4. Modal and frequency response review

Modal and frequency response analysis answers a deceptively simple question: when a vehicle system is excited by road input, powertrain torque ripple, motor cogging, or acoustic pressure, does it respond quietly and predictably, or does it amplify? The review task is to confirm that the finite element (FE) model predicts the correct natural frequencies, that the associated mode shapes are physically plausible and correctly paired with those frequencies, that damping assumptions are defensible, and that no system mode sits dangerously close to a known excitation order. This chapter focuses on how to interrogate those four items during a technical sign-off.

What the Solver Actually Produces

A normal modes (eigenvalue) solve of the form (K − ω²M)φ = 0 returns eigenvalues λ = ω² and eigenvectors φ. The eigenvalue gives the natural frequency f = ω / 2π; the eigenvector gives the mode shape, normalized arbitrarily (commonly to unit modal mass, φᵀMφ = 1). Two consequences matter for review. First, the absolute amplitude of a mode shape is meaningless — only the relative displacement pattern and the sign convention carry information. Second, the frequency is only as good as the mass and stiffness distribution, so a correct-looking mode shape at the wrong frequency usually points to a mass or boundary-condition error, not a shape error.

Frequency response (FRF) analysis then applies a harmonic load F(ω) and solves (K − ω²M + iωC)u = F. The response peak location is governed by the undamped natural frequency, while the peak height is governed almost entirely by the damping model. This is why damping assumptions must be reviewed separately from frequency accuracy.

Reviewing Natural Frequencies Against Targets

Frequencies are reviewed against three references: physical test data, subsystem-level targets, and excitation orders. For a body-in-white (BIW) or full vehicle, the first global bending and torsion modes are typically compared with a hammer or shaker test. A common acceptance band is ±5% for the first three global modes and ±8% for higher local modes, though the exact tolerance should be stated in the program's NVH requirement document rather than assumed.

Order-based targets are more actionable. For an internal combustion engine at idle, the first-order firing frequency is f = RPM × cylinders / (2 × 60). A four-cylinder engine idling at 750 rpm produces a second-order excitation at 25 Hz. If a steering column mode is predicted at 24 Hz, the separation margin is only about 4%, which is inside the typical 10–15% avoidance band and should be flagged. For electric drive units, the dominant orders are the motor's electrical order and the gear mesh order, so the review must use the correct excitation source rather than a legacy engine order table.

Mode Shape Interpretation

Mode shapes must be checked for three failure patterns. The first is local vs. global misclassification: a mode labeled "global torsion" that actually shows a local panel breathing mode indicates the model's panel stiffness is wrong or the mode was mislabeled in post-processing. The second is rigid-body contamination: if the first six modes are not near zero, the boundary conditions or inertia relief setup is incorrect. The third is mode pairing errors: symmetric and antisymmetric pairs should appear at similar frequencies; a large split between them often indicates an asymmetric mass or constraint that was not intended.

Animation is essential. A static contour plot of a mode shape hides phase reversals and node lines. Reviewers should animate at least the first ten modes and confirm that node lines fall where the structure is actually constrained or stiffened.

Damping Assumptions

Damping is the most frequently abused input in frequency response review because it is often left at a default. Three models are common:

• Modal damping (constant ζ): simplest, but applies the same damping ratio to all modes. Typical values: 2–3% for welded steel BIW, 4–6% for trimmed body with seals and interior, 8–12% for elastomer-isolated subframes.
• Rayleigh damping: C = αM + βK, where α controls low-frequency damping and β controls high-frequency damping. It is convenient but can produce non-physical damping at extreme frequencies if α and β are fitted from only two modes.
• Structural (hysteretic) damping: C = iηK, often used for rubber and adhesive joints where damping is frequency-independent.

The review question is not "which model is best" but "is the chosen model consistent with the hardware and the frequency band of interest?" A frequency response prediction that matches test peak height at 30 Hz but uses 2% modal damping on a fully trimmed body is likely compensating for a stiffness error rather than representing real damping.

Resonance Avoidance and Separation Margins

Resonance avoidance is a margin problem, not a binary check. The standard practice is to define a minimum separation between a system mode and any excitation order across the operating range. A typical rule is:

separation = |f_mode − f_excitation| / f_excitation
accept if separation ≥ 0.10 (10%) for body modes
accept if separation ≥ 0.15 (15%) for steering and seat modes
accept if separation ≥ 0.20 (20%) for safety-critical brackets

When a margin is violated, the reviewer should require one of four actions: shift the mode by stiffness or mass change, shift the excitation by changing gear ratio or motor control, add damping, or isolate the path. Simply reporting the violation without a proposed countermeasure is not a complete review.

Practical Review Workflow

A defensible modal review follows a fixed sequence. First, verify the rigid-body modes are below 0.1 Hz and that the model's total mass matches the bill of materials within 2%. Second, compare the first five elastic modes against test or subsystem targets and record the percentage deviation. Third, animate each mode and confirm the shape label matches the physical deformation. Fourth, extract the damping values actually used in the FRF deck and confirm they correspond to the hardware build state (BIW, trimmed, or fully assembled). Fifth, overlay the excitation order lines on the FRF and compute the separation margin for every mode within the operating range. Sixth, for any margin below the threshold, require a documented countermeasure with a predicted frequency shift.

Deliverable: Modal Review Summary Table

The following table format is recommended for the sign-off package. Each row corresponds to one mode within the operating frequency band.

The verdict column must be traceable to a specific margin rule, and any "Fail" row must reference a countermeasure action item with an owner and a target frequency. This table, combined with the animated mode shapes and the FRF overlay, forms the complete modal and frequency response review package.