Using the basic notation for coordinates in Section :ref:`sec-generalnotation`, we use the following quantities and symbols for equations of motion and solvers:
quantity
|
symbol
|
description
|
|---|---|---|
number of :ref:`ODE2 <ODE2>` coordinates
|
n
|
|
number of :ref:`ODE1 <ODE1>` coordinates
|
n_\FO
|
|
number of :ref:`AE <AE>` coordinates
|
m
|
|
number of system coordinates
|
n_{\SYS}
|
SYSN
|
:ref:`ODE2 <ODE2>` coordinates
|
{\mathbf{q}} = [q_0,\, \ldots,\, q_{n_q}]\tp
|
:ref:`ODE2 <ODE2>`, displacement-based coordinates (could also be rotation or deformation coordinates)
|
:ref:`ODE2 <ODE2>` velocities
|
\vel = \dot {\mathbf{q}} = [\dot q_0,\, \ldots,\, \dot q_{n_q}]\tp
|
:ref:`ODE2 <ODE2>` velocity coordinates
|
:ref:`ODE2 <ODE2>` accelerations
|
\ddot {\mathbf{q}} = [\ddot q_0,\, \ldots,\, \ddot q_{n_q}]\tp
|
:ref:`ODE2 <ODE2>` acceleration coordinates
|
:ref:`ODE1 <ODE1>` coordinates
|
{\mathbf{y}} = [y_0,\, \ldots,\, y_{n_y}]\tp
|
vector of n_y coordinates for :ref:`ODE1 <ODE1>`
|
:ref:`ODE1 <ODE1>` velocities
|
\dot {\mathbf{y}} = [\dot y_0,\, \ldots,\, \dot y_{n_y}]\tp
|
vector of n velocities for :ref:`ODE1 <ODE1>`
|
:ref:`ODE2 <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 :ref:`ODE1 <ODE1>` and :ref:`ODE2 <ODE2>` coordinates
|
data coordinates
|
{\mathbf{x}} = [x_0,\, \ldots,\, x_l]\tp
|
vector of l data coordinates in any configuration
|
{\mathbf{f}}_\SO\in \Rcal^{n_q}
|
right-hand-side of :ref:`ODE2 <ODE2>` equations; (all terms except mass matrix \times acceleration and joint reaction forces)
|
|
{\mathbf{f}}_\SO\in \Rcal^{n_y}
|
right-hand-side of :ref:`ODE1 <ODE1>` equations
|
|
{\mathbf{g}}\in \Rcal^{m}
|
algebraic equations
|
|
mass matrix
|
{\mathbf{M}}\in \Rcal^{n_q \times n_q}
|
mass matrix, only for :ref:`ODE2 <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
|