Introduction

DEtools Package and plots Package

>	restart; with(DEtools): with(plots):

Warning, the name changecoords has been redefined

Examples

Consider some examples of solutions for oscillatory systems with fixed parameters and initial conditions.

1) Free damped oscillator .

This oscillator is characterized by the frequency   , damping factor   g and initial conditions .

2) Forced damped oscillator I .

This oscillator is characterized by the   frequency   , damping factor   g , amplitude of external force , frequency   of external force and initial conditions   I .

3) Forced damped oscillator II .

This oscillator is characterized by the   frequency   , damping factor   g , amplitude of external force , frequency   of external force and initial condition s  II.  

4) Forced damped oscillator III .

This oscillator is characterized by the frequency   , damping factor   g , amplitude of external force , frequency   of external force which is equal to the frequency of damped   oscillation   and initial condition s  III.

5) Forced undamped oscillator IV .

This oscillator is characterized by the frequency   , damping factor   g , amplitude of external force , the frequency   of external force which is equal to resonance frequency and initial condition s  IV.

   Section 3.1. FREE DAMPED OSCILLATOR

Consider the free damped oscillator. The equation of oscillation and initial conditions in this situation take the forms:

where x(t) is the oscillation variable,  is the frequency, and g  is the damping factor.

MAPLE  expressions for

- fixed parameters,

- initial conditions,

- frequency of damped   oscillation,

- amplitude of free undamped oscillation;

- differential equation of oscillation,

- resulting analytical solution,

- plot

are:

>	omega0:=4; gamma0:=1/2; x0:=7/4; v0:=3; omega1:=sqrt(omega0^2-gamma0^2); Amp:=sqrt(x0^2+(v0/omega0)^2); Eqn_01:=diff(x(t),t,t)+2*gamma0*diff(x(t),t)+omega0^2*x(t)=0; sol_01:=dsolve( {Eqn_01,x(0)= x0,D(x)(0)=v0}, x(t)); plot( rhs(sol_01), t=0..15);

Resulting solution contains products of harmonic functions with the frequency of damped   oscillation w1  and exponential functions:

>	x_01_harm1(t):=sin(3/2*sqrt(7)*t); x_01_harm2(t):=cos(3/2*sqrt(7)*t); x_01_exp(t):=exp(-1/2*t);

Plots of these functions are:

>	plot([x_01_harm1(t),x_01_harm2(t), x_01_exp(t)], t=0..15,linestyle=[1,1,3],color=[red,blue,magenta], thickness=[2,2,2]);

Product of the amplitude of oscillation Amp  and the decreasing exponential function is the envelope curve for the the resulting solution x_01(t):

>	rhs(sol_01); Amp*x_01_exp(t);-Amp*x_01_exp(t);

>	plot([rhs(sol_01),Amp*x_01_exp(t),-Amp*x_01_exp(t)], t=0..15, linestyle=[1,3,3], color=[red,blue,blue],thickness=[3,2,2]);

Now we can compare solutions for the damped and undamped   oscillations. Solution for the undamped   oscillations we can obtain as follows:

>	Eqn_01:=diff(x(t),t,t)+omega0^2*x(t)=0; sol_01_UD:=dsolve( {Eqn_01,x(0)= x0,D(x)(0)=v0}, x(t));

>	plot([rhs(sol_01),rhs(sol_01_UD)],t=0..15,linestyle=[1,3],color=[red,blue],thickness=[3,2]); plot([rhs(sol_01),rhs(sol_01_UD)],t=0..2.1,linestyle=[1,3],color=[red,blue],thickness=[3,2]);

The first plot demonstrates the decreasing of damped   oscillations in comparison with undamped oscillations.

The last plot demonstrates the difference of frequencies (and periods) of undamped   and damped   oscillations, too: the period of undamped oscillations is greater than the period of damped oscillations.

   Section 3.2. FORCED DAMPED OSCILLATOR I

Consider the forced damped oscillator. The equation of oscillation and initial conditions in this situation take the forms:

where x(t) is the oscillation variable and f0  is the amplitude of the external force.

MAPLE expressions for

- fixed parameters,

- initial conditions,

- frequency of damped   oscillation,

- frequency of external force,

- amplitude of external force,

- differential equation of oscillation,

- resulting analytical solution,

- plot

are:

>	omega0:=2; gamma0:=7/20; x0:=2; v0:=0; omega1:=sqrt(omega0^2-gamma0^2); f_02:=4; Omega:=3; Eqn_02:=diff(x(t),t,t)+2*gamma0*diff(x(t),t)+omega0^2*x(t)=f_02*cos(Omega*t); sol_02:=dsolve( {Eqn_02,x(0)= x0,D(x)(0)=v0}, x(t)); plot( rhs(sol_02), t=0..30);

This solution contains products of harmonic and exponential functions:

>	x_02_harm1(t):=sin(1/20*sqrt(1551)*t); x_02_harm2(t):=cos(1/20*sqrt(1551)*t); x_02_exp(t):=exp(-7/20*t); x_02_reg1(t):=sin(3*t); x_02_reg2(t):=cos(3*t);

Plots of these functions are:

>	plot([x_02_harm1(t),x_02_harm2(t), x_02_exp(t)], t=0..25,linestyle=[1,1,3],color=[red,blue,magenta], thickness=[1,1,1]); plot([x_02_reg1(t),x_02_reg2(t), x_02_exp(t)], t=0..25,linestyle=[1,1,3],color=[red,blue,magenta], thickness=[1,1,1]);

Solution x_02(t) is the superposition of damped oscillation and forced oscillation:

where

>	x_02_pr(t):=434/414681*exp(-7/20*t)*sin(1/20*sqrt(1551)*t)*sqrt(1551)+7882/2941*exp(-7/20*t)*cos(1/20*sqrt(1551)*t); x_02_reg(t):=840/2941*sin(3*t)-2000/2941*cos(3*t); plot([rhs(sol_02), x_02_pr(t),x_02_reg(t)], t=0..25, linestyle=[1,3,4], color=[gold,blue, magenta]);

Plots of the resulting solution x_02(t) and x_02_pr(t) show the presence of transitional and steady state oscillations. Steady state corresponds to damping of oscillation of the system.

The product of the amplitude A_pr  and the decreasing exponential function is the envelope curve for the resulting solution x_02(t)

>	A_pr:=eval(subs(t=0, x_02_pr(t))): plot([x_02_pr(t),A_pr*x_02_exp(t),-A_pr*x_02_exp(t)], t=0..25, linestyle=[1,3,3], color=[red,blue,blue],thickness=[3,1,1]);

The next plot demostrates the resulting solution x_02(t):

a) the steady state of oscillation corresponds to damping of oscillation,

>	plot([rhs(sol_02),A_pr*x_02_exp(t),-A_pr*x_02_exp(t), x_02_pr(t)], t=0..25,linestyle=[1,3,3,3],color=[gold,blue,blue,magenta],thickness=[3,1,1,2]);

b) in the steady state, the resulting solution differs from the external force by the constant phase f :

>	phi:=Pi-arctan(840/2000); phi:=evalhf(Pi-arctan(840/2000));

This value of the phase is nearly to value of p . Therefore during steady state resulting solution is almost in reversed phase to external force:

>	plot([rhs(sol_02),A_pr*x_02_exp(t),-A_pr*x_02_exp(t),f_02*cos(Omega*t)], t=0..25,linestyle=[1,3,3,1],color=[gold,blue,blue,magenta],thickness=[3,1,1,1]);

   Section 3.3. FORCED UNDAMPED OSCILLATOR

Consider the forced damped oscillator with null initial position and velocity and external force with frequency equal to the resonance frequency. The equations of the oscillation take the forms:

where x(t) is the oscillation variable,    is the frequency, g  is damping factor,  is the frequency of external force, f0  is the amplitude of external force and

is the resonance frequency.

MAPLE  expressions for

- fixed parameters,

- initial conditions,

- frequency of damped   oscillation,

- frequency of external force   (resonance frequency),

- amplitude of damped   oscillation,

- differential equation of oscillation,

- resulting analytical solution,

- plot

are:

>	omega0:=5; gamma0:=1/4; x0:=0; v0:=0; omega1:=sqrt(omega0^2-gamma0^2); f_03:=4; Omega1:=sqrt(omega0^2-2*gamma0^2); Eqn_03:=diff(x(t),t,t)+2*gamma0*diff(x(t),t)+omega0^2*x(t)=f_03*cos(Omega1*t); sol_03:=dsolve( {Eqn_03,x(0)= x0,D(x)(0)=v0}, x(t)); plot(rhs(sol_03), t=0..25);

The above solution contains products of harmonic and exponential functions and harmonic function corresponding to steady state of oscillation

>	x_03_harm1(t):=sin(1/4*sqrt(399)*t); x_03_harm2(t):=cos(1/4*sqrt(399)*t); x_03_exp(t):=exp(-1/4*t); x_03_reg1(t):=sin(1/4*399^(1/2)*t); x_03_reg2(t):=cos(1/4*399^(1/2)*t);

Plots of these functions are:

>	plot([x_03_harm1(t),x_03_harm2(t), x_03_exp(t)], t=0..25,linestyle=[1,1,3],color=[red,blue,magenta], thickness=[1,1,1]); plot([x_03_reg1(t),x_03_reg2(t), x_03_exp(t)], t=0..25,linestyle=[1,1,3],color=[red,blue,magenta], thickness=[1,1,1]);

Solution x_03(t) is the superposition of damped oscillation and forced oscillation:

where

>	x_03_pr(t):=-32/399*exp(-1/4*t)*sin(1/4*399^(1/2)*t)*399^(1/2)-32/399*exp(-1/4*t)*cos(1/4*399^(1/2)*t); x_03_reg(t):=32/399*sin(1/4*398^(1/2)*t)*398^(1/2)+32/399*cos(1/4*398^(1/2)*t); plot([rhs(sol_03), x_03_pr(t),x_03_reg(t)], t=0..20, color=[gold,blue, magenta],thickness=[3,2,1]);

Plots of the resulting solution x_03(t) and x_03_pr(t) show presence of transitional and steady state of oscillation. Steady state corresponds to damping of oscillation of the system.

The product of the amplitude A_pr  and the decreasing exponential function is the envelope curve for the resulting solution x_03(t)

>	A_pr:=((32/399)^2+((-32/399)*sqrt(399))^2)^(1/2): plot([x_03_pr(t),A_pr*x_03_exp(t),-A_pr*x_03_exp(t)], t=0..25, linestyle=[1,3,3], color=[red,blue,blue],thickness=[3,1,1]);

The next plot demonstrates the resulting solution x_03(t):

a) steady state of oscillation corresponds to the damping of oscillation,

>	plot([rhs(sol_03),A_pr*x_03_exp(t),-A_pr*x_03_exp(t), x_03_pr(t)], t=0..25,linestyle=[1,3,3,3],color=[gold,blue,blue,magenta],thickness=[3,1,1,2]);

b) if the product of the amplitude A_pr  and the decreasing exponential function

are envelope curves for the solution, then the functions

are envelope curves for the resulting forced solution x_03(t)

>	plot([rhs(sol_03),A_pr*(1-x_03_exp(t)),-A_pr*(1-x_03_exp(t))], t=0..25,linestyle=[1,3,3],color=[gold,blue,blue],thickness=[3,1,1]);

This value of the phase is nearly to value of pi/2 . Therefore, during steady state, the resulting solution is out-of-phase with the external force by almost pi/2 :

>	plot([rhs(sol_03),A_pr*x_03_exp(t),-A_pr*x_03_exp(t),f_03*cos(Omega1*t)], t=0..25,linestyle=[1,3,3,1],color=[gold,blue,blue,magenta],thickness=[3,1,1,1]);

   Section 3.4. FORCED DAMPED OSCILLATOR

Consider the forced damped oscillator with null initial position and external force with frequency equal to the resonance frequency. The equations of oscillation in this situation take the forms:

where

is the resonance frequency.

MAPLE  expression for

- fixed parameters,

- initial conditions,

- frequency of damped   oscillation,

- frequency of external force   (resonance frequency) ,

- amplitude of damped   oscillation,

- differential equation of oscillation,

- resulting analytical solution,

- plot

are:

>	omega0:=5; gamma0:=1/4; x0:=0; v0:=22; omega1:=sqrt(omega0^2-gamma0^2); f_04:=4; Omega1:=sqrt(omega0^2-2*gamma0^2); Eqn_04:=diff(x(t),t,t)+2*gamma0*diff(x(t),t)+omega0^2*x(t)=f_04*cos(Omega1*t); sol_04:=dsolve( {Eqn_04,x(0)= x0,D(x)(0)=v0}, x(t)); plot(rhs(sol_04), t=0..25);

The resulting solution contains products of harmonic and exponential functions and harmonic functions corresponding to the steady state:

>	x_04_harm1(t):=sin(1/4*sqrt(399)*t); x_04_harm2(t):=cos(1/4*sqrt(399)*t); x_04_exp(t):=exp(-1/4*t); x_04_reg1(t):=sin(1/4*sqrt(398)*t); x_04_reg2(t):=cos(1/4*sqrt(398)*t);

Plots of these functions are:

>	plot([x_04_harm1(t),x_04_harm2(t), x_04_exp(t)], t=0..25,linestyle=[1,1,3],color=[red,blue,magenta], thickness=[1,1,1]); plot([x_04_reg1(t),x_04_reg2(t), x_04_exp(t)], t=0..25,linestyle=[1,1,3],color=[red,blue,magenta], thickness=[1,1,1]);

Solution x_04(t) is the superposition of damped oscillation and forced oscillation:

where

>	x_04_pr(t):=8/57*exp(-1/4*t)*sin(1/4*sqrt(399)*t)*sqrt(399)-32/399*exp(-1/4*t)*cos(1/4*sqrt(399)*t); x_04_reg(t):=32/399*sin(1/4*398^(1/2)*t)*398^(1/2)+32/399*cos(1/4*398^(1/2)*t); plot([rhs(sol_04), x_04_pr(t),x_04_reg(t)], t=0..25, color=[gold,blue, magenta]);

Plots of the resulting solution x_04(t) and x_04_pr(t) show the presence of transitional and steady state oscillations.

The product of the amplitude A_pr  and the decreasing exponential function is the envelope curve for the resulting solution x_04(t)

>	A_pr:=(((8/57)*sqrt(399))^2+(32/399)^2)^(1/2): plot([x_04_pr(t),A_pr*x_04_exp(t),-A_pr*x_04_exp(t)], t=0..25, linestyle=[1,3,3], color=[red,blue,blue],thickness=[3,1,1]);

Next plot demonstrates resulting solution x_04(t):

a) steady state of oscillation corresponds to damping of oscillation,

>	plot([rhs(sol_04),A_pr*x_04_exp(t),-A_pr*x_04_exp(t), x_04_pr(t)], t=0..25,linestyle=[1,3,3,3],color=[gold,blue,blue,magenta],thickness=[3,1,1,2]);

b) if the product of the amplitude A_pr  and the decreasing exponential function

are envelope curves for the solution, then the functions

are envelope curves for the resulting forced solution x_03(t) ( A_pr  is the amplitude of the oscillation, A_reg  is the amplitude oscillation during steady state):

>	A_reg:=sqrt((32/399*sqrt(398))^2+(32/399)^2): plot([rhs(sol_04),A_pr*(x_04_exp(t))+A_reg,-A_pr*(x_04_exp(t))-A_reg], t=0..25,linestyle=[1,3,3],color=[gold,blue,blue],thickness=[3,1,1]);

c) during steady state resulting solution differs from external force by a constant phase:

This value of the phase is nearly to value of pi/2. Therefore during steady state resulting solution is out-of-phase with the external force by almost pi/2:

>	plot([rhs(sol_04),A_pr*x_04_exp(t)+A_reg,-A_pr*x_04_exp(t)-A_reg,f_04*cos(Omega1*t)], t=0..25,linestyle=[1,3,3,1],color=[gold,blue,blue,magenta],thickness=[3,1,1,1]);

   Section 3.5. FORCED DAMPED OSCILLATOR

Consider the forced damped oscillator with null initial position and external force with frequency equal to the resonance frequency. The equations of oscillation take the forms:

where

is the resonance frequency.

MAPLE expressions for

- fixed parameters,

- initial conditions,

- frequency of damped   oscillation,

- frequency of external force   (resonance frequency),

- amplitude of damped   oscillation,

- differential equation of oscillation,

- resulting analytical solution,

- plot

are:

>	omega0:=5; gamma0:=1/4; x0:=4; v0:=0; omega1:=sqrt(omega0^2-gamma0^2); f_05:=4; Omega1:=sqrt(omega0^2-2*gamma0^2); Eqn_05:=diff(x(t),t,t)+2*gamma0*diff(x(t),t)+omega0^2*x(t)=f_05*cos(Omega1*t); sol_05:=dsolve( {Eqn_05,x(0)= x0,D(x)(0)=v0}, x(t)); plot(rhs(sol_05), t=0..25);

Resulting solution contains products of harmonic and exponential functions:

>	x_05_harm1(t):=sin(1/4*sqrt(399)*t); x_05_harm2(t):=cos(1/4*sqrt(399)*t); x_05_exp(t):=exp(-1/4*t); x_05_reg1(t):=sin(1/4*398^(1/2)*t); x_05_reg2(t):=cos(1/4*398^(1/2)*t);

Plots of these functions are:

>	plot([x_05_harm1(t),x_05_harm2(t), x_05_exp(t)], t=0..25,linestyle=[1,1,3],color=[red,blue,magenta]); plot([x_05_reg1(t),x_05_reg2(t), x_05_exp(t)], t=0..25,linestyle=[1,1,3],color=[red,blue,magenta]);

The solution x_05(t) is the superposition of damped and forced oscillation:

Plots of the resulting solution x_05(t) and x_05_pr(t) show the presence of transitional and steady state oscillation.

The product of the amplitude A_pr  and the decreasing exponential function is the envelope curve for the resulting solution x_05(t)

>	x_05_pr(t):=-4/57*exp(-1/4*t)*sin(1/4*399^(1/2)*t)*399^(1/2)+1564/399*exp(-1/4*t)*cos(1/4*399^(1/2)*t); x_05_reg(t):=32/399*sin(1/4*398^(1/2)*t)*398^(1/2)+32/399*cos(1/4*398^(1/2)*t);

>	A_pr:=sqrt((((4/57)*sqrt(399))^2+(1564/399)^2)): plot([x_05_pr(t),A_pr*x_05_exp(t),-A_pr*x_05_exp(t)], t=0..25, linestyle=[1,3,3], color=[red,blue,blue],thickness=[3,1,1]);

Next plot demonstrates the resulting solution x_04(t):

a) steady state of oscillation corresponds to damping of oscillation,

>	plot([rhs(sol_05),A_pr*x_05_exp(t),-A_pr*x_05_exp(t), x_05_pr(t)], t=0..25,linestyle=[1,3,3,3],color=[gold,blue,blue,magenta],thickness=[3,1,1,2]);

b) during steady state, the resulting solution differs from the external force by a constant phase:

This value of the phase is nearly to value of pi/2 . Therefore, during steady state, the resulting solution is out-of-phase with the external force by almost pi/2 :

>	plot([rhs(sol_05),f_05*cos(Omega1*t)],t=0..25,linestyle=[1,1],color=[gold,magenta],thickness=[3,1]);

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