Skip to content

Algebra for single and double pendulum index reduction #73

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
merged 2 commits into from
Jul 23, 2019
Merged

Algebra for single and double pendulum index reduction #73

Changes from all commits
Commits
Show all changes
2 commits
Select commit Hold shift + click to select a range
62d2c0a
Algebra for single pendulum index reduction
ranjanan Jul 22, 2019
4c3a8c5
Add algebra for double pendulum too
ranjanan Jul 22, 2019
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
80 changes: 76 additions & 4 deletions tutorials/exercises/02-workshop_solutions.jmd
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -192,6 +192,40 @@ plot(sol, xscale=:log10, tspan=(1e-6, 1e5), layout=(3,1))
```

## Part 3: Manual Index Reduction of the Single Pendulum
Consider the equation:
$$
x^2 + y^2 = L
$$
Differentiating once with respect to time:
$$
2x\dot{x} + 2y\dot{y} = 0
$$
A second time:
$$
\begin{align}
{\dot{x}}^2 + x\ddot{x} + {\dot{y}}^2 + y\ddot{y} &= 0 \\
u^2 + v^2 + x(\frac{x}{mL}T) + y(\frac{y}{mL}T - g) &= 0 \\
u^2 + v^2 + \frac{x^2 + y^2}{mL}T - yg &= 0 \\
u^2 + v^2 + \frac{T}{m} - yg &= 0
\end{align}
$$

Our final set of equations is hence
$$
\begin{align}
\ddot{x} &= \frac{x}{mL}T \\
\ddot{y} &= \frac{y}{mL}T - g \\
\dot{x} &= u \\
\dot{y} &= v \\
u^2 + v^2 -yg + \frac{T}{m} &= 0
\end{align}
$$

We finally obtain $T$ into the third equation.
This required two differentiations with respect
to time, and so our system of equations went from
index 3 to index 1. Now our solver can handle the
index 1 system.

## Part 4: Single Pendulum Solution with IDA
```julia
Expand Down Expand Up @@ -223,10 +257,46 @@ plot(sol, vars=(3,4))
```

## Part 5: Solving the Double Penulum DAE System
```julia
```

# Problem 4: Performance Optimizing and Parallelizing Semilinear PDE Solvers (I)
For the double pendulum:
The equations for the second ball are the same
as the single pendulum case. That is, the equations
for the second ball are:
$$
\begin{align}
\ddot{x_2} &= \frac{x_2}{m_2L_2}T_2 \\
\ddot{y_2} &= \frac{y_2}{m_2L_2}T_2 - g \\
\dot{x_2} &= u \\
\dot{y_2} &= v \\
u_2^2 + v_2^2 -y_2g + \frac{T_2}{m_2} &= 0
\end{align}
$$
For the first ball, consider $x_1^2 + y_1^2 = L $
$$
\begin{align}
x_1^2 + x_2^2 &= L \\
2x_1\dot{x_1} + 2y_1\dot{y_1} &= 0 \\
\dot{x_1}^2 + \dot{y_1}^2 + x_1(\frac{x_1}{m_1L_1}T_1 - \frac{x_2}{m_1L_2}T_2) + y_1(\frac{y_1}{m_1L_1}T_1 - g - \frac{y_2}{m_1L_2}T_2) &= 0 \\
u_1^2 + v_1^2 + \frac{T_1}{m_1} - \frac{x_1x_2 + y_1y_2}{m_1L_2}T_2 &= 0
\end{align}
$$

So the final equations are:
$$
\begin{align}
\dot{u_2} &= x_2*T_2/(m_2*L_2)
\dot{v_2} &= y_2*T_2/(m_2*L_2) - g
\dot{x_2} &= u_2
\dot{y_2} &= v_2
u_2^2 + v_2^2 -y_2*g + \frac{T_2}{m_2} &= 0

\dot{u_1} &= x_1*T_1/(m_1*L_1) - x_2*T_2/(m_2*L_2)
\dot{v_1} &= y_1*T_1/(m_1*L_1) - g - y_2*T_2/(m_2*L_2)
\dot{x_1} &= u_1
\dot{y_1} &= v_1
u_1^2 + v_1^2 + \frac{T_1}{m_1} +
\frac{-x_1*x_2 - y_1*y_2}{m_1L_2}T_2 - y_1g &= 0
\end{align}
$$
```julia
function f(out, da, a, p, t)
L1, m1, L2, m2, g = p
Expand Down Expand Up @@ -271,6 +341,8 @@ sol = solve(prob, IDA())
plot(sol, vars=(3,4))
plot(sol, vars=(8,9))
```

# Problem 4: Performance Optimizing and Parallelizing Semilinear PDE Solvers (I)
## Part 1: Implementing the BRUSS PDE System as ODEs

```julia
Expand Down