Library / Structural dynamics and vibration testing / SDY-12
Structural dynamics and vibration testing
Simply supported beam modal ordinate
Signed, dimensionless shape of the nth bending mode of a uniform pinned–pinned beam at a position x measured from one end.
- Ref
- SDY-12
- Rev
- v1
- Field
- Structural dynamics and vibration testing
- Result
- φn
The mode shape of a uniform pinned–pinned beam is φₙ(x) = sin(nπx/L). It is zero at the ends and at n − 1 interior nodes, and its peak magnitude is one at the antinodes. A negative value means the beam moves opposite to the reference direction at that point.
Signed, dimensionless shape of the nth bending mode of a uniform pinned–pinned beam at a position x measured from one end. Its peak magnitude is one.
Calculator
- Position as a fraction of the span, x/L0.25
- Distance between nodes, L/n0.5 m
- Slope of the shape, (nπ/L)cos(nπx/L)3.847×10−16 1/m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Mode 2 at 0.25 m of a 1 m beam has an ordinate of 1, an antinode where motion is largest.
- Sensors placed at a node cannot see that mode, so choose positions with a non-zero ordinate for every mode of interest.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Mode number | 1 to 10 | −0.9969 | 1 | 1.997 | +1 | −29.3 % |
| x Position from one end | 0 to 1 m | −0.9945 | 1 | 1.995 | +10% | −1.23 % |
| L Beam length | 0.3 to 5 m | −0.866 | 1 | 1.866 | +10% | −1.02 % |
Sweep of Position from one end
| Position from one end (m) | Modal ordinate |
|---|---|
| 0 | 0 |
| 0.125 | 0.7071 |
| 0.25 | 1 |
| 0.375 | 0.7071 |
| 0.5 | 1.225e-16 |
| 0.625 | −0.7071 |
| 0.75 | −1 |
| 0.875 | −0.7071 |
| 1 | −2.449e-16 |
Sweep one variable
Variables and how each one contributes
Modal ordinate
What the equation gives you. Everything else feeds it.
Mode number
The mode number (a whole number from 1). Higher modes have more nodes. Raising it by 1 (to 3) moves the result by −29.3 %, from 1 to 0.7071.
Position from one end
The position along the beam, in m, measured from one end. Moves the point at which the mode shape is read. Raising it by 0.05 (to 0.3) moves the result by −4.89 %, from 1 to 0.9511.
Beam length
The length, in m. A longer beam is slower to vibrate: the frequency falls with the square of the length. Raising it by 10 % (to 1.1) moves the result by −1.02 %, from 1 to 0.9898.
Worked example with the values above
- Position as a fraction of the span: x/L = 0.25 / 1 = 0.25.
- Phase: nπx/L = 2 × π × 0.25 = 1.5708 rad.
- φ = sin(1.5708) = 1.
Assumptions and limits
- A uniform Euler–Bernoulli beam: slender, constant section, small deflections, no rotary inertia or shear deformation.
- For a trial shape, it satisfies the geometric support conditions.
Common mistakes
- Using the formula for thick or short beams, where shear and rotary inertia lower the frequencies.
- Using the wrong dimensionless root for the supports.
- Mixing the mass per metre with the total mass.
Statistics and uncertainty
Frequency scales with the square root of EI/μ and inversely with L², so a 2 % error in length gives a 4 % error in frequency. Section properties and the true support conditions are the dominant uncertainties in practice.
Where it is used
- Benchmarking a finite-element beam model against a closed-form result.
- Estimating bending frequencies and wave speeds for bars, shafts, pipes and cantilevers.
Maths you need
- Second moment of area: the section property that sets bending stiffness
- Mode shape: the pattern of motion at one natural frequency
- Wavenumber: rad/m, 2π over wavelength