Inertia relief#

A structure without supports, e.g. an aircraft or a launch vehicle in flight, a spacecraft, or any free-free model, can move as a rigid body: its stiffness matrix KC0 is singular, with the rigid-body modes in its null space, and the linear static system of Linear static analyses cannot be solved as it stands. A static solution exists only for balanced loads, without a resultant force or moment. Under unbalanced loads, e.g. the aerodynamic loads and the weights of an aircraft that is not trimmed, or the thrust of a launch vehicle, the free structure accelerates; under steady loads it deforms elastically while it accelerates as a rigid body. Inertia relief balances the loads with the d’Alembert forces of this rigid-body acceleration and computes the displacements relative to a reference support. In pyfe3d this is done by apply_inertia_relief(), whose Notes give the equations and the references.

Workflow#

  1. Assemble KC0 as for a linear static analysis, see Overview and workflow, without any support.

  2. Assemble the mass matrix M with the update_M methods of the elements, consistent (mtype=0) or lumped (mtype=1), see Choosing an element, and add the point masses on the diagonal. The mass matrix must give every rigid-body mode a mass, e.g. a beam needs a rotary inertia about its axis, intrhoy2 + intrhoz2 of BeamProp.

  3. Build the load vector fext. A gravity field is M @ a_g, with a_g the uniform acceleration field of the gravity at the translational dofs.

  4. Choose the reference support, the Boolean array bref, and, when the model itself is constrained, e.g. by a plane of symmetry or by the dofs that a model in one or two dimensions does not use, the Boolean array bfix of these constraints.

  5. Solve:

    from pyfe3d.solver import apply_inertia_relief
u, acc, Phi, R = apply_inertia_relief(KC0, M, fext, bref, bfix,
                                      return_reactions=True)

    u are the displacements relative to the reference support, acc the rigid-body accelerations of the reference dofs, Phi the rigid-body modes, with which Phi @ acc is the acceleration of every dof and -M @ Phi @ acc the d’Alembert forces, and R the residual, whose values at bref are the reactions of the reference support, zero up to the round-off.

  6. Post-process u as in a linear static analysis, e.g. with update_probe_ue and update_probe_finte of the elements. The element forces of a distributed inertia load contain the element’s own share of it, as for any distributed load.

The reference support#

The reference support sets to zero as many dofs as there are rigid-body modes, six for a free structure in 3D, fewer with bfix, e.g. two, heave and pitch, for the half model of the airplane of Tutorial: inertia relief of a stick aircraft model, MSC Nastran example HA144A, whose plane of symmetry and fore-aft translation are constrained by bfix. It must be statically determinate: it suppresses every rigid-body mode without restraining any deformation, i.e. the rows of the rigid-body modes at the reference dofs form a non-singular square matrix. Typical choices are the six dofs of one node, or three translations of one node, two of a second node and one of a third node that are not aligned. apply_inertia_relief() checks the choice:

  • with too few dofs, or dofs that leave a rigid-body mode free, e.g. two translations along the line joining their nodes, the stiffness matrix of the free dofs, \(\boldsymbol{K}_{{C0}_{uu}}\), is singular, which raises a numpy.linalg.LinAlgError, as for insufficient boundary conditions in linear_static();

  • with more dofs than the rigid-body modes, the support is statically indeterminate: it would restrain the deformation and carry reactions, which raises a numpy.linalg.LinAlgError too.

With a statically determinate support, the reactions are zero and the support does not change the solution: another choice changes the displacements by a rigid-body motion only, while the strains, the stresses and the internal forces are the same. The displacements are thus always relative to the chosen support, which moves with the rigid-body acceleration; a support at a point of interest, e.g. the centre of gravity or a payload interface, makes them easier to read.

Balanced and unbalanced loads#

For balanced loads, e.g. a structure under its own weight balanced by other loads, or self-equilibrated loads, the rigid-body accelerations are zero up to the round-off and inertia relief reduces to a fictitious support of the singular static problem: the displacements are those of the structure supported by the reference support, which carries no reaction.

For unbalanced loads, the plain linear static solution with the same support would carry the resultant of the loads at the support, with unrealistic reactions and internal forces near it. Inertia relief replaces these reactions by the d’Alembert forces of the rigid-body acceleration, distributed as the mass. The result is a quasi-static state, valid when the loads are steady and the transient response has been damped out; it is not a dynamic response.

Relation to the prescribed displacements#

The reference support is a set of prescribed null displacements, the known set \(k\) of the partition of Linear static analyses, with bk = bref | bfix and uk = 0, and the reactions are computed as there, with the d’Alembert forces added to the external forces. The difference is the choice of these dofs: a support of a linear static analysis restrains the structure and carries the loads, whereas the reference support of inertia relief only fixes the rigid-body motion and carries no load.

Worked examples#

  • Tutorial: static analysis with inertia relief: springs and point masses, with the closed-form solutions, a free beam under a balanced load set, and a free beam and bar under an end force, against closed-form solutions;

  • Tutorial: inertia relief of a stick aircraft model, MSC Nastran example HA144A: the stick model of an airplane in trimmed level flight, the MSC Nastran example HA144A, against the published results. The published aerodynamic box forces, transferred to the beams with Nastran’s linear splines (SPLINE2), which carry the moments of the forces through their rigid arms, reproduce the published bar forces and displacements to the printed digits; an inverse-distance load transfer and the loads of a vortex lattice method are compared as well (see the references of the tutorial).

References#

  • Barnett, A. R., Widrick, T. W. and Ludwiczak, D. R., Closed-Form Static Analysis with Inertia Relief and Displacement-Dependent Loads Using a MSC/NASTRAN DMAP Alter, NASA TM-106836, NASA Lewis Research Center, Cleveland, OH, 1995. https://ntrs.nasa.gov/citations/19950013233

  • Bathe, K. J., Finite Element Procedures, 2nd ed., K. J. Bathe, Watertown, MA, 2014.