Mass properties from exported meshes

Mass, centre of gravity (CG) and inertia tensor of an assembly – a vehicle, a rocket, a robot arm – are needed by everybody downstream of the CAD model: flight simulation, structural analysis, balancing, transport. Getting them out of the CAD system is surprisingly painful: the assembly must be fully loaded, every part needs the right material, bought parts are modelled as envelopes with the wrong density, and the result lives in one engineer’s session. This page records how mass properties are computed from triangle meshes, what can go wrong, how other tools take them, and the design TeamCraft adopts. Date: 2026-11-09.

The method: signed tetrahedra and the divergence theorem

For a closed, consistently oriented triangle mesh, every volume integral of a polynomial can be turned into a surface integral (divergence theorem) and evaluated exactly per triangle. Equivalently: each triangle and the origin span a tetrahedron whose signed volume is a · (b × c) / 6; summed over a closed surface, the parts outside the body cancel and the sum is the volume.

• Mirtich (1996), Fast and Accurate Computation of Polyhedral Mass Properties, Journal of Graphics Tools 1(2): the classic reduction of the volume integrals of 1, x, y, z, x², y², z², xy, yz, zx to face and then line integrals (projection onto a coordinate plane).
• Eberly (2002, revised 2009), Polyhedral Mass Properties (Revisited), Geometric Tools: the same ten integrals for triangle meshes with a compact set of per-triangle sub-expressions – no projection, no case analysis, about 50 multiplications per triangle. This is the formula TeamCraft uses.
• Tonon (2004), Explicit Exact Formulas for the 3-D Tetrahedron Inertia Tensor in Terms of its Vertex Coordinates, Journal of Mathematics and Statistics 1(1): the same result written per tetrahedron; useful as an independent check in tests.

From the ten integrals with density ρ:

• mass m = ρ ∫ dV, CG c = ρ ∫ r dV / m;
• inertia about the CG I = ρ ∫ (|r|² E − r rᵀ) dV − m (|c|² E − c cᵀ) (tensor form: the off-diagonal entries are the negative products of inertia, I_xy = −∫ xy dm; CAD systems differ here, so exports must state the convention).

Numerical care. Parts exported in a vehicle frame can sit metres away from the origin while being millimetres thick. ∫ x² dV − m c² then cancels almost all digits. The integrals are therefore accumulated relative to a point inside the part (the centre of its bounding box) and only the CG is shifted back – the inertia about the CG does not depend on that point.

Combining parts – the parallel axis theorem. For bodies i with mass mᵢ, CG cᵢ and inertia Iᵢ about their own CG: m = Σ mᵢ, c = Σ mᵢ cᵢ / m, and with dᵢ = cᵢ − c: I = Σ (Iᵢ + mᵢ (|dᵢ|² E − dᵢ dᵢᵀ)). The same shift gives the inertia about any reference point (a nose tip, a launch-lug station, a gimbal point).

Accuracy against tessellation

A mesh is an inscribed approximation of curved surfaces: the volume comes out slightly low. For a cylinder of radius r tessellated with n segments the area ratio is (n / 2π) sin(2π / n) ≈ 1 − (2π/n)²/6: 36 segments lose 0.5 %, 360 segments 0.005 %. Expressed in the chordal deviation δ that CAD exporters ask for, the relative volume error of a round section is about (4/3) δ / r: δ = 0.01 mm on a 50 mm radius is 0.03 %. Inertia (∝ r²) has about twice the relative error of the volume. In practice the density – a handbook value for a material whose real density differs by a percent or two, infill of printed parts, glue, paint, fasteners – dominates the error, not the tessellation; an export with a fine chordal tolerance (0.01–0.05 mm) is ample.

Thin walls are the exception worth knowing: in a tube both surfaces are inscribed, so the outer one loses volume and the cavity loses volume too. The two errors partly cancel, but what remains is relative to the small wall cross-section, not to the whole disc – a 1.5 mm airframe wall exported with a coarse tolerance can be off by a few percent. Fine export settings matter there.

What breaks the method

• Open meshes (holes, missing faces, a partial STEP tessellation): the signed volume then depends on where the origin is – the result is not a physical quantity. Detected by counting edges used by exactly one triangle. The values are computed (relative to the part’s own bounding box centre, which limits the damage for small holes) but flagged.
• Non-manifold edges (three or more triangles on one edge – two bodies touching along an edge, duplicated faces): usually still correct when the bodies are themselves closed; flagged.
• Inconsistent orientation (some triangles flipped): the flipped ones subtract instead of add. Detected per edge (a shared edge must be used once in each direction). Cannot be repaired without a surface walk and is flagged.
• Fully inverted meshes (every triangle flipped, a common exporter mistake): the volume comes out negative – the whole mesh is inverted and the sign is simply flipped, with a note.
• Hollow bodies and cavities are correct by construction: an inner surface oriented inwards subtracts its volume.
• Several bodies in one file (fins exported together) add up correctly as long as each is closed and they do not overlap; overlapping bodies are counted twice (a mesh-boolean would be needed; out of scope).
• Point clouds have no volume.

Units

STL has no unit field at all; OBJ, PLY and OFF neither. 3MF declares a unit in the model header, glTF is metres by specification, STEP carries its unit in the file. CAD exports of STL are millimetres by convention but inch exports exist. Mass properties scale with the unit s (metres per model unit): volume and mass with s³, CG with s, inertia with s⁵ – an inch file read as millimetres is off by 25.4⁵ ≈ 10⁷ in inertia, which is obvious, but a centimetre file read as millimetres is “only” off by 1000 in mass, which can slip through. A per-file unit, shown next to every computed value, is therefore necessary; millimetres as the default, glTF as metres.

Frames: one vehicle frame or local frames with placements

Two ways to get parts into one coordinate system:

1. Every part exported in the assembly / vehicle frame. The CAD system writes each component in assembly coordinates (NX: Export STL with the assembly as work part and the component selected, or the output coordinate system; most CAD systems have an equivalent). The PLM then only adds – no transforms, no placements, no dependency on the CAD system’s assembly file. A part used several times (four fins) is either exported once with all instances, or once per instance as its own file.
2. Parts in their local frame plus placements. The BOM line stores a 4×4 transform per instance (Teamcenter’s occurrence transforms, the SOLIDWORKS component tree, NX’s structure). This is what CAD-integrated PLM does, but it needs the placements in the PLM – and keeping them in sync is exactly the CAD-integration work that is fragile.

TeamCraft’s BOM lines carry no placements today (the 3D preview of NX and SOLIDWORKS assemblies reads them from the assembly file, not from the BOM). Option 1 is the workflow that removes the CAD system from the loop, and it is the one adopted. Placements on BOM lines (per-instance transforms) are follow-up work.

Quantities in a common frame. A BOM line “fin × 4” with a file that shows one fin would put four fins on top of each other: the mass is right, the CG and inertia are not. The file therefore records how many units it shows (instances, default 1); the line contributes quantity / instances times the file’s body. When the two differ, the line is flagged (“copies stacked at the same place”).

How other tools take mass, CG and inertia

Common ground: computed values come from geometry × material density; a measured or assigned value always wins and covers the whole sub-tree when set on an assembly; and the simulation tools want one set of numbers – mass, CG and the tensor about the CG – with the coordinate system and the tensor convention stated.

Measured mass for bought and finished parts

Bought parts (motors, electronics, valves, bearings) are modelled as envelopes: geometry × density is meaningless for them. Their mass comes from a data sheet or a scale; the CG from the data sheet or a balance test, the inertia rarely. Finished sub-assemblies are weighed as a whole. So a revision gets a measured mass with an optional CG (and an optional inertia tensor about that CG): when set, it replaces everything computed for that revision and for its sub-tree, and it is shown as measured. Measured mass without a CG still counts in the total mass, but cannot place the mass; the CG and inertia then cover the placed mass only and say so.

Decision

• Pure module massprops: Eberly’s integrals relative to the bounding box centre, inertia about the CG per unit density in model units, plus an edge check (open, non-manifold, inconsistently oriented edges, welded on a tolerance grid like the shape signatures) and the fix of a fully inverted mesh; combination by parallel axis; principal moments. Bounded, no panics, tested against analytic solids.
• Computed in the isolated preview process from the cached preview mesh and cached next to it per content hash (<sha>.v<N>.m<V>.json), like the shape signatures – the server never parses an untrusted mesh. Units and density are applied afterwards (unit-free cache).
• Densities: an admin-managed material library (name → kg/m³); the revision’s material attribute is looked up there by name. A per-revision density override wins.
• Per-file settings: the length unit (mm by default, m, in; glTF defaults to m) and the number of instances the file shows. The geometry file of a revision is chosen explicitly or picked automatically (STL, 3MF, OBJ, PLY, OFF, AMF, glTF, then STEP, then other formats with a 3D preview).
• Measured values per revision: mass, optional CG, optional inertia about that CG, a note; they win over everything computed and cover the sub-tree.
• Source order per BOM line: measured → computed from the geometry file → the entered mass (no position) → missing. An assembly without a measured mass is the sum of its children (its own geometry and entered mass are not added – they would count the children twice).
• Frames: all parts in the common vehicle frame; placements on BOM lines are follow-up work.
• Roll-up over the multi-level BOM with the revision rule, quantities, and the read rights (restricted lines are left out with a note, as in the existing mass roll-up). Result: total mass, CG, inertia about the CG and about a chosen reference point, principal moments; per line the source, and warnings for open meshes, missing densities, unplaced masses and stacked copies.
• Interfaces: a page on the revision with CSV and JSON export, GET /api/v1/revisions/{id}/mass-properties, a read-only MCP tool.