The Elliptic Integral: A simple pendulum
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When the angle of swing is not small, the motion of a pendulum is no longer well approximated by simple harmonic motion. The solution, by way of the elliptic integral, cannot be expressed in a finite number of terms of elementary functions. However, [i]a differential equation is a complete model of a problem[/i]. I think, this is what Newton was saying. Writing the problem as finite differences, we can examine it numerically.... or pass it to a computer. Integration in time = Press play. |
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I assume the mass of the rod is negligible. Say, a rigid wire. The difference equation is in the update script of [i]t[/i]. It is as follows: Given step n-1: θ[n-1] = (the current angle out-of-rest of the pendulum, in radians) v[n-1] = (current velocity) F[n-1] = (the component of gravitational force affecting the pendulum at its current position). Step n: Δv[n] = (F[n-1]/m) Δt v[n] = v[n-1]+Δv[n] Δθ[n] = (v[n]\l)Δt θ[n] =θ[n-1] + Δθ[n] Update: Added damping term μ, proportional to velocity. The update script is the same, but F[n] includes a second term, F[n] = F1 + F2 = −mg sinθ −2μmg v |