Calculation
f1 = λ1²/(2πL²)·√[EI/(ρA)], λ1≈3.926602
The beam vibrates transversely; its distributed own mass together with flexural rigidity sets the first natural frequency.
Erste Biegeeigenfrequenz des gleichförmigen eingespannt-gelenkigen Euler-Bernoulli-Balkens
Calculate the First Bending Frequency of a Fixed-Pinned Beam
The beam vibrates transversely; its distributed own mass together with flexural rigidity sets the first natural frequency.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- First bending natural frequency f1 — Lowest nonzero natural frequency in Hz; compare with operating and excitation frequencies, but not as a safe operating limit without further checks.
- Beam span L — Distance between fixed and pinned ends along the straight centroidal axis, from drawing or measurement.
- Young modulus E — Linear-elastic material value from a data sheet; with I it forms flexural rigidity EI.
- Second moment of area I about bending axis — Section property for the lateral vibration plane of interest, from a profile table or section calculation.
- Material density ρ — Mass per material volume from material data; with A it gives ρA, mass per beam length.
- Cross-sectional area A — Constant material area of the same beam section as I, from drawing or section table; ρA is distributed self-mass.
Example
E·I=3,000 N·m², ρ·A=3 kg/m and L=1 m give circular frequency ω1≈487.6 rad/s and f1≈77.60 Hz using λ1=3.926602. This matches the textbook example.
Assumptions and limits
Straight prismatic Euler–Bernoulli beam with constant EI and mass per length ρA, small undamped free bending vibration. Left end ideally clamped (no transverse displacement or rotation); right end ideally pinned/laterally restrained and free to rotate. Attached point masses, shear deformation, rotary inertia, preload and damping are excluded.
Understand First bending frequency of a beam
A beam can vibrate across its axis after a disturbance even without an external force. This calculator estimates the first natural frequency of a uniform beam clamped at one end and pinned at the other.
What does this quantity describe?
A bending natural frequency belongs to a free vibration shape of the beam. The clamp prevents lateral movement and rotation at the left end; the right pin prevents lateral movement but permits rotation. Distributed own mass per length is μ=ρA, with density ρ and section area A. For constant flexural rigidity EI, f1=λ1²/(2πL²)·√(EI/μ). λ1≈3.926602 is the smallest positive eigenvalue for these exact end conditions, derived from beam vibration boundary conditions.
Formula and variables
f1 = λ1²/(2πL²)·√[EI/(ρA)], λ1≈3.926602
Mass per length: μ=ρAFirst circular natural frequency: ω1=λ1²/L²·√(EI/μ)Natural frequency: f1=ω1/(2π)Fixed-pinned: λ1≈3.926602
| Symbol / input | Meaning |
|---|---|
| First bending natural frequency f1 | Lowest nonzero natural frequency in Hz; compare with operating and excitation frequencies, but not as a safe operating limit without further checks. |
| Beam span L | Distance between fixed and pinned ends along the straight centroidal axis, from drawing or measurement. |
| Young modulus E | Linear-elastic material value from a data sheet; with I it forms flexural rigidity EI. |
| Second moment of area I about bending axis | Section property for the lateral vibration plane of interest, from a profile table or section calculation. |
| Material density ρ | Mass per material volume from material data; with A it gives ρA, mass per beam length. |
| Cross-sectional area A | Constant material area of the same beam section as I, from drawing or section table; ρA is distributed self-mass. |
Choose the inputs correctly
L is free length between the end conditions in m from drawing or measurement. E is linear-elastic Young modulus in GPa from a data sheet. I is second moment of area in cm⁴ about the axis of the chosen bending motion; A is area of the same section in cm², both from a profile table or geometry model. ρ is material density in kg/m³. All five inputs must be positive. f1 is first natural frequency in Hz for comparison with excitation, but it is not yet an approved operating speed.
How to use the calculator
Confirm that one beam end truly restrains both transverse movement and rotation, while the other is laterally restrained but free to rotate. Find L, E, I, A and ρ for the same beam. Compare the result with operating excitation and use an extended model for attached components or different supports.
Worked example
At L=1 m, E=200 GPa and I=1.5 cm⁴, EI=3,000 N·m². With ρ=7,500 kg/m³ and A=4 cm², μ=3 kg/m. Then ω1=λ1²·√(3000/3)≈487.6 rad/s and f1=ω1/(2π)≈77.60 Hz, matching the local textbook example.
Understand the result and units
At otherwise unchanged section, doubling beam length reduces f1 to one quarter. Larger E or I raises it, and greater distributed mass ρA lowers it, each with square-root scaling. Other supports have different eigenvalues and must not use λ1=3.926602.
SI basis: E in Pa, I in m⁴, ρ in kg/m³, A in m² and L in m. Then EI is N·m², μ=ρA is kg/m, ω is rad/s and f is Hz. Input units are converted automatically.
Typical applications
Initial resonance screening of beams, crossmembers and slender machine frames with distributed self-mass when these support conditions fit.
Assumptions, limits and common mistakes
Straight prismatic Euler–Bernoulli beam with constant EI and ρA, ideal fixed-pinned supports and small undamped free vibration only. Attached point masses, shear deformation, rotary inertia, preload, compliant supports, damping and higher modes are omitted. Natural frequency alone is not a resonance or safety verification.
Common mistake: The right pin is laterally restrained and free to rotate; a free cantilever tip is a different case. A and I must describe the same section. ρA is mass per length, not total beam mass.
Frequently asked questions
What is “First bending frequency of a fixed-pinned beam” used for?
Initial resonance screening of beams, crossmembers and slender machine frames with distributed self-mass when these support conditions fit.
Where do the input values come from?
L is free length between the end conditions in m from drawing or measurement. E is linear-elastic Young modulus in GPa from a data sheet. I is second moment of area in cm⁴ about the axis of the chosen bending motion; A is area of the same section in cm², both from a profile table or geometry model. ρ is material density in kg/m³. All five inputs must be positive. f1 is first natural frequency in Hz for comparison with excitation, but it is not yet an approved operating speed.
What does the result not cover?
Straight prismatic Euler–Bernoulli beam with constant EI and ρA, ideal fixed-pinned supports and small undamped free vibration only. Attached point masses, shear deformation, rotary inertia, preload, compliant supports, damping and higher modes are omitted. Natural frequency alone is not a resonance or safety verification.