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Theory and Physics
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.
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, meaning "mass × acceleration". Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind". In static analysis, this term is set to zero, based on the assumption that "forces are applied slowly, so acceleration can be ignored". It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you pull a spring, you feel a "force trying to return it", right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question — an iron rod and a rubber band, which stretches more under the same force? Obviously the rubber. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "high stiffness ≠ strong". Stiffness is "resistance to deformation", strength is "resistance to failure" — they are different concepts.
- External Force Term (Load Term): Body force $f_b$ (e.g., gravity) and surface force $f_s$ (pressure, contact force). Think of it this way — the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but it becomes "compression" — sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibration energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle — intentionally absorbing vibration energy to improve ride comfort. What if damping were zero? Buildings would continue swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is important.
Assumptions and Applicability Limits
- Continuum Assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small Deformation Assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and stress-strain relationship is linear.
- Isotropic Material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definition).
- Quasi-Static Assumption (for static analysis): Ignores inertial and damping forces, considering only equilibrium between external and internal forces.
- Non-Applicable Cases: For large deformation/large rotation problems, geometric nonlinearity is required. For plasticity, creep, and other nonlinear material behaviors, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformation) |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel) |
| Force $F$ | N (Newton) | Unify as N in mm system, N in m system |
Numerical Methods and Implementation
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.
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