Dyads, Dyadics, and Inertia

Dyads

Dyads and dyadics are a difficult concept to understand, initially. Dyads may be defined in a number of ways, but the way which makes most sense to me is that dyads are the vector-based mathematical representation for physical phenomena that contain coupled, multi-dimensional information. An example of a physical system that contains multi-dimensional information is inertia. Inertia describes the spatial distribution of mass throughout a rigid body, and this spatial distribution comes encoded in a structure in which mass is represented in two-vector pairs.

Mathematically, operating on a dyad from the left or the right produces a different answer. if ${d}=(\vec{v}_1,\vec{v}_2)$, then $\vec{v}_3\cdot d=(\vec{v}_3\cdot \vec{v}_1)\vec{v}_2$. The ordering of vectors in a dyad is fixed, meaning that $\vec{v} \cdot d \neq d \cdot \vec{v}$. Dyads may be multiplied by scalars, which are commutative. Thus $3*(v1,v2) = (3v1,v2) = (v1,3v2)$ are all equivalent statements. Thus, most often, scalar values are brought out of dyads so that only unit vectors are conatined within the dyad itself.

In pynamics, the dyad class represents a two-vector collection. Dot and cross product operations are supported between vectors and dyads. A dyad class in pynamics may be constructed by supplying any two basis vectors.

import pynamics

from pynamics.frame import Frame
from pynamics.dyadic import Dyad
from pynamics.system import System

system = System()
pynamics.set_system(__name__,system)

A = Frame('A')
d = Dyad(A.x,3*A.y)

v = A.x+A.y+A.z

d.dot(v)

3*A.x

v.dot(d)

3*A.y

Dyadics

A dyadic is a linear combination of dyads, just as a vector is a linear combination of one or more basis vectors. In pynamics, dot product, cross product, and addition operations between dyadics and vectors are supported.

d = Dyad(A.x,3*A.y)
e = Dyad(A.z,3*A.x)
my_dyadic = d+e
my_dyadic

(A.x, 3*A.y)+(A.z, 3*A.x)

type(my_dyadic)

pynamics.dyadic.Dyadic

v.dot(d+e)

3*A.x + 3*A.y

Inertia

Inertia is the term for the distribution of mass throughout a rigid body. Depending on the point about which your system rotates, that distribution of mass is different. That is why, in pynamics the inertia class needs quite a bit of information before you can use it.