Natural frequency and dynamic response
The natural frequency of a mass on a spring or of a uniform beam, and the response to a harmonic force or a pulse: dynamic amplification, phase, transmissibility, the largest displacement and the equivalent static force. Use it for machine supports, equipment platforms, a first look at a floor or footbridge frequency, and impact or blast-type pulses. It is structural dynamics, closed forms of the textbooks; no code sets it.
Inputs of the dynamic system #
- System: a mass on a spring (weight W and stiffness k), or a uniform beam: simply supported, cantilever, fixed both ends, or fixed and pinned, with E, I, the span L and the weight per length w that moves with it (for a floor, the dead load and the expected live load, not the design live load).
- Damping ratio ζ: your estimate, in percent of critical. Welded steel about 1 to 2 %, bolted steel 2 to 3 %, concrete 2 to 5 %. It is used for the harmonic response; the pulse peaks are undamped.
- Force: none (frequency only), a harmonic force F0 at the forcing frequency f, a rectangular pulse or a half-cycle sine pulse of peak F0 and duration td. On a beam the force acts at midspan, or at the tip of a cantilever.
- Limits: the lowest natural frequency you accept and the largest dynamic displacement, each optional (0 turns it off).
This is an analysis: it has no auto-design.
Checks performed #
| Check | Basis |
|---|---|
| Natural frequency, period, damped frequency | ωn = √(k/m), m = W/g; beam f = λ²/(2π L²) √(EI/m̄), the first three modes (Blevins, Table 8-1) |
| Equivalent system of a beam | Static stiffness at the load point (48, 3, 192 and 768/7 EI/L³) with the mass that gives the first-mode frequency |
| Harmonic response | Rd = 1/√((1 − r²)² + (2ζr)²), phase, transmissibility (Chopra 3.2, 3.7) |
| Rectangular and half-sine pulses | Undamped peak of the forced and free phases (Chopra 4.7, 4.8) |
| Natural frequency at least the limit | Your limit |
| Largest dynamic displacement | Rd F0 / k against your limit |
| Advisory: near resonance | r from 0.8 to 1.25, where the response depends on damping |
The equivalent static force Rd F0 is what to apply in a static calculation of the support for the peak response.
Limitations of the dynamic system #
- One degree of freedom, linear, viscous damping. A beam is reduced to its first mode at the load point; the higher modes are listed but not in the response.
- Uniform beams of constant EI and mass only; no axial load, shear deformation or rotary inertia.
- Pulses: undamped and of the two shapes given; no general load history, no impulse entered as an area (a pulse much shorter than the period acts as one, and the report says so).
- Not a floor vibration check: for walking and rhythmic excitation of floors use the floor vibration calculation (AISC Design Guide 11).
Verification examples #
Verification status: 19 values compared in 8 examples (8 published, 0 hand calculations from the cited equations): 19 within tolerance, 0 recorded deviations. Code basis: written without the code document on hand, and confirmed only in part through published examples.
| Example | Source | Values |
|---|---|---|
| Mass on a spring, harmonic force | Chopra, Dynamics of Structures, Ch. 2 and 3 (closed forms, hand values) | fn, Tn, Rd, phase, transmissibility, displacement |
| Simply supported beam | Blevins, Table 8-1; AISC Design Guide 11 Eq. 3-3 | First three modes; the simplified 0.18 √(g/Δ), within 0.5 % |
| Cantilever, fixed-fixed, fixed-pinned | Blevins, Table 8-1 | First mode, stiffness at the load point |
| Rectangular and half-sine pulses | Chopra 4.7, 4.8 | Rd at td/Tn = 1/4 and 1 |
Chopra and Blevins are not on hand: their equations are standard and quoted from knowledge, and the values are computed from them by hand. AISC Design Guide 11 is in the reference library.