Animate natural mode shapes for simply supported, cantilever, and fixed-fixed beams. Select multiple modes for superposition, and view the natural frequency spectrum.
$$EI \frac{\partial^4 w}{\partial x^4} + \rho A \frac{\partial^2 w}{\partial t^2} = 0$$
オイラー–ベルヌーイ梁の運動方程式。EI：曲げ剛性 [N·m²]、ρA：単位長さ質量 [kg/m]
$$f_n = \frac{(\beta_n L)^2}{2\pi L^2}\sqrt{\frac{EI}{\rho A}}$$
n 次固有振動数 [Hz]。βₙL は境界条件依存の特性値（単純支持: nπ、片持ち1次: 1.875、固定1次: 4.730）
$$\frac{f_n}{f_1} = \left(\frac{\beta_n L}{\beta_1 L}\right)^2$$
周波数比。単純支持梁では f₂/f₁=4、f₃/f₁=9 と整数の二乗比になる
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.
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.