At low frequency almost all force reaches the foundation, near resonance it may be amplified, and beyond the isolation threshold less is transmitted. Damping limits the resonance peak but can worsen high-frequency isolation.
Select a target and calculate.
At low frequency almost all force reaches the foundation, near resonance it may be amplified, and beyond the isolation threshold less is transmitted. Damping limits the resonance peak but can worsen high-frequency isolation.
At r = 2 and D = 0.1, TF = √(1+0.4²)/√((1−4)²+0.4²) = 0.356. A 100 N sinusoidal excitation force gives about 35.6 N transmitted force amplitude in this ideal model.
One degree of freedom, linear spring, viscous damper, fixed foundation and harmonic force on the mass in steady state. Multiple modes, nonlinear rubber mounts, run-up, foundation motion and random loads are excluded. At r = 1 and D = 0 there is no finite stationary amplitude.
A machine mount should keep vibration forces away from its foundation. How much of a regularly varying force reaches the support through spring and damper?
The model has a mass on a linear spring and viscous damper above a fixed foundation, excited by a sinusoidal force. r is excitation frequency divided by undamped natural frequency and D is viscous damping relative to critical damping. Spring force follows displacement and damper force follows velocity; their amplitudes are phase-shifted and combine geometrically. The steady displacement amplitude yields force transmissibility TF = √(1+(2Dr)²)/√((1−r²)²+(2Dr)²). TF is foundation-force amplitude divided by excitation-force amplitude.
TF = √(1+(2Dr)²) / √((1−r²)²+(2Dr)²)
Frequency ratio: r = fExcitation/fNaturalForce transmissibility: TF = √(1+(2Dr)²)/√((1−r²)²+(2Dr)²)Foundation-force amplitude: FF = TF·FExcitation| Symbol / input | Meaning |
|---|---|
| Force transmissibility TF | Amplitude ratio of force at the fixed foundation to sinusoidal input force. Values below 1 mean isolation, above 1 amplification; displayed as a decimal, not a percentage. |
| Frequency ratio r | Excitation frequency divided by undamped natural frequency of the same mass-spring system. Find both in the same unit; r = 1 is near resonance. |
| Damping ratio D | Dimensionless viscous damping relative to critical damping, from vibration test or damper data. D = 0 is undamped; typical isolation examples use values below 1. |
r is dimensionless: use the same unit for both frequencies, either Hz or rad/s; r = 1 is near resonance. D is dimensionless linear viscous damping, obtained from damper data or a vibration test. D = 0 means undamped and 0.1 is an example. TF is displayed as a decimal, for example 0.36 for 36% force amplitude.
Determine excitation and undamped natural frequencies and enter their ratio r. Enter D for the same operating condition. Multiply TF by excitation-force amplitude to estimate harmonic force amplitude reaching the foundation.
At r = 2 and D = 0.1, numerator √(1+0.4²) = 1.077 and denominator √((1−4)²+0.4²) = 3.027. TF = 0.356. A sinusoidal excitation of 100 N amplitude then transmits about 35.6 N amplitude to the foundation.
TF < 1 means isolation; TF > 1 means force amplification. At r = 0, TF = 1. At r = √2, TF = 1 for any D; isolation begins above it. Damping limits the resonance peak but generally raises transmission far above natural frequency.
All three quantities are dimensionless. Use matching frequency units before taking their ratio. TF is an amplitude ratio, not a power or energy fraction.
Early estimates for resilient machine mounts, vibration isolators and machine foundations under nearly sinusoidal force excitation.
Linear one-mass oscillator with viscous damping, fixed foundation and steady sinusoidal force on the mass. Multiple modes, run-up, shock or random loads, moving foundation, nonlinear mounts and load-dependent damping are excluded. Exact undamped resonance has no finite steady-state solution.
Common mistake: TF relates force amplitudes, not displacements. Excitation acts on the mass, not on the foundation. At r = 1 and D = 0 no finite answer exists.
Early estimates for resilient machine mounts, vibration isolators and machine foundations under nearly sinusoidal force excitation.
r is dimensionless: use the same unit for both frequencies, either Hz or rad/s; r = 1 is near resonance. D is dimensionless linear viscous damping, obtained from damper data or a vibration test. D = 0 means undamped and 0.1 is an example. TF is displayed as a decimal, for example 0.36 for 36% force amplitude.
Linear one-mass oscillator with viscous damping, fixed foundation and steady sinusoidal force on the mass. Multiple modes, run-up, shock or random loads, moving foundation, nonlinear mounts and load-dependent damping are excluded. Exact undamped resonance has no finite steady-state solution.