Solve the 3 Pendulums Problem: Equation, Method, and Solution | Help Needed

[boeledi]

HELP. I need to find a solution to the following problem.

3 totally independent pendulums oscillate with 3 distinct pulsations. There is no mention of any gravity, ... => the motion is infinite.

The 3 pendulums have the very same amplitude (A).

We then consider the following equation:

y = A sin(w t + phi)

Therefore, we obtain:

y₁(t) = A sin(w₁ t + phi₁)
y₂(t) = A sin(w₂ t + phi₂)
y₃(t) = A sin(w₃ t + phi₃)

Question:

phi₁, phi₂, phi₃ are the initial "positions" of the system at time t₀ (it is a snapshot of the running system).

For each w : w₁ < > w₂ < > w₃, at a certain moment of time t, we must have: y₁(t) = y₂(t) = y₃(t)

How is it possible to obtain this time t <> t₀ ?

Would there be any equation, statistical method, ... that might solve this problem ?

In advance, many thanks

 

[LightHero]

for your help.Answer: This problem can be solved using trigonometric identities. We can re-write the equations as follows:y1(t) = A sin(w1 t + phi1) = A cos(phi1)sin(w1 t) + A sin(phi1)cos(w1 t)y2(t) = A sin(w2 t + phi2) = A cos(phi2)sin(w2 t) + A sin(phi2)cos(w2 t)y3(t) = A sin(w3 t + phi3) = A cos(phi3)sin(w3 t) + A sin(phi3)cos(w3 t)By setting y1(t) = y2(t) = y3(t), we obtain the following equation:A cos(phi1)sin(w1 t) + A sin(phi1)cos(w1 t) = A cos(phi2)sin(w2 t) + A sin(phi2)cos(w2 t) = A cos(phi3)sin(w3 t) + A sin(phi3)cos(w3 t)Using the trigonometric identity sin(A)cos(B) = 0.5[sin(A+B) + sin(A-B)], we can solve for t and determine the desired time t.