General linear oscillator and phasors

Let us consider a linear oscillator:
x'' + 2 γ x' + ω² x = f cos(Ω t + α),
with ω > γ ≥ 0 (we are using ' for time derivatives). The main window displays the phasor z = x + i y solution of the complexified equation
z'' + 2 γ z' + ω² z = f eⁱ^{(Ω
t + α)},
as well as the elongation, i.e., the real part x = Re z. Optionally, one can display the velocity, acceleration and force phasors, as well as their real parts and the evolution x(t), x'(t), x''(t) and f cos(Ω t + α).

Use the mouse to select the initial value for z and the values for ω, γ, f, Ω and α.
To get information about one element, put over it the mouse cursor.

Activities

With the default settings you can see that with the harmonic oscillator (γ = f = 0) the elongation phasor rotates with the constant angular velocity ω.
Check Graph to see the familiar x(t) curve, which repeats with period 2π/ω.
Enable Velocity to see the velocity phasor and its evolution. Which is the angle between both phasors?
Enable Acceleration to see the corresponding phasor. Which is the angle between elongation and acceleration phasors?

How would change your answers to the previous questions with damping (say γ = 0.1)? Check your guess by means of the simulation.
What happens with f > 0? You should be able to see two very different behaviors: the initial transitory and the final steady state solution.
With Ω ≅ ω, you should get resonance.
With f > 0, set γ = 0 and N = 2000. What kind of solution will you expect when Ω/ω is a simple fraction (1/2, 3/2, and so on) and with values like Ω/ω = 1.1274265? What would happen with a truly irrational quotient?

This is an English translation of the Basque original for a course on mechanics, oscillations and waves.
It requires Java 1.5 or newer and was created by Juan M. Aguirregabiria with Easy Java Simulations (Ejs) by Francisco Esquembre. I thank Wolfgang Christian and Francisco Esquembre for their help.