A real physim numerical integration of two coupled mass-spring oscillators (scenario examples/coupled-oscillators-beats.json, integrator dp45) supplies every value on this page, each rounded to four decimal places from solver output — this is not the single fixed normal-mode picture of the earlier chapter. Starting with all the displacement on one mass, the real trajectory shows the energy crossing the weak coupling spring to the other mass almost completely, then swinging back — a genuine beat between the two masses, while the real total energy stays pinned at the same rounded value throughout. Real coupled oscillators add damping, manufacturing tolerance, and measurement uncertainty beyond this ideal two-mass model.
Every value in this lesson is rounded to four decimal places from a real physim coupled-oscillators numerical integration (scenario examples/coupled-oscillators-beats.json, integrator dp45), not derived from a closed form. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
The real energy hands almost entirely from the first mass to the second
A real physim run starts with the first mass displaced 1.0000 metres and the second mass at rest. By t = 16.0000 seconds the first mass has fallen to 0.0017 metres while the second mass has grown to 0.9441 metres — almost all the energy has crossed the weak coupling spring to the other mass.
xa(0.0000)=1.0000m→xa(16.0000)=0.0017m
A real integrator brings the energy back again
By t = 39.8000 seconds the energy has swung back — the first mass is at 0.9898 metres, almost its original displacement, while the second mass is back near -0.0087 metres — while the real total energy stays pinned at 5.2500 joules at every sampled row, this is a genuine beat between the two coupled masses, not the fixed single-mode pictures earlier in this book.