Select boundary conditions (simply supported, cantilever, fixed-fixed, free-free, or string), enter geometry and material properties, and instantly compute the first 5 natural frequencies with animated mode shapes.
Beam bending natural frequency:
$$f_n = \frac{\lambda_n^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}$$String natural frequency:
$$f_n = \frac{n}{2L}\sqrt{\frac{T}{m}}$$λ₁: SS[π], Cantilever[1.875], FF[4.730]
Mode Selection (Animation)
Mode Shapes ψ_n(x) for n = 1–5
| Mode n | λ_n | Frequency (Hz) | Period (ms) |
|---|
The natural frequency of a beam in bending is governed by the balance between its bending stiffness (EI) and its mass per unit length ($\rho A$). The boundary conditions are captured by the eigenvalue $\lambda_n$.
$$f_n = \frac{\lambda_n^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}$$$f_n$: Natural frequency of the n-th mode [Hz]
$\lambda_n$: Dimensionless eigenvalue (e.g., 1.875 for 1st mode cantilever)
$L$: Length of the beam [m]
$E$: Elastic Modulus [Pa]
$I$: Area moment of inertia of the cross-section [m⁴] — this changes with the shape you select!
$\rho A$: Mass per unit length [kg/m]
The natural frequency of a taut string is governed by the wave speed along it, which depends on tension and linear density.
$$f_n = \frac{n}{2L}\sqrt{\frac{T}{m}}$$$f_n$: Natural frequency of the n-th mode [Hz]
$n$: Mode number (1, 2, 3...)
$L$: Length of the string [m]
$T$: Tension force in the string [N]
$m$: Mass per unit length [kg/m]
Civil Engineering - Skyscraper Design: Tall buildings act like vertical cantilever beams. Engineers must calculate their natural frequencies to ensure they don't match the dominant frequencies of wind loads or earthquakes, which would cause dangerous resonance and excessive sway.
Aerospace - Turbine Blade Design: Jet engine turbine blades are essentially cantilevers spinning at high speed. Their natural frequencies must be carefully tuned to avoid matching the frequency of passing stator blades, which would lead to catastrophic high-cycle fatigue failure.
Consumer Electronics - Smartphone Vibration: The small vibration motor in your phone is designed to excite a specific vibration mode of the entire phone chassis. Engineers simulate this to ensure the vibration feels strong and crisp, not muffled.
Musical Instruments - String & Soundboard Tuning: The pitch of a guitar string is its first natural frequency, directly set by tension, length, and mass. Similarly, the wooden soundboard of a piano has complex vibration modes that shape the tonal quality of the sound produced.
First, understand that "the calculated natural frequency is not an absolute safety value." This tool assumes ideal shapes and boundary conditions. For example, a "simply supported beam" implies supports that are truly pins and rollers allowing completely free rotation. In real structures, constraints are often stronger than assumed, making the actual frequency higher than the calculated value. Conversely, if bolted joints are loose, the frequency can be lower. CAE results are merely a "guideline," and physical measurement verification with a prototype is essential.
Next, input errors for material constants are extremely common. Pay particular attention to mixing up unit systems. If you input Young's modulus E in "MPa" when it should be in "GPa," the result will be off by a factor of 1000. Density ρ is fundamentally in "kg/m³," but when calculating from mass derived from CAD data, check the volume unit (mm³ or m³). For instance, the correct input for steel density is 7850 kg/m³.
Finally, do not be satisfied with looking only at the first mode. If the frequency of the external force is higher than the first mode, resonance could occur in the second, third, or higher modes. For example, in rotating machinery, frequencies like rotational speed multiplied by the number of blades (passing frequency) often become problematic, and these correspond to higher-order modes. Use this tool's animations to see where nodes (points that don't move) form, and use that as material for considering countermeasures like placing dampers there.
This natural frequency calculation is deeply connected to "Acoustical Engineering." For example, the vibration analysis of a rectangular membrane is exactly the vibration of a speaker diaphragm or a drumhead. When specific modes are excited, they become air vibrations (sound waves), determining timbre and sound quality. A guitar body also has complex vibration modes, and their analysis is at the core of instrument design.
It is also the starting point for structural control in "Control Engineering." In modern precision machinery and space structures, "active vibration control," which actively cancels unwanted vibrations, is used. The first step in its design is accurately understanding the natural frequencies and mode shapes of the structure to be controlled. By understanding beam fundamentals with this tool, you will build a foundational ability to interpret modal analysis results for complex structures.
Furthermore, the link with "Materials Engineering" should not be overlooked. Composite materials like CFRP (Carbon Fiber Reinforced Plastic) exhibit anisotropy (stiffness varies by direction) depending on the fiber orientation. After mastering the behavior of isotropic materials with this tool, analyzing composite material beams with more advanced CAE software will allow you to deeply understand the trade-offs between weight reduction and stiffness/vibration characteristics.
The first next step is to solidify the fundamentals of "Continuum Mechanics." The Euler-Bernoulli beam theory behind this tool is an "elementary" model that ignores shear deformation and rotational inertia. To handle thicker beams or higher-frequency vibrations, you need to learn Timoshenko beam theory. To intuitively feel the difference, try setting the length L extremely short (like a thick beam) in this tool and qualitatively examine the results.
Mathematically, "Partial Differential Equations" and "Eigenvalue Problems" are key. What the tool solves is the process of solving the space and time partial differential equation using the "separation of variables" method and determining the eigenvalues $(\beta_n L)$ from the boundary conditions. To experience this by hand calculation, try substituting the boundary conditions for a cantilever beam (fixed end: deflection and slope zero, free end: bending moment and shear force zero) into the equations and derive the characteristic equation $\cos(\beta L)\cosh(\beta L) = -1$. The solutions to this transcendental equation are 1.875, 4.694...
To get closer to practical work, I strongly recommend incorporating the concept of "damping." Real structures always have damping (energy dissipation), so the amplitude at resonance does not become infinite. The next learning topic is to study "frequency response analysis" for systems with damping, based on the undamped natural frequencies and mode shapes obtained with this tool. This will enable you to evaluate the sharpness of resonance peaks and how quickly vibrations decay (Q factor), taking your practical design skills to the next level.