Formula used
fn = λn2 ÷ (2πL2) × √(EI ÷ ρA)
ω² = [EI∫(φ″)²dx + N∫(φ′)²dx + kf∫φ²dx + spring energy] ÷ [m′∫φ²dx + Σmjφj²]
fd = fn√(1 − ζ²)
The calculator normalizes each assumed mode shape. It then integrates modal mass and stiffness. Timoshenko mode applies an approximate shear and rotary correction.
How to use this calculator
- Select the beam theory and boundary condition.
- Enter beam length and cross-section dimensions.
- Choose a material or enter custom properties.
- Add distributed, tip, or concentrated masses.
- Include axial load, damping, and support stiffness.
- Enter operating speed for resonance screening.
- Calculate, review warnings, then export results.
Example data
| Case | Support | Length | Section | Material | Added mass | Purpose |
|---|---|---|---|---|---|---|
| Machine arm | Cantilever | 3 m | 100 × 200 mm rectangle | Steel | 25 kg tip | Check motor resonance |
| Floor beam | Simply supported | 6 m | I-section | Steel | 50 kg/m | Assess walking vibration |
| Laboratory bar | Free–free | 1 m | 40 mm circle | Aluminum | None | Compare modal testing |
Frequently asked questions
What is a beam’s natural frequency?
A natural frequency is a preferred free-vibration rate. It depends on stiffness, mass, geometry, and support conditions. Excitation nearby can create large resonant responses.
When should Timoshenko theory be selected?
Use it for short, deep, or higher-mode beams. It considers shear deformation and rotary inertia approximately. Slender Euler beams usually need less correction.
How do point masses affect frequency?
Point masses increase modal inertia at their locations. Mass near an antinode causes stronger frequency reduction. Mass near a node has limited influence.
Does tension increase natural frequency?
Axial tension adds geometric stiffness during transverse vibration. It generally raises calculated bending frequencies. Compression lowers frequencies and can approach instability.
What is a safe resonance separation margin?
Required margins depend on standards and operating variability. Twenty percent is a common preliminary screening value. Critical equipment needs project-specific dynamic verification.
Why can free–free modes start above zero?
A free body has rigid translation and rotation modes. Those ideal modes have zero natural frequency. The table lists only flexible bending modes.
Can this replace finite-element modal analysis?
No analytical calculator represents every structural detail. Joints, tapering, cracks, and complex supports change behavior. Use finite elements for final critical decisions.
What does modal mass represent?
Modal mass measures inertia participating in one shape. It depends on the chosen normalization. It is not always total physical mass.
Why does beam orientation matter?
Bending inertia changes with the selected section axis. Strong-axis and weak-axis frequencies can differ greatly. Enter dimensions for the actual vibration direction.
Engineering limitations
Results are analytical estimates for preliminary vibration studies. Nonuniform members, nonlinear joints, cracks, and complex equipment need detailed modeling. Confirm important designs through testing or finite-element modal analysis.