Nomenclature for system equations of motion and solvers

Using the basic notation for coordinates in Section Notation, we use the following quantities and symbols for equations of motion and solvers:

quantity	symbol	description
number of ODE2 coordinates	\(n\)	ODE2
number of ODE1 coordinates	\(n_\FO\)	ODE1
number of AE coordinates	\(m\)	AE
number of system coordinates	\(n_{\SYS}\)	SYSN
ODE2 coordinates	\({\mathbf{q}} = [q_0,\, \ldots,\, q_{n_q}]\tp\)	ODE2, displacement-based coordinates (could also be rotation or deformation coordinates)
ODE2 velocities	\(\vel = \dot {\mathbf{q}} = [\dot q_0,\, \ldots,\, \dot q_{n_q}]\tp\)	ODE2 velocity coordinates
ODE2 accelerations	\(\ddot {\mathbf{q}} = [\ddot q_0,\, \ldots,\, \ddot q_{n_q}]\tp\)	ODE2 acceleration coordinates
ODE1 coordinates	\({\mathbf{y}} = [y_0,\, \ldots,\, y_{n_y}]\tp\)	vector of \(n_y\) coordinates for ODE1
ODE1 velocities	\(\dot {\mathbf{y}} = [\dot y_0,\, \ldots,\, \dot y_{n_y}]\tp\)	vector of \(n\) velocities for ODE1
ODE2 Lagrange multipliers	\(\tlambda = [\lambda_0,\, \ldots,\, \lambda_m]\tp\)	vector of \(m\) Lagrange multipliers (=algebraic coordinates), representing the linear factors (often forces or torques) to fulfill the algebraic equations; for ODE1 and ODE2 coordinates
data coordinates	\({\mathbf{x}} = [x_0,\, \ldots,\, x_l]\tp\)	vector of \(l\) data coordinates in any configuration
RHS ODE2	\({\mathbf{f}}_\SO\in \Rcal^{n_q}\)	right-hand-side of ODE2 equations; (all terms except mass matrix \(\times\) acceleration and joint reaction forces)
RHS ODE1	\({\mathbf{f}}_\SO\in \Rcal^{n_y}\)	right-hand-side of ODE1 equations
AE	\({\mathbf{g}}\in \Rcal^{m}\)	algebraic equations
mass matrix	\({\mathbf{M}}\in \Rcal^{n_q \times n_q}\)	mass matrix, only for ODE2 equations
(tangent) stiffness matrix	\({\mathbf{K}}\in \Rcal^{n_q \times n_q}\)	includes all derivatives of \({\mathbf{f}}_\SO\) w.r.t. \({\mathbf{q}}\)
damping/gyroscopic matrix	\({\mathbf{D}}\in \Rcal^{n_q \times n_q}\)	includes all derivatives of \({\mathbf{f}}_\SO\) w.r.t. \(\vel\)
step size	\(h\)	current step size in time integration method
residual	\({\mathbf{r}}_\SO \in \Rcal^{n_q}\), \({\mathbf{r}}_\FO \in \Rcal^{n_y}\), \({\mathbf{r}}_\AE \in \Rcal^{m}\)	residuals for each type of coordinates within static/time integration – depends on method
system residual	\({\mathbf{r}}\in \Rcal^{n_s}\)	system residual – depends on method
system coordinates	\(\txi\)	system coordinates and unknowns for solver; definition depends on solver
Jacobian	\({\mathbf{J}}\in \Rcal^{n_s \times n_s}\)	system Jacobian – depends on method