Lumped Mass Element
Lumped Mass Element: Theoretical Foundations
What is a Lumped Mass Element?
Professor, what is a "lumped mass element"? Is mass ever needed in static analysis?
A lumped mass element is an element that adds mass to a single node. In static analysis, it is used for calculating self-weight (gravity loads), but its primary use is for dynamic analysis (natural vibration, time history response, impact).
Mass Matrix
The mass matrix for a lumped mass element is a diagonal matrix:
Here $m$ is translational mass, and $I_{xx}, I_{yy}, I_{zz}$ are rotational moments of inertia.
So you can set rotational moments of inertia too.
When replacing rotating components like motors or large equipment with mass elements, natural vibration modes become inaccurate without rotational inertia. It's important to correctly set both translational mass and rotational inertia.
Applications
Main applications of lumped mass elements:
| Application | Description |
|---|---|
| Adding Non-Structural Mass | Fluid in pipes, equipment weight |
| Mass Balance Adjustment | Adjusting overall FEM model mass to design values |
| Equipment Modeling | Replacing motors, valves, etc. with mass points |
| Center of Gravity Adjustment | Aligning center of gravity with actual structure |
| Mass Scaling (Explicit Method) | Increasing stable time increment |
How do you input the mass of fluid inside a pipe?
Add the fluid mass per unit length as a lumped mass to each node of the pipe element. In Abaqus, NONSTRUCTURAL MASS is convenient, allowing you to add area density (kg/m²) or line density (kg/m) to elements.
Element Names by Solver
| Solver | Element Name | Notes |
|---|---|---|
| Nastran | CONM2 | 6DOF lumped mass. Offset possible |
| Abaqus | MASS / ROTARY INERTIA | Translational and rotational defined separately |
| Ansys | MASS21 | Select presence of rotational inertia with KEYOPT(3) |
Nastran's CONM2 is "offset possible"?
CONM2 allows the center of mass to be offset from the node. This enables representing correct inertial effects even when the center of gravity is not at the node location. Useful when the mounting point and center of gravity of large equipment are misaligned.
Summary
Let me organize the key points about lumped mass elements.
Key points:
- Element that adds mass to a single node — Does not possess stiffness
- Primarily used in dynamic analysis — Natural vibration, time history response
- Translational mass and rotational inertia — Both should be set (don't forget rotational inertia)
- Adding non-structural mass — Mass of fluid, equipment, attachments
- CONM2 (Nastran) offset — Precise modeling of center of gravity position
In static analysis, it's only related to self-weight, but in dynamic analysis, it governs natural frequencies, so it's super important.
Exactly. If the mass distribution of the structure is wrong, the natural frequencies will also be wrong. Setting mass elements is fundamental to the fundamentals of dynamic analysis.
Theoretical Background of Lumped Mass
The lumped mass matrix is an approximation method that distributes element mass to nodes in a concentrated manner, as opposed to the consistent mass matrix. In 1968, Hinton et al. showed that the lumped mass matrix has less numerical damping and is more suitable for explicit time integration. Abaqus Explicit adopts this method for all elements, supporting the computational efficiency of modern explosion and impact analysis.
Computational Methods for Lumped Mass Element
Lumped Mass vs. Distributed Mass
In FEM, there are "lumped mass" and "distributed mass (consistent mass)", right? What's the difference?
Standard FEM elements (beam, shell, solid) automatically generate a mass matrix from material density $\rho$. This is the consistent mass matrix. In contrast, lumped mass elements add additional mass to specific nodes.
Furthermore, there is the lumped mass matrix, which is the diagonalized version of the consistent mass matrix.
| Mass Matrix | Characteristics | Applications |
|---|---|---|
| Consistent Mass | Has off-diagonal components. High accuracy | Dynamic analysis with implicit method |
| Diagonal Lumped Mass | Diagonal only. High computational efficiency | Explicit Method (Explicit) |
| Additional Lumped Mass (CONM2, etc.) | Mass added by user | Adding non-structural mass |
Is diagonal lumped mass mandatory for explicit methods?
Yes. Explicit time integration (Central Difference Method) uses the inverse of the mass matrix, so if it's not diagonal, efficient computation is not possible. LS-DYNA and Abaqus/Explicit automatically diagonalize the mass matrix.
Methods for Verifying Mass
How do you verify if the overall mass of the model is correct?
Check the solver's mass summary output:
- Nastran — Output grid point mass summary with PARAM,GRDPNT
- Abaqus — TOTAL MASS in *.dat file
- Ansys — MASSUM in /POST1
Items to check:
- Does the total mass match the design value (drawing value, measured value)?
- Is the center of gravity position ($X_{CG}, Y_{CG}, Z_{CG}$) reasonable?
- Are the moments of inertia in each direction reasonable?
Checking the center of gravity position is often overlooked.
Even if the total mass is correct, if the center of gravity is off, the dynamic response (especially overturning moment, eccentricity effects) will be wrong. The mass summary should always be checked at the very beginning of analysis.
Mass Scaling
What is "mass scaling"?
The stable time increment for explicit methods is determined by $\Delta t \propto L / c$ ($L$: smallest element size, $c$: speed of sound). If there are small elements, $\Delta t$ becomes extremely small. Artificially increasing mass to lower $c$ and increase $\Delta t$ is mass scaling.
Doesn't increasing mass change inertial effects?
It does. Therefore, mass scaling should only be applied to quasi-static problems, and it's necessary to confirm that the added mass (kinetic energy) is less than 5% of the total energy.
Summary
Let me organize the numerical methods for lumped mass.