• 05-sep-2025 25i20.01 Initial discussion and sketcth.

notes

• Source of the disturbance is a vibrator, which make the grains oscillating ¹.
• Problem when aligning grouped objects in Word and solution with Inkscape ².
• Inkscape installation for creating the figures ³.
• Sections eqns-a – eqns-e on 05-sep-2025.
• There is a color scheme for showing Mermaid code on 25i25 .

eqns-a

• max function $$\tag{1} \max(a, b) = \left\{ \begin{array}{rc} a, & a \ge b, \newline b, & a < b. \end{array} \right. $$
• sign function $$\tag{2} {\rm sign(x)} = \left\{ \begin{array}{rc} 1, & x > 0, \newline 0, & x = 0, \newline -1, & x < 0. \end{array} \right. $$

egns-b

• Weight of particle $i$ is ⁴ $$\tag{3} \vec{W}_{i} = m_i \vec{g}, $$ with $g$ is gravitational acceleration.

egns-c

• Base particle has mass $m_0$, radius $R_0$, vertical position $z_0$.
• Relative position of particle $i$ from particle $j$ is $$\tag{4} \vec{r}_{ij} = \vec{r}_i - \vec{r}_j. $$
• Relative distance of particle $i$ from particle $j$ is $$\tag{5} r_{ij} = |\vec{r}_{ij}|. $$
• Unit verctor of relative position is $$\tag{6} \hat{r}_{ij} = \frac{\vec{r} _{ij}}{r _{ij}}. $$
• Overlap between to adjacent particles is $$\tag{7} \xi_{ij} = \max(0, R_i + R_j - r_{ij}). $$
• Time derivative of overlap is $$\tag{8} \dot{\xi} _{ij} = - v _{ij} \ {\rm sign}(\xi _{ij}). $$
• Normal force on particle $i$ due to contact with particle $j$ is $$\tag{9} \vec{N}_{ij} = k_N \xi _{ij} \hat{r} _{ij} - \gamma_N \dot{\xi} _{ij} \hat{v} _{ij}. $$

egns-d

• Number of particles is $n$.
• Index of particle is from 1 to $n$.
• Base or particle with index 0 is not considered as particle.
• Net external force on particle $i$ is $$\tag{10} \vec{F}_i = \left\{ \begin{array}{rc} \vec{N} _{i, i-1} + \vec{N} _{i, i+1} + \vec{W}_i, & 0 < i < n, \newline \vec{N} _{i, i-1} + \vec{W}_i, & i = n. \end{array} \right. $$

eqns-e

• Newton’s second law is ⁵ $$\tag{11} \vec{a}_i = \frac{\vec{F}_i}{m_i} $$ for partile $i$ at time $t$.
• Euler integration for calculating velocity from acceleration ⁶ gives $$\tag{12} \vec{v}_i(t + \Delta t) = \vec{v}_i(t) + \vec{a}_i \Delta t. $$
• With similar way, position is obtained from velocity using $$\tag{13} \vec{r}_i(t + \Delta t) = \vec{r}_i(t) + \vec{v}_i(t) \Delta t. $$
• New position, which is $$\tag{14} \vec{r}_i \equiv \vec{r}_i(t), $$ will update Eqns (4)-(10).

flowchart

A proposed simulation pipeline is as follow.

refs