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#location of the second centre A a=function(t,R=c(2,1),om=c(1,13)){ sum(R)*c(cos(om[2]*t),sin(om[2]*t))} #location of the peripheral point M b=function(t,R=c(2,1),om=c(1,13)){ a(t,R,om)+R[1]*c(cos(om[1]*t),sin(om[1]*t))} #plot of the location of M as above draw_position=function(R=c(1,2), om=c(3,1), phi=c(0,0) ){ position=function(t,index,R=c(1,2),om=c(3,1),phi=c(0,0)){ relative_position=function(){ return(list(R[index]*cos(om[index]*t+phi[index]),R[index]*sin(om[index]*t+phi[index]))) } if(index == 1){ return(relative_position() ) } else { p1=position(t,index-1,R=R,om=om,phi=phi) p2=relative_position() return(list(p1[[1]]+p2[[1]],p1[[2]]+p2[[2]])) }} tes=seq(0,2*pi,length=10**3) xy_range=c(-1,1)*sum(abs(R)) plot(0,0,pch="x",xlim=xy_range,ylim=xy_range,axes=F,xlab="",ylab="") pA=position(tes,1,R,om,phi) pB=position(tes,2,R,om,phi) lines(pB[[1]],pB[[2]],pch=19,cex=.4,col="tomato") lines(pA[[1]],pA[[2]],pch=19,cex=.2,col="blue") } draw_position(om=c(1,30)) #angle at time t the=function(t,R=c(2,1),om=c(1,13)){ bb=b(t,R=R,om=om) bb=bb/sqrt(sum(bb^2)) theta=acos(bb[1]) if (bb[2] theta} #angular speed at time t #by very crude differenciating dthe=function(t,R=c(2,1),om=c(1,13)){ dtes=mean(diff(tes)) (the(t+dtes)-the(t-dtes))/(2*dtes)} #new plot plot(apply(as.matrix(tes),1,dthe,om=c(-55,2)),type="l",ylim=c(-2*pi,pi))
Antoine Dreyer actually contributed to improve the above code from an earlier version and he also derived the (Cartesian) components of the speed for me:
and
if O1(0) has polar coordinates (r0,φ0) and M(0) has polar coordinates (r1,φ1) with respect to O1(0).
Filed under: R Tagged: angular speed, R
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