Animate natural mode shapes for simply supported, cantilever, and fixed-fixed beams. Select multiple modes for superposition, and view the natural frequency spectrum.
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.
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.
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.
Beam vibration analysis is truly a fundamental of structural mechanics. Understanding it opens doors to a wide variety of engineering fields. Most directly connected are "Modal Analysis" and "Experimental Modal Analysis (EMA)". This involves validating CAE eigenvalue analysis results by attaching accelerometers to a real structure, striking it with a hammer (impact test), and comparing the measured Frequency Response Function (FRF). The "node" positions you learn with this tool are also locations where you shouldn't place sensors (as vibration cannot be detected there).
It also directly connects to "NVH (Noise, Vibration, and Harshness)" evaluation in automobiles and aircraft. The "buzzing" vibration of a car's side mirror or the "boom" during high-speed driving are phenomena caused by specific modes of beams or panels being excited. Engineers visualize mode shapes like the ones you see in this tool while considering countermeasures (adding ribs or masses).
Further development leads to "Wave Propagation" and "Acoustics". Sound itself is a longitudinal wave in air, but understanding elastic waves (especially bending waves) propagating in solids is essential for machinery noise reduction and building soundproofing design. By slightly extending the beam vibration equation to include a loss term, you can model how vibration decays. Thus, from a single beam model, you can gain an integrated understanding across a broad range of fields including structures, acoustics, and control.
Once you're comfortable with this simulator, try switching your mindset from a "beam as a continuum" to a "discretized model". CAE software actually calculates by dividing the beam into many small elements (finite elements). For example, if you divide this tool's cantilever beam into 10 elements and represent the mass and stiffness at each node with matrices, the equation of motion takes the form $M\ddot{u} + Ku = 0$. Here, $M$ is the mass matrix, $K$ is the stiffness matrix, and $u$ is the displacement vector. Solving for the eigenvalues and eigenvectors of this equation is precisely the core of "eigenvalue analysis" in CAE software.
Mathematically, I recommend following the process of solving the Euler-Bernoulli differential equation using the "separation of variables" method. Assuming the solution can be separated as $w(x,t) = W(x) \cdot T(t)$, the spatial term $W(x)$ becomes a combination of sine, cosine, and hyperbolic functions. Substituting the boundary conditions yields those $\beta_n L$ (characteristic values). By writing out and following this process on paper, the "magic" behind the tool should transform into solid "logic".
As a next concrete topic, learning about "Forced Vibration of Beams and Frequency Response" is a good step. Right now you're observing free vibration (how it sways naturally), but in reality, forces from sources like engines are continuously applied at specific frequencies (forced vibration). How the amplitude behaves then, especially how large it becomes near resonance points, is determined by the amount of damping. Learning models that consider damping brings you close to a perfect first step towards practical vibration analysis.