Sine Vibration Calculator: Convert Acceleration (g), Velocity and Displacement (Peak, RMS, Peak to Peak) at Any Frequency
Each row is one sine vibration point. Type any two of frequency, acceleration, velocity and displacement, and the row fills in the other two. Two amplitudes with no frequency give the crossover frequency where they are equal. Pick peak, rms or peak-to-peak and US or metric units at the top of each column.
Sine point converter
| # | Frequency Hz |
Acceleration | Velocity | Displacement | Delete row |
|---|---|---|---|---|---|
| 1 | 5 | 0.2556 | 3.142 | 0.2 | |
| 2 | 13.99 | 2 | 8.787 | 0.2 | |
| 3 | 33 | 2 | 3.724 | 0.03592 | |
| 4 | 52.12 | 5 | 5.895 | 0.036 | |
| 5 | 500 | 5 | 0.6145 | 0.0003912 |
Keyboard and paste
- Tab moves along the row, then to the next row. Tab from the last cell adds a row.
- Enter, ↓ and ↑ move down and up the same column, like a spreadsheet.
- Calculated cells (grey italic) stay editable. Type over one and it becomes an input; the cell you typed least recently is calculated instead.
- Ctrl+Delete removes the current row. Esc undoes the cell you are in.
- Paste columns from Excel or a spec into any cell. They fill from that column to the right.
4x10^-2and4E-2both work.
| Peak | RMS | Peak to peak | |
|---|---|---|---|
| Acceleration | 0.256 g2.51 m/s² | 0.181 g1.77 m/s² | 0.511 g5.01 m/s² |
| Velocity | 3.14 in/s79.8 mm/s | 2.22 in/s56.4 mm/s | 6.28 in/s160 mm/s |
| Displacement | 0.100 in2.54 mm | 0.0707 in1.80 mm | 0.200 in5.08 mm |
Peak to peak displacement is the double amplitude (DA): 200 mils, 5,080 µm. ω = 2πf = 31.4 rad/s. One cycle takes 200 ms.
Click or tab into a row to show it here. For a sine wave, rms = peak ÷ √2 and peak to peak = 2 × peak.
Sine vibration nomograph
Solid lines: constant displacement (rise 2 decades of g per decade of frequency). Dashed lines: constant velocity (1 decade per decade). The selected row is red. A sine sweep profile is a path along these lines: constant displacement at low frequency, then constant g.
Displacement and velocity for 1 g peak
Displacement falls with the square of frequency. Velocity falls in proportion to it. For any other g level, multiply the row by that level: 5 g at 20 Hz is 5 × 0.0489 = 0.245 in DA.
| Frequency, Hz | Displacement, in DA | Displacement, mm pk-pk | Velocity, in/s peak | Velocity, mm/s peak |
|---|---|---|---|---|
| 5 | 0.782 | 19.9 | 12.3 | 312 |
| 10 | 0.196 | 4.97 | 6.14 | 156 |
| 20 | 0.0489 | 1.24 | 3.07 | 78.0 |
| 50 | 0.00782 | 0.199 | 1.23 | 31.2 |
| 100 | 0.00196 | 0.0497 | 0.614 | 15.6 |
| 200 | 4.89×10⁻⁴ | 0.0124 | 0.307 | 7.80 |
| 500 | 7.82×10⁻⁵ | 0.00199 | 0.123 | 3.12 |
| 1000 | 1.96×10⁻⁵ | 4.97×10⁻⁴ | 0.0614 | 1.56 |
| 2000 | 4.89×10⁻⁶ | 1.24×10⁻⁴ | 0.0307 | 0.780 |
MIL-STD-810H examples, checked
Figure 514.8E-2 (helicopter minimum integrity) is a sine sweep from 5 to 500 Hz: 0.20 in DA, then 2 g, then 0.036 in DA, then 5 g. Its corner frequencies are the crossovers between those levels. They are built into the converter as the default points, and the page's math gives the printed values.
| MIL-STD-810H, Method 514.8 | Inputs | Printed | Here |
|---|---|---|---|
| Fig. 514.8E-2, first corner | 0.20 in DA meets 2 g | 14 Hz | 13.99 Hz |
| Fig. 514.8E-2, second corner | 2 g meets 0.036 in DA | 33 Hz | 32.96 Hz |
| Fig. 514.8E-2, third corner | 0.036 in DA meets 5 g | 52.1 Hz | 52.12 Hz |
| Fig. 514.8E-2, metric labels | 0.20 in and 0.036 in DA | 5.1 mm, 0.91 mm | 5.08 mm, 0.914 mm |
| Para. 4.1.2, loose cargo package tester | 25.4 mm (1.0 in) pk-pk at 5 Hz | not printed | 1.28 g peak, 15.7 in/s peak |
Peak, RMS and Peak to Peak
Peak (also "single amplitude", "0-pk"): from the center to one extreme.
Peak to peak (also "double amplitude", DA, "pk-pk"): from one extreme to the other. It is 2 × peak.
RMS: the root mean square. For a pure sine wave it is peak ÷ √2, or 0.707 × peak.
A mil is 0.001 in. Displacement on rotating machinery is often given in mils pk-pk.
| Quantity | Usual sine test convention |
|---|---|
| Acceleration | g peak |
| Velocity | in/s or mm/s peak; rms for machinery |
| Displacement | in DA or mm pk-pk |
The Formulas
With ω = 2πf and x the peak displacement, sine motion has peak velocity v = ω·x and peak acceleration a = ω·v = ω²·x.
In test units, with g = 386.09 in/s² = 9.80665 m/s²:
g = 0.0511 · f² · DA (DA in inches)
g = 0.00201 · f² · DA (DA in mm)
v = π · f · DA (peak velocity, same length unit per second)
Two amplitudes with no frequency meet at one frequency only. Solve the formulas above for f:
g and DA: f = √(g / (0.0511 · DA))
g and velocity: f = g · 386.09 / (2π · v)
velocity and DA: f = v / (π · DA)
Why Sine Profiles Switch from Displacement to g
At 1 g, the table moves 0.78 in pk-pk at 5 Hz but only 0.002 in at 100 Hz. A constant-g profile would need a huge stroke at the low end, and a constant-displacement profile would need huge g at the high end.
So sine profiles hold displacement at low frequency and g at high frequency, joined at a crossover. Some add a constant-velocity band between them.
An electrodynamic shaker is limited by stroke at low frequency, by velocity in the middle, and by force (g times moving mass) at high frequency. On the nomograph above those are a constant-displacement line, a constant-velocity line and a constant-g line.
Sine g Is Not Random Grms
Everything on this page is for a single sine tone. The rms column is that tone's rms, peak ÷ √2.
A random test's Grms comes from the whole spectrum at once, and its peaks are about 3 × Grms. A random ASD value (g²/Hz) cannot be converted to a sine g point by point. Use the random vibration PSD calculator for random profiles.
Sources
MIL-STD-810H, Environmental Engineering Considerations and Laboratory Tests, Method 514.8 Vibration (31 January 2019; public release). Annex E, Figure 514.8E-2 (Category 24, helicopter minimum integrity exposure). Paragraph 4.1.2 (Procedure II, loose cargo transportation: 25.4 mm pk-pk at 5 Hz).
Standard gravity g = 9.80665 m/s² (exact, CGPM 1901); 1 in = 25.4 mm (exact).
The conversions are the equations of simple harmonic motion, derived on this page. The 0.0511 and 0.00201 constants follow from them: (2π)² / (2 × 386.09) and (2π)² / (2 × 9806.65).