Parameter Settings
Near resonance region (TR > 0.5)
No isolation effect (resonance amplification region, TR > 1)
Design Guidelines
Isolation effective: r = f_exc/f_n > √2 ≈ 1.41
Typical design: r ≈ 3–5 (TR ≈ 5–15%)
Precision equipment: r > 5 (IL > 20 dB)
Results
Transmissibility TR
—
dimensionless
Natural Frequency f_n
—
Hz
Total Stiffness k_total
—
N/mm
Mount Stiffness k_mount
—
N/mm
Static Deflection δ_st
—
mm
Frequency Ratio r
—
f_exc / f_n
Transmissibility TR vs Frequency Ratio r (Log Scale)
What is Vibration Transmissibility?
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What exactly is vibration transmissibility? I see it's the main output of this simulator.
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Basically, it's a measure of how much vibration "gets through" your isolation system. A TR of 1 means 100% of the vibration is transmitted—no isolation at all. A TR less than 1 means you're successfully isolating the equipment. Try moving the "Target Natural Frequency" slider lower than the "Excitation Frequency"—you'll see the TR drop below 1, which is the goal.
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Wait, really? So if I have a big machine vibrating at 30 Hz, I want my mounts to have a lower natural frequency? That seems backwards.
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It does seem counterintuitive! But yes, for isolation to work, the mount must be "softer" than the excitation. A common case is a diesel generator. If it shakes at 30 Hz, you design rubber mounts with a natural frequency of, say, 5 Hz. In the simulator, set f_exc=30 and f_n=5. You'll see a low TR, meaning most vibration is not transmitted to the floor.
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Okay, I see the TR value change. But what's the role of the "Damping Ratio" slider? Changing it seems to have a big effect when frequencies are close.
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Great observation! Damping absorbs energy. It's crucial when starting up or shutting down a machine, because the excitation frequency briefly passes through the natural frequency—a dangerous region called resonance. For instance, in a large pump, high damping (ζ around 0.2-0.3) prevents huge vibrations during startup. Slide ζ to 0.01 and then to 0.3 while keeping r near 1. You'll see how damping tames the resonance peak.
Physical Model & Key Equations
The core model is a single-degree-of-freedom system: a mass (your equipment) supported by a spring and damper (the isolation mounts). The key parameter is the frequency ratio \( r \), which compares the disturbance frequency to the system's inherent natural frequency.
$$ r = \frac{f_{\text{exc}}}{f_n}$$
Here, \( f_{\text{exc}}\) is the forcing frequency from the equipment (e.g., motor speed), and \( f_n \) is the natural frequency of the mass-on-spring system. Isolation requires \( r > \sqrt{2}\), meaning the system must operate well above its natural frequency.
The Transmissibility (TR) equation predicts the fraction of vibratory force or motion transmitted from the equipment to its foundation.
$$ \text{TR}= \sqrt{\frac{1+(2\zeta r)^2}{(1-r^2)^2+(2\zeta r)^2}}$$
\( \zeta \) is the damping ratio (a property of the mount material). When \( r \) is large (good isolation), the equation simplifies to \( \text{TR}\approx 1/(r^2 - 1) \), showing that lower natural frequencies yield much better isolation. The simulator uses this full equation to calculate TR, Isolation Efficiency ( \( (1-\text{TR}) \times 100\%\) ), and Insertion Loss in decibels ( \( 20 \log_{10}(1/\text{TR}) \) ).
Real-World Applications
HVAC Systems on Building Roofs: Large air handling units and chillers vibrate at motor and fan frequencies. Engineers use this calculator to specify rubber or spring isolators that lower the system's natural frequency to around 3-5 Hz, preventing the transmission of annoying hum and vibration into the building structure, which could disturb occupants.
Precision Manufacturing Equipment: CNC machines and semiconductor lithography tools are extremely sensitive to floor vibrations. Here, the goal is often "dual isolation": first isolating the tool from the floor, and then isolating the delicate components from vibrations within the machine itself. The damping ratio is carefully selected to ensure stability during rapid movements.
Marine Engine Mounts: Diesel engines on ships have strong vibrations at firing frequencies. The mounts must not only provide high isolation (very low TR) but also withstand large static loads and ship motions. The calculation for required stiffness per mount (derived from \( f_n \) and total mass) is critical for this design.
Consumer Electronics: Vibration isolation principles are used in washing machines. The drum rotation creates an excitation frequency, and the machine sits on spring-damper feet. The design aims for a low natural frequency to prevent the machine "walking" across the floor during the spin cycle, a direct application of the transmissibility concept.
Common Misconceptions and Points of Caution
Model assumptions: The mathematical model used here relies on simplifying assumptions such as linearity, homogeneity, and isotropy. Always verify that your real system satisfies these assumptions before applying results directly to design decisions.
Units and scale: Many calculation errors arise from unit conversion mistakes or order-of-magnitude errors. Pay close attention to the units shown next to each parameter input.
Validating results: Always sanity-check simulator output against physical intuition or hand calculations. If a result seems unexpected, review your input parameters or verify with an independent method.
Related Engineering Fields
Structural & Mechanical Engineering: Solid mechanics, elasticity theory, and materials science form the foundation for many of the governing equations used here.
Fluid & Thermal Engineering: Fluid dynamics and heat transfer share similar mathematical structures (conservation equations, boundary-value problems) and frequently appear in multi-physics problems alongside structural analysis.
Control & Systems Engineering: Dynamic system analysis, state-space methods, and signal processing connect to the time-dependent behaviors modeled in this simulator.