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 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|
|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|
Vibration Isolators Undamped
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
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
|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:
- Provide efficient vibration isolation
- 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 X Distribution of Load on unsymmetrical supported mass Total Load Lt
- 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:
|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.
- Very high resilience
- Low damping for maximum vibration isolation efficiency
- Very low creep
- Low chemical and oil resistance
- 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)