where $\kappa (s,t)$ is the curvature/twist vector defined as the rotation of the body-fixed frame ${ai}$ per unit contour length relative to the inertial frame ${ei}$, $\omega (s,t)$ is the angular velocity of the cross section defined as the rotation of the body-fixed frame ${ai}$ per unit time relative to the inertial frame ${ei}$, $\nu (s,t)$ is the velocity of the strand cross section centroid, $ms(s)$ is the mass of the strand per unit contour length, and $Is(s)$ denotes the diagonal $3\xd73$ tensor of principal mass moments of inertia per unit contour length. The quantities $f(s,t)$ and $q(s,t)\u2009$ are the internal force vector and internal moment vector, respectively. Finally, $Fbody(s,t)$ and $Qbody(s,t)$ denote the sum of all the distributed external body forces and moments per unit contour length, respectively, and $a3$ is the unit tangent vector. (Note that $Fbody\u2009$ and $Qbody$ may also be functions of the kinematic variables $\kappa (s,t)$, $\omega (s,t)$, and $\nu (s,t).)$ In Eqs. (1) and (2), the quantities $\kappa (s,t)$, $\omega (s,t)$, $\nu (s,t)$, and $f(s,t)$ define four unknown field variables which also satisfy two additional field equations
Display Formula

(3)$\u2202\nu \u2202s+\kappa \xd7\nu =\omega \xd7a3$

Display Formula(4)$\u2202\omega \u2202s+\kappa \xd7\omega =\u2202\kappa \u2202t$