Sub-Section 1.3.1. FORCED UNDAMPED OSCILLATOR. Nonresonance situation

Consider the forced undamped oscillator with harmonic external force. The equation of oscillation in this situation takes the forms:

a)

b)

c)

d)

e)

where is the oscillation variable, w0 is the frequency, W is the frequency of external force and f_03 are amplitudes of external forces.

The MAPLE executable expression for the differential equation of oscillation and its resulting general solution are:

a)

>	Eq_03a:= diff(x(t),t,t)+omega0^2*x(t)=f_03a*cos(Omega*t); x_03a(t):=dsolve(Eq_03a);

b)

>	Eq_03b:= diff(x(t),t,t)+omega0^2*x(t)=f_03b*sin(Omega*t); x_03b(t):=dsolve(Eq_03b);

c)

>	Eq_03c:= diff(x(t),t,t)+omega0^2*x(t)=f_031c*cos(Omega*t)+f_032c*sin(Omega*t); x_03c(t):=dsolve(Eq_03c);

d)

>	Eq_03d:= diff(x(t),t,t)+omega0^2*x(t)=f_03d*cos(n*Omega*t); x_03d(t):=dsolve(Eq_03d);

e)

>	Eq_03e:= diff(x(t),t,t)+omega0^2*x(t)=f_031e*cos(Omega*t)+f_032d*cos(n*Omega*t); x_03e(t):=dsolve(Eq_03e);

The general solution of the oscillation equation depends on the frequency w0, the frequency of external force W and two arbitrary constants _C1 and _C2.

Let f_031c equal f_03a and f_032c equal f_03b:

Note that the external force is the superposition of external forces and and that the solution is the superposition of the solution and .

Let f_031e equal f_03a and f_032e equal f_03d:

Note that the external force is the superposition of external forces and and that the solution is the superposition of the solution and .

These facts reflect property of linearity of differential equation of oscillation.

  Sub-Section 1.3.2. FORCED UNDAMPED OSCILLATOR. Resonance situation

Consider forced undamped oscillator with harmonic external force of frequency w0. This is the resonance situation. The equation of oscillation in this situation takes the forms:

f)

g)

h)

where is the oscillation variable, w0 is the frequency, W is the frequency of external force and a1 and a2 are arbitrary constant factors.

The MAPLE executable expression for the differential equation of oscillation and resulting general solution are:

f)

>	Eq_03f:= diff(x(t),t,t)+omega0^2*x(t)=f_03f*cos(omega0*t); x_03f(t):=dsolve(Eq_03f);

g)

>	Eq_03g:= diff(x(t),t,t)+omega0^2*x(t)=f_03g*sin(omega0*t); x_03g(t):=dsolve(Eq_03g);

h)

>	Eq_03h:= diff(x(t),t,t)+omega0^2*x(t)=f_031h*cos(omega0*t)+f_032h*sin(omega0*t); x_03h(t):=dsolve(Eq_03h);

The general solution of the oscillation equation depends on the frequency and the frequency of external force w0 and two rbitrary constants _C1 and _C2.

Let f_031h equal f_03f and f_032h equal f_03g:

Note that the external force is the superposition of external forces and and that the solution is the superposition of the solution and .

These facts reflect the property of linearity of differential equations of oscillation.

  Sub-Section 1.4.1. FORCED DAMPED OSCILLATOR. Single-frequency external force

Consider the forced damped oscillator with harmonic external force. The equation of oscillation in this situation takes the forms:

a)

b)

c)

where is the oscillation variable, w0 is the frequency, W is the frequency of external force, g is the damping factor and f1, f2 and f3 are constant factors.

The MAPLE expression for the differential equation of oscillation and resulting general solution are:

a)

>	Eq_04a:= diff(x(t),t,t)+2*gamma*diff(x(t),t)+omega0^2*x(t)=f_04a*cos(omega0*t); assume(gamma<omega0); x_04a(t):=dsolve(Eq_04a);

b)

>	Eq_04b:= diff(x(t),t,t)+2*gamma*diff(x(t),t)+omega0^2*x(t)=f_04b*cos((sqrt(omega0^2-2*gamma^2))*t); assume(gamma<omega0); x_04b(t):=dsolve(Eq_04b);

c)

>	Eq_04c:= diff(x(t),t,t)+2*gamma*diff(x(t),t)+omega0^2*x(t)=f_04c*cos(Omega*t); assume(gamma<omega0); x_04c(t):=dsolve(Eq_04c);

The general solution of the oscillation equation depends on frequency w0, damping factor g, and frequency of external force, two arbitrary constants _C1 and _C2, the frequency of external force W and the amplitude of external forces f.

>	Eq_02:= diff(x(t),t,t)+2*gamma*diff(x(t),t)+omega0^2*x(t)=0; Eq_04c:= diff(x(t),t,t)+2*gamma*diff(x(t),t)+omega0^2*x(t)=f_04c*cos(Omega*t); assume(gamma<omega0); x_02(t):=dsolve(Eq_02); x_04c(t):=dsolve(Eq_04c);

>	x_04c(t):=exp(-gamma*t)*sin(sqrt(-gamma^2+omega0^2)*t)*_C2+exp(-gamma*t)*cos(sqrt(-gamma^2+omega0^2)*t)*_C1+f_04c*(cos(Omega*t)*omega0^2-Omega^2*cos(Omega*t)+2*gamma*Omega*sin(Omega*t))/(Omega^4+(-2*omega0^2+4*gamma^2)*Omega^2+omega0^4);

These expressions show that the general solution for the free damped oscillator differs from the solution for the forced damped oscillator by the expression (accurate within determination of constants):

describing the presence of external force for the forced oscillator.

Note that this fact reflects the property of linearity of differential equation: the solution of a heterogeneous differential equation is the superposition of the general solution of the homogeneous equation and the particular solution of heterogeneous equation.

  Sub-Section 1.4.2. FORCED DAMPED OSCILLATOR. Multifrequency-frequency external force

Consider the forced damped oscillator with harmonic external force. The equation of oscillation in this situation takes the forms:

d)

e)

where is the oscillation variable, w0 is the frequency, g is the damping factor, W, W1, W2 are frequencies of external forces and f_04d, f_04e are amplitudes of external forces.

The MAPLE expression for the differential equation of oscillation and the resulting general solution are:

d)

>	Eq_04d:=diff(x(t),t,t)+2*gamma*diff(x(t),t)+omega0^2*x(t)=f_041d*cos(Omega*t)+f_042d*cos(2*Omega*t); assume(gamma<omega0); x_04d(t):=dsolve(Eq_04d);

e)

>	Eq_04e:=diff(x(t),t,t)+2*gamma*diff(x(t),t)+omega0^2*x(t)=f_041e*cos(Omega1*t)+f_042e*cos(Omega2*t); assume(gamma<omega0); x_04e(t):=dsolve(Eq_04e);

>

The general solutions of the oscillation equation depend on the frequency w0, damping factor g, and frequency of external force, two arbitrary constants _C1 and _C2, the frequencies of external forces W, W1, W2 and amplitudes of external forces f4, f5.