I posted a much more concise derivation (and implementation, in just 16 lines) on HN a while back but didn't gain much traction: https://news.ycombinator.com/i

I posted a much more concise derivation (and implementation, in just 16 lines) on HN a while back but didn't gain much traction: https://news.ycombinator.com/item?id=35899215 https://news.ycombinator.com/item?id=35899215 Here are the 16 lines in full: function sprung_response(t,d,v,k,c,m) local decay = c/2/m local omega = math.sqrt(k/m) local resid = decay*decay-omega*omega local scale = math.sqrt(math.abs(resid)) local T1,T0 = t , 1 if resid<0 then T1,T0 = math.sin( scale*t)/scale , math.cos( scale*t) elseif resid>0 then T1,T0 = math.sinh(scale*t)/scale , math.cosh(scale*t) end local dissipation = math.exp(-decay*t) local evolved_pos = dissipation*( d*(T0+T1*decay) + v*( T1 ) ) local evolved_vel = dissipation*( d*(-T1*omega^2) + v*(T0-T1*decay) ) return evolved_pos , evolved_vel end And the embedded interactive demo in the blog post: https://www.desmos.com/calculator/ynbtai98ns https://www.desmos.com/calculator/ynbtai98ns I'm hesitant about reposting as all of my submissions had been inexplicably blocked by HN filters, by the time someone vouches the post it has already been buried.