Name

transformations — Local coordinate systems definition block

Synopsis

transformations
transformation specification ...
...
end

Description

The transformations command specifies reference systems (rotations) with respect to the (branch) global coordinate reference system. transformations only specifies the transformation, it does not transform any coordinates or DOF's, the effective transformations being executed for the respective coordinates or DOF's by the b2000++ processor. The transformation option of the nodes command specifies the DOF reference system transformation identifier to be applied to specific nodes.

The transformations can be specified more than once.

Transformation specification

id type p1x p1y p1z p2x p2y p2z [p3x p3y p3z]

Specifies a transformation by its identifier id, the type of transformation type, and the points p1, p2, p3 defining the transformation. The transformation identifier id must consist of a positive integer and it must be unique.

If type is set to CARTESIAN (change of orthogonal base), p1, p2, and p3 are required and a change of orthogonal base will be applied.

If type is set to CYLINDRICAL a cylindrical transformation will be applied, p2-p1 defining the cylinder z-axis, (p2-p1) x p3 the cylinder radial direction, and ((p2-p1) x p3) x (p2-p1) the cylinder angular direction

If type is set to SPHERICAL a spherical transformation will be applied, with p1, p2, and p3 being required.

If type is set to BASE, p1 and p2 are required, p1 defining the base vector e1, p2 the base vector e2, the base vector e3 being calculated with p1 x p2. The base vectors need not be normalized to 1.

Example

Specify a cylindrical coordinate system 123, with the cylinder axis identical to the global z-axis, i.e. by p1 = (0,0,0) and p2 = (0,0,1), such that p2-p1 = (0,0,1).

transformations
123 cylindrical 0. 0. 0. 0. 0. 1. 0. 0. 0.
end

Transformations

The Cartesian coordinate system transformation is defined as follows: The origin is equal to the node coordinates p. The local z-axis is defined by z = p2-p1. The local y-axis is defined by y = z x (p3-p1). The local x-axis is defined by x = z x y.

The cylindrical coordinate system transformation is defined as follows: The origin is equal to the point p1. The cylinder z-axis is defined by z=p2-p1. The tangential (phi) direction of a node p is then defined by t=(p-p1) x z and the cylinder's radius direction r=z x p3. Point p3, together with p1 and p2, spans a plane in which phi=0, rotating in the positive direction around the local z-axis z = p2-p1. Note: p3 is ignored for transformation, i.e. the local (radius,phi,z) system is uniquely defined by the node coordinate p and points p1 and p2.

The spherical coordinate system transformation is defined as follows: The origin is equal to the point P1. The sphere z-axis is defined by z=p2-p1. The tangential (phi) direction of a node p is then defined by t=(p-p1) x z and the sphere's theta direction r=z x p3. The declination theta rotates in a positive direction starting from z. Note: p3 is ignored for transformation, i.e. the local (phi,theta,z) system is uniquely defined by points p1 and p2.

Internal node transformations

Given the transformation definitions and the coordinates, the B2000++ input processor computes the effective DOF transformation matrices for all nodes involved. The transformation matrices are stored in data sets NLCS.br and referred to by the second column of the NODA.br sets. The transformation is defined as unl = T ubg , where unl is a vector referring to the node local system and ubg to the branch global system. The transformation T contains the 3 normalized base vectors ei row-wise. ei specifies the ith base vector of the node local system with respect to the branch coordinate system.

e11 e12 e13

e21 e22 e23

e31 e32 e33