Mass Representation & Inertial Modelling

What Is It?

Mass representation is the way the finite element model describes the mass and inertia of the structure and its attached equipment. For static analysis, the mass determines the inertia loads under acceleration — the body forces that simulate gravity, manoeuvre loads or spin loads. For dynamic analysis, the mass determines the natural frequencies, the mode shapes and the response to transient and harmonic excitation. The mass in an FE model comes from two sources: the distributed mass of the structural elements (computed from the material density and the element volume) and the concentrated mass of attached equipment (engines, electronics, payloads, fuel) represented by mass elements at the equipment centres of gravity. The mass representation must capture three things: the total mass, the centre of gravity location and the mass moment of inertia distribution. Get any of these wrong and the static inertia loads, the natural frequencies and the dynamic response will all be wrong.

Why It Matters

Mass drives the inertia loads in static acceleration analysis and the dynamic response in modal, transient and harmonic analysis. In a static acceleration case, the body force is mass times acceleration — a wrong mass gives a wrong load, and a wrong CG gives a wrong moment arm, changing the load distribution. In a modal analysis, the natural frequencies scale with the square root of stiffness over mass — a 10% mass error gives a 5% frequency error, which can move a resonance into or out of an excitation band. In a transient analysis, the mass determines the inertia that resists acceleration — a wrong mass gives a wrong dynamic amplification and a wrong peak response. The mass representation is as important as the stiffness representation — yet it is often treated as an afterthought, with equipment mass lumped approximately and rotary inertia ignored. For any analysis where mass matters, the mass representation must be as carefully considered as the stiffness.

MASS DRIVES INERTIA LOADS AND DYNAMIC RESPONSE. In static acceleration, the mass times acceleration is the load; in dynamics, the mass determines the frequencies and the response. A wrong mass, a wrong CG or a wrong rotary inertia gives a wrong result — even if the stiffness is perfect. Treat mass representation with the same rigour as stiffness representation.

Total Mass, CG and Rotary Inertia

The three quantities that must be correctly represented are the total mass, the centre of gravity and the mass moment of inertia. The total mass determines the resultant inertia force under acceleration. The CG location determines the moment arm — a CG that is offset from the structural centreline creates a moment under lateral acceleration that can dominate the loading. The mass moment of inertia (the rotational inertia about the CG) determines the resistance to angular acceleration — it matters for rotational dynamics, for pitch and yaw response and for the natural frequencies of bending modes that involve rotation. The mass moment of inertia tensor has six independent components (three principal moments and three products of inertia); for a general equipment item, all six may matter. A concentrated mass element that represents only the total mass and the CG but not the rotary inertia is adequate for translational response but wrong for rotational response.

Mass properties that must be represented:

  Total mass:
    M = ∫ ρ dV

  Centre of gravity:
    x_cg = (1/M) ∫ ρ·x dV
    y_cg = (1/M) ∫ ρ·y dV
    z_cg = (1/M) ∫ ρ·z dV

  Mass moment of inertia tensor (about CG):
    Ixx = ∫ ρ·(y² + z²) dV
    Iyy = ∫ ρ·(x² + z²) dV
    Izz = ∫ ρ·(x² + y²) dV
    Ixy = -∫ ρ·x·y dV   (products of inertia)
    Ixz = -∫ ρ·x·z dV
    Iyz = -∫ ρ·y·z dV

  For a concentrated mass element at the CG:
    M, [I] (6 independent components) must all be specified

Concentrated Mass vs Distributed Mass

Concentrated mass (a mass element at a point) represents equipment — engines, payloads, electronics — whose internal stiffness is not of interest but whose mass and inertia are. The mass element is placed at the equipment CG and connected to the structure by rigid links (RBE2) or distributed coupling (RBE3). The mass element carries the total mass and the rotary inertia tensor; the connection transfers the inertia forces to the mounting structure. Distributed mass (the material density of the structural elements) represents the mass of the structure itself — the skins, frames, spars, webs. The distributed mass is computed automatically from the element volume and the material density. The choice between concentrated and distributed depends on the component: structure is distributed; equipment is concentrated. The error arises when equipment that should be concentrated is omitted (the mass is missing) or when equipment that has significant extent is concentrated at a single point (the rotary inertia is wrong because the spatial distribution is lost).

Rotary Inertia — Why It Matters

The rotary inertia — the mass moment of inertia about the CG — is the resistance to angular acceleration. It matters whenever the dynamic response involves rotation: bending modes of a beam with a heavy end mass (the rotary inertia lowers the frequency); pitch and yaw of a structure with a long equipment arm (the rotary inertia determines the rotational natural frequency); torsional vibration of a shaft with a disc (the rotary inertia is the inertia). A concentrated mass element that specifies only the total mass and not the rotary inertia treats the equipment as a point mass with zero rotational inertia — this over-predicts the rotational natural frequencies and under-predicts the rotational response. For equipment with significant extent (a long engine, a spread-out payload), the rotary inertia can be large and must be specified. The rotary inertia is computed from the mass distribution: I = ∫ r² dm, where r is the distance from the rotation axis. The further the mass is from the axis, the larger the rotary inertia — a point mass at the CG underestimates it.

COMMON MISTAKE: Representing equipment as a point mass with no rotary inertia. The total mass and the CG may be correct, but the rotational natural frequencies are over-predicted and the rotational dynamic response is under-predicted. For equipment with significant extent, specify the full inertia tensor — not just the mass.

Mass in Static Acceleration Analysis

In a static acceleration analysis, the inertia load is the mass times the acceleration, applied as a body force. The body force is distributed: each element experiences a force equal to its mass times the acceleration, and the concentrated mass elements experience their mass times the acceleration at their CG. The resultant is the total mass times the acceleration, applied at the CG. The moment about any point is the mass times the acceleration times the distance from the CG to the point — so the CG location determines the moment arm. A wrong total mass gives a wrong resultant; a wrong CG gives a wrong moment. Both must be correct. The acceleration is applied as a gravity-like field — a constant acceleration vector over the whole model — or as a spin (centrifugal) acceleration that varies with radius. The mass representation must be correct for both: the total mass for the resultant, the CG for the moment, and the spatial distribution for the local body force distribution.

Inertia load from mass under acceleration:

  Acceleration field:  a = (a_x, a_y, a_z)

  Each element:  F_e = m_e · a   (body force, distributed)
  Each mass element:  F_m = M_m · a   (at CG)

  Resultant:  F_total = M_total · a
  Moment about origin:  M = F_total × r_cg
                      = (M_total · a) × r_cg

       r_cg (CG location)        a (acceleration)
           │                        │
           │           ┌────────────┘
           ▼           ▼
     ●─────●─────●─────●  ← structure with distributed mass
     │           │
     │    ●      │  ← concentrated mass at equipment CG
     │   (M,I)   │     connected by RBE to mount
     │           │
     ●─────●─────●

  CG offset → moment arm → inertia moment under lateral acceleration

Mass in Modal and Dynamic Analysis

In modal analysis, the mass matrix and the stiffness matrix together determine the natural frequencies and the mode shapes. The natural frequencies scale as ω = √(k/m) — the mass in the mode is the effective mass of that mode, which depends on how the mode engages the mass distribution. A heavy equipment mass at the end of a flexible beam lowers the bending frequency; a heavy disc on a shaft lowers the torsional frequency. The rotary inertia lowers the frequency of modes that involve rotation. In transient and harmonic analysis, the mass determines the inertia force that resists acceleration — the dynamic amplification depends on the ratio of the excitation frequency to the natural frequency, which depends on the mass. Getting the mass right is essential for any dynamic analysis — the frequencies, the mode shapes and the response amplitudes all depend on it.

Verification of Mass Representation

The mass representation should be verified before any analysis is believed. The total mass of the model should be checked against the known or estimated total mass of the structure — a discrepancy indicates a missing component, a wrong density or a wrong volume. The CG location should be checked against the known or estimated CG — a discrepancy indicates a mass that is in the wrong place. The mass moments of inertia can be checked by a separate calculation or by a solver utility that computes the inertia tensor. For dynamic analysis, a modal analysis of the mass on rigid supports (a "modal shake test") reveals whether the mass is distributed as expected — the modes should show the equipment masses moving as rigid bodies with the expected inertia. These checks are cheap and catch mass representation errors before they propagate into the analysis results.

VERIFICATION: Check the total mass, the CG location and the mass moments of inertia against known or estimated values before running any analysis. A mass error of 10% gives a frequency error of 5% and a load error of 10% — both significant. The mass properties are cheap to check and expensive to get wrong.

Key Takeaways

• Mass drives inertia loads (static acceleration) and dynamic response (modal, transient, harmonic)
• Three quantities must be correct: total mass, CG location and mass moment of inertia tensor
• Concentrated mass elements represent equipment — specify the full inertia tensor, not just the mass
• Distributed mass (element density) represents the structure — check density and volume are correct
• Rotary inertia matters for any rotational response — omitting it over-predicts rotational frequencies