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function calculate_force(
left_pos,
middle_pos,
right_pos,
K,
alpha = 0,
beta = 0,
)
linear_force = K * (middle_pos - left_pos + middle_pos - right_pos)
quadratic_force = alpha * (middle_pos - left_pos)^2 + alpha * (middle_pos - right_pos)^2
cubic_force = beta * (middle_pos - left_pos)^3 + beta * (middle_pos - right_pos)^3
return linear_force + quadratic_force + cubic_force
end
function tendency!(du, u, p, t)
# unpack the params
N, K, m = p
# get the positions and momenta
qs = u[1:2:end]
ps = u[2:2:end]
# go over the points in the lattice and update the state
for i in 2:N-1
mass = m
if i == 2 * Int(N / 2) - 1 || i == 2 * Int(N / 2)
mass = 10000
end
du[i*2-1] = ps[i] / mass
force =
du[i*2] = force / mass
end
force_end = K * (qs[2] - 2 * qs[1] + qs[N-1])
du[1] = ps[1] / m
du[2] = force_end / m
du[end-1] = ps[end] / m
du[end] = force_end / m
end
function get_initial_state(
N,
initial_displacement = 2,
initial_velocity = 0,
)
state = zeros(2 * N)
middle_index = 2 * Int(N / 2) - 1 # middle mass
state[middle_index] = initial_displacement
state[middle_index+1] = initial_velocity
return state
end
using DifferentialEquations
function run_simulation(
N,
K,
m,
final_time,
initial_displacement = 2,
initial_velocity = 0,
)
println("Running simulation with N = $N, K = $K, m = $m, final_time = $final_time, initial_displacement = $initial_displacement, initial_velocity = $initial_velocity\n")
s_0 = get_initial_state(N, initial_displacement, initial_velocity)
# pack the params
p = N, K, m
t_span = (0.0, final_time)
prob = ODEProblem(tendency!, s_0, t_span, p)
sol = solve(prob, Tsit5(), reltol = 1e-10, abstol = 1e-10) # control simulation
println("Done Running Sim!\n\n")
return sol
end
using Plots
function animate_positions(
states,
time_steps,
time_min = 0,
time_max = 30,
red_threshold = 2,
)
println("Animating positions")
anim = @animate for i in 1:length(time_steps)
t = time_steps[i]
if t < time_min
continue
end
if t > time_max
break
end
positions = states[i][1:2:end]
v_middle = states[i][Int(length(states[1]) / 2)]
# plot(positions, label = "t = $(round(t, digits = 3)), v_middle=$(round(v_middle, digits=3))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3))
if v_middle >= red_threshold
plot(positions, label = "t = $(round(t, digits = 3)), v_middle=$(round(v_middle, digits=3))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3),
color = :red, legend = :topright,
)
else
plot(positions, label = "t = $(round(t, digits = 3)), v_middle=$(round(v_middle, digits=3))", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", ylim = (-3, 3),
color = :blue, legend = :topright,
)
end
end
mp4(anim, "t/animate-positions.mp4", fps = 30)
println("Done animating positions")
end
function plot_starting_and_final_positions(
states,
time_steps,
)
p1 = plot(states[1][1:2:end], label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Displacement", title = "First Three Modes")
plot!(p1, states[end][1:2:end], label = "Final", marker = :circle)
# plot the vels
p2 = plot(states[1][2:2:end], label = "Initial", marker = :circle, xlabel = "Mass Number", ylabel = "Velocity", title = "First Three Modes")
plot!(p2, states[end][2:2:end], label = "Final", marker = :circle)
# save the plots
savefig(p1, "t/initial-final-positions.png")
savefig(p2, "t/initial-final-velocities.png")
end
function analyize_vels(
states,
time_steps,
threshold = 1.975,
)
println("Analyzing velocities:\n")
output = []
for i in 1:length(states)
if states[i][Int(length(states[i]) / 2)] >= threshold
push!(output, i)
println("Time: ", time_steps[i], " Position: ", states[i][Int(length(states[i]) / 2)])
end
end
# plot the first 10 seconds of Velocity
data = []
for i in 1:length(states)
if time_steps[i] > 10
break
end
push!(data, states[i][Int(length(states[i]) / 2)])
end
p = plot(data, label = "Velocity Over Time", xlabel = "Time", ylabel = "Velocity")
savefig(p, "t/velocity-over-time.png")
println("\nDone!\n\n")
return output
end
# Run the simulation
N = 10 # number of masses
beta = 0 # cubic string spring
K = 100 # spring constant
A = 10 # amplitude
final_time = 10000 # seconds
m = 1 # mass of particles
plot_data = []
my_vel = 10
sol = run_simulation(N, K, m, final_time, 0, my_vel)
println("final time: ", sol.t[end])
# s = sol.u[1:2:end]
analyize_vels(sol.u, sol.t, my_vel)
plot_starting_and_final_positions(sol.u, sol.t)
animate_positions(sol.u, sol.t, 0, 1, my_vel)
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