Apply harmonic, impulse, random or El Centro ground motion to an SDOF system and compute the response via RK4. Visualize displacement and acceleration time histories with the frequency response function (FRF).
Parameters
Excitation Type
Mass m
kg
Stiffness k
Damping ratio ζ
Excitation freq. f_ex
Hz
Amplitude F₀
N
Results
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ωₙ [rad/s]
—
Natural freq. fₙ [Hz]
—
Peak displacement [mm]
—
Dynamic amp. factor β
—
ωd [rad/s]
—
Damped period Td [s]
—
Peak acceleration [m/s²]
—
Damping ratio ζ
Time History Response
Frequency Response Function |H(ω)| vs ω/ωₙ
Frf
SDOF equation of motion (absolute displacement, base excitation adds ü_g):
Frequency response function: $|H(\omega)| = \dfrac{1}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}$, $r=\omega/\omega_n$
At resonance ($r=1$): $|H| \approx \dfrac{1}{2\zeta}$ (maximum dynamic amplification factor)
What is an SDOF System?
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What exactly is a "Single Degree of Freedom" or SDOF system? It sounds like a big simplification.
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Basically, it's the simplest model for vibrating structures. Imagine a mass that can only move in one direction, connected to a spring and a damper. In practice, we use it to represent the fundamental behavior of complex things, like the first sway mode of a building. Try moving the 'Mass (m)' and 'Stiffness (k)' sliders above—you'll see the natural frequency change instantly, which is the core concept.
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Wait, really? So the damping ratio ζ I can set... what does that do to the vibration?
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Great question. Damping is what dissipates energy, making vibrations die out. A common case is car shock absorbers. If you set ζ = 0 (no damping) in the simulator and apply an impulse, the mass will oscillate forever. Now, increase ζ to 0.1 (10% damping)—you'll see the response amplitude decay over time. If you crank it up past 1.0, you get "overdamped" behavior where it just slowly creeps back without oscillating.
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The simulator has "El Centro" as an excitation. What's special about that, and why would I use random vibration?
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El Centro is a famous earthquake ground motion record from 1940. Engineers use it as a standard test to see how a structure might respond to a real quake. When you select it, the simulator applies that exact acceleration history as the base excitation. Random vibration, on the other hand, models unpredictable forces like wind gusts or rough-road driving. Switching between these excitations shows how the same structure can have wildly different responses!
Physical Model & Key Equations
The fundamental equation governing the SDOF system is Newton's second law, balancing inertial force with the spring force, damping force, and any external excitation.
$$m\ddot{x}+ c\dot{x}+ kx = F(t)$$
Here, $m$ is the mass (kg), $c$ is the viscous damping coefficient (N·s/m), $k$ is the stiffness (N/m), $x$ is the displacement (m), and $F(t)$ is the time-dependent external force (N). The dots denote derivatives with respect to time (acceleration and velocity).
From the mass and stiffness, we derive the system's inherent dynamic properties. The damping coefficient is often expressed via the damping ratio ζ.
$\omega_n$ is the undamped natural angular frequency (rad/s), $f_n$ is the natural frequency in Hz, $\zeta$ is the damping ratio (unitless), and $\omega_d$ is the damped natural frequency. The simulator calculates these live as you adjust $m$, $k$, and $\zeta$.
Real-World Applications
Seismic Building Design: Engineers model a multi-story building as an equivalent SDOF system to estimate its fundamental period and maximum displacement during an earthquake. This is the foundation of the response spectrum method used in building codes worldwide. The "El Centro" excitation in the simulator directly mimics this analysis.
Automotive Suspension Tuning: A car's wheel assembly (unsprung mass) and body (sprung mass) are often analyzed as SDOF systems. By adjusting stiffness (spring rate) and damping (shock absorber), engineers balance ride comfort and handling. Try simulating an impulse force to see how different ζ values control the "bounce".
Machinery Vibration Analysis: Rotating equipment like pumps or turbines experience harmonic forces at the operating frequency. If this frequency matches the system's natural frequency ($f_{ex} = f_n$), resonance occurs, causing large, damaging vibrations. Use the simulator's harmonic excitation and sweep the frequency to see the dramatic peak in response.
CAE Software Verification: Before running a complex, time-consuming finite element analysis in software like Ansys or Abaqus, engineers will create a simple SDOF benchmark. The results from this simulator provide a hand-calculation check to ensure the FEA setup and solver settings are correct for linear dynamic simulations.
Common Misconceptions and Points to Note
When you start using this simulator, there are a few points that are easy to misunderstand. First, you might think "increasing the mass (m) makes it less prone to shaking, so it's safer," but that's only half true. While the natural frequency $f_n = \frac{1}{2\pi}\sqrt{k/m}$ does decrease with increased mass, the story is different for "acceleration inputs" like earthquakes. Looking at the right side of the equation of motion $m\ddot{x}+ c\dot{x}+ kx = -m\ddot{x}_g$, you'll see the external force increases proportionally with mass. In other words, an overly heavy mass block might actually experience larger inertial forces. Balance is key.
Next, thinking that a higher damping ratio ζ is always better is another pitfall. While vibrations do die out faster, since the damping force is proportional to velocity, making the damping coefficient $c$ unnecessarily large can suppress the peak displacement response, but the force from the damper itself ($c\dot{x}$) acting on the member can become very large. For example, setting ζ to 0.3 might reduce the maximum displacement under the El Centro wave, but if you don't calculate the instantaneous damper force, the support might fail. Design is about trade-offs.
Finally, don't overconfidently assume "since I calculated it with a single degree of freedom, it will behave exactly like this." Real structures have multiple modes (ways of shaking), so you're only looking at the first mode. For instance, when modeling a tall tank as a single degree of freedom, even if you represent the mass (m) as the total weight and the stiffness (k) with a foundation spring, you completely miss phenomena like sloshing (liquid oscillation) at the top, which is a separate issue. Keep in mind that this tool is for "understanding the essence of phenomena and getting a feel for parameter sensitivity," not for direct use in final design.