1. Transformation Matrix and Angle Calculations Time-series of angles may be calculated based on element transformation matrices. The transformation matrix may be stored for elements in the dynamic analysis and further post-processed to calculate the angle between local x-axis of two elements. The transformation matrix \([\Lambda\)] contains direction cosines between axes of the global coordinate system \(x,y,z\) and the local coordinate system \(x^{’},y^{’},z^{’}\). A vector \(V\) can be expressed in terns of components \(u,v,w\) in system \(x,y,z\) or in terms of components \( u^{’},v^{’},w^{’}\) in system \( x^{’},y^{’},z^{’}\). \[\begin{Bmatrix} u^{’}\\ v^{’}\\ w^{’} \end{Bmatrix} = [\Lambda] \begin{Bmatrix} u\\ v\\ w \end{Bmatrix}\] \([\Lambda\)] is an orthogonal matrix, hence the inverse transform is \[\begin{Bmatrix} u\\ v\\ w \end{Bmatrix} = [\Lambda]^{T} \begin{Bmatrix} u^{’}\\ v^{’}\\ w^{’} \end{Bmatrix}\] The cosine of an angle between a local and global axis is found directly from \([\Lambda\)].For examle, the angle between local x-axis and global z-axis (for each timestep) \(\phi_{xz}\) is \[\phi_{xz} = acos(\Lambda(1,3))\] The angle between local x-axis and global y-axis \[\phi_{xy} = acos(\Lambda(1,2))\] The angle between local x-axis and global x-axis \[\phi_{xx} = acos(\Lambda(1,1))\] The angle between the local x-axis of two elements is \[\phi_{xx} = acos(V_{x1} \cdot V_{x2} )\] where \[V_{x1}= [\Lambda_{1}]^{T} \begin{Bmatrix} 1\\ 0\\ 0 \end{Bmatrix}\] \[V_{x2}= [\Lambda_{2}]^{T} \begin{Bmatrix} 1\\ 0\\ 0 \end{Bmatrix}\] Note that the post-processor will return the smallest positive angle between the elements/axes in degrees. Currently, angle between two elements (relative element angles) may be requested in storage parameters.