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Structural Dynamics

Beam Vibration Modes

Animate natural mode shapes for simply supported, cantilever, and fixed-fixed beams. Select multiple modes for superposition, and view the natural frequency spectrum.

Beam Settings

Active Modes (multi-select)

Natural Frequencies

Visualization
Freq

What is Modal Analysis for Beams?

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What exactly are these "vibration modes" I see animating in the simulator? They look like weird, frozen waves on the beam.
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Basically, they're the beam's natural "preferred" shapes for vibrating. Think of a guitar string—it vibrates in a specific sine wave pattern when plucked. That's its first mode. A beam has similar patterns, but they depend on how it's held. Try changing the "Boundary Conditions" dropdown above from "Simply Supported" to "Cantilever"—you'll see the first mode shape change from a smooth arch to a curve that's fixed at one end.
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Wait, really? So each mode has its own frequency? And that's what the formula in the FAQ is for?
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Exactly! The formula $f_n = \frac{(\beta_n L)^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}$ gives the natural frequency for mode *n*. The tricky part is $\beta_n L$—it's a constant that changes with both the mode number and the boundary condition. For instance, a simply supported beam has $\beta_n L = n\pi$, nice and simple. But for a cantilever (like a diving board), the first mode uses $\beta_1 L \approx 1.875$. That's why the frequencies are so different. Slide the "Mode Number" control and watch how both the shape and the calculated frequency change.
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Okay, but the simulator also lets me "superimpose" modes. Why would I want to combine these weird shapes?
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Great question! In practice, a beam never vibrates in just one perfect mode. A real vibration—like a bridge shaking from wind or a car hitting a pothole—is a mix of all the modes. The "Modal Superposition" principle says we can build any complex vibration by adding up scaled versions of these basic mode shapes. Check the "Superimpose Modes" box and adjust the sliders for Modes 1 and 2. You'll see how combining them creates a much more complex, realistic motion. This is the foundation of dynamic analysis in CAE software like Ansys.

Physical Model & Key Equations

The governing equation for the free, undamped vibration of an Euler-Bernoulli beam is derived from its bending mechanics and inertia. It describes how the beam's deflection $w(x,t)$ varies with position and time.

$$EI \frac{\partial^4 w}{\partial x^4}+ \rho A \frac{\partial^2 w}{\partial t^2}= 0$$

Here, $E$ is Young's Modulus (material stiffness), $I$ is the Area Moment of Inertia (cross-section shape stiffness), $\rho$ is density, and $A$ is cross-sectional area. $EI$ is the flexural rigidity—the beam's resistance to bending.

By assuming a harmonic solution in time, we solve for natural frequencies and mode shapes. The frequency of the n-th vibration mode is given by:

$$f_n = \frac{(\beta_n L)^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}$$

The key is the dimensionless parameter $\beta_n L$, which is a root of the characteristic equation determined solely by the boundary conditions. For example: Simply Supported: $\beta_n L = n\pi$; Cantilever (Fixed-Free): $\beta_1 L \approx 1.875, \beta_2 L \approx 4.694$; Fixed-Fixed: $\beta_1 L \approx 4.730$. This is why changing the boundary condition in the simulator has such a dramatic effect on the results.

Frequently Asked Questions

When the boundary conditions change, the constraint state at the beam ends changes, altering the natural mode shapes and natural frequencies. For example, for the same mode order, a fixed beam has the highest natural frequency, followed by a cantilever beam, and then a simply supported beam with the lowest. The mode shapes also exhibit characteristic forms depending on the boundary conditions, such as zero deflection angle at a fixed end.
The actual vibration of a beam involves complex motion where multiple natural modes are excited simultaneously. By superimposing the modes and varying their amplitude ratios and phase differences, the display allows for a visual understanding of vibration waveforms close to reality. This is useful for avoiding resonance in design and simulating scenarios where specific modes are dominant.
In the spectrum graph, the horizontal axis represents frequency, and the vertical axis represents amplitude (or mode contribution). The frequencies at which peaks occur are the natural frequencies of the beam. When applying external excitation, resonance can be prevented by avoiding these peak frequencies. Comparing multiple peaks also helps determine which modes are dominant.
You can change parameters such as beam length, cross-sectional shape (width and height), Young's modulus, and density in the input fields on the screen. When these values are modified, the bending stiffness EI and mass per unit length ρA are recalculated, and the natural frequencies and mode shapes are updated in real time. By testing with actual design values, you can intuitively observe changes in vibration characteristics.

Real-World Applications

Civil Engineering - Bridge Design: Engineers must ensure a bridge's natural frequencies don't match the rhythmic loading from wind (vortex shedding) or marching soldiers to avoid catastrophic resonance. Modal analysis identifies these vulnerable frequencies so the design can be stiffened or dampened.

Aerospace - Aircraft Wings: Wing flutter is a dangerous coupling of torsional and bending vibration modes. CAE simulations using modal superposition predict flutter speeds, allowing designers to adjust wing geometry and mass distribution long before a physical prototype is built.

Automotive - NVH Reduction: Noise, Vibration, and Harshness (NVH) teams analyze the modal frequencies of car chassis, body panels, and exhaust systems. They aim to shift these frequencies away from engine firing frequencies or road noise to create a quieter, smoother ride.

Consumer Electronics - Smartphone Design: The vibration of a phone casing can affect microphone and speaker performance. Modal analysis helps place internal components and stiffeners to ensure the housing doesn't amplify certain frequencies, preserving sound quality.

Common Misconceptions and Points to Note

While experimenting with this tool, you might encounter a few points that are easy to misunderstand. First, you might tend to think that higher mode orders mean more violent vibration, but that's actually not the case. While the natural frequency does increase with higher modes, how intensely the structure actually vibrates in that mode depends on the frequency and energy of the applied external force. For example, even if Mode 5 has a frequency of 100Hz, it will hardly be excited if the external force is at 10Hz. Conversely, if resonance occurs in the fundamental mode (Mode 1), it can lead to large amplitudes and even failure, making the fundamental mode the most critical.

Next, understand that "superposition" is not just simple addition. The simulator might look like it's just adding the amplitudes of each mode, but in actual dynamic response, phase (the timing of vibration) is extremely important. For instance, even if Mode 1 and Mode 2 have the same amplitude, they can cancel each other out if they are out of phase, resulting in smaller vibration. In CAE modal superposition methods, this phase relationship is properly accounted for in the calculation.

Finally, regarding the tool's parameters, pay attention to the fact that "bending stiffness (EI)" is heavily influenced by cross-sectional shape. Changing only the material's Young's modulus (E) doesn't change the second moment of area (I). For example, the I for a rectangular beam with width 10mm and height 20mm, calculated for the height direction, is approximately 6667 mm^4. If you "lay it flat" to a width of 20mm and height of 10mm, I plummets to about 1667 mm^4. This means simply changing the beam's orientation reduces stiffness to 1/4 and halves the natural frequency. In design, this "orientation of the cross-section" can be critical, so be careful.

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How to Use

  1. Set flexural rigidity (EI) using the slider—typical values range 50,000 to 500,000 N·m² for steel beams or composite sections.
  2. Adjust mass per unit length (ρ) from 10 to 100 kg/m to represent different beam weights or added structural mass.
  3. Control animation speed with the speed slider to observe mode shapes at realistic vibration frequencies; higher speeds compress time for rapid mode cycling.
  4. Select individual modes (1st, 2nd, 3rd) or superimpose multiple modes to visualize interference patterns and combined dynamic response.
  5. Read natural frequency values (Hz) displayed for each mode—these update automatically based on EI and ρ inputs.

Worked Example

Consider a 3 m simply-supported steel beam with EI = 180,000 N·m² and ρ = 45 kg/m. The first natural frequency calculates to approximately 4.2 Hz. Setting animation speed to 1.0× shows the fundamental mode (single half-sine wave) completing one cycle in 0.24 seconds. The second mode oscillates at 16.8 Hz with a full-wave profile. When both modes are superimposed at equal amplitude, you observe beat patterns as the higher frequency periodically reinforces and cancels the lower mode—critical for predicting resonance in mechanical systems or buildings under wind or seismic loading.

Practical Notes

  1. Damping is neglected in this undamped modal analysis—real structures experience 2–5% critical damping; use results to identify resonance zones, then apply dampers to avoid excitation near natural frequencies.
  2. For cantilever or fixed-end beams, natural frequencies differ by ~25–35% compared to simply-supported; adjust EI or ρ experimentally to match your boundary condition.
  3. In aerospace composites, EI varies with fiber orientation; simulate reduced EI (80,000–120,000 N·m²) to see how anisotropy lowers frequencies and shifts mode shapes.
  4. Animation speed at 0.5× is ideal for teaching; speeds above 2.0× reveal high-frequency coupling effects relevant to vibration isolation design.