myASC-ODE Simulation Framework

myASC-ODE Simulation Framework#

Welcome to the myASC-ODE simulation framework documentation. This book contains interactive simulations and analysis of mechanical systems using C++ and Python.

Contributors

  • Constantin Pierer

  • Jonas Gastecker

  • Lukas Lechner

Features#

  • Kreisel Systems: Simulating a spinning Kreisel

  • Crane Dynamics: Realistic crane structure modeling with vibration analysis

Interactive Simulations#

This book contains fully interactive Jupyter notebooks with:

  • Real-time 3D visualizations

  • Dynamic parameter adjustments

  • Comprehensive analysis tools

Only the user-friendly interface is implemented in Python; all performance-critical computations are handled by the underlying C++ framework.

Explore the interactive notebooks to see the simulations in action! We will first briefly mention some of the theory behind the implemented methods.

Solving a Mass Spring System with a generalized \(\alpha\) Method#

In an effort to avoid the instabilities that the Newmark method leads to for non linear ODEs, we can introduce the generalized \(\alpha\) method. We recall the Newmark scheme for a second order ODE

\[ M \ddot{x} = F(x) \]

to be given as

\[\begin{split} \begin{aligned} x_{n+1} &= x_n + \tau v_n + \tau^2 \left( \left( \tfrac{1}{2} - \beta \right) a_n + \beta a_{n+1} \right), \\ v_{n+1} &= v_n + \tau \left( (1-\gamma) a_n + \gamma a_{n+1} \right) \end{aligned} \end{split}\]

where \(v_n = \dot{x}_n\) and \(a_n = M^{-1} F(x_n)\).

For a damping parameter \(\rho^\infty\), we can introduce the new variables

\[\begin{split} \begin{aligned} \alpha_m &= \frac{2 \rho^\infty - 1}{\rho^\infty + 1}, \\ \alpha_f &= \frac{\rho^\infty}{\rho^\infty + 1} \end{aligned} \end{split}\]

and set

\[\begin{split} \begin{aligned} \beta &= \frac{1}{4} (1 - \alpha_m + \alpha_f)^2, \\ \gamma &= \frac{1}{2} - \alpha_m + \alpha_f \end{aligned} \end{split}\]

to get to the generalized \(\alpha\) method formulation

\[\begin{split} \begin{aligned} x_{n+1} &= x_n + \tau v_n + \tau^2 \left( \left( \tfrac{1}{2} - \beta \right) a_n + \beta a_{n+1} \right), \\ v_{n+1} &= v_n + \tau \left( (1-\gamma) a_n + \gamma a_{n+1} \right), \\ x_{n+1-\alpha_f} &= (1-\alpha_f) x_{n+1} + \alpha_f x_n, \\ a_{n+1-\alpha_m} &= (1-\alpha_m) a_{n+1} + \alpha_m a_n \end{aligned} \end{split}\]

Systems with constraints#

We want to implement a way to model joints between two masses. For this, we introduce systems with constraints. We define the Lagrange function for a constrained system as

\[ L(x, \lambda) = -U(x) + \langle \lambda, g(x) \rangle \]

with

\[ U(x) = m g x_z, \qquad g(x) := \lVert x - x_0 \rVert^2 - l^2 \]

being the length constraint.

This leads to the second order system of ODEs

\[\begin{split} \begin{aligned} m_i \ddot{x}_i &= \frac{\partial}{\partial x_i} L(x, \lambda), \\ 0 &= \nabla_\lambda L(x, \lambda) \end{aligned} \end{split}\]

This system can be solved using the generalized \(\alpha\) method.

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