Harmonic Analysis
FEA LEARNING CENTER
Harmonic Response
Poking the System One Frequency at a Time
By Joseph P. McFadden Sr.
McFaddenCAE.com
Companion document to the FEA Learning Center
in the Abaqus INP Comprehensive Analyzer
Put your hand on a running engine and you feel a vibration. That vibration has a frequency — the firing rate of the cylinders, the rotation speed of the shaft, the hum of the bearings. It's periodic. It's sustained. And it repeats at the same frequency for as long as the engine runs.
That's fundamentally different from the broadband random vibration we covered in the Learning Center discussion on random vibration. Random vibration hits your structure with all frequencies simultaneously — a wall of noise. Harmonic excitation hits it with one frequency at a time — a single, focused poke. And by sweeping that poke across a range of frequencies, you build a complete map of how your structure responds to sustained periodic excitation at every frequency in the band.
The result is a frequency response function — amplitude and phase versus frequency. Peaks in that function are resonances. Valleys are anti-resonances. The shape of each peak — how sharp, how tall, how wide — tells you about the damping and the modal participation at that frequency.
If random vibration asks "what happens when my structure lives in broadband noise," harmonic response asks "what happens when I poke it at exactly this frequency, and hold it there?" Both questions matter. Both use the modal foundation. And as we'll see, real environments often contain both — broadband noise with discrete tonal content sitting on top.
The Why — When Harmonic Response Is The Right Tool
Harmonic response is the tool for any problem where the excitation is periodic — or where you want to understand the frequency-dependent behavior of the structure independent of any particular environment.
Rotating machinery is the classic application. Turbines, compressors, motors, pumps, fans — all produce periodic forces at their rotation frequency and its harmonics. A motor spinning at 1,800 RPM generates a fundamental excitation at 30 Hertz. The second harmonic is 60 Hertz. The third is 90 Hertz. If any of those frequencies land on a structural resonance, the response amplifies. And unlike random vibration where the energy is spread across the spectrum, a tonal excitation concentrates all its energy at one frequency. If that frequency is a resonance, the amplification is sustained, cycle after cycle, indefinitely.
The amplification factor at resonance is Q — approximately one over twice the damping ratio. A structure with 3 percent damping has a Q of about 17 — meaning the response at resonance is 17 times what it would be at a non-resonant frequency. At 2 percent damping, Q is 25. At 1 percent, Q is 50. These are the same Q values that appear in random vibration analysis, but here the consequence is different — instead of amplifying a narrow slice of broadband energy, the structure is amplifying 100 percent of the input because all the input energy is at that one frequency.
Vibration isolation is another classic application. You mount a sensitive instrument on isolators and you want to know the transmissibility — how much vibration passes through the isolators from the base to the instrument — as a function of frequency. Below resonance, the transmissibility is close to one — the isolators haven't engaged yet, and the instrument rides with the base. At resonance, the transmissibility peaks — the isolators are actually making things worse, amplifying the motion. Above resonance — specifically above the square root of two times the natural frequency — the transmissibility drops below one. This is the isolation region, where the isolators are finally doing their job, reducing the transmitted vibration. Harmonic response analysis maps this entire curve, showing you exactly where the crossover happens and how much isolation you get at each frequency.
Sine sweep qualification is another direct application. Many qualification standards — MIL-STD-810, RTCA DO-160, IEC 60068 — specify sinusoidal vibration profiles that sweep slowly through a frequency range. The product must survive the sweep without functional degradation. Harmonic response analysis predicts the structural response at every frequency in the sweep, identifying critical resonances and the peak stresses that occur when the sweep frequency passes through each one.
Real-world Harmonic Environments — Beyond The Textbook
If you design electronics for warehouses, distribution centers, and logistics infrastructure, you might think harmonic analysis is less relevant than random vibration. After all, your devices live in broadband environments — trucks, forklifts, conveyors.
But look more carefully at those environments, and you'll find discrete tonal content everywhere.
Conveyor drive motors produce strong tonal excitation at the motor rotation frequency and its harmonics. A standard AC induction motor running at 1,750 RPM generates a fundamental at 29.2 Hertz. The belt drive connecting the motor to the conveyor rollers introduces its own tonal content — belt passage frequency depends on belt length and speed, and belt tension variations modulate the amplitude. Gear reducers between the motor and the drive roller produce gear mesh frequency — the number of teeth times the shaft speed — typically in the 100 to 500 Hertz range. All of these are sustained, periodic excitations transmitted through the conveyor frame into anything mounted on it — scanner gantries, vision systems, weigh-in-motion platforms, print-and-apply stations.
Forklift engines — particularly internal combustion models — produce strong tonal vibration at the firing frequency. A four-cylinder gas engine at 2,400 RPM fires at 80 Hertz. That tone and its harmonics are transmitted through the forklift chassis into the dashboard, the cab structure, and any vehicle-mounted terminal or display bolted to the frame. If the terminal's mounting bracket has a resonance near 80 Hertz or 160 Hertz, the display will vibrate visibly at that frequency for as long as the engine runs.
Refrigerated trailers add compressor harmonics. The refrigeration unit compressor runs continuously during transport, generating tonal vibration in the 25 to 60 Hertz range depending on the unit. That vibration is transmitted through the trailer structure and into the cargo. Temperature-sensitive electronics being shipped in a refrigerated trailer — or cold-chain monitoring devices mounted to the trailer wall — experience both the broadband road vibration and the sustained compressor tone simultaneously.
Airport baggage handling systems are rich in tonal content. The large drive motors that power the main trunk conveyor lines, the diverter mechanisms that cycle at fixed rates, the carousel drives that rotate at constant speed — each contributes a discrete frequency to the vibration environment. Machine vision gantries spanning these conveyors experience those tones as sustained base excitation. If a gantry resonance coincides with a motor harmonic, the camera mount vibration at that frequency is amplified by Q — and the image blur at that frequency is sustained and systematic, not random.
The key insight is this: many real vibration environments are not purely random and not purely harmonic. They are a combination — broadband random noise with discrete tonal peaks sitting on top. Random vibration analysis captures the response to the broadband content. Harmonic response analysis captures the response to the tonal content. For environments with strong tones, you may need both analyses to fully characterize the structural response. Or you may need a combined sine-on-random test profile, which is specified in MIL-STD-810 Method 514.8 Annex D for exactly this situation.
The What — How Harmonic Analysis Works
There are two methods: modal and direct. Understanding the difference matters for choosing the right approach.
The modal method uses the results of a preceding modal analysis. The mode shapes and natural frequencies are computed once, and then at each frequency in the sweep, the solver computes the contribution of each mode to the total response. The structure's transfer function — how much it amplifies or attenuates at each frequency — is built from the superposition of individual modal contributions, each with its own natural frequency, mode shape, and damping. This is computationally efficient because the expensive modal extraction happens once and the frequency sweep simply evaluates a closed-form expression at each step. For lightly damped systems with well-separated modes, the modal method is accurate and fast.
The direct method solves the full system of equations at each frequency without decomposing into modes. At every frequency step, it assembles the stiffness, mass, and damping matrices, adds the excitation, and solves for the response directly. It's more general — it handles frequency-dependent damping, non-proportional damping, and doesn't require a separate modal step. It's also more expensive computationally, especially for large models, because every frequency step is essentially a full system solve. Use the direct method when the modal method's assumptions break down — heavy damping, closely-spaced modes, frequency-dependent material properties, or when you need accurate phase relationships between response points.
Both methods produce the same type of output: complex-valued displacement, stress, and acceleration at each point in the model at each frequency. The complex values encode both magnitude and phase — the amplitude of the oscillation and its time relationship to the input force.
Input Versus Response — The Transfer Function Made Visible
In the random vibration discussion, I spent time explaining that the PSD input characterizes the environment, and the structure's dynamics reshape that input into a response that looks very different. The input is the question. The structure is the filter. The response is the answer.
Harmonic response makes that input-to-response relationship more visible than any other analysis type. In fact, the frequency response function that harmonic analysis produces is the transfer function itself.
Think about what the analysis does. At each frequency, you apply a unit sinusoidal force — or a unit base acceleration — and you measure the response amplitude and phase. Move to the next frequency. Apply the same unit input. Measure again. Sweep across the entire range. The resulting curve — response per unit input versus frequency — is the structure's transfer function. It tells you exactly how much the structure amplifies or attenuates at every frequency.
Below the first natural frequency, the transfer function is flat and close to unity. The structure moves with the input like a rigid body. No amplification, no attenuation — the response tracks the input one-to-one.
At a natural frequency, the transfer function peaks. The height of the peak is approximately Q. A structure with 2 percent damping shows a peak of about 25 — meaning the response is 25 times the input at that frequency. The peak is sharp for lightly damped structures and broad for heavily damped ones. The width of the peak at the half-power points — 3 dB below the maximum — is directly related to the damping ratio. Narrow peak means light damping and high amplification. Wide peak means heavy damping and moderate amplification.
Between resonances, the transfer function dips — sometimes dramatically. These are anti-resonances, where the modal contributions cancel each other. At an anti-resonance, the structure may actually move less than the input, even below its isolation frequency.
Above the last significant resonance, the transfer function rolls off. The structure can't follow the rapid input oscillations, and the response attenuates with increasing frequency. This is the isolation region — the same physics that isolators exploit.
Now here's the connection to random vibration that makes the whole perturbation family click. The random vibration solver doesn't just throw the PSD at the structure and compute the response from scratch. It uses the same transfer function — the same modal superposition — that the harmonic analysis reveals. The random response PSD at any point equals the input PSD multiplied by the square of the transfer function at each frequency. The harmonic analysis gives you the transfer function. The random analysis gives you the response to a broadband input filtered through that same transfer function. They're siblings. Same math. Different inputs.
If you've done a harmonic response analysis and you see a sharp resonance peak at 150 Hertz with Q of 20, and then you look at the input PSD for your truck transport environment and see 0.01 G-squared per Hertz at 150 Hertz, you can estimate the response PSD at that location: 0.01 times Q-squared — 0.01 times 400 — equals 4 G-squared per Hertz at resonance. The harmonic analysis told you the amplification. The PSD told you the input. The product is the response.
Damping — The Parameter That Controls Everything
In harmonic response more than any other analysis type, damping is not optional. It's essential. Without damping, the response at resonance is mathematically infinite. In the real world, there's always some damping, and the peak response is always finite. In your model, if you forget to include damping — or if you underestimate it — the results at resonance will be meaningless spikes that don't correspond to physical reality.
In the modal method, you can specify damping directly for each mode — a much cleaner approach than the alternatives. If you have test data showing that mode 1 has 2 percent damping and mode 5 has 5 percent damping, you can assign those values directly. If you don't have mode-specific data, a uniform damping ratio applied to all modes is the standard assumption. Two percent for machined structures, three percent for bolted assemblies, five percent for structures with gaskets, potting, or isolation mounts. These are the same guidelines as random vibration because the damping is a property of the structure, not the loading.
In the direct method, Rayleigh damping is the most common approach. You specify two coefficients — alpha for mass-proportional damping and beta for stiffness-proportional damping. The resulting damping ratio varies with frequency. Alpha contributes damping that decreases with frequency — dominant at low frequencies. Beta contributes damping that increases with frequency — dominant at high frequencies. The combined damping matches your target ratio at exactly two frequencies and deviates everywhere else.
Choose those two anchor frequencies to bracket the range where you care most about accuracy. If your resonances of interest are at 50 and 300 Hertz, anchor there. Between the anchors, the effective damping will dip slightly below your target. Outside the anchors, it will rise — sometimes substantially. Always plot the effective damping ratio versus frequency to verify that it's physically reasonable across the entire sweep range. Unphysically high damping at the extremes will suppress real resonances. Unphysically low damping in the middle will exaggerate them.
And here's a subtlety that catches engineers in real-world problems. Damping in real structures comes primarily from joints — bolted connections, press fits, bonded interfaces, and contact surfaces that dissipate energy through friction and micro-slip. This is amplitude-dependent damping. At low vibration levels, the joints don't slip and the damping is low. At higher levels, micro-slip begins and the damping increases. A structure can have 1 percent damping at low amplitude and 5 percent at high amplitude.
Linear harmonic response analysis cannot capture this. It uses a single damping value regardless of amplitude. If you tune your damping to match low-amplitude test data, you'll overpredict the response at high amplitude because the real structure has more damping than your model at those levels. If you tune to high-amplitude data, you'll underpredict the low-amplitude response. This is one of the fundamental limitations of linear perturbation analysis for structures with significant joint damping.
The How — Practical Setup In Abaqus
The modal method requires two steps — a frequency extraction step followed by a steady-state dynamics step. The direct method requires only the steady-state dynamics step, but it's more expensive per frequency point.
Frequency point spacing matters more in harmonic analysis than in almost any other analysis type, because the resonance peaks can be extremely sharp. A structure with 1 percent damping has a resonance peak that is only 2 percent of the resonant frequency wide at the half-power points. At 100 Hertz, that's a 2-Hertz-wide peak. If your frequency points are spaced 5 Hertz apart, you'll miss the peak entirely and your results will show no resonance where a dramatic one exists.
Use biased spacing — more points near expected resonances, fewer in between. In Abaqus, the bias parameter controls this distribution. A bias of 3 or higher concentrates points logarithmically, putting more resolution where narrow peaks are likely.
Or better yet — use the modal method, and the solver automatically identifies the resonant frequencies from the modal extraction. The response at each natural frequency is computed exactly from the modal superposition, not interpolated between frequency points. You won't miss peaks regardless of spacing.
Interpreting The Output
Harmonic response output is deterministic, not statistical. This is a fundamental difference from random vibration and it changes how you read the results.
In random vibration, the output is an RMS value — the standard deviation of a fluctuating response — and you multiply by three for the design stress. In harmonic response, the output is the peak amplitude of the sinusoidal response at each frequency. There's no sigma multiplier. No probability table. If the analysis says the stress amplitude at 150 Hertz is 40 Megapascals, then the stress oscillates between plus 40 and minus 40 Megapascals every cycle at that frequency. That's the actual stress, not a statistical estimate.
But you still need to think carefully about what you're comparing that amplitude to.
For static failure — will the part yield or fracture right now — compare the peak stress amplitude to the material's yield or ultimate strength. If the peak harmonic stress plus any static preload stress exceeds yield, you have an overstress problem.
For fatigue — which is usually the governing concern for sustained harmonic excitation — you need the stress amplitude and the number of cycles. A motor running at 1,750 RPM produces 29.2 cycles per second. Over 8 hours, that's 842,000 cycles. Over a year of continuous operation, it's over 250 million cycles. Even a moderate stress amplitude can cause fatigue failure at those cycle counts. Compare the stress amplitude to the S-N curve for the material and determine the fatigue life. For most metals, if the stress amplitude is below the endurance limit, the fatigue life is effectively infinite. Above it, every cycle accumulates damage.
The field output at each frequency step shows the deformed shape at peak response for that excitation frequency. If you animate through the frequency sweep, you'll see the structure's response pattern change — at some frequencies the tip moves the most, at others the midpoint dominates, at others the structure barely responds. That animation is one of the most informative visualizations in structural dynamics because you can literally watch the mode shapes activate as the sweep frequency passes through each resonance.
History output gives you the frequency response function — amplitude versus frequency at a specific node. Plot this, and the resonances jump out as peaks. The height of each peak is controlled by damping. The width at half-power — 3 dB below the peak — gives you the Q factor directly. If you have test data, overlay the analytical and experimental frequency response functions. Agreement in peak locations validates your modal frequencies. Agreement in peak heights validates your damping. Discrepancies in peak location mean your stiffness or mass distribution is wrong. Discrepancies in peak height mean your damping is wrong. This is one of the most rigorous model validation approaches available.
The Perturbation Restriction
Harmonic response analysis — whether modal or direct — is a linear perturbation procedure. The restriction is the same as for modal, SRS, and random vibration: no contact elements, no material nonlinearity, no large deformations. The solver uses the stiffness matrix from the base state, assumes it remains constant, and computes the response as a linear function of the input.
This is especially relevant for harmonic response of bolted joints. In the real structure, a bolted joint has contact surfaces that can micro-slip under vibration, dissipating energy through friction. This friction-based damping is a major source of energy dissipation in real structures — and it's invisible to a linear perturbation analysis. The perturbation solver doesn't know the contact surfaces exist. It sees only the tied constraint or merged mesh you used to represent the joint.
If your model consistently underpredicts damping compared to test data — if the analytical resonance peaks are taller and sharper than the measured ones — friction at joints is the most likely explanation. You can compensate by increasing the modal damping ratios, but recognize that you're adding an empirical correction, not modeling the physics. The damping you add is accounting for energy dissipation mechanisms that the linear solver fundamentally cannot capture.
For problems where the nonlinearity matters — amplitude-dependent stiffness or damping, loose joints, gaps that open and close during the vibration cycle — you need nonlinear frequency response analysis or time-domain transient simulation with a harmonic input. That's beyond the scope of the standard perturbation approach, but it's sometimes necessary for accurate prediction of real-world behavior.
Sine Or Random — Choosing Your Poke
This whole discussion — and really this whole Learning Center series — comes back to a single idea. We understand systems by poking them. We choose the poke deliberately, to elicit a response that reveals the system's character. And the choice of poke determines what we learn.
A sine sweep is the most disciplined poke you can deliver. You excite the structure at one frequency. You listen to the response. You measure how much it amplified, how much it delayed, how much energy it absorbed. Then you step to the next frequency and ask again. By the time you've swept the full range, you've had a one-on-one conversation with the structure at every frequency in the band. You know where it resonates. You know how sharply. You know the Q at each peak. You know the transfer function — the complete dynamic personality of the structure, laid out frequency by frequency.
That's why sine sweep is fundamentally a diagnostic tool. Its purpose is to get to know the system. You're not trying to break it. You're trying to understand it. In a test lab, the sine sweep is the first thing you run — before qualification, before random, before anything. Because you need to know where the resonances are before you can evaluate whether the structure will survive its service environment. In analysis, harmonic response serves the same purpose — it gives you the transfer function that tells you how the structure will respond to any input, before you ever define what that input is.
Random vibration is a different kind of poke. It hits the structure with all frequencies simultaneously — a broadband wall of energy. It's not asking the system about one frequency at a time. It's asking the system to respond to everything at once, the way the real environment demands. The advantage is realism. A truck doesn't excite your device at 30 Hertz and then politely wait while it excites at 31 Hertz. It excites at all frequencies simultaneously, and the structural modes interact, sharing energy, coupling through the physical connections in ways that a single-frequency poke never reveals.
But the tradeoff is clarity. In a random vibration response, you can see that the RMS stress at a location is high — but you can't directly see which mode is responsible without decomposing the response spectrum. The broadband poke tells you the total answer. The sine poke tells you the answer frequency by frequency. Both are valuable. They serve different purposes.
Then there's sine dwell — the targeted poke. Once the sine sweep has identified the resonant frequencies, you can park the excitation at a specific resonance and hold it there. Every cycle at that frequency drives the structure at maximum amplification. This is the most severe poke you can deliver at any single mode — more severe than random, because in random the energy is spread across the entire bandwidth and no single resonance receives 100 percent of the input power. In a sine dwell, 100 percent of the input goes directly into that one mode, sustained indefinitely.
This makes sine dwell the worst-case assessment for peak stress at a known resonance. If the structure survives a sine dwell at resonance for the required duration, it has margin against any broadband environment that includes that frequency. It's the stress test after the interview — you've found the weakness, and now you're testing it deliberately.
For fatigue, the difference matters in a specific way. A sine dwell accumulates damage at exactly one stress amplitude — the resonant amplitude — every cycle. It's the most efficient way to drive fatigue at a single frequency. Random vibration accumulates damage across a distribution of amplitudes at all active modes simultaneously. A 60-second random test might inflict more total fatigue damage than a 60-second sine dwell because it's engaging multiple modes — but less damage at any individual mode because the energy is shared.
In practice, many qualification programs use both — and in a specific order that follows this logic.
First, sine sweep. Get to know the system. Identify resonant frequencies, measure transmissibilities, compare to the analytical predictions, adjust the FEA model if needed. This is the diagnostic phase.
Next, random vibration. Subject the system to a representative broadband environment — the actual service PSD or the qualification envelope. Verify that the structure survives the realistic loading with all modes excited simultaneously. This is the qualification phase.
Then another sine sweep. The same one as before. Compare the resonant frequencies before and after the random test. If the frequencies shifted, something changed — a bolt loosened, a bond degraded, a crack initiated. The structure's personality changed, and the second sweep reveals it. This is the damage detection phase.
And if needed, sine dwell at critical resonances — to demonstrate margin, to accelerate fatigue for life testing, or to satisfy a specific requirement in the qualification standard.
Often, the specification decides for you. MIL-STD-810 Method 514 defines random vibration profiles for transportation and platform environments, and Method 528 defines sine sweep and dwell for rotating machinery. RTCA DO-160 specifies sine sweep for aircraft-mounted equipment in propeller or rotor environments. IEC 60068-2-6 is sine. IEC 60068-2-64 is random. ISTA uses random for transport simulation. JEDEC uses both for electronic component qualification. The standard tells you which poke to apply. But understanding why each standard chose that particular poke — what it's trying to reveal about the system — is what lets you interpret the results intelligently instead of just checking a pass-fail box.
The Bigger Picture — Harmonic And Random As Siblings
Harmonic response is the most intuitive of the perturbation procedures. You apply a sinusoidal force or acceleration at one frequency and you see how the structure responds. Sweep through frequencies and the complete transfer function emerges. The physics are clean, the output is direct, and the connection between natural frequencies, damping, and response is visible in every plot.
It's also the procedure that connects most directly to physical testing. A sine sweep on a shaker table is the physical equivalent of a harmonic response analysis. The frequency response function measured by an accelerometer at a response point, divided by the input measured at the control point, is the physical equivalent of the analytical transfer function. The correlation between the two is one of the most rigorous validations you can perform on an FEA model.
And as I mentioned at the beginning — harmonic and random vibration are siblings. Same modal foundation. Same transfer function. Different inputs. Harmonic asks what happens at one frequency. Random asks what happens at all frequencies simultaneously. The real world often gives you both — broadband noise from road surfaces, floor vibrations, and structural resonance, plus discrete tones from motors, compressors, belt drives, and gear mesh. Understanding both analyses, and knowing when each applies, is what gives you the complete picture of your structure's dynamic behavior.
For the full discussion of perturbation limitations, see the dedicated Learning Center discussion. For the interplay between harmonic response, random vibration, and SRS — all built on the modal foundation — see the FEA Best Practices audiobook Volume 2: The System's Natural Character at McFaddenCAE.com.
This has been a Learning Center discussion on harmonic response. I'm Joe McFadden. Thanks for listening.
About the Author
Joseph P. McFadden Sr. is a CAE engineer specializing in finite element analysis, modal analysis, materials behavior, and injection mold tooling validation. With nearly four decades of experience in structural simulation, he brings a holistic perspective to engineering education — connecting how systems respond to how people think and learn.
His work at McFaddenCAE.com includes the Abaqus INP Comprehensive Analyzer — a desktop tool for analyzing, visualizing, and extracting sub-assemblies from large FEA models without requiring an Abaqus license — along with DSP tools for SRS computation, jerk extraction, velocity change analysis, and energy balance verification.
The FEA Learning Center is an integrated educational platform within the Analyzer, providing guided discussions on structural dynamics topics with working example INP files. This document series is the companion written reference for those discussions.
The four-volume FEA Best Practices audiobook series — Building the Model, The System's Natural Character, When Things Collide, and Keeping the Simulation Honest — is available at McFaddenCAE.com.
