4. Harmonic response analysis fundamentals

Modal analysis answers a passive question: which frequencies will this structure naturally amplify? Harmonic response analysis answers the operational question that follows: given a known oscillating load, how large will the vibration actually be at each frequency? This chapter covers the physics of steady-state forced response, how a frequency sweep is discretized and interpreted, and how to choose between the direct and modal superposition solution methods.

Steady-State Forced Response

In a harmonic response analysis, every applied load is assumed to vary sinusoidally at a single frequency ω, with a defined amplitude and phase. The equation of motion for a damped, linear system under harmonic excitation is:

[M]{ẍ} + [C]{ẋ} + [K]{x} = {F(ω)}·e^(iωt)

Because the excitation is harmonic and the system is linear, the steady-state solution is also harmonic at the same frequency, but with a frequency-dependent amplitude and a phase lag relative to the load:

{x(t)} = {X(ω)}·e^(i(ωt + φ))

Substituting this assumed form eliminates time and reduces the problem to a complex algebraic system solved independently at each frequency:

(-ω²[M] + iω[C] + [K]){X} = {F}

The term in parentheses is the dynamic stiffness matrix. Its real part, K − ω²M, captures the competition between elastic restoring force and inertia; its imaginary part, iωC, captures energy dissipation. Near a natural frequency, the real part approaches singularity and the response amplitude peaks — this is resonance. The phase angle φ transitions through roughly 90° across each resonance, which is why phase plots are often more diagnostic than magnitude plots alone.

Two points distinguish harmonic response from transient dynamic analysis. First, only the steady-state solution is computed; initial-condition transients are excluded by construction. Second, the analysis is performed frequency by frequency, so the result is a spectrum rather than a time history. If the real loading is broadband or impulsive, harmonic response is the wrong tool — use transient or response spectrum analysis instead.

Frequency Sweep: Discretization and Interpretation

A frequency sweep evaluates the solution at a discrete set of frequencies between a lower and upper bound. The choice of spacing is not cosmetic; it determines whether sharp resonances are captured or missed entirely.

Three sweep strategies are common:

• Linear sweep — constant Δf between f_min and f_max. Suitable when the response is smooth and resonances are well separated.
• Logarithmic sweep — constant ratio between successive frequencies. Useful when the band spans several decades, such as 1 Hz to 2000 Hz.
• Modal/cluster sweep — frequencies concentrated around known natural frequencies, with sparse points elsewhere. This is the most efficient approach when a prior modal analysis is available.

A practical rule for modal clustering: place at least 5–10 solution points within each resonance half-power bandwidth, Δf = 2ζf_n. For a mode at 120 Hz with 2% damping, the half-power bandwidth is 4.8 Hz, so points every 0.5–1.0 Hz are needed near that peak. A uniform 10 Hz linear sweep would step straight over this resonance and report a response far below the true peak — a classic and costly error.

Damping input deserves equal attention. Harmonic response requires a damping model: constant modal damping ratio ζ, Rayleigh stiffness/mass proportional coefficients α and β, or constant structural damping coefficient. Rayleigh damping relates to modal damping approximately as ζ_n = α/(2ω_n) + βω_n/2, so α dominates at low frequencies and β at high frequencies. If the sweep covers a wide band, a single constant ζ may be a poor fit; verify that the implied ζ_n is physically reasonable across the range.

Direct vs. Modal Superposition Methods

Both methods solve the same dynamic stiffness equation; they differ in how the system is reduced and solved.

Direct method solves the full coupled system at every frequency using the complete mass, damping, and stiffness matrices. No eigenanalysis is required, so any damping form — including frequency-dependent or non-proportional damping — can be used directly. The cost scales with the number of frequencies and the full DOF count, making it expensive for large models with many sweep points.

Modal superposition (mode superposition) first extracts a truncated set of eigenvectors, then projects the equations into modal coordinates. Each mode becomes an independent single-DOF oscillator, and the total response is the sum of modal contributions. This is dramatically faster when many frequencies are requested, because the expensive eigenanalysis is done once and each subsequent frequency is a cheap modal summation. Its limitations: the damping must generally be modal (or approximated as such), and the mode set must be truncated carefully — modes above the sweep range can still contribute statically and should be included or accounted for via a residual vector.

A defensible selection procedure: if the model has fewer than roughly 50,000 DOF and the sweep has fewer than ~20 points, direct is simple and safe. If the sweep is dense or multiple load cases share the same mesh, run modal superposition and verify convergence by increasing the number of extracted modes until the peak response stabilizes. For non-proportional damping (e.g., discrete dashpots), direct is the only rigorous option.

Reading the Result and Judging Adequacy

The primary output is a response spectrum: displacement, velocity, or acceleration amplitude versus frequency, often with a phase curve. The engineering judgment is not simply "is the peak large?" but whether the peak occurs near an operating frequency and whether the resulting stress or displacement exceeds an allowable.

For a worked example, consider a bracket with a first mode at 120 Hz and 2% modal damping, excited by a 10 N harmonic force applied at a mounting pad. A modal superposition run with 20 extracted modes and a cluster sweep of 0.5 Hz spacing between 80 and 160 Hz produces a peak displacement of approximately 0.42 mm at 119.5 Hz. Halving the step to 0.25 Hz changes the peak to 0.43 mm — a 2.4% change, confirming convergence. A direct run on the same model gives 0.43 mm, validating the modal truncation. If the allowable displacement at the pad is 0.30 mm, the design fails and requires either added stiffness to shift the mode above the excitation band or increased damping.

Typical failure symptoms and their causes:

• Peak amplitude far below physical expectation — sweep too coarse, resonance stepped over. Refine near known modes.
• Response grows monotonically with frequency — missing high-frequency modes in modal superposition, or incorrect load scaling. Increase extracted modes or check load units.
• Phase jumps in the wrong direction — sign convention error in the imaginary damping term or load phase definition. Verify that iωC uses a consistent sign.
• Direct and modal results diverge significantly — non-proportional damping present, or mode truncation too aggressive. Add modes or switch to direct.

In summary, harmonic response analysis converts a known oscillating load into a frequency-domain response by solving the dynamic stiffness equation at each sweep point. The sweep must be resolved finely enough to capture resonance peaks, the damping model must be physically consistent across the band, and the choice between direct and modal superposition should follow from model size, damping type, and the number of frequencies required. When these conditions are met, the resulting spectrum directly supports pass/fail decisions against displacement, stress, or acceleration allowables.