Inertia Relief

With inertia relief, the applied loads are balanced by a set of translational and rotational accelerations. These accelerations provide body forces, distributed over the structure in such a way that the sum total of the applied forces on the structure is zero. This provides the steady-state stress and deformed shape in the structure as if it were freely accelerating due to the applied loads. Boundary conditions are applied only to restrain rigid body motion. Because the external loads are balanced by the accelerations, the reaction forces corresponding to these boundary conditions are zero.

This calculation is automated.

Inertia relief boundary conditions may be defined in the Bulk Data section of the input deck or they may be determined automatically by the solver.

Use SUPORT Entries

PARAM,INREL,-1 is used to activate inertia relief.
The SUPORT and SUPORT1 Bulk Data Entries are used to define up to six reaction degrees of freedom of the free body.
SUPORT entries will be used in all relevant subcases and therefore do not need to be referenced in the Subcase Information section.
SUPORT1 entries need to be referenced by a SUPORT1 data selector statement for use within a subcase.

Automatic Support Generation

Up to six rigid body modes:

Inertia relief boundary conditions may be generated automatically by using PARAM,INREL,-2.

Greater than six rigid body modes:

Inertia relief boundary conditions may be generated automatically by using PARAM,INREL,-2.
The METHOD parameter on PARAM,INREL can reference the ID of EIGRL or EIGRA entry.
Eigenvalue subcases are internally generated to calculate the rigid body modes, inertial loads, and support points.

In OptiStruct, inertia relief can be applied to linear static, nonlinear static analyses, and modal frequency response analyses. For nonlinear static analysis with contact, by default, only freeze contact is supported with inertia relief. If non-freeze contact is present, PARAM,IR4NLCON,YES can be used to allow the model to run with inertia relief. A static subcase with inertia relief is not supported by default in a linear buckling analysis. PARAM,INRELBCK,1 or PARAM,INRELBCK,2 can be used to attempt Buckling Analysis based on Inertia relief. Inertia relief is meaningless in normal modes analysis.

PARAM,PRINFACC,1 can be used to print additional information such as the output of inertial relief rigid body forces and accelerations in the .out file.

These rigid body forces and accelerations are intermediate results from the inertia relief solution. The reference point printed in the .out file is not the point of support location for inertia relief. The reference point is used for rigid body rotation calculation and these rigid body modes are part of the equations used in the calculation of inertia relief.

Theory

For static analysis of structures with rigid body modes, inertia relief calculations can be included in the solution process. In particular, structures with non-structural masses may be significantly influenced by inertia relief effects. PARAM,INREL,-1 or PARAM,INREL,-2 can be used to allow the inclusion of inertia relief calculations.

Inertia relief forces are calculated based on the rigid body modes and the global mass matrix of the model. The corrected load vector, $f'$ is calculated as:

$f' = f - M Φ u$

Where,

$f$: Load vector
$M$: Global mass matrix
$Φ$: The set of all the rigid body modes that satisfy the boundary conditions on the model
$u$: Reduced displacement vector

The value of $u$ is calculated as:

$[Φ^{T} M Φ] u = Φ^{T} f$

Where,

$[Φ^{T} M Φ]$: Reduced mass matrix
$Φ^{T} f$: Reduced load vector

PARAM,INREL,-1 or a Constrained Structure using PARAM,INREL,-2:
If PARAM,INREL,-1 is set or for PARAM,INREL,-2, if the structure is constrained in any way, the static analysis solution under inertia relief uses:
$K' U = f'$
Where, $K'$ is calculated using the original stiffness matrix ( $K$ ), plus constraints at the SUPORT degrees of freedom to render the displacements at the SUPORTs to be zero. SUPORT degrees of freedom are specified by you (for PARAM,INREL,-1) or automatically generated by OptiStruct (for PARAM,INREL,-2).
PARAM,INREL,-2 for a Free-Free Structure:
When PARAM,INREL,-2 is used for a Free-Free structure, an alternative method is used by default. In this method, OptiStruct imposes MPCs instead of automatically generating SUPORT degrees of freedom. This modifies the equation $K' U = f'$ as:
$[\begin{matrix} K & M Φ \\ Φ^{T} M & Φ^{T} M Φ \end{matrix}] [\begin{matrix} U \\ u \end{matrix}] = [\begin{matrix} f \\ Φ^{T} f \end{matrix}]$
Equation 4 is a combination of $f' = f - M Φ u$ and $[Φ^{T} M Φ] u = Φ^{T} f$ and the additional requirement that the inertia-relieved displacement be orthogonal to the rigid body modes ( $Φ^{T} M U = 0$ ). Subsequently, this MPC augmentation has been further modified as (ignores equation $[Φ^{T} M Φ] u = Φ^{T} f$ ):
$[\begin{matrix} K & M Φ \\ Φ^{T} M & 0 \end{matrix}] [\begin{matrix} U \\ u \end{matrix}] = [\begin{matrix} f \\ 0 \end{matrix}]$