1. Notations

\(\wedge\)	vector product symbol.
\(\theta \wedge\)	multiplication operator by the vector \(\theta\) on the left.
\(q\text{'}\)	derived from \(q\) with respect to the curvilinear abscissa.
\(\dot{q}\)	derived from \(q\) with respect to time.
\(s\)	curvilinear abscissa on the line of the centers of the sections.
\(\stackrel{ˆ}{u}\)	associated 3rd order antisymmetric matrix of axial vector \(u\).
\(1\)	3rd order unit matrix.
\(\text{Df}\text{.}\Delta x\)	directional derivative of \(f\) in the \(\Delta x\) direction
\(\left[\frac{d}{\mathrm{ds}}1\right]\)	\(\text{DIAG}\left[\frac{d}{\text{ds}},\frac{d}{\text{ds}},\frac{d}{\text{ds}}\right]\) diagonal matrix.

\(A,{I}_{\mathrm{1,2}\text{ou}3}\)	area and moments of inertia with respect to the main axes 1, 2 or 3 of the right section.
\(B\)	deformation matrix.
\(C\)	behavior matrix.
\(E,G,\rho\)	Young’s modulus and shear stiffness, density.
\({e}_{i\text{}i=\mathrm{1,3}}\)	general coordinate axes.
\({E}_{i}{(s)}_{i=\mathrm{1,3}}\)	main inertia axes of the \(s\) x-axis section in the reference position.
\(\overline{f}(s,t)\)	linear external force exerted on the beam.
\(f(s,t)\)	force in the beam at x-axis \(s\) and at the instant \(t\).
\(F(s,t)\)	\({R}^{T}(s,t)f(s,t)\).
\({F}_{\text{ext}}\)	external forces given to the nodes.
\({F}_{\text{iner}},{F}_{\text{int}}\)	inertial forces and forces within the nodes.
\({I}_{\rho }\)	inertia tensor of a unit length of a beam in a deformed position, expressed in the general axes.
\({J}_{\rho }\)	inertia tensor with a unit length of a beam in the reference position, expressed in the general axes.
\(\overline{m}(s,t)\)	linear external moment exerted on the beam.
\(m(s,t)\)	moment in the beam at the x-axis \(s\) and at the moment \(t\).
\(M(s,t)\)	\({R}^{T}(s,t)m(s,t)\).
\({N}_{i}\)	form function relating to node \(i\).
\(R(s,t)\)	operator or matrix, in general axes, for rotating the right cross section of abscissa \(s\), from the reference configuration to that at time \(t\).
\({R}_{o}(s)\)	rotation passing from the general axes to the main inertia axes of the \(s\) x-axis in the reference configuration.
\({R}_{\text{tot}}(\text{s,t})\)	\(R(s,t){R}_{o}(s)\).
\(\text{SO}(3)\)	group of rotation operators in 3-dimensional space.
\({t}_{i}{(s,t)}_{i=\mathrm{1,3}}\)	main inertia axes of the \(s\) x-axis section at the time \(t\).
\({x}_{o}(s,t)\)	position, at time \(t\), of the center of the right section of the right cross section of abscissa \(s\).
\(\varepsilon\)	\({x}_{o}^{\text{'}}-{t}_{1}\).
\(E\)	\({R}^{T}\varepsilon\).
\(\Pi\)	\((\begin{array}{cc}{R}_{\text{tot}}& 0\\ 0& {R}_{\text{tot}}\end{array})\) position of the abscissa section \(s\) the instant \(t\), \(d\) finite by the position vector \({x}_{0}\) from the center and the rotation vector \(\Psi\)
\(\phi (s,t)\)	\(\{\begin{array}{c}{x}_{o}(s,t)\\ \Psi (s,t)\end{array}\)
\(\delta \phi (s)\)	\(\left\{\begin{array}{c}\Delta {x}_{o}(s)\\ \Delta \Psi (s)\end{array}\right\}\): virtual displacement at abscissa \(s\).
\(\Delta \phi (s)\)	\(\left\{\begin{array}{c}\Delta {x}_{o}(s)\\ \Delta \Psi (s)\end{array}\right\}\): correction of displacement to abscissa \(s\).
\(\chi (s,t)\)	vector defining, at abscissa \(s\) and at time \(t\), the variation in curvature with respect to the reference configuration.
\(X\)	\({R}^{T}\chi\).
\(\Psi (s,t)\)	vector rotation, at time \(t\), of the x-axis section \(s\) in relation to its reference position.
\({\Psi }_{i-\mathrm{1,}i}^{n}\)	rotation vector between time \(i-1\) and iteration \(n\) of time \(i\).
\({\omega }_{i}^{n}\)	angular speed of a beam section calculated at iteration \(n\) of time \(i\).
\(Q,{Q}^{-1}\)	operator for passing a rotation vector to the associated quaternion and its inverse.