FEA Best Practices Vol 2

FEA Best Practices

—

Volume 2: The System’s Natural Character

Joe McFadden

McFaddenCAE.com

FEA Learning Center • Audiobook Script

In Volume 1, we built the foundation -- consistent units, correct materials, appropriate elements, and a converged mesh. We talked about how a simulation is really just a mathematical poke. You build a representation of a system and you disturb it, and the response tells you about the system's nature.

Now we're going to start poking.

This volume covers four analysis types that are deeply connected -- modal analysis, harmonic response, random vibration, and shock response spectrum. They look different on the surface, but they're all members of the same family. They all ask the same fundamental question: given that this structure has a certain nature -- its mass, its stiffness, its damping -- how does it respond when the world disturbs it?

And they all share one critical characteristic: they are all linear perturbation procedures. That phrase -- linear perturbation -- is going to come up again and again in this volume. It means no contact, no material nonlinearity, no geometric nonlinearity, no large displacements. It means the solver is working with a linearized snapshot of your system and computing small oscillations around that state. This is not a limitation of the software. It's a mathematical requirement of the eigenvalue formulation that underlies everything we're about to discuss.

Keep that in your mind as a thread running through all four topics. Every time we shift to a new analysis type, I'll point out how that perturbation thread connects.

Modal Analysis -- Discovering What's Already There

Modal analysis is the starting point for everything in this volume. If you only learn one dynamic analysis type, this is the one.

So what does it do? It determines the natural frequencies and mode shapes of a structure. The natural frequencies are the frequencies at which a structure naturally wants to vibrate when disturbed. The mode shapes are the deformation patterns associated with each frequency.

Think of a guitar string. When you pluck it, it vibrates at specific frequencies -- its fundamental and its harmonics. Those frequencies aren't designed into the string by some external force. They're inherent. They come from the string's length, its tension, its mass per unit length. They're baked in by the physical properties of the system.

Structures work exactly the same way. A bracket, a circuit board, an aircraft wing -- each has a set of natural frequencies that are determined entirely by its geometry, its material properties, and how it's supported. Modal analysis discovers those frequencies. The word "natural" tells you everything. These aren't imposed. They're inherent.

Now here's something that trips up many engineers, and it's important enough that I want to be very clear about it. The mode shapes that the solver produces have arbitrary magnitudes. The solver normalizes them according to a mathematical convention, and the resulting displacement values are arbitrary in scale. You cannot look at a mode shape plot and say "this point displaces 2 millimeters." That number has no physical meaning.

What is meaningful is the pattern and the ratios. If the mode shape shows one location displacing twice as far as another, that ratio is real. The pattern tells you where the structure moves the most, where it moves the least, and where it doesn't move at all -- those are the nodes. But the absolute scale? That depends on the actual loading, how well it couples with the mode, and the damping. The mode shape gives you the pattern. The loading and damping give you the scale. Confusing relative patterns with absolute displacements is one of the most common sources of erroneous engineering conclusions.

And here's the other thing to visualize. The actual vibration of a real structure is almost never a single clean mode shape. The physical motion at any instant is built up from many modes superimposed simultaneously, each contributing with its own amplitude and phase. Think of it like music. A single mode is a pure tone -- one frequency, one pitch. But real structural vibration is a chord. Many tones playing at once, and the character of the response is determined by which modes participate and how strongly each contributes. Modal analysis gives you the individual notes. The loading determines which chord gets played.

Let me talk about the perturbation limitations specifically for modal analysis, because this is where the rubber meets the road -- or more accurately, where the rubber doesn't meet the road, because contact is prohibited.

Modal analysis requires solving an eigenvalue problem on the stiffness and mass matrices. That eigenvalue problem is inherently linear. It cannot accommodate contact -- no contact pairs, no general contact, no surface-to-surface interactions. If your model has contact definitions, the solver will either ignore them, treat the surfaces as disconnected, or throw an error.

This creates a real practical challenge because most real-world assemblies involve contact. A bolted flange, a snap-fit housing, a press-fit bearing -- these all transmit force through contact. For modal analysis, you must replace every contact interaction with a linear equivalent: tie constraints, multi-point constraints, coupling constraints, or spring elements. The choice matters. A tie constraint assumes a perfectly rigid bond -- no slip, no separation -- which typically makes the model stiffer than reality and over-predicts frequencies. If joint flexibility matters, use spring elements calibrated from test data.

Material nonlinearity is also inactive. Plasticity, damage, hyperelastic behavior -- the solver sees only Young's modulus and Poisson's ratio. Your rubber's full nonlinear curve? Ignored. Your metal's strain hardening? Ignored. The solver uses the linear elastic tangent stiffness, period.

The one important exception: pre-stressed modal analysis. You can run a static general step first to establish a stressed state, then perform the frequency extraction as a perturbation about that state. The solver linearizes the tangent stiffness at the end of the static step. This captures stress stiffening -- a pre-tensioned cable has higher frequencies than a slack one, and a spinning rotor has different frequencies than a stationary one. This two-step approach is the correct way to handle structures where the mean load significantly affects stiffness.

Now let's talk about resonance, because this is where modal analysis connects to the real world.

Resonance occurs when the driving frequency of an external excitation aligns with a natural frequency. And here's something I want to frame carefully, because it gets misrepresented constantly: resonance is not inherently bad.

Resonance is a physical state in which energy transfer between excitation and structure becomes extremely efficient. That's it. Whether that's beneficial or catastrophic depends entirely on the engineering context.

Every acoustic musical instrument depends on resonance for sound production. A guitar body, a violin, a piano soundboard, a bell -- they all function because their structures are tuned to resonate at specific frequencies. Ultrasonic welding horns are designed through modal analysis so that their natural frequency precisely matches the operating frequency -- typically 20 or 40 kilohertz -- maximizing vibrational amplitude at the welding surface. MEMS resonators, piezoelectric energy harvesters, and ultrasonic cleaning transducers all operate at resonance by design. Tuned mass dampers use resonance of a secondary mass to absorb energy from a primary structural mode.

Conversely, resonance can be catastrophic when it's uncontrolled. The Tacoma Narrows Bridge in 1940, turbomachinery blade fatigue, spacecraft component failures during launch -- all consequences of excitation aligning with natural frequencies without adequate damping or frequency separation.

The engineering question is always the same: identify the natural frequencies, understand the excitation environment, and determine whether the intersection of the two serves or threatens the design intent. Modal analysis provides the foundation for that determination.

Before we move on, a quick note on terminology. When we reduce vibration through energy dissipation, the correct word is damp -- not dampen. To dampen means to moisten or make slightly wet. We damp a system. The system has damping. We add a damper. The response is damped. The misuse of "dampen" appears even in technical journals, but it remains incorrect. Precision in engineering language reflects precision in engineering thinking.

One more critical concept. Once you have mode shapes and frequencies, how do you get actual physical quantities -- displacements in real units, strains, stresses?

Here's the chain. External loading is projected onto each mode shape through participation factors to obtain a modal force. That modal force drives a single-degree-of-freedom oscillator defined by the mode's frequency and damping. The solution to that equation yields the modal coordinate -- a time-varying scalar that tells you how much that mode actually responds. Multiply each mode shape by its modal coordinate, sum across all retained modes, and you get actual physical displacements in real engineering units. From those displacements, strains are computed through the element shape functions, and stresses follow through the material's constitutive relationship.

This recovery chain reveals where errors propagate. Too few modes means an incomplete displacement field. Wrong damping means wrong amplitudes at resonance. A coarse mesh means poor strain and stress recovery even when frequencies are correct. Understanding this chain is what separates someone who runs the software from someone who understands the analysis.

So modal analysis reveals the system's natural character. The next question is: what happens when we poke it with a sustained, periodic force?

Harmonic Response -- Poking at One Frequency

Harmonic response analysis calculates the steady-state response of a structure to sinusoidal excitation. The key word is steady-state. We're not interested in the startup transient or the initial ringing. We want the settled vibration pattern after the system has been vibrating long enough to reach a consistent amplitude at each point.

Think of a washing machine in spin cycle. After it gets up to speed, it vibrates at a constant amplitude and frequency. That settled, repetitive vibration is what harmonic response predicts.

The output is a frequency response function -- amplitude and phase versus excitation frequency. Sweep from below the first natural frequency to above the last one you care about, and you see exactly where the resonance peaks are, how high they go, and how quickly they drop off. The peaks occur at the natural frequencies we found in modal analysis. The height of each peak is controlled by the damping. The width of each peak is also controlled by damping. Low damping means tall, narrow peaks -- high amplification over a narrow band. High damping means short, broad peaks -- less amplification, spread over a wider range.

Here's the perturbation thread again. Harmonic response in Abacus uses a steady-state dynamics step, which is a linear perturbation procedure -- whether you use the direct or modal method. Same rules: no contact, no material nonlinearity, no geometric nonlinearity. If your rubber isolator mount is modeled with a hyperelastic material, the solver uses only the initial linear stiffness. If your bolted joint can slip, you can't capture that. The analysis sees a frozen, linear system.

The practical workflow starts with modal analysis -- you need the modes first if you're using the modal method. Then you define your frequency sweep. And here's a common trap: if your frequency spacing is too coarse, you'll skip right over a resonance peak and never see it. Near resonances, use fine spacing -- half a Hertz or even a tenth of a Hertz. Away from resonances, coarser spacing is fine.

And damping is absolutely required. Without it, the theoretical response at resonance is infinite. In real structures, damping always exists, but you must define it in your model. Typical values: welded steel, half a percent to 2 percent. Bolted structures, 2 to 5 percent. Rubber mounts, 5 to 20 percent. If you're uncertain, run sensitivity studies at upper and lower bound values.

One concept worth understanding for vibration isolation: transmissibility. Below the system's natural frequency, transmissibility is near 1 -- the structure passes the vibration through with little change. At the natural frequency, the response amplifies -- that's the resonance peak. Above the square root of 2 times the natural frequency, the transmissibility drops below 1 -- that's the isolation region. This is the fundamental principle behind every vibration isolator and every equipment mount. Harmonic response analysis quantifies this behavior precisely.

Now -- what if instead of poking at one frequency, the excitation contains energy at many frequencies simultaneously, and it's random?

Random Vibration -- Poking Everywhere at Once

Random vibration is fundamentally different from harmonic excitation. It's not a clean sine wave at one frequency. It's a continuous, unpredictable input containing energy spread across a broad frequency range simultaneously.

Think about what a structure experiences during a rocket launch. It's not vibrating at one frequency. It's being shaken by acoustic noise, engine vibration, aerodynamic buffeting -- all at once, all mixed together, constantly changing. Or think about a car driving on a rough road. The tires hit bumps, potholes, surface texture -- all random, all broadband. The structure sees vibration energy at hundreds of frequencies simultaneously.

Because the input is random, we can't predict exact stress values at exact times. Instead, we work with statistics. The key tool is the Power Spectral Density -- or PSD -- which describes how vibration energy is distributed across frequency. The units are typically g-squared per Hertz.

From the PSD, we compute RMS response -- root mean square -- which represents the statistical average amplitude. But here's the critical thing: RMS is the average, not the peak. For peak estimates, we use the 3-sigma rule. Multiply RMS by three, and you get the stress level that's exceeded only 0.3 percent of the time. That 3-sigma value is what you compare against material allowables. Designing to RMS instead of 3-sigma is non-conservative and dangerous.

The perturbation thread continues. Random vibration in Abacus uses a steady-state dynamics random response step, built on top of a modal analysis. Same family, same restrictions. No contact. No material nonlinearity. No geometric nonlinearity. The entire analysis chain -- from modal extraction through random response -- operates in the linear regime.

If your structure has significant nonlinear behavior under vibration -- loose components rattling, large-amplitude flexible elements, contact surfaces opening and closing -- you would need a full transient explicit analysis with a synthesized random time history. That's orders of magnitude more expensive, but it captures the actual physics.

The workflow: first, modal analysis -- mandatory. Extract modes covering the full PSD frequency range, and the rule of thumb is modes up to twice the maximum PSD frequency. If your PSD goes to 2,000 Hertz, extract modes to at least 4,000 Hertz. Then define your PSD curve, specify modal damping -- typically 2 to 5 percent -- and run the random response step.

For a quick sanity check, Miles' equation is invaluable. G-RMS approximately equals the square root of pi over 2, times the natural frequency, times Q, times the PSD level at that frequency. This gives you a back-of-the-envelope estimate that should be in the same ballpark as your FEA results. If it's wildly different, something is wrong with your model.

Post-processing is where the engineering value lives. Review RMS stress distributions. Calculate 3-sigma values. Assess fatigue damage using methods like Dirlik or Steinberg -- and don't skip this step. Random vibration causes cumulative fatigue damage even at moderate stress levels. A structure that survives a single cycle at 3-sigma stress might still fail after millions of random cycles.

Common specifications: NASA-STD-7001 for spacecraft, MIL-STD-810 for military equipment, GEVS for spacecraft qualification. A typical spacecraft PSD ramps up from 20 to 50 Hertz, holds flat at around 0.04 g-squared per Hertz from 50 to 800 Hertz, then ramps down to 2,000 Hertz. Overall level about 7 G-RMS.

So harmonic response asks what happens when we poke at one frequency. Random vibration asks what happens when we poke at all frequencies simultaneously. The next question is: what happens when a single, violent shock event hits our structure, and we need to characterize how severe it is across frequencies?

Shock Response Spectrum -- Characterizing the Violence

The Shock Response Spectrum -- SRS -- is a way to characterize a shock environment in the frequency domain. It's absolutely essential in aerospace and defense engineering, and it's one of the most elegant concepts in structural dynamics.

Here's the concept. Imagine you have a shock pulse -- some acceleration-versus-time curve from a pyrotechnic event, a stage separation, or a drop impact. Now imagine you test that shock on a whole array of tiny single-degree-of-freedom oscillators, each tuned to a different natural frequency. One oscillator at 100 Hertz. Another at 200. Another at 500. All the way up to maybe 10,000 Hertz. For each oscillator, you record its peak response.

Now plot those peak responses versus natural frequency. That curve is the SRS.

What it tells you is profoundly useful: at each frequency, the SRS shows the maximum response that any structure with that natural frequency would experience from this particular shock. It converts a complicated time-domain event into a frequency-domain severity measure. You can compare different shock environments, envelope multiple events into a single worst-case curve, and define qualification test levels -- all in the frequency domain.

When you apply an SRS as base excitation in Abacus, you use a response spectrum step. And here's our perturbation thread one more time: response spectrum analysis is a linear perturbation procedure, built on a prior modal analysis. Same restrictions. No contact. No material nonlinearity. No geometric nonlinearity.

If you need to capture nonlinear effects -- parts separating, materials yielding, large deformations -- you must run a full transient shock analysis in the time domain and compute the SRS from the time history results in post-processing. Abacus doesn't compute SRS directly from transient results -- you'll need Python, MATLAB, or a dedicated SRS tool for that.

Key parameters: the Q factor is critical. Q equals 10 is standard for most specifications, corresponding to 5 percent damping. Higher Q means more amplification at resonance and more conservative results. Always specify Q alongside any SRS data -- without it, the numbers are meaningless.

Frequency range depends on your application. Pyroshock typically covers 100 to 10,000 Hertz. Mechanical shock covers 10 to 2,000 Hertz. Standard spacing is one-third octave bands.

The severity levels can be staggering. Spacecraft pyroshock near a separation plane can reach 10,000 to 100,000 g's above 1,000 Hertz. A one-meter drop test produces SRS levels around 1,000 to 2,000 g's. These numbers sound extreme, but remember -- they represent peak responses of tiny oscillators, not necessarily the peak of the structure itself. The SRS is a characterization tool, not a direct prediction of structural response.

Common specifications: MIL-STD-810 for military equipment, NASA-STD-7003 for pyroshock, RTCA DO-160 for avionics, GEVS for spacecraft. If you work in any of these domains, you will encounter SRS requirements.

The Perturbation Family

Let me step back and connect what we've covered in this volume.

We started with modal analysis -- discovering the natural frequencies and mode shapes that are baked into the structure by its geometry, materials, and boundary conditions. That's the foundation.

Then harmonic response -- we poke the system at one frequency at a time and sweep across a range, mapping out where the resonances are and how severely they amplify.

Then random vibration -- we poke the system at all frequencies simultaneously, working with statistics because the input is unpredictable, and we assess both the RMS response and the fatigue damage.

Then SRS -- we characterize a violent shock event in the frequency domain, asking how severe it is for structures across the entire frequency range.

All four are perturbation procedures. All four are built on the eigenvalue solution. All four inherit the same restrictions: no contact, no material nonlinearity, no geometric nonlinearity. If your modal analysis model has these limitations -- and it must -- then every downstream analysis inherits them. The harmonic response, the random vibration response, the SRS response -- they all operate on the same linearized system.

This is not a weakness. For the vast majority of structural dynamics problems where vibration amplitudes are small relative to the structure's dimensions, the linear perturbation assumption is entirely valid. And the computational efficiency is extraordinary -- a random vibration analysis that would take days as a full transient explicit simulation can be completed in minutes using the perturbation approach.

But you must know where the boundary is. If your structure has bolted joints that slip, rubber mounts with highly nonlinear stiffness, components that separate under vibration, or large-amplitude flexible members -- the linear perturbation results are, at best, approximate and, at worst, misleading. In those cases, you need the nonlinear transient methods we'll cover in Volume 3.

In Volume 3, we cross that boundary. We enter the world of explicit dynamics -- where contact is not only allowed but essential, where materials yield and fail, where geometry deforms dramatically, and where the physics gets messy. Shock analysis, contact formulations, drop test workflows, and bulk viscosity for managing shock waves. It's a different world with different rules, but the philosophy is the same: understand the nature of the system and respect it.