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Unformatted text preview: ME?“ 634;. HW 11>? So [ddCums PROELEM 9.3
Part a Longitudlnsl dllplacaucntt on tho rod Iatialy thc wave (quarto. 2 2  ‘
DuEumdchuunu‘fKgi. (I)
ac2 a 2 x "51919,!rlzer6(x52)  Rc[3(x)05uzl to; aghusotdal cxcltaztonl. ﬁgs!) :19 ha vfﬁ5jen is 3(x) 9 Ctstn ﬁx + 929q5 ﬁg Uherc 8 §HQJB71L The two conniantl It.
lound (tom ch. boundary condition , ,
‘ n 3—3 (m)   NHL!) + Ht) ' Ch)
2 3t ,
50".!) " 0 (c)
Than conditions bacon.
~Hu2 Eu.)   AK f}: m + to (d)
and
3(0)  o (a)
for tinusoidll cxcttnttons. “
Now we find C2  O and
l
c I ———°—~____ .
1 AE3co¢31~ .Vuzllnal, a)
Hancc.
60m)  ——'&————_ mu .3“) .
“swanmanna: " °  m
and
T‘x't) . E g% . W Ref! .30“) AEBcosSlHu’ttnﬁl
Part b
A: x  l.
6(£.:)  ——L— net: .3‘“) '
AEScotﬁlHuz ° (t) wherescotsl  wJE7§ cot (ml/07:);
For "all u. COMM/(V?) ‘ l a
tale l— ((t) A! 2 (J)
'i ’ H“ . and 6(l.t)  p, PROBLEH 9.) (Continued) This cqulrion 1151 uqod to describe a has: on tho Ind o! I nasalor spring: 2
d
No ;:;   K: + {(r) (R)
and '
x  Ic(§05ucl.'
 H uzi   X; + f
o .' ' (I)
or
x  ——1—,. an m
KH u'
o
Coupring (J) and (L) we note that
.. A! . '
x t and to t. (n) Our comparison is cemplctr and sinco H >> 0A! we can us. tho narrlcss aprin; nodal
with $.noli No  H on the 0nd. ‘M nonim 9 .5
A rclponso tho: can be rcprclcntcd purely ll l vnvo trnvoiin; in tho negativ‘ x diroczion implies tho: thorn ho no vsvo rofloction at tho lollhand boundary.
U. nus: hove l
«0.0 + — No.0  0
If m or seen in Soc. 9.1.1b. This condition can to ouristiod by a viscous dopor alone: “10.0 + lv(o.:)  o ' (b)
Honco, we con write a  Alp? H  o (C) K  0. “031.84 a. 2’:
This problo lakes tho nan. point on Proba. 8.16 and 8.17. with tho
ndditionnl effect 0! material notion included. Regardlesl of tho nation. with the curron: constrained an given. the nagnotic ficld intonaity is sore
to the right of tho block and uniforl into rho papor (3 direction) :9 tho 10!: of the block. uhero   9
"1"1 , V (a)
The only contribution to an intgrarion of the otroan tonsor ovor I ourfnco enclosing the block in on ch. 11!: surlaco. Thus I 2 '
(x f d: fxx   d1 5 no": (b)
2
d: Io . ( )
"'z"“o(r) ‘ Tho nagnetic fore. in to the right Ind indoptndon: of tho Iaznncio layuoldl
number. PROBLE‘ 9.6 , I _.. . . .  tint. we can celculete the force a! umetic origin. 1". on the rod. I!
we deiine 6(l.t) to he the e.c. deflection of the rod et 3  i. then uein; Anpete'e leu end the Maxwell etreee teneor (tq. 8.5.‘1 with negnetoetrictiol ignored) we find
Hokﬂzlz ( . (I)
‘ 2(45(z.¢)15 1hie result cen eleo he obtained uein; the energy nethode 9! Chap. 3 (See
Appendix E. Table3.1). Since d >> 6(l.t). we nay linearize Ix: voluzx’ quﬂzlz  .
(x I 2‘: + T 50.0 0) The first tern represente e constant force which is balanced by e stetic deflection
on the rod. If He eaeue that thin etatlc deflection is included in the
equilibrium length t, then we need only use the last tern of I‘ to conpute the dyneic deflection “2.0. In the bulk of the rod we have the veve equetion:
to: sinusoidal variations 6(x.t)  n.(3(x)&3“‘1 (c)
we can write the conplex eplitude‘ZH) u 3(x)  C1 ein Bx + e, cos 8: ' ' . (d) where 8 u%. At I  0 we have etfixed end. eo 6(a)  0 end 62 5 0. At I I l
the boundary condition in e 38 '
I O  I:  A! 5; (Mt). . (C)
o: 2 2. ,
ll N3 !
o * d3
0  J 6(xl)  M: a: (P1) ' (I) d PROM)?! 9.6 (Continued) Substitutin; vie obtain ' ' u a;:311
° c sin e:  c Asa cos Bl _  t.)
d! 1 1 .
Our solution is 30:)  cl ein 8x and for s nontrivial solution we trust have C1 5‘ 0. So, divide (g) by C1 and obtain the resonsnce condition! u A8212
° 3 J sin 61  ans eos a: (h)
d
Substituting 3 II «mfgsud restun'ing. 9e heve
J ,
is? (thgj I tsnOnlJE) (1)
U N'I‘l
o vhich. when solved (or u. yields the eigenfrequencies. firsphicslly. the first
We eigenfrequencies ere found [ton the sketch. . the slope of the straight line decreases Notice that es the current I is increased
end the first eigenfrequency (denoted by ml) goes to zero and then seemingly “appears for still higher currents. Attuslly m1 now bacones innEiner and an
be found from the equation 3 {gm1124355)  my. (mind?) not! (1) Just as there are negative solutions to (i). w 1. an: .. etc., so there en now
solutions 3 Hull. Thus. because «)1 is bassinety. the systel is unstsble, (smplitude of one solution eraring in‘ tile). Hence when the slope at the "night line becomes less than unity. the sync.
ie unstable. This condition an be stated as: J
STABLE ———> E: , > 1 (k)
u N‘I‘J
O
or
ad’  1)
UKSTABLZ ———+ 2 3 < l . Q "on X j PRGELEM 0.2“
Part I The (tn and at x  0 input: that mm  0 and using equations 9.1.21 through 9.1.:ﬁ we can onuily find that veiocitv pulse. "ounce of!” x I 0
boundarv with the same sign and magnitude. the value! for v(x.t): 7or the xt plane no can indicate Part 5 u. can make use of part (a) if we use superposition. Consider the super.
position oi boundarv and initial conditions: e free end. 1(0.t)  0 91th the
initial conditions in part (a) and the ?(D.t) 1: Shaun in '13. ﬂ.P20b with
initial conditions on T and v zero. Since the syste in linear, we can add
the velocities the: result tron the tuo situations end thus have the not
velocity. For the response to the second act of conditions we have 1 0/4, Add this velocity set to the net in putt (I) and we obtein: 11 "mu; 0.:3 Pair 3 mu par: 's xlilar to Hal». “.M ulzh tvo'unnltllcuimn
V0  0 1rd :hc mu 1: Wu: 1: width (My) Instead of 2:1. The No "Mutt uhle yinldln; :hc natural (rc uenctu an "3"
1a(uL «453). n ' (ﬂ)
and
20 .
~45?»  nu(u!. 0" V42 (1) yields UL W?  9 when n  l. 2. ... and corresponds to ulucim which an "odd". or ﬁx)   Efx). 0:) can In solvul graphically and correspond:
to aoluzlons vhtch an "even". or {.(x)  ((3). I Part h The effect of taking ‘3 u to reduce the unenfrcuucnctu of tho.— "evnn"
mom's. The "odd" loluttonu predicted hy (s) an: Sudpanda: of the mu :4.
This» is physically. réasonahlc sine: than 13 a node at (ha WI, and “net
the man doom': lovq than 1: no Mann! force. For do. “oven” uoluums
prodicua by (h). u. node: that 1!."  0 we have annually the {annual
frequencies of a umbranc of length 21.. A: H  . tho syn: responds 11k. two “Hugs; ncuhnnu o! lcnxth L. Thu infinite nu net: 111:. c
rigid boundary. .
. ...
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 Fall '08
 WonjongKim

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