Vibration, Isolation and Noise

Natural frequency, damping, resonance, transmissibility, rotor unbalance, critical speed and machinery limits.

On this page

  1. Why machines vibrate
  2. The single degree of freedom model
  3. Resonance and damping
  4. Worked example: mounting the pump unit
  5. Rotor unbalance and critical speed
  6. Measuring and judging vibration
  7. Noise
  8. FAQ

Why machines vibrate

Every machine has masses and elasticities, so it has natural frequencies at which it likes to oscillate. Rotating and reciprocating parts, impacts, gears, bearings and flow all supply forces at definite frequencies. When a force frequency matches a natural frequency, resonance follows: the vibration grows, shakes the foundation, loosens bolts, fatigues parts and makes noise. The design tasks are to know the natural frequencies, to keep forces away from them, to reduce the forces (balance), to isolate what is left and to damp what cannot be avoided.

The single degree of freedom model

ωn = √(k ÷ m)    fn = ωn ÷ 2π    δst = m g ÷ k = g ÷ ωn²    ζ = c ÷ (2 √(k m))

Rule of thumb: a 1 mm static deflection means 15.8 Hz; 10 mm means 5 Hz; 25 mm means 3.2 Hz. A soft mounting (low fn) therefore needs a large deflection under load.

Resonance and damping

The amplification at resonance of a lightly damped system is Q = 1 ÷ (2 ζ): 10 times for ζ = 0.05. The force or motion transmitted through a spring-damper mounting is the transmissibility T = √(1 + (2 ζ r)²) ÷ √((1 − r²)² + (2 ζ r)²), with r the ratio of the forcing to the natural frequency (see the figure). There are three regions: r < 1.41 (the mounting amplifies, worst at r = 1), r = 1.41 (T = 1, no benefit whatever the damping) and r > 1.41 (the mounting isolates, better the higher r and the lower the damping).

Vibration transmissibility011.412340123456frequency ratio r = forcing frequency ÷ natural frequencytransmissibilityζ = 0.05ζ = 0.20ζ = 0.50isolation starts at √2T = √(1 + (2ζr)²) ÷ √((1 − r²)² + (2ζr)²). Above r = √2 the mounting isolates; light damping isbetter there, and heavy damping only helps near resonance.
Transmissibility against the frequency ratio for three damping ratios.

Worked example: mounting the pump unit

Worked example: the 250 kg motor and pump unit of the press at 1 500 rpm

  • The excitation: the rotor (15 kg) is balanced to ISO 1940 grade G6.3 (see below): residual eccentricity e = G ÷ ω = 6.3 ÷ 157 = 40.1 µm and a rotating force F0 = m e ω² = 15 × 0.040107 × 24 674 = 14.8 N at 25 Hz. Bolted rigidly, the foundation feels all of it.
  • The mounting: choose fn = 5 Hz: k = m ωn² = 250 × 987 = 247 kN/m in total (about 62 kN/m at each of four mounts); the static deflection is δst = g ÷ ωn² = 9.9 mm.
  • Isolation: frequency ratio r = 25 ÷ 5 = 5; with ζ = 0.05: T = 0.047, so the transmitted force is 0.69 N: 95.3 percent isolation.
  • Starting up: the motor passes through the mount’s resonance at 300 rpm (5 Hz) on its way up; there the amplification is 10 and the unbalance force there is only F0 (5 ÷ 25)² = 0.59 N (it grows with ω²), so the amplitude is X = (F ÷ k) Q = 0.024 mm: brief and tiny. Use limit stops for large pumping reaction loads.
  • A bad mounting: mounts of 30 Hz natural frequency (stiff rubber, r = 0.83) would amplify the force by 3.2 and a mounting with fn = 20.8 Hz (r = 1.2) by 2.2: always check the operating speed range against the natural frequency of the mounts and the frame.
  • Frame natural frequency: the C-frame (Machine Design Case Study: the 80 kN Press Frame) has a stiffness of about 80.4 kN ÷ 0.588 mm = 137 kN/mm and an effective mass of about 254 kg: f = (1 ÷ 2π) √(k ÷ m) = 117 Hz, well above the 25 Hz excitation (r = 0.21).
IsolatorTypical natural frequencyNotes
Rubber / elastomer pads8 to 15 Hzcheap; damping 0.05 to 0.15; deflection to 10 mm; aging
Steel coil spring2 to 5 Hzlarge deflection, very low damping (add dampers), no aging
Air spring1 to 3 Hzlowest frequency, adjustable level; needs air supply
Cork, felt, wire rope8 to 25 Hz; 3 to 20 Hzlight loads, compact; wire rope handles shock
Active isolation< 1 Hzprecision instruments: sensors and actuators

Rotor unbalance and critical speed

A rotor with its mass center off the axis by an eccentricity e produces a rotating force m e ω². The balance quality grade G (ISO 21940-11, formerly ISO 1940) is the product e ω in mm/s: G6.3 for pump and fan rotors, G2.5 for turbines, motors and machine-tool spindles, G1 and G0.4 for precision grinders (see Flywheels, Balancing and Rotors for balancing). A rotating shaft also has a critical speed at which its speed equals its lateral natural frequency, and it whirls violently. For a 45 mm steel shaft 0.5 m long between pinned bearings with a 20 kg disc at mid-span: k = 48 E I ÷ L³ = 16.2 MN/m, ωc = √(k ÷ m) = 901 rad/s = 8 603 rpm. The operating speed of 1 500 rpm is 17 percent of it: a stiff, sub-critical shaft (keep below 70 percent, or above 130 percent for a super-critical design that runs through the critical quickly).

Measuring and judging vibration

  • Sensors: accelerometers (piezoelectric) for high frequencies, velocity sensors for general machinery, proximity probes for shaft motion.
  • Quantities: overall velocity (RMS, mm/s) for the severity, acceleration for bearing and gear faults and the spectrum (FFT) to identify the source by its frequency: 1× running speed (unbalance), 2× (misalignment), a multiple of the line frequency (electrical), blade pass or gear mesh frequency, and bearing fault frequencies.
  • Limits: ISO 20816 classes: for medium machines (15 to 75 kW) on rigid foundations, below about 2.3 mm/s RMS is good, up to 4.5 mm/s acceptable and above 7 mm/s unsatisfactory (typical zones).
  • Logging: trend the readings over months; a rise of 2.5 times from the baseline warrants investigation (Data Logging covers sampling and aliasing).
  • Damping from a decay record: two successive peaks of 10 and 6 units give the logarithmic decrement δ = ln(10 ÷ 6) = 0.51 and ζ = δ ÷ √(4π² + δ²) = 0.081.

Noise

Vibrating surfaces radiate sound. Sound pressure level is on a logarithmic scale, so levels add as 10 log₁₀ Σ 10L/10: three sources of 78, 75 and 72 dB(A) combine to 80.4 dB(A) (only 2.9 dB more than the loudest). Doubling the distance from a point source in free field lowers the level by 6 dB: from 80 dB at 1 m to 68 dB at 4 m. Exposure limits are 85 dB(A) for 8 hours with a 3 dB exchange rate (NIOSH, EU) or 90 dB(A) with 5 dB (OSHA): each +3 dB halves the allowed time, so at 88 dB(A) the limit is 4 hours. Reduce noise at the source (balance, soft mounts, quieter gears), along the path (enclosures, damping treatment) and last at the receiver (hearing protection: see Workshop Safety).

Related

FAQ

What is a natural frequency?

The frequency at which a structure oscillates freely after a disturbance. Every elastic structure has several (modes); the lowest is usually the most important.

Do rubber mounts always help?

Only above 1.41 times their natural frequency. A stiff rubber mount under a low-frequency machine can amplify the vibration. Compute the natural frequency and check the machine’s speed range.

What is damping for, if isolation prefers little damping?

Damping limits the amplitude at resonance (starting and stopping, shock) and in the amplification region. A compromise of ζ = 0.05 to 0.2 is usual.

How do I find the cause of vibration?

Take a spectrum and see which frequency is large: 1× running speed means unbalance, 2× misalignment or looseness, bearing frequencies mean bearing damage, line-frequency multiples mean electrical faults.

References

  • Rao, S. S., Mechanical Vibrations, Pearson; Inman, D. J., Engineering Vibration, Pearson; Harris, C. M., Piersol, A. G., Harris’ Shock and Vibration Handbook, McGraw-Hill.
  • ISO 21940-11 (balance quality), ISO 20816 (evaluation of machine vibration), ISO 10846 (isolator transfer stiffness), ISO 3744 (sound power).
  • Den Hartog, J. P., Mechanical Vibrations, Dover; Bies, D. A., Hansen, C. H., Engineering Noise Control, CRC Press.

Last reviewed: October 2026. Mount, damping and balance values are typical; check measured values on the real machine.