Amortiguamiento de Coulomb

Datos

m = 3.0;

k = 1000;

Μ = 0.5;

g = 9.81;

x0 = 0.1;

v0 = 1;

t0 = 0;

Ecuaciones

Ωn =

k

m

Τ =

2 Π

Ωn

Primer Sección

Sol1 = DSolve@8m x''@tD + k x@tD -Μ m g, x@t0D x0, x'@t0D v0<, x@tD, tD

x01t = x@tD . Sol1@@1DD

PlotB:x01t, -

Μ g m

k

>, 8t, 0, Τ<, PlotStyle ® 8Blue, 8Blue, Dashed<<F

v01t = D@x01t, tDSol11 = FindRoot@v01t 0, 8t, 0<Dt1 = Re@tD . Sol11@@1DDx1 = Re@x01tD . t ® t1

Segunda Sección

Sol2 = DSolve@8m x''@tD + k x@tD +Μ m g, x@t1D x1, x'@t1D 0<, x@tD, tDx12t = x@tD . Sol2@@1DDPlot@x12t, 8t, 0, Τ<D

PlotB:x01t, x12t, +

Μ g m

k

>, 8t, 0, Τ<,

PlotStyle ® 8Blue, 8Red, Dashed<, 8Red, Dashed<<F

v12t = D@x12t, tD

Sol21 = FindRootBv12t 0, :t, t1 +

Τ

2

>F

t2 = t . Sol21@@1DDx2 = x12t . t ® t2

Tercera Sección

Sol3 = DSolve@8m x''@tD + k x@tD -Μ m g, x@t2D x2, x'@t2D 0<, x@tD, tDx23t = x@tD . Sol3@@1DDPlot@x23t, 8t, 0, Τ<D

PlotB:x01t, x12t, x23t, -

Μ g m

k

>, 8t, 0, Τ<,

PlotStyle ® 8Blue, Red, 8Green, Dashed<, 8Green, Dashed<<F

v23t = D@x23t, tD

Sol31 = FindRootBv23t 0, :t, t2 +

Τ

2

>F

t3 = Re@tD . Sol31@@1DDx3 = Re@x23tD . t ® t3

Cuarta Sección

Sol4 = DSolve@8m x''@tD + k x@tD +Μ m g, x@t3D x3, x'@t3D 0<, x@tD, tDx34t = x@tD . Sol4@@1DD

PlotB:x01t, x12t, x23t, x34t,Μ g m

k

>, 8t, 0, 2 Τ<,

PlotStyle ® 8Blue, Red, Green, 8Magenta, Dashed<, 8Magenta, Dashed<<F

v34t = D@x34t, tD

Sol41 = FindRootBv34t 0, :t, t3 +

Τ

2

>F

t4 = Re@tD . Sol41@@1DDx4 = Re@x34tD . t ® t4

Quinta Sección

Sol5 = DSolve@8m x''@tD + k x@tD -Μ m g, x@t4D x4, x'@t4D 0<, x@tD, tDx45t = x@tD . Sol5@@1DD

PlotB:x01t, x12t, x23t, x34t, x45t, -

Μ g m

k

>, 8t, 0, 2 Τ<,

PlotStyle ® 8Blue, Red, Green, Magenta, 8Blue, Dashed<, 8Blue, Dashed<<F

v45t = D@x45t, tD

Sol51 = FindRootBv45t 0, :t, t4 +

Τ

2

>F

t5 = Re@tD . Sol51@@1DDx5 = Re@x45tD . t ® t4

Gráfica

2 Amortiguamiento de Coulomb.nb

Gráfica

xTot = Piecewise@88x01t, t < t1<, 8x12t, t < t2<, 8x23t, t < t3<, 8x34t, t < t4<, 8x45t, t < t5<<D

PlotB:xTot,Μ m g

k

, -

Μ m g

k

>, 8t, 0, 3 Τ<, PlotRange ® All,

AxesLabel ® 8"t", "x@mD"<, PlotLabel ® "Amortiguamiento de Coulomb"F

Envolventes

t1

x1

m1 =

-2Μ m g

k

Τ

2

Sol6 = FindRoot@v12t m1, 8t, t1<Dtenv = t . Sol6@@1DDxenv = x12t . t -> tenv

xEnv1 = m1 Ht - tenvL + xenv

xEnv2 = -xEnv1

PlotB:xTot,Μ m g

k

, -

Μ m g

k

, xEnv1, xEnv2>, 8t, 0, 3 Τ<, PlotRange ® All,

AxesLabel ® 8"t", "x@mD"<, PlotLabel ® "Amortiguamiento de Coulomb"F

Amortiguamiento de Coulomb.nb 3