Calculate natural frequencies and mode shapes of a multi-DOF spring-mass system with animation visualization. real-time rendering of frequency response function (FRF) with resonance warning.
System Definition
DOF N :
Mass m₁…mₙ (kg)
Spring constants k₁…kₙ₊₁ (N/m) — k₁: left ground spring; kₙ₊₁: right ground spring; intermediate: inter-mass springs
Damping ratio ζ
K[i][i] = k[i] + k[i+1] K[i][i±1] = −k[i+1]
ModeShapeAnimation
Click mass blocks in the animation to switch modes
What exactly is a "natural frequency" in this simulator? I see the masses start to wiggle when I hit "Calculate".
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Great observation! Basically, every structure, like a car chassis or a building, has specific frequencies at which it "likes" to vibrate if you give it a nudge. These are its natural frequencies. In this simulator, when you click "Calculate", it solves the math to find these special frequencies for your specific spring-mass setup. Try changing the mass `m` sliders on the left—you'll see the calculated frequencies update instantly.
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Wait, really? And what are those weird wiggling patterns it shows after? Sometimes all masses move together, sometimes they move opposite each other.
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Exactly! Those animated patterns are called "mode shapes." Each natural frequency has its own unique pattern of motion. The first mode is often all masses moving in sync—that's the fundamental frequency. The second mode might have them moving opposite, like a seesaw. This is crucial for engineers. For instance, if a machine's operating speed matches one of these frequencies, it can vibrate violently and break. Slide the "Damping ratio ζ" up from zero and re-calculate. See how the animation changes? Damping absorbs energy and reduces the vibration amplitude.
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So the tool is solving a big math problem to find these? How does connecting springs and masses create that matrix `K` I see in the theory?
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Precisely! The simulator builds a mathematical model of your physical system. The stiffness of each spring (`k`) tells us how forces are transferred between masses. The rules `K[i][i] = k[i] + k[i+1]` and `K[i][i±1] = −k[i+1]` are the recipe for assembling the "stiffness matrix" `K` based on your spring connections. The masses form the "mass matrix" `M`. The core eigenvalue problem we solve is `(K - ω²M)φ = 0`, where the solutions `ω` are the natural frequencies (in rad/s) and `φ` are the mode shape vectors you see animated.
Physical Model & Key Equations
The fundamental equation of motion for an undamped multi-degree-of-freedom (MDOF) system is derived from Newton's second law. For free vibration, it takes the form of a generalized eigenvalue problem.
Here, $\mathbf{M}$ is the mass matrix (diagonal for point masses), $\mathbf{K}$ is the stiffness matrix (built from spring constants `k`), $\omega$ is the natural frequency in radians per second, and $\boldsymbol{\phi}$ is the mode shape vector (eigenvector).
For a more realistic model that includes energy dissipation, we add damping. Using proportional (Rayleigh) damping, the damped natural frequency $\omega_d$ is calculated.
$$
\omega_d = \omega_n \sqrt{1 - \zeta^2}
$$
In this equation, $\omega_n$ is the undamped natural frequency from the eigenvalue solution, and $\zeta$ is the damping ratio, which you control directly with a slider in the simulator. This ratio determines how quickly oscillations die out.
Frequently Asked Questions
Yes, you can adjust the mass of each point mass and the stiffness of each spring by entering numerical values or using sliders. After changes, the eigenvalue analysis is automatically re-executed, and the vibration modes and frequency response graph are updated.
By default, they are displayed in Hz (hertz). Internally, the angular frequency ω (rad/s) is used for calculations, but the conversion f = ω/(2π) is performed automatically, outputting in Hz for intuitive understanding.
A warning is displayed when the external excitation frequency approaches any natural frequency (typically within ±5%). This allows you to immediately try frequency adjustments or parameter changes to avoid resonance during the design phase.
This simulator supports up to 10 degrees of freedom. This upper limit is set considering the balance between computational load and visualization, and it is within the range where eigenvalue analysis using the Jacobi method operates in real time.
Real-World Applications
Automotive NVH (Noise, Vibration, and Harshness): Engineers use eigenvalue analysis to predict the natural frequencies of car frames, engine mounts, and suspension components. They ensure these frequencies are far from the engine's firing frequency or road-induced vibrations to prevent uncomfortable buzzing or shaking felt by passengers.
Aerospace & Aircraft Design: The wings and fuselage of an aircraft are subjected to constant aerodynamic forces. Modal analysis is critical to avoid flutter, a dangerous instability where a structural mode couples with aerodynamic forces, potentially leading to catastrophic failure. Designers shift natural frequencies away from excitation ranges.
Civil Engineering & Earthquake Engineering: Skyscrapers and bridges have complex mode shapes. Engineers calculate these to understand how a structure will respond to an earthquake or strong winds. The goal is to design structures where the dominant, energy-absorbing modes can safely dissipate seismic forces without collapsing.
Consumer Electronics & Manufacturing: From preventing a washing machine from "walking" across the floor during spin cycle to ensuring a smartphone's circuit board doesn't resonate with the vibration motor, eigenvalue analysis is used to design products that are quiet, stable, and durable throughout their operating life.
Common Misconceptions and Points to Note
When starting this type of analysis, there are a few common pitfalls, so let's cover them upfront. First is the assumption that "there is only one natural frequency." If you increase the degrees of freedom to 3 or 5 in this simulator, you'll quickly see that there are as many natural frequencies and modes as there are degrees of freedom in the system. Five degrees of freedom means five modes. In design, you can't just focus on the lowest frequency (the first mode); you need to check all higher-order frequencies that might be present in the excitation source. For example, even if you avoid the primary engine rotation speed, its second or third harmonic components might perfectly match a higher-order natural frequency.
Next, a pitfall in parameter setting. If you randomly change the "mass" and "spring constant" values in the simulator, the overall behavior becomes hard to interpret. Remembering the fundamental principle that natural frequency is proportional to the square root of (spring constant / mass) is very helpful. For instance, if you quadruple the mass, you must also quadruple the spring constant to maintain the same frequency. Doubling all masses and springs uniformly leaves the natural frequencies unchanged (scaling law). Having this intuition will make your parameter tuning much faster.
Finally, don't forget that a world with "damping ζ=0" is unrealistic. While it simplifies calculations and resonance peaks become theoretically infinite, real structures always have damping. You can confirm in this tool that slightly increasing ζ from 0.01 to 0.05 dramatically reduces the peak height. In practice, estimating "how much damping exists" is often the most challenging and crucial part of the design.
With damping ratio ζ = 0.05, resonance magnification (Q factor) ≈ 1/(2ζ) = 10. Setting the excitation frequency to each f_n produces large-amplitude response.
Design Criteria: ISO 10816 (vibration evaluation of rotating machinery), ISO 2631 (whole-body vibration). In buildings, a sway angle of 1/200 or less is the habitability criterion.