Natural Frequency of Beam Calculator – Vibration Analysis (Steel & Wood)
The Natural Frequency of Beam Calculator evaluates vibration and dynamic serviceability for steel and wood beams. Calculate fundamental natural frequency (fn in Hz) across simple, cantilever, fixed, and continuous supports.
Input section properties, span, and mass — receive frequency results, angular frequency, and practical guidance against human comfort limits (e.g., floor vibration 4–5 Hz).
Essential for floor systems, bridges, or sensitive structures. Complements static checks from our Ultimate Steel Beam Calculator.
Natural Frequency of Beam Calculator
Euler-Bernoulli & Timoshenko Theory · Multi-Mode Analysis · Resonance Checker
| Mode n | βnL | fn (Hz) | ωn (rad/s) | Tn (s) | fd,n (Hz) | Resonance Risk |
|---|
For a uniform prismatic beam, the natural frequency of the n-th mode is:
Equivalently in angular form: \(\omega_n = \frac{(\beta_n L)^2}{L^2} \sqrt{\frac{EI}{\rho A}}\)
- \(f_n\) = natural frequency of mode n (Hz)
- \(\beta_n L\) = eigenvalue / frequency constant (depends on BC and mode)
- \(E\) = Young's modulus (Pa)
- \(I\) = second moment of area (m\u2074)
- \(\rho\) = material density (kg/m\u00B3)
- \(A\) = cross-sectional area (m\u00B2)
- \(L\) = beam length (m)
| BC | Mode 1 | Mode 2 | Mode 3 | Mode 4 | Mode 5 |
|---|---|---|---|---|---|
| Simply Supported (SS) | \(\pi\) | \(2\pi\) | \(3\pi\) | \(4\pi\) | \(5\pi\) |
| Cantilever (CF) | 1.8751 | 4.6941 | 7.8548 | 10.9955 | 14.1372 |
| Fixed-Fixed (CC) | 4.7300 | 7.8532 | 10.9956 | 14.1372 | 17.2788 |
| Fixed-Pinned (CP) | 3.9266 | 7.0686 | 10.2102 | 13.3518 | 16.4934 |
| Free-Free (FF) | 4.7300 | 7.8532 | 10.9956 | 14.1372 | 17.2788 |
For SS: \(\beta_n L = n\pi\). For CF and others, values are roots of the characteristic equation.
where \(\zeta\) = damping ratio (0 for undamped, 1 for critically damped)
P > 0: tension (increases \(f_n\)); P < 0: compression (decreases \(f_n\)). Critical buckling when P = \(-P_{cr}\).
r = radius of gyration = \(\sqrt{I/A}\); \(\kappa\) = shear correction factor; G = shear modulus
Schematic mode shapes for first 3 modes. Select a boundary condition to update the diagram.
Dashed centerline indicates beam neutral axis; curve shows approximate deflection shape.
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Natural Frequency of Beam Calculator — Complete User Guide & Formula Reference
Step-by-step instructions • All calculation formulas explained • Common mistakes • FAQ • Worked examples
Table of Contents
- What Is a Natural Frequency of Beam Calculator?
- Key User Pain Points & How This Calculator Solves Them
- Visual Overview — Beam Boundary Conditions & Mode Shapes
- Step-by-Step User Guide
- All Calculation Formulas — Explained in Detail
- Input Units & Accepted Values (Validation Reference)
- Eigenvalue (βnL) Reference Table for All Boundary Conditions
- Material Property Reference Table
- Accuracy Note & Theory Limitations
- Frequently Asked Questions (FAQ)
What Is a Natural Frequency of Beam Calculator?
Understanding beam vibration is fundamental to safe structural and mechanical engineering design.
The Natural Frequency of Beam Calculator is an engineering tool that computes the resonant (natural) frequencies at which a structural beam will oscillate when disturbed by an external force, such as wind, machinery vibration, foot traffic, or seismic ground motion. These are the frequencies at which the beam wants to vibrate when left to itself — no sustained energy input required.
If an external force repeatedly applies energy at or near one of these natural frequencies, resonance occurs. Resonance causes vibration amplitudes to grow rapidly, potentially leading to fatigue cracking, excessive deflection, noise, occupant discomfort, or catastrophic structural failure — as famously illustrated by the Tacoma Narrows Bridge collapse (1940).
This calculator uses Euler-Bernoulli and Timoshenko classical beam theory to compute up to 10 vibration modes instantly, eliminating hours of manual calculation.
Key User Pain Points & How This Calculator Solves Them
Real engineers and students face these frustrations daily — here's how the calculator addresses each one.
Complex Manual Calculations Prone to Error
The Euler-Bernoulli frequency formula involves square roots, fourth-power geometry terms, and mode-dependent eigenvalues. A single unit mistake invalidates the entire result.
✓ Auto-computes all terms from raw dimensionsDifferent Formulas for Each Boundary Condition
Cantilever, simply supported, and fixed-fixed beams all use different βnL eigenvalues. Looking these up from Roark's or textbook tables is slow and error-prone.
✓ 5 boundary conditions with built-in eigenvalue tablesUnit Inconsistency Causing Order-of-Magnitude Errors
Mixing GPa with mm⁴, or kg/m³ with inches, is an extremely common source of catastrophic errors — even for experienced engineers.
✓ One-click SI ↔ Imperial toggle with full auto-conversionOnly Fundamental Mode Computed
Most online calculators only compute f₁. Resonance can occur at higher harmonics (f₂, f₃…), which rotating machinery, pedestrians, or wind can easily excite.
✓ Up to 10 modes calculated simultaneouslyNo Moment of Inertia Calculator for Standard Sections
Computing I for an I-beam or hollow pipe by hand requires additional sub-calculations, adding setup time and extra steps before reaching the frequency answer.
✓ 5 cross-section types with auto I & A calculationNo Resonance Risk Warning Against Operating Speeds
Engineers often design a beam without checking whether its natural frequency overlaps with the RPM of nearby motors, pumps, or HVAC fans — until vibration becomes a problem.
✓ Built-in resonance checker with Safe / Caution / Danger flagsNo Support for Timoshenko Theory (Thick Beams)
Short, stocky beams (L/h < 10) violate the slender beam assumption. Euler-Bernoulli over-predicts frequency for these cases. Timoshenko theory corrects for shear deformation and rotary inertia.
✓ Full Timoshenko correction with shear modulus G and κNo Damping or Added-Mass Analysis
Real structures have damping (reduces effective frequency) and often carry equipment, people, or pipes. Ignoring these gives overly optimistic (unsafe) frequency predictions.
✓ Damping ratio ζ and point mass inputs with Dunkerley correctionVisual Overview — Boundary Conditions, Mode Shapes & Beam Terminology
Understanding the physical setup before entering numbers prevents the most common input errors.
Fig. 1 — Five boundary condition configurations with 1st mode shapes and key beam parameters for a rectangular cross-section.
Step-by-Step User Guide — How to Use the Calculator
Follow these steps in order for a complete and accurate calculation. Each step maps to a specific section of the calculator interface.
Choose Your Unit System
Before entering any values, click either SI (Metric) or Imperial at the top of the calculator. This switches all unit labels simultaneously — meters ↔ feet, GPa ↔ ksi, kg/m³ ↔ lb/ft³.
- SI: Length in m, Modulus in GPa, Density in kg/m³, Force in N, Mass in kg
- Imperial: Length in ft, Modulus in ksi, Density in lb/ft³, Force in lbf, Mass in lb
Select the Boundary Condition (Support Configuration)
Use the Support Configuration dropdown. This is the most critical input — it determines the eigenvalue βnL used in the formula. Choose based on how the beam ends are physically supported:
- Simply Supported (SS): Both ends rest on pins/rollers — free to rotate, no end moment. Common in simple spans and bridge girders.
- Cantilever (CF): One end rigidly bolted to a wall, other end completely free. Common in shelf brackets, balconies, and crane booms.
- Fixed-Fixed (CC): Both ends are fully welded or embedded — no rotation permitted. Gives highest natural frequency.
- Fixed-Pinned (CP): One end fully fixed, one end pinned. Occurs in continuous beam spans.
- Free-Free (FF): Both ends free — represents floating beams, e.g., a beam in mid-air during transport or a rocket fuselage in flight.
Select Beam Theory
Use the Beam Theory dropdown:
- Euler-Bernoulli (default): Accurate for slender beams where L/h > 10. Assumes plane sections remain plane and shear deformation is negligible.
- Timoshenko: Use when L/h < 10 (deep beams, short spans). Accounts for shear deformation and rotary inertia. Requires Shear Modulus G and Shear Correction Factor κ.
If Timoshenko is selected, two additional fields appear: Shear Modulus G and Shear Correction Factor κ.
Enter Beam Length (L)
Type the total span length in the Beam Length (L) field. Accepted range: any positive value > 0.
- SI: Enter in metres (m) — e.g., 3.5 for a 3.5 m span
- Imperial: Enter in feet (ft) — e.g., 12 for a 12 ft span
- Internally, all values are converted to SI (metres) before calculation
Note: Natural frequency scales as 1/L² — doubling the beam length reduces f₁ by a factor of 4.
Select Cross-Section Type & Enter Dimensions
Use the Cross-Section Type dropdown, then enter the required dimensions:
- Rectangular: Width b (horizontal) & Height h (vertical — in the bending direction). Make sure h is the dimension in the direction of vibration.
- Circular Solid: Outer diameter d
- Hollow Pipe: Outer diameter OD & Inner diameter ID. OD must be > ID.
- I-Beam: Flange width bf, flange thickness tf, web height hw, web thickness tw
- Custom: Directly enter cross-sectional area A and second moment of area I if you have pre-computed values from CAD or section tables.
Select Material or Enter Custom Properties
Click the Material Preset dropdown to auto-fill Young's Modulus E and Density ρ. Available presets:
- Structural Steel (E = 200 GPa, ρ = 7850 kg/m³)
- Aluminum 6061 (E = 69 GPa, ρ = 2700 kg/m³)
- Concrete (E = 30 GPa, ρ = 2400 kg/m³)
- Timber/Douglas Fir (E = 12 GPa, ρ = 600 kg/m³)
- Titanium Ti-6Al-4V, Carbon Fiber CFRP, Copper
For non-standard materials, select Custom and manually type E and ρ values.
(Optional) Open Advanced Options — Damping, Added Mass, Axial Load, Resonance Check
Click the Advanced Options toggle to expand additional inputs:
- Point Mass (kg or lb): A concentrated mass added at midspan (equipment, motor, person). Uses Dunkerley's approximation to lower the effective frequency.
- Damping Ratio ζ (0–0.999): Fraction of critical damping. Typical values: steel = 0.01–0.02, concrete = 0.04–0.07, timber = 0.05–0.10. Used to compute the damped natural frequency fd.
- Axial Load P (N or lbf): Positive = tension (increases frequency), Negative = compression (decreases frequency, approaches buckling). Enter 0 if no axial load.
- Operating Frequency (Hz): The excitation frequency from machinery or pedestrians. The calculator flags any natural frequency within ±15% as a resonance risk.
Click "Calculate Natural Frequency"
The calculator instantly computes all results and switches you to the Results tab, showing:
- Fundamental frequency f₁ (Hz), angular frequency ω₁ (rad/s), period T (s), damped frequency fd (Hz)
- Auto-computed section properties: A, I, μ, r, slenderness ratio, flexural rigidity EI
- Full multi-mode frequency table for modes 1 through N
- Resonance risk assessment for each mode (if an operating frequency was entered)
Export, Copy, or Print Your Results
Use the buttons at the bottom of the Results tab:
- Copy Results — Copies a plain-text summary of all inputs and results to your clipboard for pasting into reports or emails.
- Print / Export PDF — Opens the browser print dialog. Use "Save as PDF" in the destination to create a portable PDF report.
All Calculation Formulas — Detailed Derivation & Explanation
Every formula used by the calculator is documented below with variable definitions, units, and physical interpretation.
Formula 1 — Euler-Bernoulli Natural Frequency (Core Formula)
This is the fundamental formula for the n-th mode natural frequency of a uniform prismatic beam, derived from the Euler-Bernoulli beam differential equation of motion:
fn in Hz; ωn in rad/s; relationship: ωn = 2π fn
| Symbol | Quantity | SI Unit | Imperial Unit | Physical Role |
|---|---|---|---|---|
| fn | Natural frequency of mode n | Hz (= 1/s) | Hz | Oscillations per second; primary output |
| ωn | Angular frequency of mode n | rad/s | rad/s | ω = 2π f; used in vibration dynamics equations |
| βnL | Eigenvalue / frequency constant for mode n | dimensionless | dimensionless | Root of characteristic equation; depends on BC & mode number. See Table in Section 7. |
| L | Beam length (span) | m | ft → converted to m | Frequency ∝ 1/L² — most sensitive parameter |
| E | Young's Modulus (elastic modulus) | Pa (entered as GPa) | psi (entered as ksi) | Material stiffness; frequency ∝ √E |
| I | Second moment of area (about bending axis) | m⁴ | ft⁴ → converted to m⁴ | Geometric stiffness; frequency ∝ √I |
| ρ | Material mass density | kg/m³ | lb/ft³ → converted to kg/m³ | Inertia term; frequency ∝ 1/√ρ |
| A | Cross-sectional area | m² | ft² → converted to m² | Combined with ρ → mass per unit length μ = ρA |
Formula 2 — Cross-Section Properties (Automatically Computed)
h is the dimension parallel to the bending axis. For a beam bending vertically, h is the vertical depth.
OD must be strictly greater than ID. If ID ≥ OD, the calculator will report a geometry error.
This is the standard parallel-axis / subtraction method for doubly-symmetric I-sections about the major (strong) axis.
Formula 3 — Damped Natural Frequency
When a structure has internal material damping (or added dampers), the effective frequency at which it actually oscillates is slightly lower than the undamped natural frequency:
For ζ = 0.05 (5% damping, typical concrete): fd = 0.9987·fn — only 0.13% lower. For ζ = 0.2 (20%): fd = 0.98·fn. Damping has a minor effect on frequency but a major effect on vibration amplitude at resonance.
| Symbol | Quantity | Typical Range | Notes |
|---|---|---|---|
| fd | Damped natural frequency | Hz | Always ≤ fn |
| ζ | Damping ratio | 0 to <1 | ζ = c / ccr; ζ = 1 is critically damped (no oscillation) |
| fn | Undamped natural frequency | Hz | From Euler-Bernoulli / Timoshenko formula above |
Formula 4 — Timoshenko Beam Correction for Short/Thick Beams
The Euler-Bernoulli theory over-estimates frequency for short beams because it ignores shear deformation and rotary inertia. The Timoshenko correction factor reduces the EB frequency:
where r = √(I/A) is the radius of gyration, κ is the shear correction factor, and G is the shear modulus. The correction factor is always ≤ 1, meaning Timoshenko frequency ≤ Euler-Bernoulli frequency.
| Symbol | Quantity | SI Unit | Typical Values |
|---|---|---|---|
| G | Shear modulus | Pa (entered as GPa) | Steel: 77 GPa | Alum: 26 GPa | Concrete: ~12 GPa |
| κ | Shear correction factor | dimensionless | Rect: 5/6 ≈ 0.833 | Circular: 0.9 | I-beam: ~0.44 |
| r | Radius of gyration = √(I/A) | m | Indicates how mass is distributed about bending axis |
Formula 5 — Axial Load Effect on Natural Frequency
An axial tensile force stiffens a beam (increases frequency), while a compressive force softens it (decreases frequency, approaching zero at the buckling load):
P > 0 = tension (increases f); P < 0 = compression (decreases f). When P = −Pcr,1, frequency → 0 (buckling onset). Critical buckling load Pcr,1 for Simply Supported = π²EI/L² (Euler buckling).
Formula 6 — Dunkerley's Method for Added Point Mass
When a concentrated mass is added to the beam (e.g., a motor or equipment), the effective natural frequency drops. Dunkerley's approximation combines the beam's own frequency with the frequency as if the beam were massless but carried the point load:
μL = total beam mass (kg); mpoint = added point mass (kg). This is applied to the fundamental mode only. For higher modes with added mass, full eigenvalue analysis (FEA) is recommended.
Formula 7 — Period of Vibration
Period T is the time for one complete oscillation cycle. Used in earthquake engineering (response spectra) and fatigue life estimation (cycles per second).
Input Units & Accepted Values — Validation Reference
Use this table to verify your inputs are in the correct units before calculating. All values must be positive and non-zero unless noted.
| Parameter | SI Input Unit | Imperial Input Unit | Valid Range | Common Error |
|---|---|---|---|---|
| Beam Length L | metres (m) | feet (ft) | > 0 | Entering mm instead of m (×1000 error) |
| Section dimensions (b, h, d, etc.) | metres (m) | feet (ft) | > 0; OD > ID for pipes | Entering mm instead of m (×1000 error in I) |
| Young's Modulus E | GPa (e.g., 200 for steel) | ksi (e.g., 29,000 for steel) | > 0 | Entering Pa or psi without unit conversion |
| Density ρ | kg/m³ (e.g., 7850 for steel) | lb/ft³ (e.g., 490 for steel) | > 0 | Entering g/cm³ instead of kg/m³ |
| Shear Modulus G | GPa (e.g., 77 for steel) | ksi (e.g., 11,200 for steel) | > 0 (Timoshenko only) | Leaving at steel default when using aluminum |
| Shear Correction Factor κ | Dimensionless (no unit) | 0.1 – 1.0 | Using 0.833 (rectangular) for circular sections | |
| Damping Ratio ζ | Dimensionless (decimal, NOT %) | 0 to 0.999 | Entering 5 instead of 0.05 for 5% damping | |
| Point Mass m | kg | lb | ≥ 0 | Entering kN load instead of mass in kg |
| Axial Load P | N | lbf | any real number; P < 0 = compression | Entering kN without converting (×1000 error) |
| Operating Frequency | Hz (always, regardless of unit system) | > 0 (optional) | Entering RPM instead of Hz (divide RPM by 60) | |
| Custom Area A | m² | ft² | > 0 | Entering mm² instead of m² |
| Custom Inertia I | m⁴ | ft⁴ | > 0 | Entering mm⁴ or cm⁴ instead of m⁴ |
Eigenvalue (βnL) Reference Table for All Boundary Conditions
The βnL values (also called frequency factors or modal constants) are the most critical input to the natural frequency formula. They are roots of the characteristic equation for each boundary condition.
| Boundary Condition | Mode 1 | Mode 2 | Mode 3 | Mode 4 | Mode 5 | Higher Modes (n ≥ 3) |
|---|---|---|---|---|---|---|
| Simply Supported (SS) | π = 3.1416 | 2π = 6.2832 | 3π = 9.4248 | 4π = 12.5664 | 5π = 15.7080 | nπ |
| Cantilever (CF) | 1.8751 | 4.6941 | 7.8548 | 10.9955 | 14.1372 | (2n−1)π/2 ≈ exact for n ≥ 5 |
| Fixed-Fixed (CC) | 4.7300 | 7.8532 | 10.9956 | 14.1372 | 17.2788 | (2n+1)π/2 ≈ exact for n ≥ 3 |
| Fixed-Pinned (CP) | 3.9266 | 7.0686 | 10.2102 | 13.3518 | 16.4934 | (4n+1)π/4 ≈ exact for n ≥ 3 |
| Free-Free (FF) | 4.7300 | 7.8532 | 10.9956 | 14.1372 | 17.2788 | Same as CC (symmetric) |
Note: For Free-Free beams, there are two rigid body modes at f = 0 Hz (pure translation and pure rotation) before the first elastic mode at β₁L = 4.7300. The calculator excludes these rigid-body modes.
Frequency Ratios Relative to Mode 1
| Boundary Condition | f₂/f₁ | f₃/f₁ | f₄/f₁ | f₅/f₁ |
|---|---|---|---|---|
| Simply Supported (SS) | 4.0 | 9.0 | 16.0 | 25.0 |
| Cantilever (CF) | 6.27 | 17.55 | 34.39 | 56.84 |
| Fixed-Fixed (CC) | 2.76 | 5.40 | 8.93 | 13.34 |
| Fixed-Pinned (CP) | 3.24 | 6.76 | 11.53 | 17.64 |
| Free-Free (FF) | 2.76 | 5.40 | 8.93 | 13.34 |
Example: For a simply supported beam with f₁ = 5 Hz, the second mode f₂ = 20 Hz and third mode f₃ = 45 Hz. A machine running at 18–22 Hz would be near resonance with Mode 2.
Material Property Reference Table
Pre-loaded values used by the material presets. Use these as defaults for custom materials when manufacturer data is unavailable.
| Material | E (GPa) | ρ (kg/m³) | G (GPa) | ν (Poisson) | Typical ζ | Notes |
|---|---|---|---|---|---|---|
| Structural Steel | 200 | 7850 | 77 | 0.30 | 0.01–0.02 | E430, S275, A36, S355 |
| Aluminum 6061-T6 | 69 | 2700 | 26 | 0.33 | 0.01–0.02 | High strength-to-weight ratio |
| Concrete (Normal) | 25–35 | 2400 | 12 | 0.20 | 0.04–0.07 | E depends on f'c; use 30 GPa for C30 |
| Timber / Douglas Fir | 12 | 600 | 0.75 | 0.37 | 0.05–0.10 | Parallel-to-grain; highly variable |
| Titanium Ti-6Al-4V | 116 | 4430 | 44 | 0.34 | 0.01 | Aerospace alloy; high E/ρ ratio |
| Carbon Fiber CFRP (UD) | 135 | 1600 | 7 | 0.30 | 0.005–0.01 | Unidirectional laminate, fiber direction |
| Copper | 110 | 8940 | 41 | 0.34 | 0.02 | High density limits frequency |
| Steel (Imperial) | 29,000 ksi | 490 lb/ft³ | 11,200 ksi | 0.30 | 0.01–0.02 | ASTM A36 / A992 |
⚙ Accuracy Note & Theory Limitations — Read Before Relying on Results
- Euler-Bernoulli accuracy: Results are within ~1–2% of exact solutions for beams with slenderness ratio L/h > 10. For L/h < 10, switch to Timoshenko theory. For L/h < 5, use FEA.
- Uniform beam assumption: This calculator assumes a prismatic (constant cross-section) beam along its full length. Tapered, stepped, or composite beams require numerical FEA methods for accurate results.
- Boundary condition idealization: Real support conditions are never perfectly rigid or perfectly pinned. Partial fixity, foundation flexibility, and joint stiffness can shift actual frequencies by 5–25% from theoretical values.
- Dunkerley for added mass: The point-mass correction using Dunkerley's method is an approximation (conservative — it under-estimates frequency). For multiple added masses or off-center masses, results may differ from FEA by up to 10–15%.
- Axial load formula validity: The axial load correction is valid for P < 0.5·Pcr. Approaching the buckling load, the formula becomes less accurate and FEA or specialized stability analysis should be used.
- Not a substitute for FEA or code compliance: This tool is for preliminary design, education, and quick checks. Critical structural designs should be verified by a licensed engineer using full FEA and applicable codes (AISC, Eurocode, ASCE, etc.).
Frequently Asked Questions (FAQ)
Answers to the most common questions from engineers, students, and designers using this calculator.
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