Detailed Vibration Isolation Theory

For effective vibration isolation the isolator natural frequency fn should be less than 50% the lowest disturbing frequency fe. Elastomeric rubber-metal isolators are used to prevent transmission of vibration from (active) or to (passive) supported equipment. Rubber based anti vibration mounts offer good isolation of disturbing frequencies ƒe of 12 Hz and above at reasonable cost. To isolate frequencies below 12 Hz low frequency isolators should be used.

Vibration Transmissibility

Vibration transmissibility T (i.e. % or fraction) of the vibration which the isolators transmit to the supported equipment (passive) or from the supported equipment (active) is calculated using the formula:

• ƒe – disturbing frequency can be determined by measurement. The isolator natural frequency ƒnd is given by:
• Ktd = Sum of Isolator Dynamic Spring Constants (K1+K2+K3…) N/m
• M = Supported system mass kg

For natural rubber and coil spring isolators static and dynamic spring constants are the same.

Examples Natural Frequency ƒn Hz Isolate above Disturbing Frequency ƒe Hz
Air systems 1.5+ 3+
Helical Coil spring systems Compare 2.5+ 5+
Rubber Metal AV Mounts 6.0+ 12+

Sources of vibration in rotating machines

Source Disturbing Frequency fe Hz
Primary out of balance 1 x rpm x 0.0167
Secondary out of balance 2 x rpm x 0.0167
Shaft misalignment 2 x rpm x 0.0167
Bent Shaft 1 & 2 x rpm x 0.0167
Gears(N=number of teeth) N x rpm x 0.0167
Drive Belts (N=belt rpm) N,2N,3N,4N x 0.0167
Aerodynamic or hydraulic forces (N=blades on rotor) N x rpm x 0.0167
Electrical (N=synchronous frequency) N x rpm x 0.0167

Force Equations:

##### M A + Kz = F(t)
• M = Mass Kg
• A = Acceleration m/s2
• K = Spring Constant N/m
• F = Applied force N
• z = Deflection m
• ω = 2π ƒ
Vibration Isolators with Damping

MA +CV+ Kz = F(t)

• A = acceleration m/s2
• V = velocity m/s
• D = displacement m
• C = damping coefficient Ns/m
• K = spring constant N/m
• P = Peak vibration force N
• f = frequency Hz

Significant problems occur when the disturbing frequency fe is near to or coincident with the natural frequency of the supporting structure (floor, foundation or subsoil). Damping Factor frequency ratio R fe/fn

C/Cc 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.05 20 66 80 87 91 93 94 95
0.10 19 64 79 85 89 91 93 94
0.15 17 62 76 83 87 90 91 93
0.20 16 59 74 81 85 87 89 91
0.30 12 52 67 75 80 83 85 87
Percentage Isolation Efficiency

Damping is expressed as a ratio C/Cc (ζ) which is a fractional measure of vibration energy absorbed by the isolator and not given back to the isolated equipment but dissipated into heat within the isolator. Rubber metal anti vibration mountings are generally made using NR Natural Rubber which has low damping. This is to:

1. Provide efficient vibration isolation
2. Avoid excessive heat build up when isolating active vibration sources. Where oil and other contamination is present the anti vibration mount must be designed so as to prevent the contaminants coming in contact with the rubber. Alternatively NBR (Nitrile) rubber isolators can be used, which have high oil and chemical resistance.

Phase lag (angle) When a damped isolator is subject to an input vibration the reactive response lags behind the input which can be expressed as a phase lag (angle). The greater the phase lag, the greater the damping and dissipated energy.Phase Lag between response and excitation is given by:

• Phase lag (angle) φ = tan-1 (1/Q(ω /ω0 -ω0/ω) )
• K = Isolator Spring Constant N/m
• M = Supported Mass kg
• ƒe = disturbing frequency Hz

Damping is required where movement of the supported equipment must be minimised especially at resonance. Damping is also required when shock is to be absorbed.

Relationships between vibration units:

• RMS = √ 2* (Peak)
• ω = 2π ƒ
• A = ωV
• V = A/ω = A/2πf
• V = ωD
• D = V/ωThe above formula are valid for both vertical and horizontal vibrationsVertical Axis Z
• Longitudinal Axis Y
• Transverse Axis Distribution of Load on unsymmetrical supported mass Total Load Lt
• L.ALt*((b-c)/b)*(d/a)
• L.BLt*(c/b)*(d/a)
• L.CLt*(c/b)*((a-d)/a)
• L.DLt*((b-c)/b)*( ((a-d)/a)

It is important to aim for as near as possible the same static deflection for each isolator by selecting suitable sizes and stiffnesses to match loads at each point.

• Static deflection at A mm = L.A/K.A etc.
• K.A= Vertical Spring Constant of isolator at A N/mm
• L.A= Static Load N at A

Vertical Natural Frequency:

• M= total equipment mass kg
• K.T = K.A + K.B + K.C + K.D N/mm

When specifying Isolators it is important to ensure that the vertical and horizontal isolator natural frequencies are less than 50% of the lowest significant disturbing frequencies (determined by rotating speeds) or by measurement.

Isolators with C/CcProduct:

 Isolators with C/Cc Product No damping 0 Helical Springs Low damping 0.01 NR Natural Rubbers Moderate damping 0.05 CR Neoprene / Chloroprene Rubbers Good damping 0.1 NBR Nitrile Rubbers High damping 0.2- 0.3 Helical spring with viscous damping

Against receipt of full information free advice and proposals will be given for the use of all Farrat products. A charge may be made where a detailed site survey is required involving vibration measurements.

Natural Frequencies and Coupled Modes: In most applications the vertical natural frequency of an isolation system is considered to be the most important. However the position of the isolators in relation to the equipment Centre of Gravity (C/g) should be taken into account.

Uncoupled Modes: Isolators are in the same horizontal plane as the C/g. Vertical, horizontal and rotational modes are uncoupled.

Coupled Modes: Isolators below the C/g The motion of the system is a combination of vertical, horizontal and rotational motion coupled with rocking about a lower or upper rocking centre.

Rubber Metal Vibration Isolators: Rubber is produced in natural, synthetic or thermoplastic forms.

NR Natural

• Very high resilience
• Low damping for maximum vibration isolation efficiency
• Very low creep
• Low chemical and oil resistance

NBR Nitrile

• Moderate resilience
• Damping ratio C/Cc=&ζ= ca 0.10
• Low creep
• High oil and chemical resistance

CR Chloroprene / Neoprene

• High resilience
• Damping ratio C/Cc=ζ= ca 0.05
• Low creep
• Moderate oil and chemical resistance
• Fire retardant properties

Typical Applications: Low frequency anti vibration mountings Structural bearings

Typical Applications: Anti vibration mountings in hydraulic and other chemical environments. Shock absorption pads Anti vibration pads for plant and machinery

Typical Applications: Acoustic damping pads Floating floors Structural bearings Anti vibration pads Sensitive equipment isolation

Optimum support design for Rubber Isolators: For maximum elastically supported stability, positioning rubber metal isolators at an angle to the vertical loads the rubber in a combination of shear and compression. Ideally the shear and compression deflection should be almost the same. To achieve this the angle should be 30. To calculate vertical deflection δt (mm):

• G = Shear Modulus (N/mm2)
• Ec = Compression Modulus (N/mm2)
• H = Rubber height (mm)
• A= Loaded rubber area (mm2)
• Ft = Load (N)