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Unformatted text preview: 270 Chaoter 4 Higher—Order Linear Differential Equations (ii) When b3 < Emir, we have A’twf) m 0 when (of = 0, which now
gives a relative mininnnn, or when CU] 2": i"— w 33w, which gives the
.  ' ' "I 21,11 I I. . nrcttimnm of Ahoy). _ '  . . .
' '_ 7 _' _Su_bsti__tating the nonzero Critical valne into Atrof) gives themairiw ' mum amplitude, .. . '  ' ' =  ..  .  _. Amax b in m  "1 4mg.
Pe n d u I u ms In Figs 4.6.7 the black curves exhibit property (i) and the colored curves illustrate
' property (ii).
Chooseforced pendulum [053“ Under certain initial conditions, forced damped oscillators can also lead to
chaotic phase poman arise before chaotic motion2 We postpone discussion of this aspect to Sect 71.5 your eyes, __ "1e hernotion:qu a forcedmassspring system using a secondorder differential equation with constant  coefﬁcients W "bladedtaftypical real~world nonhomogeneous' forcing term'that is amenable to the method  ofgiiiideteinnned_ coefﬁcients; Ether: combined the techniques of Secs“ 4.2.4.4 to solve the resnlting initialvalue ' " o. _othfthi:idemped_'_and_undamped cases We introduced theconcepts of resonance, beats, transient, ._ "
"istate solutions orderth interpret the results” I I ' I I ' I 7 ' 4.6 Problems MassSpring Problems Find the position function x0) for 8. Puling Down A iiilb weight stretches a spring 1 ft to
each of the forced mass~5pring systems in Problems 1W6. equilibzinmi Then the weightstarts fiomrest 1 it below the
Find the amplitude and phase shift for x”, the steady—state equilibrium position. The damping force is 62'; An extcn
solution nal force of'ticos 4t is applied at! m 0‘ Find the position
function x0) for r > 0‘ 1. 171:1, b222, k=l, Fmdcost
9. MassSpring Again A mass of 100 kg is attached to a
2. m = 1, b m 2. k = 3, F n cos 3r long spring suspended from the ceiling When the mass comes to rest at equilibrium, the spring has been stretched 3 "I m 2’ b = Oi k 3 3: F = 4503 8’ 20 cm. The mass is then puiled down 40 cm below the
1 5 equilibrium point and released Ignore any damping or
4. m m , b = 2, k m 2—, F = Ecosr extetnal forces.
” I (:1) Verify that it n 4900 zit/in,
mmmi, b=2, k=2, F=2rzost I ,
(b) Solve for the motion of the mass
5. m = 11 b m 4, k = 5, F m 2 COS 3r (0) Find the amplitude and period ofthe motion
(d) Now add damping to the system with damping co
7. Pushing Up An 8lb weight stretches a spring 4/ 3 ft to efﬁcient given by b m 500 nt sec/m. Is the system
equilibrium. Then the Weight starts from rest 2 ft above underdamped, critically damped, oz overdamped?
' the equilibrium position. The damping force is 25x An (8) Sogve the damped System with the same Emma;
external force of Zoos 2t 15 apphed at r = 0 Find the conditions. position function xm f'ort > 0 1See 1. Hi Hubbard. “What it Means to Understand a Differential Equation,” College Mathematics Journal 25 no 5 (i994), 372684. 19‘ Adding Forcing Suppose the mass—spring system of 11. 12. 1.3. 14. 15. Probiein 9 {ie, mmEOO kg, k249th tit/m. bmSOO
nt sec/m) is forced by an oscillatory function f(t}m
IGOCOSLUIL {a} What value oi'wf will give the largest amplitude for
the steadystate solution? (in) Find the steadystate solution when (of x 7, ( c) Now consider the system with no damping, b = 0, and
any 2 7'. What is the form of the particular solution to
the systein? Do not solve for the constants. Electric Analog Using a tiohm resistor, construct an
[ARC—Circuit that is the analog of the mechanical system in
Problem 10, in the sense that the two systems are governed
by the same differential equation That is, what values for
L, C, and Wt) will give a multiple of the following? 100.? + 5001': + 4900.1: 2 100 cos (of: Damped Forced Motion I Find the steadystate motion
of a mass that vibrates according to the iaw + 85c + 36x 2 72 cos 6: Damped Forced Motion II A 324%) weight is attached to
a spring Suspended from the ceiling, stretching the spring
by 1.6 ft before coming to rest. At time t = 0 an exter—
nal force of f (r) = 20 cos 2: is applied to the system
Assume that the mass is acted upon by a damping three
of 45c, where Ii? is the instantaneous velocity in feet per
second Find the dispiacementofthe weight with MD) as 0
and 5cm) = 0‘ Calculating Charge Consider the series circuit shown in
Fig. 4.6.3, for which the inductance is 4 henries and the
capacitance is 0.01 iarads There is negligible resistance,
The input voltage is 10 cos 41‘. At timer m 0, the current
and the charge on the capacitor are both zero Determine
the charge Q as a function of rt 1! w 4 henries = lOcos 4: volts C=U.01 farads
FIGURE 4.6t8 Circuit for Problem 14 Charge and Current A resistor of 12 ohms is connected
in series with an inductor of one henr'y, a capacitor of Section 4 6 Forced Oscillations 271
O {)1 farads, and a voitage source supplying 12 cos 101 . At
t : D, the charge on the capacitor is zero and the current
in the circuit is also zero (a) Determine Qtr), the charge on the capacitor as a
function of time tort > 0 (1)} Determine [{t), the current in the circuit as attraction
of'time for r > O. True/False Questions For" Problems 16 and 17, give CtjltSIi
ﬁcntion for your answer of true or false 16. True or f'aise‘? For a {or'ced damped massspring system
with sinusoidai forcing, the frequency of the steadystate
soiution is the same as that of the forcing function 17. True or false? For a ferced damped mass—spring system
with sinusoidal forcing, the amplitude of the steadystate
solution is the same as that of the forcing function" 18. Beats Express cos 3: — cost in the form A sin oer sin 5!,
and sketch its graph 19. The Beat Goes On Express sin .3: — sint in the form
A sin or: cos ﬁt, and sketch its graph Steady State For Problems 20—22, ﬁnd the steadymstnte .50
lm‘iau having the form x” = C costtor w» (3), for the damped
.sysreni 20. +45; +4): m cos:
2]. 5t+2i+2x = 2cost 22. +:'c + .r = 4cos3t Resonance A mass of one slug is hanging at rest on a spring
whose constant is 1.2 lb/ﬁ. A! time t = 0, an external force of
ft!) m 16 cos a): [b is applied to the system. 23. What is the frequency of the forcing function that is in
resonance with the system? 24" Find the equation of motion of the mass with resonance 25. Ed’s Buoy3 Ed is sitting on the dock and observes a cylin
drical buoy bobbing vertically in calm water: He observes
that the period of oscillation is 5 sec and that 4 ft of the
buoy is above water when it reaches its maximum height
and 2 ft above water when it is at its minimum height An “ old seaman tells Ed that the buoy weighs 2000 lies (a) How wiii this buoy behave in rough waters, with the
waves 6 ft from crest to trough, and with a period of
7 sec it you negiect damping? (b) Will the buoy ever be submerged? 3Tin's probiern is based on a problem taken from Robert E. Gaskeli, Engineering rlzfnn'remnricas {Dryden Press, 1958) 272 Chapter 4 HigherOrth Linear {litterential Equations 26. General Solution of the Damped Forced System Con
sider the damped forced massspring equation mi + bi: + kx : F0 cos tuft (3) Verify, using the method of undetermined coefﬁ—
cients, that equations (14) and (15) give the particutat
solution. (in) Using (15), verify equations (17) and (18), and show
that in the damped caso the transient solution will go
to zero, so the particutat soiution Witt be the long—term
or steadystate tesponse of the system Phase Portrait Recognition For" Problems 2760, match the
phaseplane diagram: shown in Fig 4.6.9 to the appropriate
diﬁer‘ential equations 27, it +0.33% +x = cost 28. + x = 0 29. it wln'c =cost .30. +0.35: +x :0 (C) (Di
Fl G U R E 4. 6 . 9 Phase poﬁraits to match to Problems 2730 31. Matching 3!) Graphs Another way of showing the in—
tetactiOn 0f the independent votiable r and the dependent
variables .x and it, for a mass—spring system modeled by
mi + bi: + kit = F(t) with initial conditions .x(0) = .x0
and 32(0) m 1:9, is by means of‘an exit graph. Associate the
properties (a)(g) with the gt‘aphs shown in Fig 4 610.
Some properties may be associated with more than one
graph, and vice VBE‘SEL (a) Pure tesonance
(b) Beats (c) Forced damped motion with a sinusoidai forcing
function (ti) .x(0) > O; .ic(0) m 0 (e) x(0) m 0; 32(0) = 0 (t) Steadystate periodic motion
(g) Untorced damped motion F i G U R E 4.6.1 0 Trajectoxies in tryspace for l’tobiem 31 33. MassSpring Analysis I Suppose that
x(t) m 4cos 4t  3 sin 4t §« 5t sin 42‘ is the solution of a massspting system mic' + bit + k3: 2 F0), x(0) = x0, 36(0) 3 vol Assume that the homogeneous solution is not identically
Zero. (a) Determine the part of the solution associated with the
homogeneous DE. (b) Cateuiate the amplitude of the oscillation of the ho—
mogeneous solution. (c) Determine the amplitude of the particular solution“ 3.3. 34. 35. (d) Which part of the soiution will be unchanged if the
initial conditions are changed? (e) If the mass is 1 kg, what is the Spring constant? (1’) Describe the motion of the mass according to the
solution. Eiectrical Version Suppose that
Q0) 2 4cos at ~ 5 sin 4r + (it cos a: is the soiutioh of a given LRC system .. a l .
LQ + R9 + EQ W V0), (2(0) = Q0, 9(0) = 10‘ Assume that the homogeneous solution is not identically
zero (a) Determine the part of the solution that is the transient
solutions (b) Calculate the amptitude of the oscillation described
by the transient solution. (c) State which part of the solution is the steadystate
solution. (d) Which part of the solution will be unchanged if the
initiai conditions are changed? (e) If the inductance is I henry, what is the capacitance? (f) Describe what happens to the charge on the capacitor
according to the solution. MassSpring Analysis 11 Suppose that
.x(t) m 3424' cost ~ It!” sint + ﬁcoswt — 6) is the solution of a massspring system mi + bi + kx 2 F0 coswfr, .r(0) = x0, .i(0) a: no Assume that the homogeneous solution is not identically
earn. (a) What part of the solution is the transient solution?
(b) If the mass is i lcg, what is the damping constant 1;? (c) Is the system underdarnped, critically damped, or
overdarnped? (d) What is the time—varying ampiitude of the transient
solution? (e) What part of the solution is the steadystate solution? (f) Find the angular frequency a); of the forcing function.
Find the forcing amplitude F0. Perfect Aim Neglecting air friction, the planar motion
(x(t), y(t)) of'an object in a gravitational ﬁeld is governed Section 4.6 Forced Oscillations 273
by the equations 53:0 and Tim—g, where g is acceleration due to gravity (See Fig. 4.611.)
Suppose that you ﬁre a dart gun, iocated at the origin, di—
rectly at a target located at (x0, ya). Suppose also that the
dart has initial speed so, and at the exact moment the dart
is ﬁred the target object starts to f'aii. (a) Find the vertical distance the dart and target as a func—
tion of time. (b) Show that the dart wiii always hit the target.
(0} At what height wiil the dart hit the target? (as) :5 FIGURE 4.6.11 Paths of target and
dart for Problem 35 36. Extrema of the Amplitude Response Verify the ex— trema! properties of the amplitude response function
A(wf) in Fig 4.6.7, 37. Suggested Journal Entry Sometimes resonance in a physical system is desirable, and sometimes it is unde—
sirable. Discuss whether the resulting resonance would
be heipfui or destructive in the following systems (a)m(e).
Then give at ieast one additional example on each side. (a) Soldiers marching on a bridge with the same fre—
quency as the natural frequency of the bridge (b) A person rocking a car stuck in the snow with the same
frequency as the natural frequency of the stuck car (c) A chiid pumping a swing (d) Vibrations caused by air passing over an airplane wing
having the same frequency as the natural ﬂutter of the
wing (e) Acoustic vibrations having the same frequency as
the natural vibrations of a wine glass (the “Memorex
experiment”) ...
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 Fall '07
 RickRugangYe
 homogeneous solution, steadystate solution, damped forced motion

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