# Rotation Convention in ANSYS®

In a variety of situations when using ANSYS software, whether through the Mechanical APDL interface, or the Workbench the convention adopted in ANSYS for rotations, such as the rotational degrees of freedom of a node or of a coordinate system, is the 3-1-2 “Euler Angle” sequence. In this sequence, a first rotation is taken about the original Z axis, a second rotation is taken about the X′ axis that results from the first rotation, and a third rotation is taken about the Y″ axis that results from the second rotation. Using updated axes for the second and third rotations is called “intrinsic rotation”. Nodal degree of freedom rotations are expressed in radians in ANSYS. Rotations of coordinate systems are expressed in degrees.

The term R3 is used to refer to a rotation about a Z axis, R1 refers to a rotation about an X axis, and R2 refers to rotation about a Y axis. The 3-1-2 rotation sequence means rotations about Z, then X′, then Y″.

The right hand rule is used for rotations in ANSYS. In the following rotation matrix figure:

θ1 is the size of the first rotation about the original Z axis (called THXY in ANSYS)

θ2 is the size of the second rotation about the updated X′ axis (called THYZ in ANSYS)

θ3 is the size of the third rotation about the recently updated Y″ axis (called THZX in ANSYS)

Using the following notation:

c2 = cos(θ2) s2 = sin(θ2)

c3 = cos(θ3) s3 = sin(θ3)

R3 = first rotationθ 1 about the original Z axis

R1 = second rotation θ2 about the new X axis (X′) that results from rotation R3

R2 = third rotation θ3 about the newest Y axis (Y″) that results from rotation R1

A rotation matrix can be created for transformations via these three rotations:

Figure 1: Rotation Matrix for the 3-1-2 Euler Angle Sequence

Users will recall that rotations do not commute.

## Extra information:

Note that a variety of 3-rotation sequences could have been chosen when creating ANSYS, which chooses to use the 3-1-2 convention.

Various disciplines (vehicle engineering, flight dynamics, robotics, computer graphics, and others) and software packages choose from several 3-rotation angle possibilities. Some authors distinguish Euler angles, in which the (updated) first axis used is repeated in the third rotation, from Tait-Bryan angles, in which each axis is used once only (an example is *roll-pitch-yaw*).

**Euler angles**

1-2-1 1-3-1 2-1-2 2-3-2 3-1-3 3-2-3

**Tait-Bryan angles**

1-2-3 1-3-2 2-1-3 2-3-1 3-1-2 3-2-1

The convention chosen can depend on where the singularity of the inverse rotation is located, and whether it is inconvenient for the discipline in which the convention is adopted. If there must not be a singularity in an inverse rotation, a common choice is to use Euler Parameters (Quaternions), a 4 term rotation convention, in which the 4 terms include a non independent parameter that constrains the other 3 parameters. In the ANSYS help system there is brief mention of ANSYS using quaternions internally.

An image of the Euler Rotation Angles is given in the *Mechanical APDL Modeling and Meshing Guide*:

Figure 2: ANSYS Definition of the Three Rotations