The following notations are employed throughout the present documentation set.

Vectors are denoted by boldface type, such as \(\mathbf{v}\). The length or magnitude is denoted \(\left\| \mathbf{v} \right\|\) or simply \(v\). The addition of a hat denotes a unit vector, such that \(\hat{\mathbf{v}} = \mathbf{v} / \left\| \mathbf{v} \right\|\). The addition of a dot denotes a time derivative, such as \(\dot{\mathbf{v}} = \partial \mathbf{v} / \partial t\).

There is a global coordinate system (\(xyz\)). The vector \(\mathbf{v}\) can be expressed in the global coordinate system by the relations:

\[ \begin{align}\begin{aligned}:label: notations1\\\begin{split}\begin{array}{l} \mathbf{v} = \mathbf{v}(x,y,z) = v_x \hat{\mathbf{i}} + v_y \hat{\mathbf{j}} + v_z \hat{\mathbf{k}} \\ \mbox{with } v_x = \mathbf{v} \cdot \hat{\mathbf{i}},\; v_y = \mathbf{v} \cdot \hat{\mathbf{j}},\; v_z = \mathbf{v} \cdot \hat{\mathbf{k}} \end{array}\end{split}\end{aligned}\end{align} \]

where \(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\), and \(\hat{\mathbf{k}}\) are unit vectors directed along the positive \(x\), \(y\), and \(z\) axes, respectively. The global coordinate system is used by both the 2D and 3D models. The 2D model is oriented to lie in the \(xy\) plane, resulting in the following constraints in (see Figure 1):

  1. Particles and contacts remain centered in the \(xy\) plane.

  2. Particles consist of unit-thickness disks.

  3. Forces act in the \(xy\) plane such that

    (1)\[\mathbf{F} = \mathbf{F}(x,y,z) = F_x \hat{\mathbf{i}} + F_y \hat{\mathbf{j}} + F_z \hat{\mathbf{k}} \quad (F_z \equiv 0).\]
  4. Moments act perpendicular to the \(xy\) plane such that

    (2)\[\mathbf{M} = \mathbf{M}(x,y,z) = M_x \hat{\mathbf{i}} + M_y \hat{\mathbf{j}} + M_z \hat{\mathbf{k}} \quad (M_x \equiv M_y \equiv 0).\]

There are no such constraints on the 3D model.


Figure 1: Global coordinate system and orientation of the 2D model, which consists of unit-thickness disks centered in the \(xy\) plane.