Electromagnetic Force Harmonics & Structural Vibration

Why the Mean Electromagnetic Force Is Often Not the Governing Load

Electrical equipment can carry a modest mean electromagnetic load while experiencing a much more important oscillatory component. Slotting, pole interaction, current harmonics, switching, phase imbalance and geometric asymmetry can all create force components at discrete frequencies. If one of those components aligns with a lightly damped structural mode, the resulting vibration and alternating stress can exceed the response implied by the static mean load. The engineering task is therefore not simply to integrate force over the electromagnetic model; it is to preserve the frequency content, spatial distribution and phase information that determine how the structure is excited.

Represent the Excitation as a Spectrum of Physical Force Components

A periodic electromagnetic load is conveniently represented by a mean component plus harmonic terms. Each harmonic has an amplitude, frequency, phase and spatial pattern. Two force components with the same frequency but different phase can reinforce or cancel at a support, and two harmonics with similar global resultants can excite completely different modes because their spatial distributions differ. The receiving structural model should therefore retain the force description needed for the response quantity being assessed. A single equivalent RMS force is rarely sufficient for resonance, fatigue or noise-sensitive problems.

F(x,t) = F₀(x) + Σ Fₙ(x) cos(ωₙt + φₙ)

Modal response for mode r:  q̈ᵣ + 2ζᵣωᵣq̇ᵣ + ωᵣ²qᵣ = Qᵣ(t)/Mᵣ

Spatial Force Shape Matters as Much as Frequency

A mode is strongly excited only when the electromagnetic load has meaningful modal participation with that mode shape. A circumferential pressure wave may couple strongly to one ovalisation mode but weakly to a rigid-body-like housing mode. A local coil-end force can drive a bracket mode that is invisible in a global force resultant. Modal generalised force, obtained by projecting the mapped force field onto each mode, is therefore a powerful diagnostic. It explains why a large electromagnetic force can produce little vibration and why a much smaller harmonic can dominate response.

Preserve Phase When Multiple Sources Act Together

In polyphase machines, multiple coils, teeth or conductor groups contribute simultaneously. Their phase relationships are part of the physics. Summing magnitudes before structural transfer can destroy cancellation and create a nonphysical envelope; transferring each source independently but combining structural amplitudes incoherently can do the same. The analyst should identify whether loads are deterministic and phase-related, statistically unrelated or conservatively enveloped by requirement. For deterministic harmonic response, complex force vectors are usually the cleanest representation because amplitude and phase travel together through the modal solution.

Choose the Structural Solution Method to Match the Excitation

For steady sinusoidal content, harmonic response is usually the most efficient method. A modal superposition solution is effective when the structure is approximately linear and the retained mode set spans the excitation range. Direct harmonic analysis may be preferable when contact, frequency-dependent material behaviour or other effects make modal reduction unsuitable. Transient response is appropriate when force amplitude or frequency changes rapidly, when switching produces a non-periodic event, or when nonlinear contact and clearance alter the load path. The electromagnetic solver output should not dictate the structural method; the temporal character of the load and mechanical nonlinearity should.

Damping Can Control the Margin Near Resonance

At resonance, predicted amplitude is often highly sensitive to damping. Numerical default damping is therefore not an acceptable substitute for evidence. Sources may include material damping, joint friction, winding impregnation, bearings, mounts and surrounding assemblies. Where test data are unavailable, the analysis should show response over a justified damping range rather than present one precise number. The most useful sensitivity plot is usually response versus frequency for several damping assumptions, because it exposes whether the design is robust to both modal-frequency error and uncertain damping.

Thermal State Can Shift the Dynamic Problem

Electromagnetic vibration is often assessed at operating temperature, not at room temperature. Thermal growth can change bearing preload, joint stiffness, air gaps and contact state; modulus can change with temperature; winding or potting materials can soften; and rotor speed or electrical frequency may move across a structural resonance during warm-up. A sequential thermal-preload modal analysis can therefore be necessary before harmonic response. The key question is whether the operating state changes mode frequency, damping or load distribution enough to alter the resonance conclusion.

Stress Recovery and Fatigue Need More Than Displacement Plots

A low vibration amplitude can still create a local alternating stress at a bracket root, weld, conductor support or bonded interface. Stress recovery should use a mesh and output method appropriate to the fatigue assessment, with care around singularities and local contact edges. For multiple deterministic harmonics, stress histories can be reconstructed from complex responses or assessed component-by-component where linear superposition is valid. Mean stress from thermal preload or steady electromagnetic force may need to be combined with the alternating component. The fatigue method belongs to the material/joint detail, not to the electromagnetic solution itself.

Verification Strategy

Verify electromagnetic force harmonics first by checking integrated force/torque and known symmetry or order relationships. Then verify structural modes and modal effective mass, followed by harmonic mesh/modal truncation and damping sensitivity. Check that force mapping preserves the complex resultant at each frequency. Correlation options include accelerometers, laser vibrometry, strain gauges, microphone data where structural-acoustic coupling is understood, and order-tracked vibration measurements. A matched overall RMS acceleration is weak evidence if the model predicts the wrong dominant order or mode shape.

Engineering Outcome

A defensible electromagnetic vibration assessment explains which electrical phenomena create the important force orders, which structural modes they couple into, how damping and operating temperature affect amplification, and how response is converted into a structural acceptance quantity. The strongest result is not a colourful harmonic plot but a traceable chain from electromagnetic order through modal participation to measured or independently checked vibration and stress.

For electromagnetic vibration, preserve frequency, spatial shape and phase. A correct total force with the wrong harmonic structure can produce the wrong structural conclusion.

Key takeaways

• Treat electromagnetic excitation as ordered harmonic fields, not merely a static resultant.
• Check modal participation, damping and thermal-state sensitivity near resonance.
• Correlate dominant orders and mode shapes, not only overall vibration level.