Procedures

Computes the acceleration transformation matrix relating the position of a point expressed in a rotating and translating body relative to its parent frame.

Applies a displacement constraint to the specified degree of freedom.

Evaluates a linear chirp function.

Computes the modal damping factors $\zeta_i$ given the proportional damping terms $\alpha$ and $\beta$ where $\alpha + \beta \omega_{i}^2 = 2 \zeta_{i} \omega_{i}$, $\lambda_{i} = \omega_{i}^2$, and $\lambda_i$ is the $i^{th}$ eigenvalue of the system.

Creates a connectivity matrix for the element.

Computes the cross-product of a vector.

Employs the method of fractional overshoot to estimate the damping ratio from the response of a system to a step input. This method is useful for cases where the damping ratio is between approximately 0.5 to 0.8. In such range, the logarithmic decrement approach becomes less precise.

Computes the damping ratio from the logarithmic decrement $\delta$. The damping ratio is related to the logarithmic decrement by the following relationship.

Determines the nature of stability/unstability near the point at which the dynamics matrix was computed.

Computes the Denavit-Hartenberg transformation matrix for the specified DH parameters.

Computes the Denavit-Hartenberg matrix for a local x-axis rotation.

Computes the Denavit-Hartenberg matrix for a local z-axis rotation.

Computes the Denavit-Hartenberg matrix for a local x-axis translation.

Computes the Denavit-Hartenberg matrix for a local z-axis translation.

Tests to see if two lines intersect.

Estimates the bandwidth of the resonant mode of a vibratory system. The bandwidth is the width of the range of frequencies for which the energy is at least half its peak value and is computed as $\Delta f = \frac{f_n}{Q}$.

Evaluates the response of an underdamped single-degree-of-freedom, linear system to a step function of amplitude $X_s$.

Given a free-response time history, this routine attempts to find the logarithmic decrement and resonant frequency of a vibratory system. The logarithmic decrement is estimated by finding successive peaks by means of peak detection.

Estimates the settling amplitude for a step response.

Fits an experimentally obtained frequency response by model for either a receptance model:

Computes the frequency response functions for a system of ODE's.

Computes a 4-by-4 identity matrix.

Initializes a rigid_body object.

Computes the inverse of a quaternion.

Tests to see if a point lies on a line.

Tests to see if a point lies on a plane.

Computes a single Jacobian generating vector given the position vector of the link origin, $\vec{d}$, and the joint axis unit vector, $\vec{k}$.

Returns the common normal line between two lines pointing from ln1 to ln2. In the event that the two lines are parallel within the specified tolerance, there exist an infinite number of common normals; therefore, a line will be chosen that runs from ln1 to ln2 with the point at t = 0 coincident with the point at t = 0 on ln1.

Constructs a new line from a point (defines the point where t = 0) and a direction vector.

Computes the logarithmic decrement given the value of two successive peaks in the time history of the free vibratory response of the system. The logarithmic decrement is calculated as follows.

Solves the LTI system given by the specified state space model.

Computes the modal frequencies and modes shapes for multi-degree-of-freedom system.

Gets the line parameter for the point on the line nearest the specified point.

Normalizes mode shape vectors such that the largest magnitude value in the vector is one.

Returns the normal vector of a plane.

Generates a Poincare map by determining the intersections of the supplied trajectory with the specified plane.

Projects a point onto a plane.

Computes the shortest distance between a point and a line.

Computes the shortest distance between a point and a plane.

Estimates the Q-factor for a vibratory system. The Q-factor is computed $Q = \frac{1}{2 \zeta}$.

Reports an array index-out-of-bounds error.

Reports an array size error.

Reports an error associated with an incorrect number of constraints.

Reports a convergence error.

A generic error reporting routine.

Reports a matrix size error.

Reports a mismatch in matrix sizes.

Reports a memory allocation error.

Reports a nonmonotonic array error.

Reports an error relating to a non-square mass matrix.

Reports an error relating to a non-square matrix.

Reports an error relating to a non-square stiffness matrix.

Reports a null forcing routine pointer error.

Reports a null solver routine error.

Reports an overconstraint error.

Reports a zero-difference between two variables where a non-zero difference was expected.

Reports an error associated with a zero-valued frequency value.

Restores the constrained degrees-of-freedom from the boundary conditions applied by apply_boundary_conditions.

Computes the rise time for an underdamped, second-order system. The rise time is the time it takes for the system response to go from 0% to 100% of its final value and is given by the following relationship.

Constructs the rotation matrix describing a rotation about an x-axis such that $\overrightarrow{r_2} = \textbf{R}_x \overrightarrow{r_1}$.

Constructs the rotation matrix describing a rotation about a y-axis such that $\overrightarrow{r_2} = \textbf{R}_y \overrightarrow{r_1}$.

Constructs the rotation matrix describing a rotation about a y-axis such that $\overrightarrow{r_2} = \textbf{R}_z \overrightarrow{r_1}$.

Computes the projection of vector x onto vector y. The scalar projection is defined such that $s = \frac{\vec{x} \cdot \vec{y}}{||\vec{y}||}$.

Computes the derivative of the shape function with respect to the natural coordinate specified.

Computes the second derivative of the shape function with respect to the natural coordinate specified.

Solves the inverse kinematics problem for a linkage. An iterative solution procedure is utilized.

Extracts the equivalent rotation angle and axis of rotation given a 3-by-3 rotation matrix.

Converts a 3-element vector to a 3-by-3 skew-symmetric matrix. A skew-symmetric matrix is defined as follows.

Computes the angle between two vectors.

Projects a vector onto a plane.

Computes the vector pojection of vector x onto vector y. The vector projection is defined such that $proj_{y} \vec{x} = \frac{\vec{x} \cdot \vec{y}}{||\vec{y}||^{2}} \vec{y}$

Computes the velocity transformation matrix relating the position of a point expressed in a rotating and translating body relative to its parent frame.