Forced Vibration: Excitation Frequency Is Not the Natural Frequency

Separate excitation and response

For a linear mass-spring-damper under sinusoidal force, steady displacement amplitude is F₀/√((k−mω²)²+(bω)²). The imposed ω is a forcing input. Primary reference: forced oscillator equation.

Use the formula only with its defined linear, steady-state assumptions. A structural natural frequency is derived from the model rather than set by the drive command. A single observed amplitude does not identify the forcing magnitude, mode or damping by itself.

A fictional response comparison

Assume m = 1 kg, k = 1,000 N/m, b = 20 N·s/m and F₀ = 10 N. At ω = 10 rad/s, the denominator is √850,000, so amplitude is approximately 10.85 mm. At ω = 30 rad/s, it is √370,000, giving approximately 16.44 mm.

The larger second response does not mean the assembly acquired a new natural frequency. Both rows use the same model. The example compares two specified forcing frequencies; it does not locate a universal resonance peak or establish a permissible displacement. No real machine response is claimed.

Document what changes

Prepare a comparison row for forcing amplitude, frequency, mass, stiffness, damping and response coordinate. If speed changes also alter the forcing amplitude, update that input explicitly instead of attributing the whole change to the denominator.

Request the applicable response model and evidence before making a vibration diagnosis. Preserve transient versus steady-state conditions and the location of the observation. Model uncertainty, multiple modes and nonlinear contacts can require a more detailed assessment than this illustrative calculation.

Customer Questions

Primary References

These references support the technical principles discussed in this guide. The worked examples and review questions are educational.

• Primary reference: forced oscillator equation