### 10.2.1. Translational Equations of Motion

The equation of motion for a translating center of mass , Equation 10–1, can be expressed as:

| (10–3)

where is the mass and represents the sum of all forces which include aerodynamic, weight of rigid body, spring and/or explicit external force.

Expanding the sum of all forces, Equation 10–3 may be written as:

| (10–4)

where is the aerodynamic force, is the gravity, is the linear spring constant and is all other external forces acting on the body.

#### 10.2.1.1. Newmark Integration

The linear momentum solver makes use of the Newmark integration scheme. The conventional Newmark integration scheme [203] can be expressed as:

| (10–5)

| (10–6)

where is time, is representative of an entity such as spatial or angular position, and the subscripts and represent the known solution at time and unknown solution at time , respectively.

The Newmark integration depends on two real parameters and . These parameters are directly linked to accuracy and stability of the Newmark time integration scheme.

In typical applications of the Newmark method and are chosen to be ¼ and ½, respectively. This choice of parameters corresponds to a trapezoidal rule that results in a second order accurate scheme that is also unconditionally stable in linear analyses. The linear momentum equations are solved using these parameters.

For fixed values of the external forces, the linear momentum solver advances the solution over time using the following procedure:

1. Expressing Equation 10–4 as

   | (10–7)

   and substituting for into Equation 10–5 where is replaced by and rearranging, is determined from:

   | (10–8)

2. From Equation 10–5 where is replaced by and rearranging, is determined from:

   | (10–9)

3. From Equation 10–6 where is replaced by and rearranging, is determined from:

   | (10–10)

   where is time.

Note that iteration of the above procedure is required to account for dependencies of the forces on the position of the rigid body.

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