Derived Types

Defines the instantaneous state and scalar force of an axial element.

Defines an extensible force element between two attachment points. The element force acts along the line from point 1 to point 2. Body index zero denotes a fixed ground point whose coordinates are expressed in world coordinates. Subtypes override evaluate_force to compute a force from the current length and length rate.

Defines a two-dimensional Bernoulli-Euler beam element.

Defines a three-dimensional Bernoulli-Euler beam element.

Defines a link consisting of only two joints. The coordinate system of this link is situated at the distal joint with it's z-axis coincident with the axis of the joint. The link utilizes a Denavit-Hartenberg convention in order to express its geometry.

Defines a 3D Cartesian coordinate system.

Defines a linear, axial viscous damper element. When evaluated with nodal velocities, the element "strain" is the elongation rate and the element "stress" is the damper force (positive in tension).

Defines a 2D linear, axial viscous damper element with x and y translations at each node.

Defines a 3D linear, axial viscous damper element with x, y, and z translations at each node.

Defines a generalized alpha integrator for dense matrix systems.

Describes a set of Denavit-Hartenberg parameters for a single joint.

Describes a Denavit-Hartenberg table.

Defines a discrete (lumped-parameter) element. Discrete elements have no spatial extent; therefore, the stiffness, mass, and damping matrices default to zero and are overridden only where the element contributes to the system. The inherited material is unused.

Describes one joint constraint in maximal-coordinate body indexing. Body index zero denotes the fixed ground link. Each frame is a homogeneous transform from its associated body frame to the joint frame; for a ground body, the frame is expressed directly in world coordinates. The parent and child joint frames coincide when the joint constraint is satisfied.

A container of a single measurement data set.

Defines an element.

A container for a frequency response function, or series of frequency response functions.

Defines an undirected multigraph. Multiple edges between the same pair of vertices, as well as self-loops, are permitted.

Defines an undirected edge connecting two vertices.

Describes an independent loop within the graph. The loop is defined by an edge that is not part of the spanning tree; the loop itself is closed by the tree paths leading to each of the edge's vertices.

Describes a path through a graph.

Defines a kinematic joint connecting two links. The joint attaches to a coordinate frame on each of the two links; the frames are identified by their index within each link. The joint is coincident with both frames when all of its joint variables are zero.

Defines the constraint reaction exerted by a joint on its child link. Both vectors are expressed in the world coordinate frame. The reaction exerted on the parent link is equal and opposite.

Defines the behavior common to all multi-link mechanisms, including those containing closed kinematic loops. The joint variables of the mechanism are collected into a single array; the variables belonging to each joint appear in the order in which the joints were supplied.

Defines the parametric form of a line $$\boldsymbol{r}(t)=\boldsymbol{r}_0+t\boldsymbol{v}.$$

Defines a line element type.

Defines a linear viscous damper acting only along the element axis.

Defines a linear axial spring supporting tension and compression.

Defines the base class for a link within a mechanism. A link is a rigid body that carries one coordinate frame for each joint that may attach to it.

A container allowing a collection of differing link types to be stored within a single array.

Defines the variational-integrator representation of a linkage. Body index zero in a joint descriptor denotes the fixed ground link.

Defines a translational point mass element attached to a single node.

Defines a 2D translational point mass element with x and y translations.

Defines a 3D translational point mass element with x, y, and z translations.

Defines a linear-elastic-isotropic material.

A container for the frequency responses of a system of multiple inputs and multiple outputs (MIMO).

A container for model information.

Defines a link carrying an arbitrary number of joint coordinate frames. This type is suitable for the ternary and higher-order links found in closed-loop mechanisms.

Defines a node.

Defines a spatial, closed-loop linkage. Each independent loop within the mechanism contributes six loop-closure constraint equations.

Defines a planar, closed-loop linkage operating within the x-y plane. Each independent loop within the mechanism contributes three loop-closure constraint equations, avoiding the redundant equations that arise when a planar mechanism is analyzed in three dimensions.

Defines a plane as $$ax+by+cz+d=0,\qquad \hat{n}=\frac{(a,b,c)}{\sqrt{a^2+b^2+c^2}}.$$

Defines a line in 3D Euclidean space using Plücker coordinates. For a point $\boldsymbol{r}_0$ and direction $\boldsymbol{v}$, the coordinates are $L=(\boldsymbol{v},\boldsymbol{m})$ with $\boldsymbol{m}=\boldsymbol{r}_0\times\boldsymbol{v}$ and $\boldsymbol{v}\cdot\boldsymbol{m}=0$.

Defines a point in 3D, Cartesian space.

Defines a quaternion of the form $$q=w+x\,\mathbf{i}+y\,\mathbf{j}+z\,\mathbf{k}.$$ Its norm is $\lVert q\rVert=\sqrt{w^2+x^2+y^2+z^2}$; unit quaternions represent three-dimensional rotations.

Defines a four-node flat shell element.

Defines a rigid body with mass properties about its center of mass. The translational and rotational kinetic energies are represented by $$T=\frac{1}{2}m\boldsymbol{v}_{cg}^{T}\boldsymbol{v}_{cg} +\frac{1}{2}\boldsymbol{\omega}^{T}I_{cg}\boldsymbol{\omega}.$$

Defines a serial linkage.

Defines a flat shell element with six degrees of freedom per node (three translations and three rotations).

Describes a spanning tree of a graph as generated by a breadth-first traversal.

Defines a generalized alpha integrator for sparse matrix systems.

Defines a linear, axial spring element. The element "strain" is the spring elongation and the element "stress" is the spring force (positive in tension).

Defines a 2D linear, axial spring element with x and y translations at each node.

Defines a 3D linear, axial spring element with x, y, and z translations at each node.

Defines a state-space representation of a dynamic system. This implementation takes the form:

Base class for linear structural time integrators.

Defines a linear damper opposing only joint-axis twist rate.

Defines the instantaneous twist state and torque of a torsional element.

Defines an extensible torsional element bound to a revolute joint.

Defines a linear torsional spring about a revolute-joint axis.

Defines a transfer function for a continuous system of the form $H(s) = \frac{Y(s)}{X(s)}$.

Defines a three-node flat shell element.

Defines a pin-jointed 2D bar with x and y translations per node. Use the parent mass_matrix and the global axial stiffness override.

Defines a pin-jointed 3D bar with three translations per node. The local transverse axes are arbitrary; only the longitudinal direction influences axial strain and stiffness.

Defines a discrete element that connects two nodes and acts along a single axis.

Implements the maximal-coordinate variational integrator in Brudigam et al. (2023), equations (18)-(20). The nonlinear system is solved with damped Newton iterations and either dense LU or the paper's graph-structured block factorization.

Reports convergence diagnostics for a step or trajectory solve.

Defines numerical settings for the nonlinear variational step.

Defines the maximal-coordinate state of a collection of rigid bodies. Angular velocities and inertia tensors are expressed in each body's frame. Positions, translational velocities, forces, and orientations use the world frame.