Coupled oscillations and normal modes

• One may select initial positions (x1 = x₁ and x2 = x₂), damping coefficients (γ₁ and γ₂) elastic constants for the springs (k1 = k, kk = k₂ and k3 = k), and the amplitude (F/m) and frequency (Ω) of the external force applied on the left mass.
• The unit mass is that of both block (m=1).
• You may also choose initial positions with the mouse.
• If desired, the displacements from the equilibrium points can be shown, by means of vectors or/and in a graph. The external force is shown along with the displacement vectors.
• It is also possible to display the evolution in the configuration plane (x₁,x₂) or in normal coordinates (Q₁,Q₂), as well as the time evolution of those coordinates.
• To get information on an element, put over it the mouse pointer to see the corresponding tooltip.

Activities

1. Put the system in its first normal mode.
2. Put the system in its second normal mode.
3. What happens, with a small value for F/m, if the frequency Ω is a bit higher (lower) than the first (second) normal frequency?
4. What changes in the previous cases when the damping coefficients are not zero?
5. Use the simulation to analyze beats by setting a small value for the stiffness kk of the middle spring.
6. Compute the normal coordinates and guess the evolution in the (Q₁,Q₂) plane. Check your guess with the simulation in the different cases.