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CS-7863-Sci-Stat-Proj-4/Schrick-Noah_Homework-4.R

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# Project 4 for the University of Tulsa's CS-7863 Sci-Stat Course
# Higher Order Differential Equations and Shooting Method
# Professor: Dr. McKinney, Spring 2023
# Noah L. Schrick - 1492657
## 1. Transform the second order ODE into a system of two first order ODEs
# use ode45 or rk4sys to solve for the motion of the plane pendulum
if (!require("pracma")) install.packages("pracma")
library(pracma)
tmin <- 0
tmax <- 7
theta0 <- pi/4
omega0 <- 0
y.init <- c(theta0, omega0)
g <- 9.81
L <- 1.0
## Substitution Tricks
# theta_1 = theta_prime
# theta_2 = theta
# theta_1_prime = theta_pp
# theta_2_prime = theta_prime = theta_1
## Equation Replacements
ode_1.f <- function(t, y, k){
theta <- y[1]
omega <- y[2]
g <- k[1]
L <- k[2]
omega_p <- -k[1]/k[2] * sin(theta)
theta_p <- omega
as.matrix(c(theta_p, omega_p))
}
p.params <- c(g, L)
pend.sol <- ode45(f=function(t,y){ode_1.f(t,y,k=p.params)},
y=y.init, t0=tmin, tfinal=tmax)
# a. Plot
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(pend.sol$t,pend.sol$y[,2],type="l",col="red")
abline(h=0)
legend("topleft", # coordinates, "topleft" etc
c("angle","omega"), # label
lty=c(1,1), # line
lwd=c(2.5,2.5), # weight
#cex=.8,
bty="n", # no box
col=c("blue","red") # color
)
# b. Overlay a plot of the small-angle approx
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=length(pend.sol$t)),
theta0*cos(sqrt(g/L)*seq(tmin,tmax,len=length(pend.sol$t))),type="l",col="red")
abline(h=0)
legend("topleft", # coordinates, "topleft" etc
c("angle","small angle approx"), # label
lty=c(1,1), # line
lwd=c(2.5,2.5), # weight
#cex=.8,
bty="n", # no box
col=c("blue","red") # color
)
## 2. Use ode45 to solve the damped harmonic oscillator
tmin <- 0
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
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]
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
# a. IVP
tmin <- 0
tmax <- 2.04
xo <- 0
vo <- 10
projectile.f <- function(t,y){
g <- 9.81
v <- y[2] # y1dot
a <- -g # could be any f, y'' = f(t,y)
matrix(c(v,a))
}
y.init <- c(xo, vo)
ivp.sol <- ode45(f=function(t,y){projectile.f(t,y)},
y=y.init, t0=tmin, tfinal=tmax)
ivp.sol.rk4 <- rk4sys(projectile.f, 0, 2.04, c(0, 10), 100)
plot(ivp.sol$t, ivp.sol$y[,1] ,type="o",col="blue",
ylim = c(-0.5, 5.5), xlab="time",ylab="height")
par(new=T)
lines(ivp.sol.rk4$x, ivp.sol.rk4$y[,1] ,type="o",col="red",
xlab="time",ylab="height")
legend("topright",
c("ODE45, default","RK4Sys, 100 steps"), # 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. BVP
yo <- 0
ymax <- 0
proj.obj <- function(v0, y0=0, tfinal){
# minimize w.r.t. v0
proj.sol <- ode45(projectile.f,
y=c(y0, v0), t0=0, tfinal=tfinal)
final_index <- length(proj.sol$t)
yf <- proj.sol$y[final_index,1] # want equal to right boundary
log(abs(yf)) # minimize this
}
# user specifies tfinal and yfinal for BVP
v_best <- optimize(proj.obj,
interval=c(1,100), #bisect-esque interval
tol=1e-10,
y0=0, tfinal=10) # un-optimized obj params
v_best$minimum # best v0
best.sol <- rk4sys(projectile.f, a=0, b=10, y0=c(0, v_best$minimum),
n=20) # 20 integration stepstmax
plot(best.sol$x, best.sol$y[,1], type="l")
computeHeights <- function(xo, vo, tmin){
height <- xo + vo*tmin -0.5*9.81*tmin^2
}
height <- computeHeights(yo, v_best$minimum, seq(0,10,len=21))
plot(seq(0,10,len=21), height, xlab="time", ylab="height")
# c. Shooting method for damped oscillator with perturbation parameter
yo <- 0
y1 <- 1
tmin <- 0
tfinal <- 2
dho.pert.encap.f <- function(ep){
dho.pert.f <- function(t, y){
yn <- y[1]
ydot <- y[2] - yn
#ep <- 0.5
ypp <- (-(1+ep)*yn - ydot)/ep
as.matrix(c(ydot,ypp))
}
proj.obj <- function(v0, y0=yo, tfinal){
# minimize w.r.t. v0
proj.sol <- ode45(dho.pert.f,
y=c(yo, y1), t0=tmin, tfinal=tfinal)
final_index <- length(proj.sol$t)
yf <- proj.sol$y[final_index,1] # want equal to right boundary
log(abs(yf)) # minimize this
}
# user specifies tfinal and yfinal for BVP
ydot_best <- optimize(proj.obj,
interval=c(1,100), #bisect-esque interval
tol=1e-10,
y0=0, tfinal=2) # un-optimized obj params
ydot_best$minimum # best ydot
best.sol <- rk4sys(dho.pert.f, a=0, b=2, y0=c(0, ydot_best$minimum),
n=20) # 20 integration stepstmax
}
ep <- 0.5
best.sol <- dho.pert.encap.f(ep)
plot(best.sol$x, best.sol$y[,1], type="l")
ep <- 0.05
best.sol <- dho.pert.encap.f(ep)
plot(best.sol$x, best.sol$y[,1], type="l")
# analytical sol
# yprime = -y
# Same as e^x
## 4. Position of the earth and moon
# a. Plotly
# t0: jan 1, 1999 00:00:00am
x0_earth <- c(-27115219762.4, 132888652547.0, 57651255508.0)/1e3 # km
v0_earth <- c(-29794.2199907, -4924.33333333,-2135.59540741)*(86400)/1e3
# m/s -> km/day
x0_moon <- c(-27083318944, 133232649728, 57770257344)/1e3 # km
v0_moon <- c(-30864.2207031, -4835.03349304, -2042.89546204)*(86400)/1e3
# m/s -> km/day
sunearth.f <- function(t,y){
G <- 6.673e-11 # m^3 kg^-1 s^-2
M_S <- 1.9891e30
M_E <- 5.98e24
M_m <- 7.32e22
mu_sun <- G*M_S*(86400^2)/1e9 # km^3/days^2
x_earth <- y[1:3] # xyz position of earth wrt sun
v_earth <- y[4:6] # xyz velocity of earth, also part of ydot
r_earth <- sqrt(sum(x_earth^2)) # distance earth to sun
acc_earth <- -mu_sun*x_earth/r_earth^3 # part of ydot
ydot <- c(v_earth, acc_earth)
return(matrix(ydot))
}
y.init.earth <- c(x0_earth, v0_earth)
sunearth.sol <- rk4sys(f=sunearth.f, a=0, b=365.25, y0=y.init.earth, n=365)
sunearth.df <- data.frame(x=sunearth.sol$y[,1],
y=sunearth.sol$y[,2],
z=sunearth.sol$y[,3])
y.init.moon <- c(x0_moon, v0_moon)
sunmoon.sol <- rk4sys(f=sunearth.f, a=0,b=365.25, y0=y.init.moon, n=365)
sunmoon.df <- data.frame(x=sunmoon.sol$y[,1],
y=sunmoon.sol$y[,2],
z=sunmoon.sol$y[,3])
if (!require("plotly")) install.packages("plotly")
library(plotly)
fig <- plot_ly(sunmoon.df, x = ~x, y = ~y, z = ~z, name="moon", type = "scatter3d", mode="markers")
fig <- fig %>% layout(scene = list(xaxis = list(title = 'x'),
yaxis = list(title = 'y'),
zaxis = list(title = 'z')))
fig <- add_trace(fig, x = 0, y = 0, z=0, mode="markers", color = I("red"), name="sun")
fig <- add_trace(fig, x = sunearth.df[1]$x, y = sunearth.df[2]$y, z = sunearth.df[3]$z, mode="markers", color = I("green"), name="earth")
fig
mean(sqrt(rowSums(cbind(sunearth.sol$y[,1]^2,
sunearth.sol$y[,2]^2,
sunearth.sol$y[,3]^2))))
# 150 million km
sunmoon.f <- function(t,y){
G <- 6.673e-11 # m^3 kg^-1 s^-2
M_E <- 5.98e24
M_m <- 7.32e22
mu_sun <- G*M_E*(86400^2)/1e9 # km^3/days^2
x_moon <- y[1:3] # xyz position of moon wrt earth
v_moon <- y[4:6] # xyz velocity of moon, also part of ydot
r_moon <- sqrt(sum(x_moon^2)) # distance moon to earth
acc_moon <- -mu_sun*x_moon/r_moon^3 # part of ydot
ydot <- c(v_moon, acc_moon)
return(matrix(ydot))
}
x0_moonearth <- x0_moon - x0_earth
v0_moonearth <- v0_moon - v0_earth
y.init.moonearth <- c(x0_moonearth, v0_moonearth)
earthmoon.sol <- rk4sys(f=sunmoon.f, a=0,b=365.25, y0=y.init.moonearth, n=365)
earthmoon.df <- data.frame(x=earthmoon.sol$y[,1],
y=earthmoon.sol$y[,2],
z=earthmoon.sol$y[,3])
mean(sqrt(rowSums(cbind(earthmoon.sol$y[,1]^2,
earthmoon.sol$y[,2]^2,
earthmoon.sol$y[,3]^2))))
fig <- plot_ly(earthmoon.df, x = ~x, y = ~y, z = ~z, name="moon", type = "scatter3d", mode="markers")
fig <- fig %>% layout(scene = list(xaxis = list(title = 'x'),
yaxis = list(title = 'y'),
zaxis = list(title = 'z')))
fig <- add_trace(fig, x = 0, y = 0, z = 0, mode="markers", color = I("green"), name="earth")
fig
# b. Find eclipses
norm_vec <- function(x) sqrt(rowSums(cbind((x^2))))
eclipses <- norm_vec(sunearth.sol$y) %*% norm_vec(sunmoon.sol$y)
# c. keep orbiter at L2 Lagrange point for a year, ignoring the moon's effect
findZeroRelax <- function(g, x.guess, tol=1e-6, maxsteps=1e6){
steps <- 0
x.old <- x.guess
isConverged <- FALSE
while (!isConverged){
x.new <- g(x.old)
if(steps >= maxsteps || (abs(x.new-x.old)<tol)){
isConverged <- TRUE
}
else{
x.old <- x.new
}
steps <- steps + 1
}
return(c(x.new, g(x.new), steps))
}
T <- 365.25
G <- 6.673e-11 # m^3 kg^-1 s^-2
M_S <- 1.9891e30
M_E <- 5.98e24
x_earth <- c(-27115219762.4, 132888652547.0, 57651255508.0)/1e3 # km
r_earth <- sqrt(sum(x_earth^2)) # distance earth to sun
orbiter.f <- function(l) {
tmp1 <- ((T^2)/(4*(pi^2))) * G
tmp2 <- (M_S/((r_earth+l)^2)) + (M_E/(l^2))
tmp3 <- tmp1*tmp2
}
orbiter.estimate <- findZeroRelax(orbiter.f, 1000000)
l <- orbiter.estimate[1]-r_earth
x0_orbiter_tmp <- c(l, orbiter.f(l), 0)/1e3 # km
x0_orbiter <- x0_orbiter_tmp + x0_earth
y.init.orbiter <- c(x0_orbiter, v0_earth)
sunorbit.sol <- rk4sys(f=sunearth.f, a=0, b=365.25, y0=y.init.orbiter, n=365)
sunorbit.df <- data.frame(x=sunorbit.sol$y[,1],
y=sunorbit.sol$y[,2],
z=sunorbit.sol$y[,3])
fig <- plot_ly(sunorbit.df, x = ~x, y = ~y, z = ~z, name="orbiter", type = "scatter3d", mode="markers")
fig <- fig %>% layout(scene = list(xaxis = list(title = 'x'),
yaxis = list(title = 'y'),
zaxis = list(title = 'z')))
fig <- add_trace(fig, x = 0, y = 0, z=0, mode="markers", color = I("red"), name="sun")
fig <- add_trace(fig, x = sunearth.df[1]$x, y = sunearth.df[2]$y, z = sunearth.df[3]$z, mode="markers", color = I("green"), name="earth")
fig
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