Modal analysis finds the natural frequencies of a structure and the shape it vibrates in at each one, known as its mode shapes. Every object with mass and stiffness has them. If something drives the structure at or near one of those frequencies, such as an out-of-balance rotor, a pump’s vane pass, an engine or road input, the vibration can build up to many times the deflection the same force would cause if applied steadily. That is resonance, and modal analysis is how you find out where it lies.
In FEA, a modal analysis is an eigenvalue solution of the structure’s stiffness and mass, with its real supports but no applied load. It tells you which frequencies to keep clear of and, through the mode shapes, where to add stiffness or remove mass to move them. It does not tell you how large the vibration will be. That needs a harmonic, random or transient analysis with the actual excitation and damping.
Natural frequency
The simplest vibrating system is a mass on a spring. Its natural frequency is:
f = (1 / 2π) √(k / m)
where f is the natural frequency in hertz, k the stiffness in N/m and m the mass in kg. The angular natural frequency is ω = √(k / m) in rad/s, and f = ω / 2π.
Consider a 50 kg motor on a bracket with a stiffness of 2 × 10⁶ N/m (2 kN/mm). Then ω = √(2 × 10⁶ / 50) = 200 rad/s, and f ≈ 31.8 Hz. A four-pole motor on a 50 Hz supply runs at just under 1,500 rpm, about 25 Hz: close enough to the bracket’s frequency to cause trouble. A two-pole motor at around 2,950 rpm, about 49 Hz, would run above it, but would pass through resonance at every start and stop.
The square root has practical consequences. Doubling the stiffness raises the frequency by only √2, about 41%. Doubling the mass lowers it by about 29%. To double a natural frequency you need four times the stiffness or a quarter of the mass.
A useful shortcut applies to a mass resting on springs under gravity. If the mass deflects its supports by δ under its own weight, its natural frequency in that direction is approximately:
f ≈ 15.8 / √δ (δ in mm)
A machine that sinks 1 mm into its mounts has a natural frequency of about 16 Hz; one that sinks 4 mm, about 8 Hz.
Mode shapes
A real structure has many natural frequencies, each with its own mode shape: the pattern of relative motion at that frequency. The first mode of a cantilever is a simple bend with the tip moving most; the second has a point of zero motion part way along. A plate has drumming modes, a frame has bending and twisting modes, and an assembly can have modes in which one component rocks on its mounts.
Mode shapes are normalised: they show the pattern of motion, while the actual amplitude depends on the excitation. Their value is diagnostic. Mass added where a mode moves most lowers that mode’s frequency most; stiffness added where the structure bends most in that mode raises it most. Mass participation factors show how much of the structure’s mass moves in each mode and direction, which identifies the modes that base excitation, such as a vibrating vehicle floor, will drive hardest.
Resonance and damping
At resonance, the response is limited only by damping. For a single mass on a spring driven at its natural frequency, the dynamic amplification is:
Q = 1 / (2ζ)
where ζ is the damping ratio, the fraction of critical damping. A welded steel structure may have a damping ratio of 0.02 or less, giving an amplification of 25 or more: a force that deflects the structure by 0.1 mm when applied steadily can drive 2.5 mm of motion at resonance. Bolted joints, rubber mounts and damping treatments raise ζ, but most structures remain lightly damped, which is why keeping natural frequencies away from the excitation is the first line of defence.
Common sources of excitation
| Source | Frequency | Example |
|---|---|---|
| Rotating out-of-balance | 1 × running speed | A fan at 1,450 rpm excites about 24 Hz |
| Shaft misalignment | 1 × and 2 × running speed, often strongest at 2 × | A motor coupled to a pump |
| Blade or vane pass | Blade count × running speed | A six-vane impeller at 2,950 rpm excites about 295 Hz |
| Gear mesh | Tooth count × shaft speed | A 30-tooth pinion at 1,500 rpm excites 750 Hz |
| AC magnetic forces | Twice the supply frequency | Transformers on a 50 Hz supply hum at 100 Hz |
| Four-stroke engine firing | (Cylinders ÷ 2) × crank speed | A four-cylinder engine at 3,000 rpm fires at 100 Hz |
| Walking | About 1.5 to 2.5 Hz | Footbridges, mezzanine floors |
Modal analysis and harmonic response
Modal analysis is the foundation of a family of dynamic analyses. Each answers a different question.
| Analysis | Inputs | Outputs | Question answered |
|---|---|---|---|
| Modal | Stiffness, mass and supports | Natural frequencies and mode shapes | Where are the resonances, and what do they look like? |
| Harmonic response | Sinusoidal force or base motion across a frequency range, plus damping | Amplitude, acceleration and stress at each frequency | How large is the response at running speed, or across a sweep? |
| Random vibration | Power spectral density of the input, plus damping | RMS displacement, acceleration and stress | Will it survive transport or vehicle vibration? |
| Transient | Load or acceleration against time | Response over time | What happens during start-up, shock or impact? |
Harmonic and random analyses are usually built on the modal results, combining the contribution of each mode to find the total response. Where vibration repeats for millions of cycles, the stresses they produce feed a fatigue analysis.
How to move a natural frequency
- Add stiffness where the mode bends. Ribs, gussets and extra supports work best where the mode shape curves most. The bending stiffness of a rectangular section rises with the cube of its depth (I = bh³/12), so depth does far more than width.
- Close open sections. An open channel is weak in torsion. Closing it into a box section raises its torsional stiffness many times over, which lifts the twisting modes of a frame.
- Shorten spans and change supports. The first frequency of a uniform cantilever falls with the square of its length, so halving the length raises it fourfold. Clamping both ends of the same beam raises its first frequency to about 6.4 times the cantilever value.
- Remove mass where the mode moves most. Lighten overhanging parts, or move heavy items such as motors and batteries closer to the supports.
- Add mass deliberately. Where a frequency needs to come down below the excitation, an inertia base under a pump or fan does exactly that.
- Isolate. Resilient mounts set a low system frequency so that less vibration is transmitted. Isolation only begins once the excitation frequency exceeds √2 times the mounted natural frequency, so mounts are chosen to put that frequency well below the running speed.
Changing material often does less than expected. Natural frequency depends on stiffness over mass, and the ratio E/ρ is almost the same for steel (about 210 GPa over 7,850 kg/m³) and aluminium (about 70 GPa over 2,700 kg/m³). An aluminium structure of the same geometry, carrying no other mass, weighs about a third as much yet has nearly the same natural frequencies. Fibre-reinforced composites, with a much higher E/ρ along the fibres, are one of the few material changes that move frequencies substantially.
What a modal study needs
- CAD of the structure, in STEP or a native format.
- Materials, with density as well as stiffness.
- Masses not modelled in detail, such as motors, batteries, tanks and their contents, with their positions.
- How the structure is supported: bolted to a stiff floor, on rubber mounts, or free. Supports often change frequencies more than any detail of the part.
- The excitation: operating speeds and speed range, blade or tooth counts, or a vibration profile.
- Any preload that stiffens the structure, such as bolt tension or rotational speed.
Where modal analysis earns its place
Pump and fan skids, where a frame mode can sit close to running speed. Circuit boards in vehicles and aircraft, where an extra support near the middle of the board can lift its first mode well clear of the input. Drone arms and frames, where rotor speed and blade pass sweep through a range as the aircraft manoeuvres, covered in drone simulation. Camera and sensor mounts, where small vibrations blur images. Rotating equipment generally, where shafts, casings and supports all need their frequencies placed with care, as in rotating machinery simulation.
When to bring in a specialist
Vibration problems are cheapest to solve before a frame has been welded or a tool cut. SixGate™’s vibration and modal analysis finds the natural frequencies and mode shapes of your design, sets them against the excitation it will see and, where they clash, gives you the specific changes in stiffness, mass or mounting that move them clear. Harmonic response follows where the size of the vibration matters. Because the study is part of detail design and simulation, the stiffened frame or new mounting arrangement can go straight into the detailed model and the fabrication drawings, then into production, with the same team.