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.
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:
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)
From Equation 10–5 where
is replaced by 
and rearranging, 
is determined from:
(10–9)
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.