Mass, Stiffness & Dynamic Behaviour

What Is It?

Natural frequency is created by the relationship between structural stiffness and mass distribution. The simple formula fn ∝ √(k/m) captures the essential relationship, but real structures cannot be reduced to one scalar stiffness and one scalar mass. The distribution of mass and stiffness across the structure — where mass is concentrated, where stiffness is high or low, how they interact through the mode shape — determines the dynamic behaviour.

Why It Matters

Understanding the mass-stiffness relationship is essential for both analysis and design. In analysis, it explains why a model's frequencies are what they are and helps identify whether a frequency is sensitive to a particular modelling decision. In design, it provides the lever for shifting frequencies away from excitation — adding stiffness to raise a frequency, adding mass to lower one, or relocating mass or stiffness to alter the modal response without changing the total.

Where the mass is can matter as much as how much mass there is.

The Fundamental Relationship

For a single-degree-of-freedom system, the natural frequency is determined by a single stiffness and a single mass. For a continuous structure, each mode has its own generalised stiffness and generalised mass, determined by the mode shape and the distributions of stiffness and mass across the structure.

fn  ∝  √(k / m)

For a specific mode n:

fn  =  (1 / 2π) · √(k_n / m_n)

where:
k_n  =  generalised stiffness for mode n
m_n  =  generalised mass for mode n

Distributed vs Concentrated Mass

Real structures have a combination of distributed mass (the mass of the structural material itself) and concentrated mass (equipment, payload, fasteners, electronics). Distributed mass contributes to the generalised mass of each mode according to the mode shape. Concentrated mass contributes according to the mode shape amplitude at the location of the mass. This is why the location of concentrated mass has such a strong effect on natural frequencies.

Mass Location and Modal Participation

A mass placed at a modal antinode — the point of maximum motion in a mode shape — has the greatest effect on that mode's frequency. The same mass placed at a modal node — a point of minimal motion — has almost no effect. This is because the generalised mass depends on the square of the mode shape amplitude at the mass location.

Generalised mass contribution from a concentrated mass m at location x:

Δm_n  =  m · φ_n(x)²

where:
φ_n(x)  =  mode shape amplitude at location x for mode n

At an antinode:  φ_n(x) is large  →  Δm_n is large  →  strong effect on fn
At a node:       φ_n(x) ≈ 0       →  Δm_n ≈ 0       →  minimal effect on fn

A small mass added at a modal antinode may affect frequency more than the same mass at a nodal region. Where mass is placed matters as much as how much is added.

Stiffness Placement

The same principle applies to stiffness. Adding stiffness at a location of high modal deformation has a strong effect on frequency; adding it at a location of low deformation has little effect. A stiffener placed at a point of maximum bending will raise the frequency more effectively than the same stiffener placed near a mode shape node. This is a powerful design lever for shifting frequencies.

• Stiffness added at high-modal-deformation regions strongly raises frequency
• Stiffness added at low-modal-deformation regions has little effect
• Removing stiffness at high-deformation regions strongly lowers frequency
• Joint stiffness can be as important as member stiffness — a flexible joint can dominate modal behaviour

Design Implications

The mass-stiffness-location relationship gives the engineer several levers for managing dynamic response. These levers can be used to shift frequencies away from excitation, to reduce modal participation in a sensitive direction, or to alter the mode shape to reduce stress in a critical region.

Joint Stiffness

Joints — bolted connections, bonded interfaces, bearings, mounts — often have stiffness that is much lower than the connected members. A joint that is modelled as rigid when it is actually flexible can produce frequencies that are significantly higher than the real structure. Joint stiffness is one of the most common sources of discrepancy between FEA modal predictions and test measurements. Where joint flexibility matters, it should be represented explicitly in the model.

MODELLING CONSIDERATION: Local flexible joints may significantly alter natural frequencies even when the connected components themselves are very stiff. Model joint stiffness explicitly where it matters.

Common Mistakes

• Omitting non-structural mass — cables, equipment, thermal protection — from the modal model
• Modelling joints as perfectly rigid when they have finite stiffness
• Adding mass without considering where in the mode shape it is placed
• Assuming that increasing global stiffness always raises the critical frequency — local stiffness changes may target the relevant mode more efficiently
• Ignoring rotary inertia of equipment when rotational modes are relevant

When a Simpler Method May Be Better

For preliminary design, a simple hand calculation using an equivalent SDOF model can provide a quick frequency estimate and identify whether a more detailed analysis is needed. A beam or plate with known boundary conditions has tabulated fundamental frequencies. These quick estimates can guide design decisions before a full FEA model is built, but they should not replace FEA for final analysis.

Key Takeaways

• Natural frequency depends on both the quantity and distribution of mass and stiffness
• Mass at a modal antinode affects frequency far more than the same mass at a node
• Stiffness placement follows the same principle — add stiffness where modal deformation is highest
• Joint stiffness is a common source of model-test discrepancy and should be modelled explicitly where relevant
• Understanding mass-stiffness-location relationships gives the engineer design levers for managing dynamic response