Schwingungs_DGL.sagews

# Uebung 11, Aufgabe 5a

x = var('x');
y = function('y')(x)
DE = diff(y,x,2) == y+exp(x)+x^2
DE.show()

s=desolve(DE, [y(x),x],contrib_ode=True,show_method=True)
show(s)

$\displaystyle \frac{\partial^{2}}{(\partial x)^{2}}y\left(x\right) = x^{2} + e^{x} + y\left(x\right)$

/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py:1230: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...) See http://trac.sagemath.org/5930 for details. exec(

[$\displaystyle -x^{2} + K_{2} e^{\left(-x\right)} + K_{1} e^{x} + \frac{1}{4} \, {\left(2 \, x - 1\right)} e^{x} - 2$, variationofparameters]

# Uebung 11, Aufgabe 5b

x = var('x');
y = function('y')(x)
DE = diff(y,x,2) == -5*diff(y,x,1)+15*x^2
DE.show()

s=desolve(DE, [y(x),x],contrib_ode=True,show_method=True)
show(s)

$\displaystyle \frac{\partial^{2}}{(\partial x)^{2}}y\left(x\right) = 15 \, x^{2} - 5 \, \frac{\partial}{\partial x}y\left(x\right)$

/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py:1230: DeprecationWarning: Substitution using function-call syntax and unnamed arguments is deprecated and will be removed from a future release of Sage; you can use named arguments instead, like EXPR(x=..., y=...) See http://trac.sagemath.org/5930 for details. exec(

[$\displaystyle x^{3} - \frac{3}{5} \, x^{2} + K_{2} e^{\left(-5 \, x\right)} + K_{1} + \frac{6}{25} \, x - \frac{6}{125}$, variationofparameters]

# freier harmonischer Oszillator
t = var('t') # unabhängige Variable
x = function('x', t) # abhängige Variable
var('omega0') # Kreisfrequenz
var('v0 x0')
omega0=1
x0=1

v0=0

DE = diff(x,t,2) == -omega0^2*x
DE.show()
fun= desolve(DE,dvar=x,ivar=t, ics=[0,x0,v0])
show(fun)


plot(fun, (t,0, 10), ymin=-1, ymax=1,gridlines=True,  axes_labels=['Zeit $t$','Amplitude $x(t)$'],legend_label='$x_0=1,\, v_0=0,\, \omega_0=1$', legend_color=None)

Error in lines 2-2 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1230, in execute exec( File "", line 1, in <module> File "sage/calculus/var.pyx", line 133, in sage.calculus.var.function (build/cythonized/sage/calculus/var.c:1623) def function(s, **kwds): TypeError: function() takes exactly 1 positional argument (2 given)

# gedämpfter harmonischer Oszillator, schwache Dämpfung (Schwingfall)
t = var('t') # unabhängige Variable
x = function('x', t) # abhängige Variable
var('omega0') # Kreisfrequenz
var('delta') # Dämpfung
var('v0 x0')
omega0=2 # Kreisfrequenz
delta=0.2 # Dämpfung
x0=1 # Anfangswerte

v0=0  # Anfangswerte

DE = diff(x,t,2) == -omega0^2*x-2*delta*diff(x,t)
DE.show()
fun= desolve(DE,dvar=x,ivar=t, ics=[0,x0,v0])
show(fun)
p=plot(fun, (t,0, 10), ymin=-1, ymax=1,gridlines=True,  axes_labels=['Zeit $t$','Amplitude $x(t)$'], legend_label='$x_0=1,\, v_0=0,\, \omega_0=2,\, \delta=0,2$', legend_color=None)
e1=plot(exp(-0.2*t) ,(t,0, 10),color='red')
e2=plot(-exp(-0.2*t), (t,0, 10),color='red')
show(p+e1+e2)

omega0 delta (v0, x0)

$\displaystyle D[0, 0]\left(x\right)\left(t\right) = -4 \, x\left(t\right) - 0.400000000000000 \, D[0]\left(x\right)\left(t\right)$

$\displaystyle \frac{1}{33} \, {\left(\sqrt{11} \sin\left(\frac{3}{5} \, \sqrt{11} t\right) + 33 \, \cos\left(\frac{3}{5} \, \sqrt{11} t\right)\right)} e^{\left(-\frac{1}{5} \, t\right)}$

# gedämpfter harmonischer Oszillator, starke Dämpfung (Kriechfall)
t = var('t') # unabhängige Variable
x = function('x', t) # abhängige Variable
var('omega0') # Kreisfrequenz
var('delta') # Dämpfung
var('v0 x0')
omega0=2 # Kreisfrequenz
delta=3 # Dämpfung
x0=1 # Anfangswerte
v0=0  # Anfangswerte
D=sqrt(delta^2-omega0^2)
show(D)

DE = diff(x,t,2) == -omega0^2*x-2*delta*diff(x,t)
DE.show()
fun= desolve(DE,dvar=x,ivar=t, ics=[0,x0,v0])
show(fun)
p=plot(fun, (t,0, 10), ymin=0, ymax=1,gridlines=True,  axes_labels=['Zeit $t$','Amplitude $x(t)$'], legend_label='$x_0=1,\, v_0=0,\, \omega_0=2,\, \delta=3$', legend_color=None)

show(p)

omega0 delta (v0, x0)

$\displaystyle \sqrt{5}$

$\displaystyle D[0, 0]\left(x\right)\left(t\right) = -4 \, x\left(t\right) - 6 \, D[0]\left(x\right)\left(t\right)$

$\displaystyle -\frac{1}{10} \, {\left(3 \, \sqrt{5} - 5\right)} e^{\left(-t {\left(\sqrt{5} + 3\right)}\right)} + \frac{1}{10} \, {\left(3 \, \sqrt{5} + 5\right)} e^{\left(t {\left(\sqrt{5} - 3\right)}\right)}$

# Frequenzgang bei erzwungener Schwingung
var('omega')# A
var('omega0') # Kreisfrequenz
var('delta') # Dämpfung
var('F0, m')
A(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))
show(A)
omega0=2 # Kreisfrequenz
delta=0.2 # Dämpfung
F0=1 #
m=1 #
delta=0.05
A0(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))
delta=0.2
A1(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))
delta=0.5
A2(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))
delta=1
A3(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))

p0=plot(A0, (omega,0, 5), ymin=-0.5, ymax=1.5,gridlines=True, color='red',  axes_labels=['Frequenz $\omega$','Amplitude $A$'], legend_label='$ \delta=0,05$', legend_color=None)
p1=plot(A1, (omega,0, 5), ymin=-0.5, ymax=1.5,gridlines=True,  color='blue', axes_labels=['Frequenz $\omega$','Amplitude $A$'], legend_label='$ \delta=0,2$', legend_color=None)
p2=plot(A2, (omega,0, 5), ymin=-0.5, ymax=1.5,gridlines=True, color='green',  axes_labels=['Frequenz $\omega$','Amplitude $A$'], legend_label='$ \delta=0,5$', legend_color=None)
p3=plot(A3, (omega,0, 5), ymin=-0.5, ymax=1.5,gridlines=True, color='yellow',  axes_labels=['Frequenz $\omega$','Amplitude $A$'], legend_label='$ \delta=1$', legend_color=None)

#Max=diff(A,omega)
#show(Max)

var('wr delta')


wr(delta)=sqrt(omega0^2-2*delta^2)


#Amax(wr)=F0/(m*sqrt((omega0^2-wr^2)^2+2*(omega0^2-wr^2)*wr^2))# Maximumkurve in Abhängigkeit von wr
Amax(wr)=F0/(m*sqrt(omega0^4- wr^4))# Maximumkurve in Abhängigkeit von wr


p4=plot(Amax, (wr,0.1, 2), ymin=-0.5, ymax=1.5,gridlines=True, linestyle='--',color='black',  axes_labels=['Frequenz $\omega_r$','Amplitude $A$'], legend_label='Kurve der max. Amplitude', legend_color=None)

show(p0+p1+p2+p3+p4)
#show(p4)

omega omega0 delta (F0, m)

$\displaystyle \omega \ {\mapsto}\ \frac{F_{0}}{\sqrt{4 \, \delta^{2} \omega^{2} + {\left(\omega^{2} - \omega_{0}^{2}\right)}^{2}} m}$

(wr, delta)

/ext/sage/sage-8.0/local/lib/python2.7/site-packages/matplotlib/font_manager.py:273: UserWarning: Matplotlib is building the font cache using fc-list. This may take a moment. warnings.warn('Matplotlib is building the font cache using fc-list. This may take a moment.')

# Frequenzgang bei erzwungener Schwingung
var('omega')# A
var('omega0') # Kreisfrequenz
var('delta') # Dämpfung
var('F0, m')
A(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))
show(A)
omega0=2 # Kreisfrequenz
delta=0.2 # Dämpfung
F0=1 #
m=1 #
delta=0.05
A0(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))
delta=0.2
A1(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))
delta=0.5
A2(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))
delta=1
A3(omega)=F0/(m*sqrt((omega0^2-omega^2)^2+4*delta^2*omega^2))

p0=plot(A0, (omega,0, 5), ymin=-0.5, ymax=1.5,gridlines=True, color='red',  axes_labels=['Frequenz $\omega$','Amplitude $A$'], legend_label='$ \delta=0,05$', legend_color=None)
p1=plot(A1, (omega,0, 5), ymin=-0.5, ymax=1.5,gridlines=True,  color='blue', axes_labels=['Frequenz $\omega$','Amplitude $A$'], legend_label='$ \delta=0,2$', legend_color=None)
p2=plot(A2, (omega,0, 5), ymin=-0.5, ymax=1.5,gridlines=True, color='green',  axes_labels=['Frequenz $\omega$','Amplitude $A$'], legend_label='$ \delta=0,5$', legend_color=None)
p3=plot(A3, (omega,0, 5), ymin=-0.5, ymax=1.5,gridlines=True, color='yellow',  axes_labels=['Frequenz $\omega$','Amplitude $A$'], legend_label='$ \delta=1$', legend_color=None)

#Max=diff(A,omega)
#show(Max)

var('wr delta')


wr(delta)=sqrt(omega0^2-2*delta^2)


#Amax(wr)=F0/(m*sqrt((omega0^2-wr^2)^2+2*(omega0^2-wr^2)*wr^2))# Maximumkurve in Abhängigkeit von wr
Amax(wr)=F0/(m*sqrt(omega0^4- wr^4))# Maximumkurve in Abhängigkeit von wr


p4=plot(Amax, (wr,0.1, 2), ymin=-0.5, ymax=1.5,gridlines=True, linestyle='--',color='black',  axes_labels=['Frequenz $\omega_r$','Amplitude $A$'], legend_label='Kurve der max. Amplitude', legend_color=None)

show(p0+p1+p2+p3+p4)
#show(p4)

omega omega0 delta (F0, m)

$\displaystyle \omega \ {\mapsto}\ \frac{F_{0}}{\sqrt{4 \, \delta^{2} \omega^{2} + {\left(\omega^{2} - \omega_{0}^{2}\right)}^{2}} m}$

(wr, delta)