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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 30 "Harmonic oscillator anima
tions" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "
Martha Abell and James Braselton in their textbook " }{TEXT 277 35 "Di
fferential Equations with Maple V" }{TEXT -1 243 " (Academic Press) pr
ovide some lines of code that allow to show movies of the mass-spring \+
system by connecting straigh-line segments to depict a spring. The ide
a is to use the solution to the differential equation for the harmonic
 oscillator, " }{TEXT 257 1 "y" }{TEXT -1 1 "(" }{TEXT 258 1 "t" }
{TEXT -1 180 "), which provides the location of the moving endpoint of
 the spring, i.e., the attached mass as a function of time. We begin b
y providing the solution to the differential equation." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 218 "The parameters are \+
the mass and the spring constant, as well as the gravitational acceler
ation. We work implicitly in MKSA (SI) units, and pick values after th
e set-up of the equation and solution for the general case:" }}{PARA 
0 "" 0 "" {TEXT -1 1 "[" }{TEXT 260 1 "g" }{TEXT -1 25 " is measured i
n m/sec^2, " }{TEXT 259 1 "m" }{TEXT -1 8 " in kg, " }{TEXT 261 1 "k" 
}{TEXT -1 8 " in N/m]" }{TEXT 262 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 23 "g:='g': m:='m': k:='k':" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 55 "The forces acting on the mass are gravity (in the form " 
}{TEXT 272 1 "F" }{TEXT -1 1 "_" }{TEXT 273 1 "y" }{TEXT -1 4 " = -" }
{TEXT 274 2 "mg" }{TEXT -1 149 ", i.e., position-independent at the su
rface of the Earth), and the spring force given by Hooke's law, which \+
counteracts the motion from equilibrium, " }{TEXT 271 1 "F" }{TEXT -1 
6 "_H =- " }{TEXT 270 1 "k" }{TEXT -1 1 "(" }{TEXT 269 1 "y" }{TEXT 
-1 1 "-" }{TEXT 268 1 "y" }{TEXT -1 7 "0). If " }{TEXT 267 1 "y" }
{TEXT -1 17 " is greater than " }{TEXT 266 1 "y" }{TEXT -1 41 "0, the \+
force accelerates in the negative " }{TEXT 265 1 "y" }{TEXT -1 27 " di
rection, and vice versa." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 972 "Note that one can make different kinds of springs
: the usual springs encountered occasionally are tightly wound, and ar
e to be stretched only (e.g., a slinky), but we know coil springs from
 car/motorcycle/bicycle suspensions that can be both stretched and com
pressed. With gravity present a tightly wound spring with a mass attac
hed acquires an equilibrium point from which the spring can be both co
mpressed and further stretched, i.e., the equilibrium point is no long
er given as the length of the spring. The equilibrium point is determi
ned as the point at which the gravitational attraction to the Earth's \+
surface is balanced by the spring force which results from expanding t
he spring from its original equilibrium point to suspend the mass. We \+
assume for simplicity that we have a stretch-only spring, which is ide
alized to be of zero length when totally compressed (natural state). T
he equilibrium position for the mass-spring system under gravity becom
es (we label " }{TEXT 275 1 "y" }{TEXT -1 112 "1 now as the height at \+
which the system balances, the original balancing position that enters
 Hooke's law being " }{TEXT 276 1 "y" }{TEXT -1 5 "0=0):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "y0:=0; y1:='y1':" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%#y0G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "EQL:=-m*g-k*(y1-y0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EQL
G/,&*&%\"mG\"\"\"%\"gGF)!\"\"*&%\"kGF)%#y1GF)F+\"\"!" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 18 "y1:=solve(EQL,y1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#y1G,$*&*&%\"mG\"\"\"%\"gGF)\"\"\"%\"kG!\"\"!\"\"" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "We measure the displacement of th
e mass starting at " }{TEXT 263 1 "y" }{TEXT -1 107 "=0 (this being th
e equilibrium point without any mass attached) using the conventional \+
direction (positive " }{TEXT 264 1 "y" }{TEXT -1 58 " is upwards). The
 gravitational acceleration therefore is " }{TEXT 280 1 "F" }{TEXT -1 
1 "_" }{TEXT 279 1 "y" }{TEXT -1 2 "=-" }{TEXT 278 2 "mg" }{TEXT -1 
32 ". The Newton equation now reads:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "NE:=m*diff(y(t),t$2)=-m*g-k*(y(t)-y0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#NEG/*&%\"mG\"\"\"-%%diffG6$-%\"yG6#%\"tG-%\"$
G6$F/\"\"#F(,&*&F'\"\"\"%\"gGF(!\"\"*&%\"kGF(F,F(F8" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "sol:=dsolve(NE,y(t));" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%$solG/-%\"yG6#%\"tG,(*&*&%\"mG\"\"\"%\"gGF.\"\"\"%
\"kG!\"\"!\"\"*&%$_C1GF.-%$sinG6#*&-%%sqrtG6#*&F1F0F-F2F0F)F.F.F.*&%$_
C2GF.-%$cosGF8F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 659 "While ever
y physics student definitely should be able to derive this general ans
wer to the differential equation by hand, we assume that s/he can at l
east verify that the solution found by Maple is correct by substitutio
n. We have two integration constants to be determined by the initial c
onditions. One typical condition is to release the system from rest at
 an arbitrary height. We could take the general solution, find the vel
ocity expression by differentiation, and determine _C1 and _C2. Altern
atively, we can specify the initial conditions, so that dsolve comes u
p with the special solution of interest. We pick some arbitrary height
 and zero velocity." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "IC:=
y(0)=-0.2,D(y)(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6$/-%\"y
G6#\"\"!$!\"#!\"\"/--%\"DG6#F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "sol:=dsolve(\{NE,IC\},y(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$solG/-%\"yG6#%\"tG,&*&*&%\"mG\"\"\"%\"gGF.\"\"\"%\"k
G!\"\"!\"\"*&*&,&*&F-F0F/F0$\"\"&\"\"!F1$F3F:F.-%$cosG6#*&-%%sqrtG6#*&
F1F0F-F2F0F)F.F.F0F1F2$\"+++++?!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 33 "To plot we specify the constants:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "g:=10; m:=1; k:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%\"gG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"#5" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "Y_s:=unapply(rhs(sol),t):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 72 "We graph a stretch of 10 seconds together with the equili
brium position:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "y1;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "plot([Y_s(t),y1],t=0..10);" }}{PARA 13 "" 1 "" 
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{PARA 0 "" 0 "" {TEXT -1 229 "We recognize the simple harmonic motion \+
about the equilibrium position. To graph a cartoon side view of the ma
ss-spring system we need a sequence of straight-line segments. We pick
 a radius for the spring and the number of turns:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 15 "w:=0.2: nT:=10:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 145 "Given a position for the mass y(t) we divide this length
 into 2 nT segments, and generate a list of points which alternate on \+
the left and right." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "DL:=
proc(t) local i,L,d; global w,nT;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "d:=evalf(Y_s(t)/(2*nT+1)); L:=[[0.,0.],[w,d*0.5]]:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 216 "for i from 1 to nT do: L:=[op(L),[-w,d*(2*i-0
.5)],[w,d*(2*i+0.5)]]; od: L:=[op(L),[0.,evalf(Y_s(t))],[w/4,evalf(Y_s
(t))],[w/4,evalf(Y_s(t))-w/8],[-w/4,evalf(Y_s(t))-w/8],[-w/4,evalf(Y_s
(t))],[0.,evalf(Y_s(t))]]; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "PL:=seq(plot(DL(j_t/10)),j_t=1..32):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "display(PL,insequence=true,axes=boxed);" }}
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07$F5$!1,++?*pB%))F07$F.$!1+++`;F6'*F07$F5$!1+++Qt,Q5Fcdl7$F.$!1+++7v!
\\6\"Fcdl7$F5$!1+++&o(z\">\"Fcdl7$F.$!1+++eyoo7Fcdl7$F5$!1+++J!ybM\"Fc
dl7$F.$!1+++0#oCU\"Fcdl7$F5$!1+++y$e$*\\\"Fcdl7$F.$!1+++^&[id\"Fcdl7$F
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FaqFeq-%*AXESSTYLEG6#%$BOXG" 1 2 0 1 0 2 9 1 2 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "There are many way
s how one can make this problem more interesting:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "1) introducing a friction
 term to make the problem realistic" }}{PARA 0 "" 0 "" {TEXT -1 44 "2)
 to consider a friven harmonic oscillator." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Let us start with problem 1:" }
}{PARA 0 "" 0 "" {TEXT -1 61 "we need to add a term which is proportio
nal to the velocity d" }{TEXT 282 1 "y" }{TEXT -1 2 "/d" }{TEXT 281 1 
"t" }{TEXT -1 265 ". Whenever the velocity gets appreciable, we want a
 force that reduces it. For positive velocity it has to be negative, f
or negative velocity (downward) it has to be positive, i.e., we want a
 friction constant times the negative of the velocity added to the for
ces:" }{TEXT 283 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "b:='b': m:='m': g:='g': k:='k':" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 53 "NE:=m*diff(y(t),t$2)=-m*g-k*(y(t)-y0)-b*diff(
y(t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NEG/*&%\"mG\"\"\"-%%dif
fG6$-%\"yG6#%\"tG-%\"$G6$F/\"\"#F(,(*&F'\"\"\"%\"gGF(!\"\"*&%\"kGF(F,F
(F8*&%\"bGF(-F*6$F,F/F(F8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 594 "Usi
ng mathematical jargon we can say that this is still a non-homogeneous
 (due to the constant gravity-related term), second-order ordinary dif
ferential equation with constant coefficients. The method of solution \+
is still the same [find the general solution to the homogeneous proble
m, (find and ) add a special solution to the inhomogeneous equation to
 the general homogeneous solution, and that is the general solution to
 the inhomogeneous problem with the boundary conditions still to be sp
ecified]. Maple still can do all the math work for us so we concentrat
e on understanding the physics:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 21 "sol:=dsolve(NE,y(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$so
lG/-%\"yG6#%\"tG,(*&*&%\"mG\"\"\"%\"gGF.\"\"\"%\"kG!\"\"!\"\"*&%$_C1GF
.-%$expG6#,$*&*&,&%\"bGF.*$-%%sqrtG6#,&*$)F=\"\"#F0F.*&F-F0F1F.!\"%F0F
.F.F)F.F0F-F2#F3FEF.F.*&%$_C2GF.-F76#,$*&*&,&F=F.F>F3F.F)F0F0F-F2FHF.F
." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 278 "The solution is expressed i
n a 'weird' way if we expect oscillatory answers, but keep in mind tha
t the exponentials may turn out to have imaginary arguments. In fact, \+
the damped HO admits both oscillatory and non-oscillatory solutions (d
epending on the strength of the damping)." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Let us put in the same initial con
ditions as before:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sol:=
dsolve(\{NE,IC\},y(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG/-%
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}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "We can think of real-life applic
ations:" }}{PARA 0 "" 0 "" {TEXT -1 507 "The shock absorbers in a car'
s (or other vehicle's) suspension system cooperate with the coils (or \+
other springs, such as multi-link suspension, or leaf springs) to prov
ide a damped oscillator. In fact, one is not interested in the kind of
 motion displayed above, but would rather want to absorb the oscillato
ry energy in one-two swings. From the analytic form of the solution in
 the case with unspecified constants we can see when a character chang
e occurs in the solution. This character change occurs as " }{TEXT 
295 1 "b" }{TEXT -1 4 "^2-4" }{TEXT 294 2 "km" }{TEXT -1 123 " turns z
ero, and the exponential has no longer complex-valued arguments [we ad
d a tiny number to avoid a zero denominator]:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 46 "g:=10; m:=1; k:=10; b:=sqrt(4*k)+10^(-Digits);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG
\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,&*$-%%sqrtG6#\"#5\"\"\"
\"\"##\"\"\"\",+++++\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"Y_s:=unapply(rhs(sol),t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
26 "plot([Y_s(t),y1],t=0..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 692 299 
299 {PLOTDATA 2 "6&-%'CURVESG6$7in7$\"\"!$!1:k6++++?!#;7$$\"1mmmT&)G\\
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[lF^_l7$Fh[lF^_l7$F]\\lF^_l7$Fb\\lF^_l7$Fg\\lF^_l7$F\\]lF^_l7$Fa]lF^_l
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45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "This cas
e is called the critically damped solution." }}{PARA 0 "" 0 "" {TEXT 
-1 145 "There are interesting questions to be asked, such as: what hap
pens if the damping is increased further? Is it desirable to do so in \+
applications?" }}{PARA 0 "" 0 "" {TEXT -1 26 "Let us double the dampin
g:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "g:=10; m:=1; k:=10; b
:=2*sqrt(4*k)+10^(-Digits);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG
\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"kG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG
,&*$-%%sqrtG6#\"#5\"\"\"\"\"%#\"\"\"\",+++++\"F." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 25 "Y_s:=unapply(rhs(sol),t):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 26 "plot([Y_s(t),y1],t=0..10);" }}{PARA 13 "" 
1 "" {GLPLOT2D 692 299 299 {PLOTDATA 2 "6&-%'CURVESG6$7X7$\"\"!$!1)***
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F(Fd\\lF(-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F(F\\\\l%(DEFAULTG" 1 2 
0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 161 "We see that it takes longer for the spring-mass sys
tem to reach equilibrium! One has introduced so much friction that it \+
takes forever to reach the final height." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 706 "There is an interesting represent
ation that allows to gain further insights. In a so-called phase-space
 diagram one plots the velocity versus the position with time playing \+
the role of the parameter. This curve [an abstract object called the p
hase-space trajectory] allows to read off how energy is dissipated in \+
the system. For the case without friction it results in a closed curve
. Solutions with different initial conditions are characterized by sim
ilar-looking phase-space trajectories whose size is a measure of the a
mount of total energy which is conserved and transformed between kinet
ic and potential energy forms. For the case with damping we obtain spi
rals, as energy is removed from the system." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Let us begin with undercritical
 damping." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "g:=10; m:=1; k
:=10; b:=1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"#5" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"kG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG#\"\"\"\"\"#
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Y_s:=unapply(rhs(sol),t
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "V_s:=D(Y_s);" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$V_sGR6#%\"tG6\"6$%)operatorG%&arrow
GF(,&*(,&$\"+)*******R!#5\"\"\"%\"IG$\"+Uj?sJ!#6F2,&#!\"\"\"\"%F2*&-%%
sqrtG6#!$f\"\"\"\"-F=6#F:F@#F9\"\")F2-%$expG6#,$*&,&#F2\"\"#F2F;#F2F:F
29$F2#F9FLF2F2*(,&F/F2F3$!+Uj?sJF6F2,&F8F2F;#F2FDF2-FF6#,$*&,&FKF2F;F8
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amount of damping it may not be immediately obvious, but we wish to po
int out that the reduction in mechanical energy (reduction of radius) \+
occurs during those phases when the magnitude of the velocity is large
st. This can be inferred from a careful inspection of the graph, or fr
om a detailed plot of the total energy as a function of time:" }}}
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}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "We leave it as an exercise to
 look at the diagrams for the critically and overcritically damped cas
es." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "N
ow we turn to the case of a driven HO. We choose a harmonic driving fo
rce with small amplitude to drive the HO away from the equilibrium. A \+
circular frequency nu characterizes the driving force." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "b:='b': m:='m': g:='g': k:='k': nu:
='nu':" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "NE:=m*diff(y(t),t$2)=-m*g
-k*(y(t)-y0)-b*diff(y(t),t)+(1/10)*sin(nu*t);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#NEG/*&%\"mG\"\"\"-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F/\"
\"#F(,**&F'\"\"\"%\"gGF(!\"\"*&%\"kGF(F,F(F8*&%\"bGF(-F*6$F,F/F(F8-%$s
inG6#*&%#nuGF(F/F(#F(\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "For \+
the undamped case the internal frequency of the oscillator is:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "omega:=sqrt(k/m);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%&omegaG*$-%%sqrtG6#*&%\"kG\"\"\"%\"mG!\"\"
F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "We need to investigate wha
t happens as nu is varied over a wide range, and particularly what hap
pens as nu approaches omega (resonance)." }}{PARA 0 "" 0 "" {TEXT -1 
101 "For a small amplitude of the driving force we can expect dramatic
 results only for nu close to omega." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "g:=10; m:=1; k:=10; b:=1/2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"gG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"#5" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"bG#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "omega;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$-%%sqrtG6#
\"#5\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "y1;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "nu:=3.1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nuG$\"#J!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "sol:=dsolve(\{NE,y(0)=y1,D
(y)(0)=0\},y(t)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "The result i
s long, and perhaps worth looking at." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "Y_s:=unapply(rhs(sol),t):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "plot([Y_s(t),y1],t=0..20);" }}{PARA 13 "" 1 "" 
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45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 373 "It is worthwhile \+
to repeat the calculation for different values of nu, and to observe t
he more complicated transient behaviour. Eventually the system respond
s with oscillations at the driving frequency nu. This latter fact is c
learly evident when choices are made for nu quite different from omega
. On the other hand, we obtain appreciable oscillations only near reso
nance." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 
"The phase-space diagram shows that the abstract phase-space trajector
y approaches a limit cycle:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
12 "V_s:=D(Y_s):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot([Y_s(t),V_
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" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 394 "The emergence of a limit cycle tells us that t
here comes a point when as much energy is dumped into the mass-spring \+
system by the driving force, as there is energy lost due to friction. \+
We encourage the student to try out cases where the friction is smalle
r, and therefore, larger-amplitude motion can be observed. We also lea
ve it as an exercise to explore different driving force frequencies." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 "To com
plete the understanding of the driven HO with damping one has to figur
e out the dependence of the final amplitude on the frequency of the dr
iving force." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "b:='b': m:=
'm': g:='g': k:='k': nu:='nu':" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "N
E:=m*diff(y(t),t$2)=-m*g-k*(y(t)-y0)-b*diff(y(t),t)+(1/10)*sin(nu*t);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NEG/*&%\"mG\"\"\"-%%diffG6$-%\"
yG6#%\"tG-%\"$G6$F/\"\"#F(,**&F'\"\"\"%\"gGF(!\"\"*&%\"kGF(F,F(F8*&%\"
bGF(-F*6$F,F/F(F8-%$sinG6#*&%#nuGF(F/F(#F(\"#5" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 3 "y1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&*&%
\"mG\"\"\"%\"gGF'\"\"\"%\"kG!\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "sol:=dsolve(\{NE,y(0)=y1,D(y)(0)=0\},y(t)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Y_s:=unapply(rhs(sol),t);" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$Y_sGR6#%\"tG6\"6$%)operatorG%&arro
wGF(,(*&,0*&-%$sinG6#*&%#nuG\"\"\"9$F5F5)%\"kG\"\"#\"\"\"!\"\"**F4F:-%
$cosGF2F5%\"bGF5F8F5F5*()%\"mG\"\"$F:%\"gGF5)F4\"\"%F:\"#5*(FBF5FDF:F7
F:FG**)FBF9F:FDF:)F4F9F:F8F:!#?**FBF:FDF:)F?F9F:FKF:FG**F0F:FKF:FBF:F8
F:F5F:*&F8\"\"\",**&FEF:FJF:F5*&FNF:FKF:F5*(FBF:F8F:FKF:!\"#*$F7F:F5\"
\"\"!\"\"#F;FG*&*(,**$FNF:F5*&-%%sqrtG6#,&FhnF5*&FBF:F8F:!\"%F:F?F:F;*
&FKF:FJF:F9F^oFVF5F4F:-%$expG6#,$*&*&,&F?F5*$FjnF:F5F5F6F:F:FBFY#F;F9F
5F:*&-F[o6#F]oF:FR\"\"\"FY#F;\"#?*&*(F4F:,*FhnF5FinF5F^oFVF`oF9F5-Fbo6
#,$*&*&,&F?F5FhoF;F5F6F:F:FBFYFioF5F:*&-F[o6#F]oF:FR\"\"\"FY#F5F_pF(F(
F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 140 "Intuitively it is clear that for undercritical da
mping the internal oscillations die out on a time scale determined by \+
the damping constant " }{TEXT 285 1 "b" }{TEXT -1 187 ". What remains \+
is the first term in the above expression whose time variation is just
 connected to the inhomogeneous term in the Newton equation, i.e., jus
t to the periodic driving force." }}{PARA 0 "" 0 "" {TEXT -1 79 "Let u
s assume that we are in the asymptotic regime, which is reached when e
xp(-" }{TEXT 286 2 "bt" }{TEXT -1 2 "/2" }{TEXT 293 1 "m" }{TEXT -1 
118 ") becomes negligible compared to the first term. We simply pick o
ut the first term in the solution. For our choice of " }{TEXT 287 1 "b
" }{TEXT -1 43 "=1/2 (a strong, but undercritical damping) " }{TEXT 
288 1 "t" }{TEXT -1 148 "_a=20 should be appropriate. We can verify fr
om a graph, that we have reached the limit cycle, i.e., driven harmoni
c motion with constant amplitude." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "Y_as:=unapply(op(1,Y_s(t)),t);" }}{PARA 11 "" 1 "" 
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F]cq$!1/1+M-&e.\"F-7$Fbcq$!1$\\U8(*\\1/\"F-7$Fgcq$!1\"QeeNRX/\"F-7$F\\
dq$!1ug4&GJu/\"F-7$$\"1</,Hxp#z$F1$!1VJ\"3%Hh[5F-7$Fadq$!1O'H1jO%\\5F-
7$$\"1]7.ZcB)z$F1$!1d\"H#*G'*)\\5F-7$Ffdq$!1cH(=`))*\\5F-7$F[eq$!1Ju%G
wq!\\5F-7$F`eq$!1c))41.rY5F-7$Feeq$!1mlSfl(H/\"F-7$Fjeq$!16%yaHzz.\"F-
7$F_fq$!1>ohMa'=.\"F-7$Fdfq$!1o+CDZ\"[-\"F-7$F^gq$!19\"*)pt$)y,\"F-7$F
hgq$!1o-qG>`55F-7$F]hq$!1a$z[HKH+\"F-7$Fbhq$!1dh%ehOE&**F[o7$Fghq$!1%>
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[o7$Fjjq$!1Y.]'3h1^*F[o7$$\"1eRA1Ei(*QF1$!1!)4$p=-R]*F[o7$F_[r$!1oR!=g
l/]*F[o7$$\"1T&)3HP!H!RF1$!1_acXVP+&*F[o7$Fd[r$!1'y[$H!HO]*F[o7$Fi[r$!
1(Ha/xm+_*F[o7$F^\\r$!1S44N#R$\\&*F[o7$Fc\\r$!1$GUXPe')e*F[o7$Fh\\r$!1
9,%Gzc!Q'*F[o7$F]]r$!14__fSK'p*F[o7$Fb]r$!13K(pUK?w*F[o7$F\\^r$!15@5:=
dL)*F[o7$Ff^r$!1[%\\QE*=4**F[o7$F[_r$!1i%)p')=.()**F[o7$F`_r$!1vWFF#>l
+\"F-7$Fe_r$!1#)[R4`t95F-7$Fj_r$!1o=O\\X`A5F-7$F_`r$!1&*)HrJ'pH5F-7$Fd
`r$!1Tv)e,=g.\"F-7$F^ar$!1*R'=+ZXX5F-7$Fhar$!1fWM\\Vy\\5F--F]br6&F_brF
*F*F`br-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$;F(Fhar%(DEFAULTG" 1 2 0 1 
0 2 9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 110 "The graph shows that the harmonic motion associated with
 the limit cycle is centered on the equilibrium point " }{TEXT 289 1 "
y" }{TEXT -1 52 "1, and is phase shifted compared to a simple sin(nu*
" }{TEXT 290 1 "t" }{TEXT -1 95 "). The phase shift is achieved by the
 combination of sines and cosines present in the solution." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 128 "Our interest is in extracting the amplit
ude as a function of the parameters. First we have to subtract the equ
ilibrium position:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "assum
e(nu,real);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "A:=combine(Y
_as(t)-y1,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG*&,(*&-%$sin
G6#*&%$nu|irG\"\"\"%\"tGF-F-%\"kGF-F-*(F,\"\"\"-%$cosGF*F-%\"bGF-!\"\"
*(F(F1)F,\"\"#F1%\"mGF-F5F1,**&)F,\"\"%F1)F9F8F1\"#5*&)F4F8F1F7F1F?*(F
9F1F/F1F7F1!#?*$)F/F8F1F?!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 26 "A0:=simplify(subs(b=0,A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#A0G,$*&-%$sinG6#*&%$nu|irG\"\"\"%\"tGF,\"\"\",&%\"kG!\"\"*&%\"mGF,)
F+\"\"#F.F,!\"\"#F1\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "We no
te that in the case without friction the asymptotic amplitude blows up
 on resonance [note that omega^2=" }{TEXT 292 1 "k" }{TEXT -1 1 "/" }
{TEXT 291 1 "m" }{TEXT -1 231 "]. This means that no limit cycle is re
ached, but that the amplitude grows linearly with time [the latter det
ail is obtained from an analysis of the solution to Newton's equation,
 we leave it as an exercise to extract this result]." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 295 "For the case with fric
tion the resonance occurs at a shifted frequency, and the mentioned ph
ase shift occurs. Let us look first at the asymptotic solution when nu
 is chosen to be on  resonance in the frictionless (undamped HO) case.
 This answer should be reasonable for small friction constants " }
{TEXT 296 1 "b" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 34 "simplify(algsubs(nu=sqrt(k/m),A));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*&*(-%$cosG6#*&-%%sqrtG6#*&%\"kG\"\"\"%\"mG!\"\"F/%\"
tG\"\"\"F3F*F/F0F3F/*&%\"bG\"\"\"F.\"\"\"F1#!\"\"\"#5" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 172 "So let us assume that near resonance the
 amplitude is given by looking just at the cosine term (as the two sin
e terms nearly cancel on resonance in the weakly damped case)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A1:=simplify(nu*b*cos(nu*t)/
denom(A));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G,$*&*(%$nu|irG\"\"
\"%\"bG\"\"\"-%$cosG6#*&F(F+%\"tGF+F+F),**&)F(\"\"%F))%\"mG\"\"#F)F+*&
)F*F7F))F(F7F)F+*(F6F+%\"kGF+F:F)!\"#*$)F<F7F)F+!\"\"#F+\"#5" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "For the amplitude of oscillation w
e have:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "A1a:=simplify(nu*b/denom(A));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$A1aG,$*&*&%$nu|irG\"\"\"%\"bGF)\"\"\",**&)F(\"\"%F+)
%\"mG\"\"#F+F)*&)F*F2F+)F(F2F+F)*(F1F)%\"kGF)F5F+!\"#*$)F7F2F+F)!\"\"#
F)\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Now we specify " }
{TEXT 298 1 "k" }{TEXT -1 5 " and " }{TEXT 297 1 "m" }{TEXT -1 39 ", w
ith the resonance at omega=sqrt(10)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 25 "A1as:=subs(m=1,k=10,A1a);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%A1asG,$*&*&%$nu|irG\"\"\"%\"bGF)\"\"\",**$)F(\"\"%F+
F)*&)F*\"\"#F+)F(F2F+F)*$F3F+!#?\"$+\"F)!\"\"#F)\"#5" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "P0:=plot([sqrt(10),t,t=0..1.5],color=blac
k):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P1:=plot(subs(b=1/5,
A1as),nu=3..3.3,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "P2:=plot(subs(b=1/15,A1as),nu=3..3.3,color=blue):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "P3:=plot(subs(b=1/45,A1as),nu=3..3.
3,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display(
P0,P1,P2,P3,axes=boxed,labels=[frequency,Amplitude]);" }}{PARA 13 "" 
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{PARA 0 "" 0 "" {TEXT -1 122 "The graph shows that the damped oscillat
or is capable of absorbing energy from the driving mechanism in the fo
llowing way:" }}{PARA 0 "" 0 "" {TEXT -1 229 "1) very weak damping: th
e oscillator can only absorb energy effectively if the mechanical ener
gy is offered at a frequencies close to the natural frequency, which i
s very close to the natural frequency of the undamped oscillator;" }}
{PARA 0 "" 0 "" {TEXT -1 205 "2) for moderate damping the range over w
hich energy can be absorbed broadens somewhat; the resonance condition
 is shifted slightly towards lower frequencies when compared to the un
damped natural frequency." }}{PARA 0 "" 0 "" {TEXT -1 329 "3) for stro
nger damping the resonance is broadened: the harmonic oscillator is ab
le to absorb frequencies over a wider range equally, but the amplitude
 of oscillation is reduced considerably. This is the result of the dri
ving force not being able to be in phase with the internal oscillatory
 motion even in the asymptotic regime." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 84 "For a more detailed analysis the read
er may wish to consult the book by Ron Greene: " }{TEXT 299 30 "Classi
cal Mechanics with Maple" }{TEXT -1 20 " (Springer-Verlag). " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "71 1 0" 0 }
{VIEWOPTS 1 1 0 1 1 1803 }