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Circular Motion VS Mass on Spring.

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Philosophy Of Physics

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104,383 görüntüleme • 5 ay önce •via X (Twitter)

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A Triple Spring-Mass Pendulum Just for The Fun of Physics We model the mass positions as r₁ = (x₁, y₁), r₂ = (x₂, y₂), r₃ = (x₃, y₃) with y positive downward in the simulation. The three spring segments are r₁ = r₁ − (0,0) r₁₂ = r₂ − r₁ r₂₃ = r₃ − r₂ Each spring applies a Hooke force along its own direction, F(r; k, L) = −k (|r| − L) (r / |r|) and each mass also feels gravity, Fgᵢ = (0, mᵢ g) So the equations of motion are m₁ r̈₁ = Fg₁ + F(r₁; k₁, L₁) − F(r₁₂; k₂, L₂) m₂ r̈₂ = Fg₂ + F(r₁₂; k₂, L₂) − F(r₂₃; k₃, L₃) m₃ r̈₃ = Fg₃ + F(r₂₃; k₃, L₃) We integrate these ODEs forward with RK4 using a small dt. Different scenes come from changing the mass ratios and from giving the third mass a small initial kick. The yellow trail is the path of the third mass. The inset phase portraits show the same motion in state space: (x₃, vₓ₃) and (−y₃, vᵧ₃) When those curves stay tight and loop cleanly, the motion is more regular. When they spread, fold, and smear, energy is being redistributed across the whole chain between gravity, spring stretch, and kinetic motion. #NonlinearDynamics #CoupledOscillators #SpringPendulum #ChaosInMotion #ComputationalPhysics #MathematicalVisualization
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