Procedures

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

Evaluates a linear chirp function. The instantaneous frequency varies linearly, $$f(t)=f_1+\frac{f_2-f_1}{T}t,$$ giving phase $\phi(t)=2\pi(f_1t+(f_2-f_1)t^2/(2T))$ and response $x(t)=A\sin(\phi(t))$.

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. Equivalently, $\zeta_i=(\alpha+\beta\omega_i^2)/(2\omega_i)$.

Creates a connectivity matrix for the element, stored in CSR format. The matrix contains exactly one non-zero (unity) entry per row; therefore, it is well-suited to a sparse representation.

Computes the cross-product of two three-dimensional vectors. The result is orthogonal to both inputs and is defined by $$\boldsymbol{x}\times\boldsymbol{y} = \begin{bmatrix}x_2y_3-x_3y_2\\x_3y_1-x_1y_3\\x_1y_2-x_2y_1\end{bmatrix}.$$

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 $\Delta f = \frac{f_n}{Q}$. Combining this relation with $Q = 1/(2\zeta)$ gives $$\Delta f = 2\zeta f_n.$$

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. If peaks are separated by $N$ cycles, the damped frequency estimate is $$f_d = \frac{N}{t_2-t_1}.$$

Estimates the settling amplitude for a step response. The final-value estimate is the zero-frequency Fourier coefficient, $$x_f \approx \frac{1}{N}\sum_{k=0}^{N-1}x_k.$$

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

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

Constructs the 4-by-4 homogeneous transformation matrix describing a rotation about an x-axis.

Constructs the 4-by-4 homogeneous transformation matrix describing a rotation about a y-axis.

Constructs the 4-by-4 homogeneous transformation matrix describing a rotation about a y-axis.

Initializes a rigid_body object.

Allocates and initializes a maximal-coordinate state. Positions and velocities are set to zero, and every orientation is set to the identity quaternion.

Computes the inverse of a quaternion. For a nonzero quaternion, $$q^{-1}=\frac{\overline{q}}{|q|^2}.$$

Tests to see if a point lies on a line.

Tests to see if a point lies on a plane.

Tests to see if a matrix is symmetric.

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

Gets the number of degrees of freedom permitted by the requested joint type.

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.

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

Computes nodal strain results by averaging the strain contributions from each element incident upon a node.

Computes nodal stress results by averaging the stress contributions from each element incident upon a node.

Computes the normal vector to a line defined by pt1 and pt2 assuming some point (pt) not on the line.

Normalizes mode shape vectors such that the largest magnitude value in the vector is one. For each column $\boldsymbol{\phi}_i$, the operation is $$\boldsymbol{\phi}_i\leftarrow \frac{\boldsymbol{\phi}_i}{\phi_{i,k}},\qquad k=\arg\max_j|\phi_{i,j}|.$$

Returns the normal vector of a 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 $Q = \frac{1}{2 \zeta}$. For a lightly damped mode, $Q$ is also approximately the ratio of the resonant frequency to its half-power bandwidth.

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 $$s = \frac{\boldsymbol{x}\cdot\boldsymbol{y}}{\|\boldsymbol{y}\|}.$$

Computes the derivative of the shape function with respect to the natural coordinate specified. The derivative is approximated centrally as $$N_{,i}(\boldsymbol{s})\approx \frac{N(\boldsymbol{s}+h\boldsymbol{e}_i)- N(\boldsymbol{s}-h\boldsymbol{e}_i)}{2h}.$$ The second derivative uses the centered finite difference $$N_{,ii}(\boldsymbol{s})\approx \frac{N(\boldsymbol{s}+h\boldsymbol{e}_i)-2N(\boldsymbol{s})+ N(\boldsymbol{s}-h\boldsymbol{e}_i)}{h^2}.$$

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. For a proper rotation, the angle is recovered from $$\theta = \cos^{-1}\left(\frac{\operatorname{tr}(R)-1}{2}\right),$$ while the skew-symmetric part satisfies $$R-R^T = 2\sin(\theta)[\hat{u}]_\times.$$

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

Computes the inverse of a 4-by-4 homogeneous transformation matrix.

Computes the unsigned angle between two nonzero vectors. $$\theta = \cos^{-1}\left(\frac{\boldsymbol{x}\cdot\boldsymbol{y}} {\|\boldsymbol{x}\|\,\|\boldsymbol{y}\|}\right),\qquad 0\leq\theta\leq\pi.$$

Projects a vector onto a plane.

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

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