Preliminaries Subroutines - Embry–Riddle Aeronautical...

Preliminaries

Subroutines phasePlot

phasePlot@q_, p_, tList_List, opts___D :=

ParametricPlot@ 8q, p< Evaluate, tList, opts, PlotRange -> All,

Axes -> False, PlotLabel -> "Phase Diagram", Frame -> True,

FrameLabel -> 8"q", "q

"<, RotateLabel ® False, AspectRatio -> 1 D

H*Example*L phasePlot@Sin@2 Π tD, D@Sin@2 Π tD, tD, 8t, 0, 10<D

-1.0 -0.5 0.0 0.5 1.0

-6

-4

-2

0

2

4

6

q

q

Phase Diagram

diffSeriesOne

diffSeriesOne@eq_, z_, t_, initialx_, initialv_, order_D :=

Module@ 8tempser, tempeq, tempvar, tempsol, tempinit<,

tempser = z ® Series@z, 8t, 0, order<D;

tempeq = HHeq@@1DD - eq@@2DDL . tempserL 0;

tempvar = HTable@D@z, 8t, j<D, 8j, 2, order + 2<D . t ® 0L Flatten;

tempsol = Solve@tempeq, tempvarD Flatten;

tempinit = 8Head@zD@0D ® initialx, Head@zD'@0D ® initialv<;

z . tempser . tempsol . tempinitD

H*Example*L diffSeriesOne@y''@tD + Γ y'@tD^2 + Ω^2 y@tD 0, y@tD, t, 1, 0, 2D

1 -

Ω2 t2

2+ O@tD3

H*Internals of diffSeriesOne*L

wrap

wrap@Θ_D := Mod@Θ, 2 Π, -ΠD;

H*Example*L Plot@wrap@ΘD, 8Θ, -4 Π, 4 Π<, AspectRatio ® 0.25,

Ticks ® 8Table@n Π, 8n, -4, 4<D, Table@n Π 2, 8n, -4, 4<D<D

-4 Π -3 Π -2 Π -Π Π 2 Π 3 Π 4 Π

-Π

-Π

2

Π

2

Π

solnPend

solnPend@8Θ0_, dΘ0_<, 8Β_, f0_, Ω0_, Ωd_<, tstart_, tend_, tstep_, opts___D :=

HNDSolve@8dΘ'@tD + Β dΘ@tD + Ω0^2 Sin@Θ@tDD == f0 Cos@Ωd tD,

dΘ@tD == Θ'@tD,

Θ@0D Θ0,

dΘ@0D dΘ0<,

8dΘ@tD, Θ@tD<, 8t, tstart, tend<, MaxSteps ® tstepD FlattenL@@2DD@@2DD

anglePend

anglePend@8Θ0_, dΘ0_<, 8Β_, f0_, Ω0_, Ωd_<, tstart_, tend_, tstep_, opts___D :=

Module@8temp<,

temp = NDSolve@8dΘ'@tD + Β dΘ@tD + Ω0^2 Sin@Θ@tDD == f0 Cos@Ωd tD,

dΘ@tD == Θ'@tD,

Θ@0D Θ0,

dΘ@0D dΘ0<,

8dΘ@tD, Θ@tD<, 8t, tstart, tend<, MaxSteps ® tstepD Flatten;

Plot@Θ@tD . temp, 8t, tstart, tend<, opts, AxesLabel ® 8"t", "Θ"<DD

anglePendw@8Θ0_, dΘ0_<, 8Β_, f0_, Ω0_, Ωd_<, tstart_, tend_, tstep_, opts___D :=

Module@8temp<,


dΘ@tD == Θ'@tD,

Θ@0D Θ0,

dΘ@0D dΘ0<,


Plot@wrap@Θ@tD . tempD, 8t, tstart, tend<, opts, AxesLabel ® 8"t", "Θ"<DD

2 nonlinear_oscillations.nb

phasePend

phasePend@8Θ0_, dΘ0_<, 8Β_, f0_, Ω0_, Ωd_<, tstart_, tend_, tstep_, opts___D :=

ModuleA8temp<,


dΘ@tD == Θ'@tD,

Θ@0D Θ0,

dΘ@0D dΘ0<,


ParametricPlotA8Θ@tD, dΘ@tD< . temp, 8t, tstart, tend<, opts, AxesLabel ® 9"Θ", "Θ

"=EE

phasePendw@8Θ0_, dΘ0_<, 8Β_, f0_, Ω0_, Ωd_<, tstart_, tend_, tstep_, opts___D :=

ModuleA8temp<,


dΘ@tD == Θ'@tD,

Θ@0D Θ0,

dΘ@0D dΘ0<,


ParametricPlotA8wrap@Θ@tD . tempD, dΘ@tD . temp<, 8t, tstart, tend<, opts,

Ticks ® 8Table@n Π 2, 8n, -2, 2<D, Automatic<, AxesLabel ® 9"Θ", "Θ

"=EE

poincarePend

poincarePend@8Θ0_, dΘ0_<, 8Β_, f0_, Ω0_, Ωd_<, tstart_, tend_, step_, opts___D :=

ModuleA8temp1, temp2<,

temp1 = NDSolve@8dΘ'@tD + Β dΘ@tD + Ω0^2 Sin@Θ@tDD == f0 Cos@Ωd tD,

dΘ@tD == Θ'@tD,

Θ@0D Θ0,

dΘ@0D dΘ0<,

8dΘ@tD, Θ@tD<, 8t, 0, tend<, MaxSteps ® stepD Flatten;

temp2 = Table@8Θ@tD, dΘ@tD< . temp1, 8t, tstart, tend, 2 Π Ωd<D;

ListPlotAtemp2, PlotStyle -> [email protected], opts, AxesLabel ® 9"Θ", "Θ

"=EE

poincarePendw@8Θ0_, dΘ0_<, 8Β_, f0_, Ω0_, Ωd_<, tstart_, tend_, step_, opts___D :=

ModuleA8temp1, temp2<,

temp1 = NDSolve@8dΘ'@tD + Β dΘ@tD + Ω0^2 Sin@Θ@tDD == f0 Cos@Ωd tD,

dΘ@tD == Θ'@tD,

Θ@0D Θ0,

dΘ@0D dΘ0<,

8dΘ@tD, Θ@tD<, 8t, 0, tend<, MaxSteps ® stepD Flatten;

temp2 = Table@8wrap@Θ@tD . temp1D, dΘ@tD . temp1<, 8t, tstart, tend, 2 Π Ωd<D;

ListPlotAtemp2, PlotStyle -> [email protected], opts, AxesLabel ® 9"Θ", "Θ

"=EE

nonlinear_oscillations.nb 3

solnDuff

solnDuff@8x0_, v0_<, 8a_, b_, c_, f0_, Ωd_<, tstart_, tend_, tstep_, opts___D :=

HNDSolve@8v'@tD + a x@tD + b x@tD^3 + c v@tD == f0 Cos@Ωd tD,

v@tD x'@tD,

x@0D x0,

v@0D v0<,

8v@tD, x@tD<, 8t, tstart, tend<, MaxSteps ® tstepD FlattenL@@2DD@@2DD

posDuff

posDuff@8x0_, v0_<, 8a_, b_, c_, f0_, Ωd_<, tstart_, tend_, tstep_, opts___D :=

Module@8temp<,

temp = NDSolve@8v'@tD + a x@tD + b x@tD^3 + c v@tD == f0 Cos@Ωd tD,

v@tD x'@tD,

x@0D x0,

v@0D v0<,

8v@tD, x@tD<, 8t, tstart, tend<, MaxSteps ® tstepD Flatten;

Plot@x@tD . temp, 8t, tstart, tend<, opts, AxesLabel ® 8"t", "x"<DD

phaseDuff

phaseDuff@8x0_, v0_<, 8a_, b_, c_, f0_, Ωd_<, tstart_, tend_, tstep_, opts___D :=

Module@8temp<,

temp = NDSolve@8v'@tD + a x@tD + b x@tD^3 + c v@tD == f0 Cos@Ωd tD,

v@tD x'@tD,

x@0D x0,

v@0D v0<,


ParametricPlot@8x@tD, v@tD< . temp, 8t, tstart, tend<, opts, AxesLabel ® 8"x", "v"<DD

poincareDuff

poincareDuff@8x0_, v0_<, 8a_, b_, c_, f0_, Ωd_<, tstart_, tend_, tstep_, opts___D :=

Module@8temp<,

temp1 = NDSolve@8v'@tD + a x@tD + b x@tD^3 + c v@tD == f0 Cos@Ωd tD,

v@tD x'@tD,

x@0D x0,

v@0D v0<,


temp2 = Table@8x@tD, v@tD< . temp1, 8t, tstart, tend, 2 Π Ωd<D;

ListPlot@temp2, PlotStyle -> [email protected], opts, AxesLabel ® 8"x", "v"<DD


psdPlot

psdListPlot@data_, fs_, opts___D :=

Module@8specraw2, fvec, datlist<,

specraw2 = Fourier@dataD;

specraw = Hspecraw2@@1 ;; Round@Length@specraw2D 2DDDL Chop Flatten;

fvec = Range@0, fs 2, fs 2 HLength@specrawD - 1LD;

datlist = Table@8fvec@@ssDD, Abs@specraw@@ssDDD<, 8ss, 1, Length@specrawD<D;

ListLogPlot@datlist, opts, Joined ® True, PlotRange ® AllDD

H*example*L Module@8dt, Npts, sampledata, freqlist, amplist, sig, sigsig, i<,

dt = 0.01; H*sec*LNpts = 1000;

freqlist = 810, 13, 25<;

amplist = 81, 3, 0.5<;

sig = 0.0;

For@i = 1, i £ Length@freqlistD, i++,

sig = sig + Table@amplist@@iDD * Sin@2 * Π freqlist@@iDD * n * dtD, 8n, 1, Npts<D;

D;

sigsig = Table@8n * dt, sig@@nDD<, 8n, 1, Npts<D;

psdListPlot@sigsig, 1 dt, PlotRange ® 885, 40<, All<DD

10 15 20 25 30 35 40

0.5

1.0

5.0

10.0

50.0

100.0

Physics Pendulum Equation

[email protected] Π, 0<, 80.2, 0, 1, 1<, 0, 100, 10 000D . 8t ® Range@0, 100, 10D<

82.82743, 0.49932, -0.0862968, -0.0275841, 0.0302764, -0.0156024,

0.00583821, -0.00160537, 0.000231931, 0.0000696012, -0.0000756445<


[email protected] Π, 0<, 80.2, 0, 1, 1<, 0, 100, 10 000, PlotRange ® AllD

20 40 60 80 100t

-1

1

2

Θ

[email protected] Π, 0<, 80.2, 0, 1, 1<, 0, 100, 10 000, PlotRange ® AllD

-1 1 2Θ

-1.5

-1.0

-0.5

0.5

1.0

Θ

phasePend@80, 4.5<, 80.2, 0, 1, 1<, 0, 100, 10 000, PlotRange ® AllD

2 4 6 8 10 12 14Θ

-1

1

2

3

4

Θ


plotlist = phasePend@80, ð<, 80, 0, 1.5, 1<, -17, 17, 50 000,

PlotRange ® 88-3 Π, 3 Π<, 8-5, 5<<, AspectRatio ® 0.5D & Range@-5, 5, 0.5D;

Show@plotlistD

-5 5Θ

-4

-2

2

4

Θ

plotlist = phasePend@80, ð<, 80.4, 0, 1.5, 1<, -17, 17, 50 000,

PlotRange ® 88-3 Π, 3 Π<, 8-5, 5<<, AspectRatio ® 0.5D & Range@-5, 5, 0.5D;

Show@plotlistD

-5 5Θ

-4

-2

2

4

Θ


dt = 0.1; tstart = 0; tend = 2000;

angledata1 = [email protected] Π, 0<, 80, 0, 2 Π, 1<, tstart, tend, 200 000D .

8t ® Range@tstart, tend, dtD<;

angledata2 = 1000 * [email protected] Π, 0<, 80, 0, 2 Π, 1<, tstart, tend, 200 000D .

8t ® Range@tstart, tend, dtD<;

f1 = psdListPlot@angledata1, 1 dt, PlotRange ® 880, 4<, 80.0001, 1000<<, Joined ® TrueD;

f2 = psdListPlot@angledata2, 1 dt,

PlotRange ® 880, 4<, 80.0001, 1000<<, PlotStyle ® Red, Joined ® TrueD;

ShowBf1, f2, AxesLabel ® :"f", "cts Hz ">F

0 1 2 3 4f10-4

0.01

1

100

cts Hz

pt1 = anglePend@80, ð Π<, 80.5, 1, 1, 2 3<, 0, 50, 10 000, PlotStyle -> [email protected] * ðD,

Ticks ® 8Π Range@0, 6, 1D, Automatic<,

AspectRatio ® 0.5, PlotRange ® AllD & 81 2, 3 2, 2, 3<;

Show@pt1D

Π 2 Π 3 Π 4 Π 5 Π 6 Πt

5

10

15

20

Θ


pt2 = phasePend@80, ð Π<, 80.5, 1, 1, 2 3<, 0, 400, 50 000,

AspectRatio ® 0.5, PlotStyle -> [email protected] * ðD,

Ticks ® 8Π Range@0, 6, 1D, Automatic<, PlotRange ® 88-3 Π, 7 Π<, 8-3, 9<<D & 81 2, 3 2, 2, 3<;

Show@pt2D

Π 2 Π 3 Π 4 Π 5 Π 6 ΠΘ

-2

2

4

6

8

Θ

pt3 = poincarePend@80, ð Π<, 80.5, 1, 1, 2 3<, 123, 400, 50 000,

AspectRatio ® 0.5, Ticks ® 8Π Range@0, 6, 1D, Automatic<,

PlotRange ® 88-3 Π, 7 Π<, 8-3, 9<<D & 81 2, 3 2, 2, 3<;

Show@pt3D

Π 2 Π 3 Π 4 Π 5 Π 6 ΠΘ

-2

2

4

6

8

Θ


pt4 = anglePend@8Π 2, 0<, 80.2, ð, 0.694< Flatten, 0, 500, 20 000,

Ticks ® 8Automatic, Π Range@-1, 10, 1 4D<, PlotStyle -> [email protected] * ðD,

PlotPoints ® 150, AspectRatio ® 0.3,

PlotRange ® AllD & 880, 1<, 80.1, 1<, 80.3, 1<<;

pt5 = anglePend@8Π 2, 0<, 80.2, ð, 0.694< Flatten, 0, 1000, 20 000,

Ticks ® 8Automatic, Π Range@-10, 10, 2D<, PlotStyle -> Hue@2 * ðD,

PlotPoints ® 150, AspectRatio ® 0.2,

PlotRange ® AllD & 880.69, 1<, 80.74, 1<, 80.83, 1<<;

Show@pt4DShow@pt5D

100 200 300 400 500t

-Π

4

Π

4

Π

2

Θ

200 400 600 800 1000t

-10 Π-8 Π-6 Π-4 Π

2 Π4 Π6 Π8 Π

Θ


pt6 = phasePend@8Π 2, 0<, 80.2, ð, 0.694< Flatten, 0, 500, 20 000,

Ticks ® 8Π Range@-10, 10, 1 4D, Automatic<, PlotStyle -> [email protected] * ðD,

PlotRange ® All

D & 880, 1<, 80.1, 1<, 80.3, 1<<;


Ticks ® 8Π Range@-10, 10, 1D, Automatic<, PlotStyle -> Hue@2 * ðD,

AspectRatio ® 0.5, PlotRange ® All

D & 880.69, 1<<;


Ticks ® 8Π Range@-10, 10, 1D, Automatic<, PlotStyle -> Hue@2 * ðD,

AspectRatio ® 0.12, PlotRange ® All

D & 880.74, 1<, 80.83, 1<<;


PlotStyle -> Hue@2 * ðD, AspectRatio ® 0.12, PlotRange ® All

D & 880.982, 1<<;

Show@pt6DShow@pt7DShow@pt8DShow@pt9D

-Π

4

Π

4

Π

2

Θ

-1.0

-0.5

0.5

Θ

Π 2 ΠΘ

-2

-1

1

2

Θ

-10 Π-9 Π-8 Π-7 Π-6 Π-5 Π-4 Π-3 Π-2 Π-Π Π2 Π3 Π4 Π5 Π6 Π7 ΠΘ

-3

3

Θ

-700 -600 -500 -400 -300 -200 -100Θ

-3

Θ


pt10 = phasePendw@8Π 2, 0<, 80.2, ð, 0.694< Flatten, 1000, 2000, 60 000,

Ticks ® 8Π Range@-1, 1, 1 4D, Automatic<, PlotStyle -> [email protected] * ðD,

PlotRange ® All

D & 880.1, 1<, 80.69, 1<, 80.83, 1<, 80.982, 1<<;

Show@GraphicsGrid@Partition@pt10, 2DDD

Θ

-0.10

-0.05

0.05

0.10

Θ

-Π -3 Π

4-

Π

2-

Π

4

Π

4

Π

2

3 Π

4Π

Θ

-2

-1

1

2

Θ

-Π -3 Π

4-

Π

2-

Π

4

Π

4

Π

2

3 Π

4Π

Θ

-2

-1

1

2

Θ

-Π -3 Π

4-

Π

2-

Π

4

Π

4

Π

2

3 Π

4Π

Θ

-2

-1

1

Θ


pt11 = poincarePendw@8Π 2, 0<, 80.2, ð, 0.694< Flatten, 1000, 2000, 60 000,

Ticks ® 8Π Range@-1, 1, 1 4D, Automatic<, PlotRange ® 88-Π, Π<, 8-Π, Π<<D &

880.1, 1<, 80.69, 1<, 80.83, 1<, 80.982, 1<<;

Show@GraphicsGrid@Partition@pt11, 2DDD

-Π -3 Π

4-

Π

2-

Π

4

Π

4

Π

2

3 Π

4Π

Θ

-3

-2

-1

1

2

3

Θ

-Π -3 Π

4-

Π

2-

Π

4

Π

4

Π

2

3 Π

4

-3

-2

-1

1

2

3

Θ

-Π -3 Π

4-

Π

2-

Π

4

Π

4

Π

2

3 Π

4Π

Θ

-3

-2

-1

1

2

3

Θ

-Π -3 Π

4-

Π

2-

Π

4

Π

4

Π

2

3 Π

4

-3

-2

-1

1

2

3

Θ

Duffing Equation

Undriven

[email protected], 0<, 81, 1, 0, 0, 1<, 0, 100, 10 000D . 8t ® Range@0, 100, 10D<

80.9, 0.89718, 0.888755, 0.874823, 0.855545,

0.831137, 0.801864, 0.768029, 0.729964, 0.688025, 0.642574<


posDuff@80, 0.4525<, 80.5, -0.61, 0, 0, 1<, 0, 100, 10 000, PlotRange ® AllD

20 40 60 80 100t

-0.5

0.5

x

[email protected], 0<, 80.5, -0.61, 0, 0, 1<, 0, 100, 10 000, PlotRange ® AllD

-0.5 0.5x

-0.4

-0.2

0.2

0.4

v

plotlist = phaseDuff@80, ð<, 80.5, -0.61, 0, 0, 1<, -17, 17, 50 000, PlotRange ®

88-1, 1<, 8-0.5, 0.5<<, AspectRatio ® 0.5D & Range@0, 0.4525, 0.4525 4D;

Show@plotlistD

-1.0 -0.5 0.5 1.0x

-0.4

-0.2

0.2

0.4

v


plotlist = phaseDuff@80, ð<, 80.5, 0.61, 0, 0, 1<, -17, 17, 50 000, PlotRange ®

88-1, 1<, 8-0.5, 0.5<<, AspectRatio ® 0.5D & Range@0, 0.4525, 0.4525 4D;

Show@plotlistD

-1.0 -0.5 0.5 1.0x

-0.4

-0.2

0.2

0.4

v

Table@8n 0.1, n 0.2<, 8n, 0, 5<D

880., 0.<, 80.1, 0.2<, 80.2, 0.4<, 80.3, 0.6<, 80.4, 0.8<, 80.5, 1.<<

plotlist =

phaseDuff@ð, 8-0.5, 0.61, 0, 0, 1<, 0, 20, 50 000, PlotRange ® 88-2, 2<, 8-1, 1<<,

AspectRatio ® 0.5D & Table@8n 0.1, 0<, 8n, -20, 20, 1<D;

Show@plotlistD

-2 -1 1 2x

-1.0

-0.5

0.5

1.0v

Driven

posDuff@80, 0.2<, 8-0.5, 0.61, 0.01, 0.1, 1<,

0, 500, 20 000, PlotRange ® All, AspectRatio ® 0.25D

100 200 300 400 500t

-1.5

-1.0

-0.5

0.5

1.0

1.5

x


phaseDuff@80, 0.2<, 8-0.5, 0.61, 0.01, 0.1, 1<, 0, 500, 20 000, PlotRange ® AllD

-1.5 -1.0 -0.5 0.5 1.0x

-0.5

0.5

v

ptd1 = posDuff@80, ð<, 8-0.5, 0.61, 0.01, 0.1, 1<, 0, 1500, 20 000,

PlotRange ® All, AspectRatio ® 0.2

D & 80.2, 0.2000001<;

Show@ptd1D

200 400 600 800 1000

-1.5

-1.0

-0.5

0.5

1.0

1.5

x

[email protected], -1.5<, 8-0.2, 0.1, 0, 1 4, 2<, 0, 500, 20 000, PlotRange ® AllD

100 200 300 400 500t

-3

-2

-1

1

2

3

x


[email protected], -1.5<, 8-0.2, 1, 0, 1 4, 2<, 0, 300, 20 000, PlotRange ® AllD

-2 -1 1 2x

-4

-2

2

4

v

[email protected], -1.5<, 8-0.2, 1, 0, 1 4, 2<, 0, 300, 20 000, PlotRange ® AllD

1.0 1.5 2.0 2.5x

-4

-2

2

4

v


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