diff --git a/Project_01j.jl b/Project_01j.jl new file mode 100644 index 0000000..d4282e5 --- /dev/null +++ b/Project_01j.jl @@ -0,0 +1,3324 @@ +### A Pluto.jl notebook ### +# v0.20.21 + +using Markdown +using InteractiveUtils + +# ╔═╡ 0d9be664-d7c5-4084-add2-25e5418742d6 +using Symbolics, ModelingToolkit, DifferentialEquations, Plots + +# ╔═╡ f17103ea-06bf-11f1-a2b0-79e68ed152eb +md"""# Project_01 - Spinning Pendulum and the Lagrange equations + +In this project, a pendulum is attached to a spinning frame. The frame has dimensions, + +$h_1 = 0.2~m$ + +$w_1 = 0.1~m$ + +and the pendulum length is $L=0.15~m$ with a $m=0.1~kg$ point mass at the end of the system. The pendulum swings in the $x'-z'-plane$ as it rotates at a constant speed, $\Omega$ + +![Spinning pendulum with rotating and fixed coordinate systems.](https://raw.githubusercontent.com/cooperrc/me5180-project_01/refs/heads/main/spinning_pendulum.svg) + +Your team's goal is to + +- build the equations of motion using Lagrange and least action $L=T-V$ +- solve for the motion for a slow rotation speed and a fast rotation speed +- visualize the solution with plots and animations + +""" + +# ╔═╡ 548d67a6-4e91-4d1f-a407-18818a85708a +import Latexify + +# ╔═╡ e2dfc44a-dd64-4527-af83-f7e5296c0768 +default(fontfamily = "Computer Modern", linewidth = 2, size = (800, 600)) + +# ╔═╡ 910c7d40-5d28-448e-aaf7-486bb000f478 +# definining variables and basic equations. +begin + @parameters L h1 w1 m g Ω + @independent_variables t + @variables θ(t) + D = Differential(t) + θ_dot = D(θ) + θ_dot_dot = D(D(θ)) + φ = Ω * t +end + +# ╔═╡ cbd246ac-cbc8-4301-b224-87e92f7912b5 +r_rot = [L*sin(θ)*cos(φ), L*sin(θ)*sin(φ), -L*cos(θ)] + +# ╔═╡ 89408ad5-0f4f-4bf9-bfaf-5ef3fdf1791d +md""" +$\vec{r}_{0} = \begin{bmatrix} w_1\cos(\phi) \\ w_1\sin(\phi) \\ h_1 \end{bmatrix}$ +""" + +# ╔═╡ 342f30ae-5a57-4330-8a18-8a9b1eabd74f +r_0 = [w1*cos(φ), w1*sin(φ), h1] + +# ╔═╡ d8d43e4b-a976-4535-a3f0-c2230fd55631 +md""" +Position vector: + +$\vec{r} = \vec{r}_{rot}+\vec{r}_0$ +""" + +# ╔═╡ 21a2c22e-99a7-46b5-b898-22028948e114 +r = r_rot .+ r_0 + +# ╔═╡ 2c97d2eb-070c-499f-8bbc-9f650953ccd1 +md""" +The Lagrangian is: + +$L = T - V$ + +$T = \frac{1}{2}mv^2$ + +$V = mgh$ +""" + +# ╔═╡ 9a46c115-04c9-42cc-9592-5f9d08014735 +md""" +$h = z = h_1 - L\cos(\theta)$ + +$v = \frac{d\vec{r}}{dt}$ + +$T = \frac{1}{2}m\left(\frac{d\vec{r}}{dt} \cdot \frac{d\vec{r}}{dt}\right)$ +""" + +# ╔═╡ 61500b9e-061e-4412-9fa7-9931ad27bee4 +# defining lagrangian stuff +begin + h = r[3] + v = D.(r) + T = m/2*sum(v .* v) + V = m*g*h + Lag = expand_derivatives(T - V) +end + +# ╔═╡ 107838fd-dc24-4b35-bc0f-531cfc6362f2 +md"""Euler-Lagrange equation: + +$\frac{d}{dt}\left( \frac{d L}{d \dot{\theta}} \right) - \frac{d L}{d \theta}= 0$ + +""" + +# ╔═╡ 242302eb-fe7e-4c20-83a2-f5a976fbbac9 +begin + dL_dθ = Symbolics.derivative(Lag, θ) + dL_dθ_dot = Symbolics.derivative(Lag, θ_dot) + el_eq = expand_derivatives(D(dL_dθ_dot) - dL_dθ) + sol = simplify(solve_for(el_eq ~ 0, θ_dot_dot)) +end + +# ╔═╡ 28406b28-a701-4b46-8620-5df4a5dc70ae +begin + @variables x1(t) x2(t) + @parameters c # damping coefficient + + # original RHS from Euler–Lagrange + original_rhs = substitute(sol, Dict(θ => x1, θ_dot => x2)) + + # add viscous damping: τ_damp = -c * θ̇ + # EoM: θ̈ = original_rhs - (c/(m*L^2))*θ̇ + damped_rhs = original_rhs - (c/(m*L^2)) * x2 + + eqs = [ + D(x1) ~ x2, + D(x2) ~ damped_rhs + ] + + @named sys = ODESystem(eqs, t, [x1, x2], [L, g, Ω, m, w1, h1, c]) + sys = structural_simplify(sys) +end + +# ╔═╡ 8776d0cc-f9fb-43ee-babe-b77ec4f189ce +# setting up initial conditions and running ODE solver + +begin + u0 = [x1 => 0.5, x2 => 0.0] # initial θ and θ_dot + tspan = (0.0, 10.0) + + p_vals = Dict(L => 0.15, g => 9.8, Ω => 2.0, m => 0.1, w1 => 0.1, h1 => 0.2, c => 0.002) # damping coefficient + p_slow = merge(p_vals, Dict(Ω => 0.5)) + p_fast = merge(p_vals, Dict(Ω => 8.0)) + + prob_slow = ODEProblem(sys, merge(Dict(u0), p_slow), tspan) + prob_fast = ODEProblem(sys, merge(Dict(u0), p_fast), tspan) + + sol_slow = solve(prob_slow, Rodas5(); reltol=1e-6, abstol=1e-8, saveat=0.0333) + sol_fast = solve(prob_fast, Rodas5(); reltol=1e-6, abstol=1e-8, saveat=0.0333) + + + solution = sol_slow # or sol_fast +end + +# ╔═╡ 1a7f4f86-3dc5-4bd3-a25a-4a8f6a96e14e +Plots.plot(solution, idxs=[x1], xlabel="t", ylabel="θ (rad)", title="Angle vs. Time") + +# ╔═╡ 3b72fdac-c439-4755-b501-dc1de46f4414 +Plots.plot(solution, idxs=[x2], xlabel="t", ylabel="θ/s (rad/sec)", title="Angular Velocity vs. Time") + +# ╔═╡ fcd29939-01e8-42ae-8fef-f70ca60eebef +begin + θ_vals = solution[x1] + t_vals = solution.t + p_use = p_slow + φ_vals = p_use[Ω] .* t_vals + + + r_x = p_vals[L] .* sin.(θ_vals) .* cos.(φ_vals) .+ p_vals[w1] .* cos.(φ_vals) + r_y = p_vals[L] .* sin.(θ_vals) .* sin.(φ_vals) .+ p_vals[w1] .* sin.(φ_vals) + r_z = -p_vals[L] .* cos.(θ_vals) .+ p_vals[h1] + + + frame_top_x_values = p_vals[w1] .* cos.(φ_vals) + frame_top_y_values = p_vals[w1] .* sin.(φ_vals) + frame_top_z_values = fill(p_vals[h1], length(t_vals)) +end + +# ╔═╡ ce3510bf-40ff-40fb-8394-a22fe3eebc48 +begin + max_radius = p_vals[L] + p_vals[w1] + + pendulum_animation = @animate for index in eachindex(t_vals) + # Static vertical frame: from (0,0,0) up to (0,0,h1) + vertical_x = [0.0, 0.0] + vertical_y = [0.0, 0.0] + vertical_z = [0.0, p_vals[h1]] + + # Rotating horizontal arm: from top of frame to moving pivot + arm_x = [0.0, frame_top_x_values[index]] + arm_y = [0.0, frame_top_y_values[index]] + arm_z = [p_vals[h1], frame_top_z_values[index]] + + # Pendulum rod: from moving pivot to bob + rod_x = [frame_top_x_values[index], r_x[index]] + rod_y = [frame_top_y_values[index], r_y[index]] + rod_z = [frame_top_z_values[index], r_z[index]] + + + trace_x = r_x[1:index] + trace_y = r_y[1:index] + trace_z = r_z[1:index] + + plot3d( + trace_x, trace_y, trace_z; + label = "Trace", + linestyle = :dot, + linecolor = :blue, + linewidth = 2, + xlim = (-max_radius, max_radius), + ylim = (-max_radius, max_radius), + zlim = (0.0, p_vals[h1] + p_vals[L] + 0.1), + aspect_ratio = :equal, + xlabel = "x", + ylabel = "y", + zlabel = "z", + title = "3D Pendulum on Rotating Frame (t = $(round(t_vals[index]; digits = 2)) s)", + ) + + + plot3d!( + rod_x, rod_y, rod_z, + label = "Pendulum Rod", + linecolor = :green, + linewidth = 3, + ) + + scatter3d!([r_x[index]], [r_y[index]], [r_z[index]], + label = "Bob", + markercolor = :red, + markersize = 6, + ) + + plot3d!(vertical_x, vertical_y, vertical_z, + label = "Frame", + linecolor = :black, + linewidth = 3, + ) + + plot3d!(arm_x, arm_y, arm_z, + label = "Rotating Arm", + linecolor = :black, + linewidth = 3, + ) + + + end +end + +# ╔═╡ 077d3892-d766-4011-8ed4-c04969e8acb8 +gif(pendulum_animation, "pendulum_3d.gif", fps=30) + +# ╔═╡ a64ae2a4-e481-402b-8ea8-c36de5620973 +begin + begin + # Faster swinging initial conditions (bigger angular velocity) + initial_angle_fast_swing = 0.6 # radians + initial_speed_fast_swing = 6.0 # rad/s (makes the bob swing faster) + + # Initial state vector for faster swinging solution + u0_fast_swing = [ + x1 => initial_angle_fast_swing, + x2 => initial_speed_fast_swing + ] + + # Use the SAME rotation speed Ω as your slow case (same pole rotation) + p_fast_swing = Dict( + L => 0.15, + g => 9.8, + Ω => p_slow[Ω], # << SAME ROTATION SPEED + m => 0.1, + w1 => 0.1, + h1 => 0.2, + c => 0.00001 + ) + + # Solve time interval + time_span_fast_swing = (0.0, 12.0) + + # Build and solve the ODE + problem_fast_swing = ODEProblem( + sys, + merge(Dict(u0_fast_swing), p_fast_swing), + time_span_fast_swing + ) + + sol_fast_swing = solve( + problem_fast_swing, + Rodas5(); + reltol = 1e-6, + abstol = 1e-8, + saveat = 0.05 + ) + end + `` +end + +# ╔═╡ e393d30a-eac2-4658-9113-319747ae36ca +begin + # Extract θ(t) and time + theta_fast_swing = sol_fast_swing[x1] + time_fast_swing = sol_fast_swing.t + + # Compute rotation angle φ(t) using SAME Ω + phi_fast_swing = p_fast_swing[Ω] .* time_fast_swing + + # Read physical parameters for readability + pendulum_length = p_fast_swing[L] + frame_width = p_fast_swing[w1] + frame_height = p_fast_swing[h1] + + # Compute pendulum bob position in 3D + bob_x_fast = frame_width .* cos.(phi_fast_swing) .+ + pendulum_length .* sin.(theta_fast_swing) .* cos.(phi_fast_swing) + + bob_y_fast = frame_width .* sin.(phi_fast_swing) .+ + pendulum_length .* sin.(theta_fast_swing) .* sin.(phi_fast_swing) + + bob_z_fast = frame_height .- + pendulum_length .* cos.(theta_fast_swing) + + # Pivot point (top of pendulum rod) + pivot_x_fast = frame_width .* cos.(phi_fast_swing) + pivot_y_fast = frame_width .* sin.(phi_fast_swing) + pivot_z_fast = fill(frame_height, length(time_fast_swing)) +end + +# ╔═╡ f266ff36-94ca-4a79-9fd1-34d37f5a6ade +begin + max_radius_fast = frame_width + pendulum_length + + fast_swing_animation = @animate for i in eachindex(time_fast_swing) + + # Trace of bob path + plot3d( + bob_x_fast[1:i], bob_y_fast[1:i], bob_z_fast[1:i]; + label="Trace", linecolor=:blue, linestyle=:dot, linewidth=2, + xlim=(-max_radius_fast, max_radius_fast), + ylim=(-max_radius_fast, max_radius_fast), + zlim=(0, frame_height + pendulum_length + 0.1), + xlabel="x", ylabel="y", zlabel="z", + title="Fast Swinging Damped Pendulum (t = $(round(time_fast_swing[i], digits=2)) s)", + aspect_ratio=:equal + ) + + # Pendulum rod + plot3d!( + [pivot_x_fast[i], bob_x_fast[i]], + [pivot_y_fast[i], bob_y_fast[i]], + [pivot_z_fast[i], bob_z_fast[i]]; + label="Rod", linecolor=:green, linewidth=3 + ) + + # Bob + scatter3d!( + [bob_x_fast[i]], [bob_y_fast[i]], [bob_z_fast[i]]; + label="Bob", markercolor=:red, markersize=6 + ) + + # Fixed vertical support + plot3d!([0,0], [0,0], [0,frame_height]; + label="Frame", linecolor=:black, linewidth=3 + ) + + # Rotating frame arm + plot3d!( + [0, pivot_x_fast[i]], + [0, pivot_y_fast[i]], + [frame_height, pivot_z_fast[i]]; + label="Arm", linecolor=:black, linewidth=3 + ) + end +end + +# ╔═╡ 36d683c2-0415-4ef4-bf75-ad7f66f5eeaf +gif(fast_swing_animation, "pendulum_fast_swing.gif", fps=30) + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +DifferentialEquations = "0c46a032-eb83-5123-abaf-570d42b7fbaa" +Latexify = "23fbe1c1-3f47-55db-b15f-69d7ec21a316" +ModelingToolkit = "961ee093-0014-501f-94e3-6117800e7a78" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +Symbolics = "0c5d862f-8b57-4792-8d23-62f2024744c7" + +[compat] +DifferentialEquations = "~7.17.0" +Latexify = "~0.16.10" +ModelingToolkit = "~11.7.2" +Plots = "~1.41.4" +Symbolics = "~7.8.0" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This 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