# Applications cheatsheet

By Afshine Amidi and Shervine Amidi

## Physics Laws

Gravitational force A mass $m$ is subject to the gravitational force $\vec{F}_g$, which is expressed with respect to $\vec{g}$ of magnitude $9.81\textrm{ m}\cdot \textrm{s}^{-2}$ and directed towards the center of the Earth, as follows:

Spring force A spring of constant $k$ and of relaxed position $\vec{x}_0$ attached a mass $m$ of position $\vec{x}$ has a force $\vec{F}_{s}$ expressed as follows:

Friction force The friction force $F_{f}$ of constant coefficient $\beta$ applied on a mass of velocity $\vec{v}$ is written as:

Mass moment of inertia The mass moment of inertia of a system of mass $m_i$ located at distance $r_i$ from point $O$, expressed in point $O$ is written as:

Torque The torque $\vec{T}$ of a force $\vec{F}$ located at $\vec{r}$ from the reference point $O$ is written as:

Newton's second law A mass $m$ of acceleration $\vec{a}$ to which forces $\vec{F}_i$ are applied verifies the following equation:

In the 1-D case along the $x$ axis, we can write it as $mx''=\sum_{i}F_i$.

In the rotationary case, around point $O$, we can write it as $J_0\theta''=\sum_iT_i$.

## Spring-mass system

### Free oscillation

Free undamped motion A free undamped spring-mass system of mass $m$ and spring coefficient $k$ follows the ODE $x''+\frac{k}{m}x=0$, which can be written as a function of the natural frequency $\omega$ as:

Free damped motion A free damped spring-mass system of mass $m$, of spring coefficient $k$ and subject to a friction force of coefficient $\beta$ follows the ODE $x''+\frac{\beta}{m}x'+\frac{k}{m}x=0$, which can be written as a function of the damping parameter $\lambda$ and the natural frequency $\omega$ as:

which has the following cases summed up in the table below:

Condition |
Type of motion |

$\lambda>\omega$ | Over damped |

$\lambda=\omega$ | Critically damped |

$\lambda<\omega$ | Under damped |

### Forced oscillation

Forcing frequency A forcing function $F(t)$ is often modeled with a periodic function of the form $F(t)=F_0\sin(\gamma t)$, where $\gamma$ is called the forcing frequency.

Forced undamped motion A forced undamped spring-mass system of mass $m$ and spring coefficient $k$ follows the ODE $x''+\frac{k}{m}x=F_0\sin(\gamma t)$, which can be written as a function of the natural frequency $\omega$ as:

which has the following cases summed up in the table below:

Condition |
Type of motion |

$\gamma\neq\omega$ | General response |

$\gamma\approx\omega$ | Beats |

$\gamma=\omega$ | Resonance |

Forced damped motion A forced damped spring-mass system of mass $m$, of spring coefficient $k$ and subject to a friction force of coefficient $\beta$ follows the ODE $x''+\frac{\beta}{m}x'+\frac{k}{m}x=F_0\sin(\gamma t)$, which can be written as a function of the damping parameter $\lambda$ and the natural frequency $\omega$ as:

## Boundary Value Problems

Types of boundary conditions Given a numerical problem between $0$ and $L$, we distinguish the following types of boundary conditions:

Name |
Boundary values |

Dirichlet | $y(0)$ and $y(L)$ |

Neumann | $y(0)$ and $y'(L)$ |

Robin | $y(0)$ and $\alpha y(L)+\beta y'(L)$ |

Numerical differentiation The table below sums up the approximation of the derivatives of $y$ at point $x_j$, knowing the values of $y$ at each point of a uniformly spaced set of grid points.

Order of derivative |
Name |
Formula |
Order of error |

First derivative | Forward difference Backward difference Central difference |
$y_{j}'=\frac{y_{j+1}-y_j}{h}$ $y_j'=\frac{y_{j}-y_{j-1}}{h}$ $y_j'=\frac{y_{j+1}-y_{j-1}}{2h}$ |
$O(h)$ $O(h)$ $O(h^2)$ |

Second derivative | Central difference | $y_j''=\frac{y_{j+1}-2y_j+y_{j-1}}{h^2}$ | $O(h^2)$ |

Direct method The direct method can solve __linear__ ODEs by reducing the problem to the resolution of a linear system $Ay=f$, where $A$ is a tridiagonal matrix.

Shooting method The shooting method is an algorithm that can solve ODEs through an iterative process. It uses a numerical scheme, such as Runge-Kutta, and converges to the right solution by iteratively searching for the missing initial condition $y'(0)$.

*Remark: in the linear case, the shooting method converges after the first two initial guesses.*