### 10.2.2. Rotational Equations of Motion

The equation of motion for a rotating rigid body can be rewritten in body-fixed coordinates (about the center of mass) as:

| (10–11)

where the mass moment of inertia matrix, , is defined as:

| (10–12)

In the mass moment of inertia matrix shown in Equation 10–12, the center of mass is given by , and is a differential element of mass.

Equation 10–11 represents the spatial coordinate form for the classical Euler’s equation for the rigid body in body-fixed coordinates.

Also, in Equation 10–11 is the total moment from all the separate contributions including spring and other external moments:

| (10–13)

where is the aerodynamic torque, is the rotational spring constant and is all other external torques acting on the body.

#### 10.2.2.1. First Order Backward Euler

From Equation 10–11 the first order backward Euler algorithm begins with:

| (10–14)

Using the first order backward Euler algorithm, is given by:

| (10–15)

where is determined from Equation 10–14.

Equation 10–15 can be expressed in terms of , angles in the global frame (see 206), as:

| (10–16)

where

| (10–17)

and where and are Euler Angle Y and Euler Angle X, respectively. For details on Euler Angles see Rigid Body Motion in the CFX-Solver Modeling Guide.

Integrating in global coordinates using the first order backward Euler algorithm:

| (10–18)

where is determined from Equation 10–16.

Note that in addition to providing only a first order integration of the rotational momentum the first order backward Euler method does not exactly conserve total angular momentum and is subject to potential of encountering Gimbal lock [205].

#### 10.2.2.2. Simo Wong Algorithm

The Simo Wong (ALGO_C1 variant) algorithm is a second order time stepping algorithm that exactly conserves energy and enforces conservation of total angular momentum. It uses a modified Newmark integration scheme with constants of the Newmark method given by ½ and to realize the second order accurate integration. A Quaternion representation is employed to avoid the potential of Gimbal lock and the use of convected incremental body fixed coordinates is adopted to ensure conservation of total angular momentum. For more information, see Simo Wong [204].

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