Damped Harmonic Oscillator

Schrick-Noah_Homework-4.R

@@ -57,8 +57,8 @@ legend("topleft", # coordinates, "topleft" etc
 plot(pend.sol$t,pend.sol$y[,1],type="l",col="blue", 
      ylim=c(-2.3,2.3), xlab="time",ylab="amplitude")
 par(new=T)
-lines(seq(tmin,tmax,len=50),
+lines(seq(tmin,tmax,len=length(pend.sol$t)),
-      theta0*cos(sqrt(g/L)*seq(tmin,tmax,len=50)),type="l",col="red")
+      theta0*cos(sqrt(g/L)*seq(tmin,tmax,len=length(pend.sol$t))),type="l",col="red")
 abline(h=0)
 legend("topleft", # coordinates, "topleft" etc
        c("angle","small angle approx"), # label
@@ -75,19 +75,78 @@ tmax <- 10
 yo <- 1
 ybaro <- 0
 
+## Substitution Tricks
+# y1 = y_prime
+# y_2 = y
+# y_1_prime = y_pp
+# y_2_prime = y_prime = y_1
+
+# Eqs:
+# m*y_1_prime + c*y1 + k*y_2 = 0
+# y_1_prime = (-c*y_1 - k*y_2)/m
+# y_2_prime = y_1
+
 # a
 m <- 1
 c <- 1
-k <- 2
+kvar <- 2
+
+dho.f <- function(t, y, kpass){
+  yn <- y[1]
+  ydot <- y[2]
+  #         c         k         m
+  ypp <- (-kpass[2]*yn - kpass[3]*ydot)/kpass[1]
+  #ypp <- -kpass[2]*y - kpass[3]*ydot
+  as.matrix(c(ydot,ypp))
+}
+
+y.init <- c(yo, ybaro)
+dho.params <- c(m,c,kvar)
+dho.sol <- ode45(f=function(t,y){dho.f(t,y,kpass=dho.params)},
+                  y=y.init, t0=tmin, tfinal=tmax)
 
 # plot pos vs time
+pos <- (-m*dho.sol$y[,2] - c*dho.sol$y[,1])/kvar
+plot(dho.sol$t, pos ,type="o",col="blue", 
+     ylim=c(-1.0,1.0), xlab="time",ylab="amplitude")
+par(new=T)
+lines(dho.sol$t,dho.sol$y[,1],type="o",col="red")
+lines(dho.sol$t,dho.sol$y[,2],type="o",col="green")
+legend("topright", # coordinates, "topleft" etc
+       c("y","yprime", "ydoubleprime"), # label
+       lty=c(1,1), # line 
+       lwd=c(2.5,2.5), # weight
+       #cex=.8, 
+       bty="n", # no box
+       col=c("blue","red", "green") # color
+)
 
 # b
 m <- 1
 c <- 2
 k <- 1
+dho.params <- c(m,c,k)
+
+y.init <- c(yo, ybaro)
+dho.params <- c(m,c,kvar)
+dho.sol <- ode45(f=function(t,y){dho.f(t,y,kpass=dho.params)},
+                 y=y.init, t0=tmin, tfinal=tmax)
 
 # plot pos vs time
+pos <- (-m*dho.sol$y[,2] - c*dho.sol$y[,1])/kvar
+plot(dho.sol$t, pos ,type="o",col="blue", 
+     ylim=c(-1.0,1.0), xlab="time",ylab="amplitude")
+par(new=T)
+lines(dho.sol$t,dho.sol$y[,1],type="o",col="red")
+lines(dho.sol$t,dho.sol$y[,2],type="o",col="green")
+legend("topright", # coordinates, "topleft" etc
+       c("y","yprime", "ydoubleprime"), # label
+       lty=c(1,1), # line 
+       lwd=c(2.5,2.5), # weight
+       #cex=.8, 
+       bty="n", # no box
+       col=c("blue","red", "green") # color
+)
 
 ## 3. Use ode45/rk4sys to solve for the traj of a projectile thrown vertically