.he cli~k: js = - k ~(t) M
II. = - 1 0 F=
(L/2)
= - k
7
,.
<;>l.
-rnl..zz" = - - - - ~(t)
8 ~(tl (L../2)
1..-
=_..!--mL~(t)
1F
"'t
=.
F (LIZ)
( conhnu«d on
jol1ow;'j I"'j')
Problem 8-5 L
IE
L
'IE
)IE
pit)
~/z
~l<'
~/Z
'I
0) l~=~~:
R. jid massl."" bay~jiJ ~njorm ba ... ( To1al lT1a ",,= m)
~etT pH: I
v,
t
lIYm
I
A
VI:
<'1..
.5s~
SSI
Vz :
1SO Tvz
~
>
Xl
acl i llj
):" ~(tl LiZ
.
TorclZ. S
~ !jet)
0'.=
~(()
.L/Z
ovo- i~'l ~j5hm :
pit) Jo5l = - k 1.11 (X I "=' L. If) = - k ~
Jsz =-k [VI 51) L
"=' -
~A = 0:
(X,"=' 2.l..,
t)
+ 1J~
Cj(-I:)
(X- z =.L/z
c ilz (;x.z =-L/Z, -l:) =-c
~ P = Fet)
lSI (L)
0=
(-I) tIt)
L
~ hI =
1f~z =].sz
,-oJ = -k[ ~ (t) +~(-:)J
= - k ~ J(+.)
(2L)
p(-i:\L ..
i
L
=-kCj(t)-t {(-fl]e,u.. !
k
!J(f:)L -
~k [3(tl+ ~(t\J L
(conhnued on fonowin3 paje)
0
Problem 8-5 (con'd)
f = .MI~.z. 11
J (L/Z) = - k
m= in
h-orn
8:
CD:
11)
~
. . 111
Fr obl
=
(-L/Z)
fr1.r = - I I 0 0=
[J (f) T ~(iJ]
5Z
.;<
=-
ZCC:)(-L../Z-)
C
!:-<. ii..aJ
171
f<'
0
L../Z
0)
-~ k (Y(-I:)+~(4:)1.z. - ~ CZ(-I:)L -~111iilt)L
j(t) =:
-
Z(i) +
;
Pt\
~
r-;-
f
6/
if F(i)+~ k ~(-t)+ Z C~J-i:) +b1nZ(t-) ~ 111, c'" ~ e,k ±k'F f' 0= - ;
1<.
ZC-t) +
~
=:
(LIZ)
-=
1< (t-)
=:
-= -
(-IJ
8-& L
et
'1
:z
L
111~=l-mp<)[)I'():)]dx=l-m[(Zt(~-Z:)Jd/'<
(8-14):
--- - - L
111;r=rrll [:t(zf- ~GJ + Hzt]J~ 't11~-=m[~(~L)-~(~L)+~(~L)]=:
k~ -= tk:l(]C)[f'(~>fdx
=
fLU [L f( ~r(~ -z:)}f
k"=:£I[L1-[I-z. x + (X)ZJd?
J..
"Tt-o
Ilt·
(8- B): F'4< (t) -= [ I
F(X' t) f
(L
p'tW=Plt)}o .. m""=(33/140)mL,
r
-mL
~:LIL[~2 -3 ~ fdx
9EI(L_L+!..I..)= 3£1 L'"
L
L lTl
'=-
=:
1:
(~)dx2
.3
_
L
3
1P(t)( ~r (J - fLJ d"l<-L
-
-"
-(
\
~L3(~) -(~tJdx= FzL(L-~L)= ~
k~-=3EI/J..~, p·U)= (3/B)LpU11
F(t)L
R-obk..n 8- 8 Co) +i.. ",~
0flev~.
(8-.20):
j1 t( 4
111-1<
=
o
m*=j! -mC"k'J) [YC>:'J)] b
Q
::>1'1
"If:
:5;'7
0
111~ = a jQ 5ih z
71/
r
"-
d>:.J
y
d"dj
If: dx lCl 5ih"7f/ dJ =1 (f.~Sin" -rr; dj"<-
~Y=1'(; [:it1"~
d[7fq
X
l)=o
1(;r[(:~- ~in~¥)[r
m~=r(;r(;r=r~"<-
DI l b{[
2
Q
o
0
0
fL'(x,y.) + 0""
a","
_ (--V) [ 2. I
'f/ (x, .
..
"w dX
_O_T_
. 7f
'" _
0"'1'
OXaj C}2
JP
.. - - -tdX2.
q
L{ ) j
ax"
7r)<.
Q .
7l~
c o 5 - 5111-·_
q
Cl
OJ "-
0" PCX,}) a"-JVc:JCIJ.) _(
= sin 70< .5 in .".J.
a
Y!C)<'Jl)] '1.
~
OJ"'-
a"-jP(X'.}J)2.]}dxd aXa)
'j.
k1'
l" Cal
0/< 0=-
=:
hOI"T1
..
ez·
k - kG
(B - 1" )
3€.I
L
3
12<- (8-14-)
~
N
5
L
--
(, =
k: k
oj:
"I
l
LN
(,-
~) :<[( ~r-e~r+ ~(~tJd~
L
0=-
==
9:<1 [(zr-G(~r+
~(~r-~(~rJd~ 9 tJ (~L - -.: L+ ..!. L__ L) = ~ rJ L<:3 42:4 8 L I
:2
.,
Prahl.,,, 8-5 (co.,·cl)
f
a.
sir,
~.
rr;t.
d
-0-
t
f dJ =/ o Q
a
COS
o
o
k
x
4-
~L- D (1"
, k«=7f4~1 a" -> ._.
2
!lJaclx
~-
0
.
x
'i-'r -r-t ;
:3 "/:3
-,-I
I--+-.t" 1'()C) -== I L
/
Concrete;
I
d=
~
O_ISO
;:. == 0.43(",
-+-- -- -~ - - -
1
v
/
z= 100
-++----1'--'--1-11-- u
>" 10'"
I !:. =100 <.
-I:
qt _~, p(xj
o
X: i'ex) ~ (r* _lOb) ~
me,,)=_ (' A (x) =
1
, I
! i
0
2iMo~oh'!> I
X"
11'
=
(1'-< - t'b) : - t 1'b
\5*)
(5-l- )
0
8_co"'' ' '
I
100
2
ZOO
t11"
=
I
R.uh: n'\i
21t-t r (
h=
=:.
J
(k;,....'«: Yft')
-
co-' (,,,,1
0.1<'0331
3. "'(,"'1
0.01151-8
mi:=
I-C05~ Zl..
rb
J[2.1f-t rex))~
ex) TT-l:r"3ex l[1+ [.....i-J}
Iu(x)=
-
--j-
11I
-==
3
fV/':=
"" tei= n-tt';"E G'SC~5~ Elie~"t €oIi::
1Tli yt"(,:;""50('/.r,') (k;",,:f-t»
(II}!')
(~if'5/fl')
0
o.<'~28q
1
-
o
0.011548
[~ex) VJ~x)ch- =jT~J[t I iCX)](I- cos ~~P;t: =ZlTfrb;(0.D4015Z + 0.18"003 ;J=4J;~::l
k== [£I(X) [)!J'b:) rd
X
==[-{
dEl1'"~(x)
+ ~\ex)lH(:Jcos:~
fdx
Probh,." 8 - II
tn, '11111/11 '1111
t:.
vex)
~_
PL" = 1< EI
I
r-- {~}..----' /
I
z:
(8-25):
I
L
Y'(.~) ~ ~il] 1
~
-
r:. r
\
/
III I II
qr,d
(3.-) "'J
lI(X,-t)=
.'. ]V(X) = 3
1/
Tro~,
L...
"
e
L
[3 ~ -
/ 1/ I I
of..:.
(B-Z~): Tma~= ~
ez.:( 8-IB):
[m c):) [~OC) reI",
111 4( =-
+
L 111; ~< + .E I Oi (~;. t
'- . '.'
4-
Tlnnk=i~~~~{~I m[3~-(~r] dx.+ I1nOk .
,[3(1)-1"rl
~ ~"/ {z mf'- [g( ~r- 0(-:j\ (~r1dx +~ mr} o
I 2 Tmo)<=~~kJ
T-rnox=
m
a[Zm-(
3L.- 5G:,
1
L +f"L. ' ) +4-m,
l ~2 -z,} [Z(::)fi1L+4-tn,)
FJ( (31.< - 4,,');\48£1),
o~X'; ~
vex) =
. { ::>Jmrn~1:"'i~ n~l«ct to
;!:=
z/<.
"{.,(8-.<5) X :. }V(x) .: .(.
f,o,"" Cz.-
l3- (\2.1 Z) r
lI1n~X
(B-27:):
"l-
=0
~
-I 1/
max
Z
=:>:0 £1 -<.....:
I
LIZ
x't
':
o
TlTlal' =-ZI Zo2 1V2
[
Z
5;ZG,
1,-)3
1 3\2.
Z
51"~d;t.=;z. £1 - - ( ' - ="0
2. m
(
L""
'-.:
3' -L - -24 , L -
8
5.32
L"
L-) t + -,e.-'<..8 f
1'Tl
~
I
] = -Z.I ~ tvZ ( -r'f tn_ i- + m ) (•) "7.2
35
I I
(Ql
~T=
7r
z
jl'fTTiL'" "flO'S E.. I
7f
r122 mL+
=YV'10 5
£I
lr
ProblOl1
8 -" (con' d)
L...
3(31-_ 3"5tnL
)
+ m1
T=2.7r
.c (-;;-mL +4ffiL)
Cal
(, ,E I
",EI
( b)
is
~tnL...=O .3
-no"",,;
I
mL.-+ 2'J . _ _
35
. ''1
T=
-;'7 col~tiTn5
and
I
£1
L 3(3+-/ 35 mt... + ml ) 2..7f
" lSI -
4mL
+
0<. T1lT
-mL1-' _
£1
2+0( :3
=<7{
/Zq
-. 35
Probl.m 3 -
/2.
" 'oJ
vz, = 2'/ 3
u.
(0) ( I ) T '=
I~·
Vi
-l L
r,
'I~'t]
'u/ r2
(o)
~ 'lowinq
=( ,
', L r ZO' 0S1f1W
(o)
(0/
= Z/3,
(j/
~' (8- 33)
, (0)
/3
= 1/3
(.J
-<:... =
,
J
£. 8- G. :
J
Co)
(0\ Tmol<.
I 2
=---l/J
T mal: (o)
'[
1 [ 'Ol]~ 7.. Z '-
= -
V1'nQ.'L= ~ <-0 I [
(0)
1
Z.'01J2
)CO;
U mOk
CO\]'
-u./
f' [, ( 01J2 17, ki lJ. y 1
Ti=
,=...!..[':7;o,I]' ~ 'r." 9.
C01J2(-I I I ') . K. 1fT Z k ""i+ 3k If
Z ~~ I [
(<<) ..
.
Loadi~ f~e ~~rv
. Fl
(0) -
7,,2 171' '-'V
_,,'01 =
IV.
7. (o) W 2 -rn' CU(D) ~
, rl'
by
O? ,
: 8-~;r)
:
Y/77/77i77i~---,
, I)
V I·'
-Tn
W'-
--
lU ('\
T;
:?,
3
=
I
L: lTJ··. 41(°'''11(1) 3
( b)
Uma ): = .-ZI
Umax
- (I) Z
1
,(0)-(,\
7 '" -; ",-0
Z?
I·,
1
1
= - I Z.{o)-tlJ 7 0 111 2 ~ - '.:-
1
(I + -._ Z
3
4-)
3+ 1 _._ 5 :3 15
G7_ 4-5' , ,
4
10
~(O)
2.8 k
70
T R,OI
~ T"iOl (C)
('\ T mOle
T
(Il 1>10 le
=
f =-l(}
2..
= _1
z.
1U
o. (;,81'
,,[-til] ~.
s~c I :z.
,,[t(f)]~
3
~.,.,..,.[UJ L "/1 Ii
'" o. ~89 5ee
:z.
]
25
<..
322
Z~5
I
, :z. &[;z(\J q 1(,) =-;;-"W "-... tl1 ( /+-+ __ I
;;,
1<)
~ Tf\1I
(I)
111
(:) .,
... <. ~II
(0'
Zo'
=--.
... T1>.11 =
2.ll
~35
c,+ 7-. =--
32<'
v
~r,,;
/10
;
i
r
1r)
= 0.G8') sec
:<25
Frob1....,.,
8-/3
T
(0\
mll"j<
= -I w 2. [ "Zo'0)]2. -20 'l
Z.
v(O) =_1 [ mC\x Z
~
111
;01]2. 1<.(..1+..1.-+-') 9
9
'J
ZU
<:.
k.
3
=-R.oo 20 m
., T.~oo =47r-~ y ;1::.- = ~
T 1\00 -
0.5'7'4
5
0.51'4- '::>"'~
•
. w (II = I rI J
UfO! = ~
IZ
l.1J(I~--,-
2 0 ' 13 - Z
-
'Z·
(II
20""
== .3 1<.
P,.-ob1~,.." 8-/3
( b)
(I \
;)rna -.l.
7U
(l)
rro"
(CO.,'J) I
4
=-z,~' 2. = -
I
4-
W
Z
(0)
'70
_
(,) [
m(l)
70
~-'-
(0)_ (I)
~o}(o
-
-
7q
30
3m
(~)(~)]
111 (0)
<. ZU
I 10 k =.zo - - -.200 -=.-_ - .zoo -= _ _
-ff/'
ROj
(cj
T
(I) 11101<
(I)
:0
.1-1
I
Tmo;Jc :::. Z kJ
[if. ('I]2.(rr, + 2m o
" [ -
~
\'
(I)J<'
Z89
.2 31
.23,
20 1t1
19
1T1
31.
+
3m
1.)
4
400
(~)
f.,39 zoom
;
z
~(O)
tUDJI = ~ ~
'\
?9 (eo) 3("3'1) .
20m
3"1<. , <. 79 k Z
T =:<7f ~II
.. TI\'I -
O. "32
",~c I
1"39 7-9
tn' = o. ~32 sac
k
Prob/un
10-1
kj
J
L
=-
1: IX) j I{
EI. I>.:)
I
II
("X) c1x.
•
j
k~3 =
l..
EL (x.)
~
If
(;>:)
~
1/ (;lee)
dX
o
£9- (10-/,-,6) :
-<--
f: ('):) = x (f _~)<-
k 23
=- /-2 E I. (_ Z L"
.. 4. 23
ProbllZ.111
""Z
--?
L
3
+
=-
5 Z
Cu"
I~ (X) .3
1 L
= -1- +- 02=.L~
+ ~ - ~) L =- _ 3
4
8El•
LZ-
10 - 2..
(10-28)
_._----------------'
~z-
(
10 - 34
f jL ;t
fi (f) = (-l: )
a) :
p~ U) ~. (10-.34):
=
(:t)
f (+)fL /((x) •
f(x,t)= X(X)jU)=
~. (:t) d;x. JO<.(X)
d;x.
F(Z +
:)~ilJwt
X(X)= <. [
.
~(X) =3(:r-Z(~r L
.. F«-I:)
=
FSiniJ-tj (2.+ o
F'" (of) = F
.
51n
-L (
"IV,
.<. -
,
4- --
5<.) '-- =.
<'9
I
-1- "
3 - - L_ 2/
L
=. -
F'-
<."'f -
-
5
No
.
-
1
511-) ""J"("
+ ):: <..
-+
.f (f) = F ~i'l wI.
P.-cblem
10-5
+
II
I
II 'I I
I' Go,
-~
2£1
- (,
Cc
1..3
31..
-31..
3L
-31..
=--
1~
k
~f
-
r
31..
31..
or
1),= I
--
~
k,,-+~ I
r--------,-------,
I I I I I
I
.
l~ ::".:J ... :~....-.)."?
~-
- ')
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10-5
(cotl'd)
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Pl
(-l: )
=
i
L
548L2.
F' (~ , -t) St;.
(lI:) ci x.
lAJ~QH
, i = 1,",:3.-+
z' (
e
(l(,
fO -Iha)
(IO-lh6)
ee.(JO -Ih<)
eZ'
11, = ,
r-->l r-"T-------.----.
I~I 'f,
:
pCf)=j5L")Ctl, 'I
I I
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If(
r
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(10 -
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PraLle,,-, :0-7 (con'd)
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I
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"-
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0')(" -
3
)
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I
FL )(-t)~ + B f
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x '''] ( :>:)~ - __ ",
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,
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-3
4 t'
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=-
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4
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(>':,-1) -
3 '2 I )
(-l: ) (-----P,-,(t ' I)?JL 3 .,.. \( r 4 3 Z -t I 1 \4
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o
3
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(X t t \=
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o
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[kitJ [kiil-]
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[k.]=-" 20- 7 [-10,£1
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11- t
.,
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(Q)
Z!~
Tro,.,..,
I
0
o
I
:
(.
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OIei'1h1 n a
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( 1/ - (., ) :
lI[k]- ",'[",]11 A _
•
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k,Z =-1<.
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t"
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A
, ==
0-411i, ;
r
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o
l!t"' ; j
VZI r ~
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4,5&4
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[0)° 0
A Z = 2,2'j: -!
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[
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A
V'Z =
1,
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1,2.9,
A
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-I
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o
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- I.
zq
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(c.)
To
l.~e oriJ-,-=U oI1O 7iiy ('ol1di{ions, t~e rnodlZ '::>hQFQS e~ 5, (II - 3&) a 11 d (II - 3:;) :
-to
[ -U m
f
[m H vn}
=
0
{vmf [k] {V.,L=
0
r
, 'W m
have.
W'l
5~nc,,- {ArCC]{5} ={5rCcf{A} and [c]=[Cr,jor t~o5(l.l'
i-t.;s 110-t l1~C~SSQt',!
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make.
CD -mo.iriX
T11ur!iFlica.hon:> .
.I
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T
m I {
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(con'eJ)
11- 1
0,;
-== m
-< {
1 0.584- 1 0.<..55>- { -
J.~9
}
=: 0.003881'11
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O;;-==1T1<::/,O ..584,0.255>
m.
O.23 -==
-rn
<" -
1.29 ,-0.952
>
-~.Z9
{
)
=
0.001(,4
rn
B.20
{
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8.Z0
Ok!
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O~l1 = f -Om k'
O'l.
=:;
k
0'3
=:.
< /t
<
r [k]{ -0~}
. 0.584,0.255>-
I, O.S84-, 0.255
k
> k
0.00191<,
k.
0.0/3<' k
Oz\=:< 1,-I.zq,-O.q5<-> k
okl
,,- z
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To170Wi~
P 11- I ,
n1
k 2m
k 3m
'k /
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r·-)
def,r -I L0 .:. w,
z
C~~A J -~ -I
Z-3A
] = - ( (,,)..2. -
/(" A
~+
IDA - 1 )
A=-W
2 111
T
-=- - c;, (A- 0, (:<3 r)(.:t- D,t580)(,H.t855)=oi
!
=
( 6) -I ;«1-)..)
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2-"'-" -
~:
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o}
Pmble.m 11-<.. (con 'd)
Jar
A, = 0.1<31 :
-I
o
/. 7'538
-I
-I
1.(,301
0.81 b 9 [
-1
o
]( V') VII
, --V2.1 3
(] C>
0
., , J or -J" = I , V21 = 0.81-" q , V3, = 0.53 i' iT
0
,
o. ?'586 :
;lz =
0.24:2.0 -1 0.484-0 [ -:-1
-I
o
J1V'~22z)
0
-1
-0.2740
<.132.
=
fO'1 Jar jo [0
,)1:<.=1,1)22= 0.2-120,V3Z = -0.883Z
)..3 = ,. 't855 : - 0·1855 [
-I
o
~ [V] [0.s'77-
0.24<-
:=
0.538
(c.)
-0.883
Ort~oq:J>-:Q1it~ wil~ '-
0,;
rfl5f
fa [1T11:
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[~ ~]f 0:"}~ ° ~ ]{-o~'"}. c [~ [~ ~]{ -0:8' }. - 00002'''0 0
=<
',0.817
7
0.538>
"Tn
2 0
0/;
0:" == {V.ny [111][-Jn }
:=
< 1, a. 8Tl',
0.538> TTl
-- 0.8&3
2 0
o~ = <.
OOOO.04m
3
O®%8m
:3
O.23"l
~
0.234-
Cl
: , 0.242., -O.883;>m
Z 0
ok! --
ProblQm 11-2. (con'd)
O.-lhC:J0r,c7ity W~:l ,.-e:sFec+ 10 [1
[ 0 '3k
= < I , 0.8:;'1, 0.53B /' 1<.
[
_I, - ~ o
-~I
0;" =c[v mr[kl[v,,}
_~]{ o. ~.qz} =
-1
Z
-I
O]!
-I
Z
Z -I
0.0004-15
k
-0.883
I
-0.1"80
0.234
lr =
0.000<10
k
j
f - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ...- - _ - l
Probkm
1/- 3
..-.l
,../1. ---I
rt- II!- ~..
r"' r-
1"11 L/2.
j =
111
--'h-
0.3
£I L/Z
bps/jt
-1 I
W",
3 KirS
v~,:.!.. Un Z 11 _ (I)L3
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L
.. [mJ= m4
-
[ :
1
[ f~] =
(.3 [4z
In £I
ZJ 5
-
L
3
lIz~: - -
48,,:r
+-I !I,l. 2
I
'£r"CII-/B):
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+ _ /,('2+50J '_ -VI Zj
4811
(V~(lr2
Q.=Z+~=2.+ (.<'13!<;r:> -4in LS (O."ldF/.fl) (loft)
:. ;\. =
ICo
:!: 8 =
8, 2.4
-'/z
5;nc~
_ IT5.8A
W,
= ~5,59 lj.:jcC , t'h =
"'Z.'" l/:sec B
15-A
... Jo>-
[
]{-V,} _{a} -Vl. -
A r = 204- : - 12-
8
to
-9
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jor
/\
vII :=
I
/0
I V2.1
= 2/3
J.. z = 8
[
4 .:;,
8 "1
]{ ~~:1
=[ :}
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Jar
~
V'Z =' 1 , Vn. = -
_,
~
~0;d sIal:,:
lola] mass, m
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horn Tj.
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rf
r
I
s
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--,
V, = I
~_ _~-=,::,-_ _..;I\
(VI = 1 )
~-\
"J\
'tnz I :: - !!.!
<-lr ~ 1 _----!I '. __ k T
~
k"" -
k
__ r:>. m/2.
k .._-J kll =4k_ _
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k
->fIr--
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tTlfl1:3: m 3
I
of
H . .
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k
,
.
kilT , kif
.
. '=
:<.K
=
TTl II L
L
11111 - m:z.1 Z Z
Vz = I : (ii, = I)
b
/ - ..... ~_m 5 1:.
.
-
~zl:-z1c
,------'----..,--7--7
\
\
lI t
t1<3'=0
IlL..
5
=-
12.
mL
ttTl 3Z ::O
t
k3 z.=o k. kzz.=3k , - - -...., E-'------, --'> --'> 1'T1zz:::: .!. m :5
",-\~mL ~--
-mlz
I..
k,~=-2k
! - ,- - - - - - - - - ' -
+k
-
,
Probkm
II - 4
(
co.,'d ) -n-J,2
+
=
1112.2
tn 2-
m,2 -L + m 22 -L = - -5 m L Z
Z
_
1J3
(V 3
~
I :
"
I)
t :----n.---i k
J
17133=m
'k33=3k. f,-----.-----;;.
I~3='
"32=0
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(U3= J)
1----
Ii?.
~
t
~
'tT13Z=O
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f1 -----
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k3 ,=o
k.
1 - - - - - - - - - 1 _ --i>
1<
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m31 =0
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12£ J
-20+)" 3-4A
o
-4-4 A
1< EI
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0
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0
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0
.3
-c,).
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v3
Preblem Tot"
AI
=
2. JOB . - I.
[
(con'J)
1/-4
0.47'3 :
521
/. 108
o
a
2.0
-1.5
-1.5
1.0
[
1)22
=
I
V".31 =
0
"
I
V 21
=
I. 37'9
-
0
A
o
1132
0
-0.508
- 0.8'f3
~
- 0.873
- 1.508
a
a
o
~ fw} =
A
VII
o-j!~lzl-!o]
o
o
o
[
for
-1.527
]! "1(}
0 jOt""
~13 V 23
- 3. 7'c>Z
=
A
"
V'3=1, V"3=-O.580
0
V33
0
{5.841~ ~
(.,
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mL
Probl~tT1 11-5
(el)
_ v, (ii,
= 1 = I)
;
--..jl~
,
I
k.
I I
I
-->11k-
--lIb--
,
1<
~k
31
I
k
11 = 3k
I
11
,
I
1<21
=
0
~
1<-
k
( cOhiinlJed on Jolla witlj p 3 e ) Q
l7]1I=m
I
k <:1 =-k.!:z
17121 = 1'11 31 = 0
R-o bletTl -
11- 5 (con' d)
V",- = 1 : (ii z = I )
lkZz. ~
~3z.f '~ k,2.
k
I -L
k
I
T
- V2! = I : (V 3 =1)
:'- ] , [ml=-rn L. 2. .
(6)
=
lZ£l /3 ~
Jd
o
o
I::>0." _
1
L/Z
3- A
(1- ;
~)(L/zf
]
e \.: --'
,
= 0./67'
1-1). <
{!fi l
,
-rnL3
,. L.Z
V,
0
3-,)",
0
Vz
-z
0
L.-
.cor-
2:.10283
(
3
\( L
- {o} ,
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1J:3
"2/-)\ Z} -
Liz
,
0
0.3805
,
,.A
~
VII
=
A
fOr-
0
V"ZI
0
2.':'283
L.-12
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0
5
1-
= 0.31'11"
Al
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i!frl nu.
= 0./10'
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3- h
5inciZ
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~
0
A
= I , V3 1= -5.Z51jl
11 11
1)"1
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0
A
(92(',
V31
0
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0
L../Z
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V,<-
1\ ~
/.
o
0
0
Vzz
... Liz
0
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A3
=
1)3Z
=0
1\
l~f
0
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V'Z =
V ZZ
= I
0
= 3.228:
r
LIZ
- 0.228
0
o
-0.228
LI2
0
~
A
0
V'3 A
{w}=
- 4.38 (LIz. )'~
r4S3}~ O.'CO.
o.
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3
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V 23
0
.
,
0
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1J33
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1
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-
0
f
0
1
0
mL
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5.2 r.. -Z-
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0.4510 L
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~
In
2£1
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=-
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1q[3
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11,
3£I
4
4,]
=1 ~---r-
j,x = 1.Z iii (2 L)
11111
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=
3-
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=
I
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7Dzz
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m = ~L
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[~ ~ ]
11-'" (con'c!)
P.ro hI em
d~t [ ~z ~ 9£1
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[
9£1
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A" :
[-S'r [
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A
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1.~4857'
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1.00
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1.00
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o.zz.
V
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u'u !l>'
1.54 85'{'
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r
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r
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0.434 {11;I _ <1 'l1'1L
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rc9'
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J
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5.2.131 '"
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0. 05 94
{
234.3 24-15
~ ~ir'lJt J .
sit]" ( /2.1 t)
I
r I. 2<;'~ Sl~: ' ' ' ( I. 85.t)' 0 , 3
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f
V:=(t) =
L
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0.0594 Sit," (IZ.,t)
r
h=1
- O.OOIB"<'
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. ",~!. BOc_;-.1-' 0 · ...~ -e"'i - --:, ~1h . ~ (I :C. 1') I.J:l8~''1 'C
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11- I
m
I
k
m
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1<.
m
-k=k.
3
i(
m
__
'1
-I
I -I
0
0
0
.2 -I :2 -I
-I
-I
"2
0
k~,= 0 -n1
1<.
=
0
mI
///////////
'1'
-
rro..-n
.. t,' ~ r~ 'f "
l0;' O~"] 0.50
0.25
0.25
0.0"
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ep-~ =
m
¥
=
T
'f
T 111 _
=
[-03:"8 - r.2~75J
m
[b. 25
0.25J
0.25
'" ~ f}
0.3291-
O.34B-/k/m '
rl
. -0.238
0.539"
.0.0570
0.40<0
o. 190"
0.2<'1
0.1157
1.00
=
0.3294-
rz~1(l n va llA.fl Frob1e.m [k"'-r../mt]~=
t =
'<},
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0.25
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~
k[ 0."
1.00
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=
r.
034- /
kim'
4B
1.034-
0
0
P..-o61'ltn
14 -
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1
... 'f(1)
<.. \b ( 0)
~
t1l --
k
L-
P..-oblcu'l1
JanOW5
1.50
10.24-
10.50
5.24
4.15
3."8
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as
P 1"1 - I ,
k' = [ t
+
=
'teO)
1 -I
-~!!l_;
=-f!!21
['t(l)r
,!!:!,
cv (0)
l
'f JI-om
= _
P 14-1
bu~ u 5 i'j
171
'f
111
t(l)
T
1. K.
(0)
r;
.:,
: .:
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0.,,35
- 0·23100
0.545
0.057iO
0.409
0.1909
0.ZZ8
o. /158
,
zu=f[' {"'4,}
-
tn
'.oc5
l0- f
k /1
d C06
tl- == .E.2-
,,'
It---'\!
y
/
d.si'1 tl- ==
.k.z
51;;
R.'
=: -
0<
q
-.,..
0<.
==
.
~,
Qrc.s1'l-
.
Q
I
~I
o<-==-=~=~
i'-(~r
Q
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T == -'m Vo2. +...!... I 2. 2.
v=
k
...!... Z
1J2. Ie
;t.== deo,"" ~ ~.
.
:1: 0
j
0
II
a
== -:2
=
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(iT -o<)
a.
.
51'10( OC
26
.
.
b
+
:2
.
COSO
Q.
+ "2 COSo<
51 "0'0<.
.z 11·z = X. •
+J.·z
4
cos ex. -
d
cos
-e- 51 n ex
• 0<..
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+ (b)" -2 cos z
+(b\<'5inZo(~2.
+
2)
~2+bz • >. ~ V. = -=.:--,---
dsin-e-sino<
a 51n . 0( "2
coso< -
b.
+
51n ex
d 5;11 (-e- "';cx) = d sin -e-
jo = 2 "
= d cos-&- coso<.
+ -:2b
CO" - 0<.
Z
= -
0<'
0
• ,,(
0<
• "
\ 51Y1 0<.
.z -2 -a b51" . 0( 4
0<0<.
."
COSO(.O<
(a.)<'C06Zo
') + cos "ex =
Q
"+bz 4
4-
•
0<
<-
Probl~m
(con'd)
/(,-1
"dV
;:)~ '" k
. 1-(dT)_ OIT +~=_I ra6(a\b .. dt d~1 OIt, dZ' 3
2
Q.
.since
='-
(b)l: Q Z!
ii, (a~-:ln + 21.,iJ
2
)
(Ql: -,~n2
(0 ~
smQll;
.
+ k t~)~
F(-t)
(a2-tf)i,+{,jf +
f ~J
-~I~~
2
-
Zn
z
k{-~tt, = r(t)
'2.
:CJ
:Co, ~I ~
~
fa6(a +b )
2
2
0
~ + k(-~t~
=p(t)
Q
= p(.t)
/
a, a,; SFr;~ It~h5
I
I
T= Z m,
I
r- ~3
v, =
X z = ~,+ (a-l-fz).sin.t> ~
j.
.<
• ."2.
'l.
,2.
.2
2.
= x? + J? =~, + ~2. ':'in.(-3
"V<'.
COS'?'3
;< ~I
+
I
A:, ~ ~, ~
. Z
m.Vc
.
. . v, ={,
+
.
(Q-l-~z)
- (Q +,e.z)s';" t3~3
~
<
co:,
. .<
"..
•
.f.t,,_ + Z~2..5''lZ'' (Q+«) C05<"~3
<.z..2.
<:
'<'
+ Z<
c
=il+.tz5i,,~,,+(Q-l-{")COS~3~3
-+.!J.2. = ie COS.~_3
+R.z) c05f3
(Q
=
x?
•
v, + Z
-+ (Q+Z") ~'''' t3~3 - Z~".<>;n~3(Cl+t.. ;
.
C05Z. 3
<."
".
-l-<.(I~cSi"(3
(Cl +,(,,) C05{i{:!.
0; t," +
sinZ3 +
+~_~ + (Q+f".{-i.,3"
2trCCl+l:") COS(3i..
T = i 171,
i./ + ~ m 2 [ Z/ +
T ==- ~
+1-Ji~) ~ ," + ~ Tnz [:: i I~" .::rin{'.3 + -< ~I
tf,Qtl
(m I
t3
(a
-I-
~<) co!.Z__~ + N + (M~? )\:]
J
i
"l712tc~in~3 +m2.(at~.)coSt3(3
c1(0T\ -- -:. -. \ =
dI
oJ
1":1
7, ' . I
.. ~I
,.
-f
-m~ t, +
,
l'T1
2. [
~z sin (,3 + f."- C05Z3~3 + i< COSZ"?3 - (Cl + (. ;6;'1 t3 + (Q -I- t,,) Z.3 ~3! J C 05
;:;'T -.-:=. 1T'I:(
agv:
(.. I
9,
' ".
61tl
{z +
«
. , J'
( con{';~:I"C:
tiC' : _ ..,
f]
J
i
zt I
Problem
1(,--2
(con'd)
d: = 7l1:1.[~I(Qt{Z)C05Z3+ (Q+tzt~a] ~( d: ~ '" 1-l1:<'[~1 ~<)co5t3+ Z' ~z dial dZ:?
coSts -
(at
df.
t, (Q+ Z.2.)~in Z.3&
+ Z(C\+Z")~<.t.3 + (a+~
and aT = 0
a~1
-m,,[ ~I
:; =
aT 3 dt
Vk z
h"
[~1~" COS ~
" TIl" ..
Vl( I = X I =0
=0
t
COS z3 3
=0
ZJ -
+ (a+~z)~:] -
~, (Cl + z") 6inz3Z3]
C/o
,e" -
Jz '" - (
Q
:. V =~[I(~-qot+
+-
iz) cos ~3
1 kz~~
-7T')ZJ(q+~z)
CDSb
t'Jen
dV
--=0
,;) ~3
rz.~.(1('-15)
Problam 1(;,-.2 (coh'd)
111<
'tT1
(~<. r ~, ~3 COS~3+ ~, .5;ht3)_111C[(Cl+~<.)i: -~, t3COS{'3] + kcZ< -
Z{ (o.+tz)" i..3
+
1TlzJ cost3 = 0
Z(Cl+tz)~zi3 T[(Q+tz)~, + ~,tzJ COSZ3 - (Q+iZ)~'t3~in~>}
- "tl1<..[ ~, tz. C05t3 -
(q + Zz),t,
~3 ~ill.(3] + 7r12.j (C\ + ~<) ~ih~3
=0
,. =0
,, .
. t,j
= 1, 2, 3
~ (m, +m2)~, +mzq~3+kJ(~,-qo) = pet) 1112
iz t,
+ 1cz.tz +
=
m
= 0
-
51" 0(,::
3
. /3L)" ( €.'
..
. T ="2 mL
T:: ~ [ iH
m
- . _. . -.5L (::J:: d-i: Clnd
~J
Z
I - (~-' /3'-J2.
1-
3 (f!1/3 L j"
3] t,"
+
= ;: "-
.
(~1/31..)" Z!
{[5 -3(iJ!3L {)i,-(2/3L')g,
tn
10
[ I -
inLI- (~'/3L)") +[5-3(N34]t'(Z;9l)l,g~
C~I / 3L
n~
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oZ, -JE-!l:-.-",
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3L
[1-C.t)/31..)"-]'-
12
I
n
j
z
V '" T KVK
=
liZ.
Z
"-~,
~
av
--= a~,
i
Kl t
-) r2.L l
.
pm I :: (~"2
4L - c ~l
pet
::
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3L
] bi'
o
2.
-1 6~,
-
1
C(),npo,-;~ wi-l.~
:. Q, C'f'plllinq J
J
= .32
rz.t.
((G,
- ( ) F -{
H', (110-/5): Z.
-lie):
s ~(n ( = [~ p (t ) L - c,tJ b i I <5 'Nne = Q, Sz, -t Q" 62.z -t .. , -+
QIJ cS~/>J
Ct, .
L -
(dT) dT
d - . - - - -+ av- = Q I d-
al,
-{
at,
a~,
m ls- 3(,f,13L)'][ 1- U"!3L)2J~, -(Z/3L2)[I~(M3L)"]g"i./ -+ [ 5 - 3 CN3L)2)(z/9/) [I g~ GO
[
(J/9L'Jf.'~" -mo
Tn
[S-3(i J /3 LJ"]
1_
(~'/3L)"r
- (1/3L"-),g, i~ [1-ti'/3L)"]
[ )- (.e.'/3L)"]'
[5 - 3( ~1/3l.fJ[ 1- (t'/3L)"]
if - [I - (£'/3L)2J (~, i,2/3L') -+ [5-3(~/3L)2] (lfi,'l9L") [1-fl f /3L)"Y
iO
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if
~I
srnall:
Probkrn
110 - 4
bl.t
L~
Ie =-m IZ
6111 =<.
if
:0 - - _ _
.L../2
f.
,
T =
1
mL~
<
12
-2
1'.1
I -
but
V, =
~I + :<
V, = $<
(ztr
"' <
'-2
Froblllm
1" - 4-
or-->ImFihJ
(con'd)
~'vJnc =
f(-l:)
<5 ""Inc
Q,
b.t • aZ<: '
COtnf'QY;~ wit~
Ill.: ('''-lie):
=
,st, + Qz St~ t-- - + <{II ¢~'J
QI
= - C
Q<
=
tl
F (-0
l-(ClT) _ eJT a~i
d-!:
i, [,- (2Z t/L)"] + it~ (8/L2.) £1
111
[ I - (2~1/.l..)2.]"
3
•
111
[1-(.2~,IL..{H, +
3
+
aR.i
av
=
a~.i
Qi
2
-
k( 2(+202 )=-CQ,,-' [ 1-(2Z,/L)2] 2. + Iv.c "'-
111·" 2
co
Belli..
(g,t~/Lz)
[1-,(2Z,/L)')"
;! ~. ~ ~I ~mo i! ---.,... .tn"
-. 3
4
6(z - c~.!
t,
'k
+"5
~I
+
.z z'z,2.1 2.
~
""5 f (t) = -
0 C
.
2../ 5·
+T
jZl
6
tnall:
'l
c~ + Tl(t'=-F(-t)
aT a~2
- L (I J. ~ ~I
=Tn
2.'
d faT) =m : I" L / - ", d-i: \ d{"-I \ fo Z
I. )
+ T Z,,- + "8 Z.
---'r - ( - ,
eo, (1"-2.4):
ki; = "-~' I
"
!
II
EIex.)f[ (.tl'J Y/.
d+
/I
C<:)
I ...
f.
0
I
f.-
. •
oy(t+J-Z)d
~
..
:+)-1
1'J - ..:...=-
rJ L . I' . -;0. ():V~ :"".1
,
_ 1< -
(j-I)
'J
(HiIOt) ;' "
1tr J
0
I .. _ (Iti.l(ft)) .' £I
/{" -
2. 93 .. )
(itl)j X d;lO C1'1-1) U+ I) L . L
((+)+2)'-
L
.
T
J:t =fLEI(iJ-l)t, ;lOci-I) ,
'.'
" = (' )/'+I'Jii c I 1 t I IJ J ~ j
('aT) - ('" 8" 9 '= at3 =111£" -9z, + -Q2.+-
-d -
-
, .. ) + -;Zz...z<:>2 + --'" 8 z·
: f a;.'i,! -' /
i) .
L (itj-'J "
L (1+J+2l
£1
Frable".,
10-5
(cor;:d)
(Iti)(ltj) 00£1
J
I{ti -
"
1.j
i+j-I
L3
(Iti)(l+j) -
i+j +
IJ L
J
v(o,-i:)=o and v(o,t)=O,-fhl1X~ i5I1on~cll5Si~ toimFo5~ TIl5tric-fiolls,
I AI) 1', + .c---_ z. L '\.
av= 3 (£13 1...
aiz
av _ a~3-8 QJjo ( ItO -30) :
....
Q.t ( 110 - 3 I) :
~
(£1 4---9 - - Ai-) e.s IJ) "-z +2. (U e 3
L
g.
(£ ' IJJ (£1 tJ ) ,-3 __ 5 L/~1+2 9 /..3-T: ~z pi'"
Cj =
J
QIJ
+3Z
(it I)
p(X,.f:) ~o(x)dx = pCt)(1)
EI(x)
'l.3
1..."
I
{9 £1. "U L3 --=rLj{3
t5
== p(f)
,
1= I,Z,3
~/(XJ~"(XJd;J.< = 0
IJ
Ci\
=
pi -
4: c.j ~j
'"
F (I:)
J.~I·
to = 1, z., 3
Prohlem
(con'd)
(&-5
IOOB ''!20
.. -mL 2520
3<00
t,ff)
420
HO
3/5
~'(i)
3&0
315
5&0
f3 rtl
-4 3
+
zE.I 3
L
3
4 L
-4
- -IJ-
12
')
9
144
2101,.
5",0 3/5 3%
.NO
315 1'51;, 4<0
t~(i)
331;," 420 'j60
~lt)
1
= \ F(t)
I
Problan 1& - CO
XI = -(F\I+I\2.)5in~2~ X," - (1\.,+R~)C05~-9z I3Ql1 :>u~attac1 to lwo rotClttons
j,=
(1\,+F\,,)C05-e-,,~j,:-(f\.'+F\,,)sin-ez.-e.
- c.<3 rotates 1n -oz 1-e<>Fc-t to 0 - 5011 rotote5 ;n -&-, nrF"cl -1:0 cCi
I
- -52.m ,1\,Z
01 -
Problqm
(10-0
(co'1'd)
hi == .!f I
1(
V = mlj hi'
+hQ'l
V=m'j(RttRz)cos-9-z
..
ae:=-m'j(R,4R?;.sih1T2
= (fi', +- R::?) cos"tJz
av
Z
(el
). ft(/"-3g):
V==V(JlljzI""JC)-(A.ljl+Azfz+"'+AmIm) V
= t11lj ( R,
JI == .. If
~ I -Bj _.
+ ~z) COS-&-2
"2 tr
z =
0
== rnq(RI+fk)C05-e-~ - A, (Rdt·,- ~2-&,J ...J
i:/,<"",c
z. cC"-Gj-t--9z") -A\R. 1 =0 -mfR., .5
"
m I ( ~ I + R." {&-2. + ~
171 I ~~
(-ii; -t- 42.) - 171,J (" I + ~ 2. hi n -&" + A 11\" = 0
~ y{, f\'((1t+-&z) f\< + 1Jf, (~I+ ~2.)L~~ + ~ y1', ~~ (-8; +~z)'
-y(j (1\, t 1'\,,) 5il1-e-2.~
= 0
=0
111, (~l-t~2.)-ii-Z '2.-
m, (I<-, + lh)~L
I -&1 2N
.'. F =
A.
(d)
5.3
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lTo m e{. of mO'..ill.mlmt
7' (~I + Ih)
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.. ,
2.
I .2 --{},=
Z
2.
p.f:O, iT-&'=O. 2' 2.
0
HRI+Rz)
7(~t+
fh)
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.
COS
.
.51 n
-ez
-&-2
-5j
-5.3 ;Z(~I+Rz)
/0 j
--=--- ( r-
5.'3
6i 11 -&z
COS 11-2.
+A
+ A
.3
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" =-. 10 ----'.>- -e: = areca,; - /0 "IT, 11 r1' 2.
~
10
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PrDbl~m
,,, -
f
o
-----7' X
k
tn, , ~o~C11
(L)2
L
Z
(10)1 = Ie,,", +tn'lz =7?l,!Z:
..•
Xz
=tz. 5il1~:
~Xz=
j"-
= f.;e co:,; Z'
--7
.2.
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+
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iz 5in
L
z,-+ t.z.i, C05~
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J"-
L..
MeG =- 2' Z
= -
zt
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= f., + Zz. Zl I
h2. = - J
Z
I
+mI4=311lIL
T=~17I,L
= -
Z
jz= Zz co-; (, - i z t, 5';n
·2..2
=
moss
Z2.
I '2. 2.'2.) £'1 + 21'TJ2.(.e,,+z"-R..I
2.·z
cos Z, C05
Z,
Q,
( contInued
ot)
= Qz.
=0
jo/1 olUi r:J fa.Je)
Prohln'1
1(, -1'
(con'cl )
jor
Probln'1
"Fc>r-
i.::. 1,Z
'10- 8
Sma 11
amplitude.
{i ~.J' ~J ~j ~ ~
L
05C;
0,
..
3,e, + 1'l14~~
+
llo hons:
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Zi,
I
=0
2~1
cos
k {Z- =mz. j
f...i ~
I
(i,j=I,Z.)
... '
