Vibration Theory

Active & Passive Shock & Vibration Isolation
Vibration Sources & Control
Shock Sources & Control
Theory Behind Isolated Foundations
Detailed Vibration Isolation Theory

Active & Passive Shock & Vibration Isolation

Vibration Control/Isolation

Vibration Control involves the correct use of a resilient mounting or material in order to provide a degree of isolation between a machine and its supporting structure. A condition should be achieved where the amount of vibration transmitted from, or to, the machine is at an acceptable level.

To achieve efficient vibration isolation it is necessary to use a resilient support with sufficient elasticity so that the natural frequency fn of the isolated machine is substantially lower that the disturbing frequency fe of vibration. The ratio fe/fn should be greater than 1.4 and ideally greater than 2 to 3 in order to achieve a significant level of vibration isolation

Damping provides energy dissipation in a vibrating system. It is essential to control the potential high levels of transient vibration and shock, particularly if the system is excited at, or near, to its resonant frequency.

Active Shock and Vibration Isolation

A foundation block for a dynamic machine should be isolated in order to reduce the effects of vibration and shock on nearby machines, people and the building structure. Controlling the source of a structural disturbance is known as active isolation.

Applications include: isolation of foundations for: power presses, pumps, drop hammers, forging machines, metal forming and cutting machines, compressors, gensets, engines and test rigs, printing machines and rolling roads.

Passive Shock and Vibration Isolation

When it is not possible to prevent or sufficiently lower the transmission of shock and vibration from the source a resiliently supported foundation block can be used for the passive isolation of sensitive equipment.

Applications include: isolated foundations for: machining centres, grinding machines, measuring and inspection equipment, laser cutters and microscopes.

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Vibration Sources & Control

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

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).

Typical support natural frequencies (fn)
Structures	Natural Frequency fn Hz	Isolator Frequency fni Hz	Isolator type
Suspended concrete floor	10-15	3-5	Helical, Air Springs
Ground Floor	12-34	6-8	Helical, Air Springs, Elastomeric
Soft Clay	12	6-8	Helical, Air Springs, Elastomeric
Medium Clay	15	6-8	Elastomeric Isolators
Stiff Clay	19	8-10	Elastomeric Isolators
Loose Fill	19	8-10	Elastomeric Isolators
Very dense mixed grain sand	24	10-12	Elastomeric Isolators
Limestone	30	10-12	Elastomeric Isolators
Hard Sandstone	34	10-12	Elastomeric Isolators

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Shock Sources & Control

Reasons for isolating the effects of shock

Shock is created by impact of one mass against another e.g. during operation of power presses, forging machines, drop hammers etc. The shock impulse caused by the impact travels through the machine structure as a deflection wave. If the machine is rigidly connected to its foundation this deflection wave enters the foundation and the surroundings. The shock will generally cause the affected masses to vibrate at their own natural frequencies.

Reduction in shock severity by use of suitable isolators is achieved by the isolators storing the energy of the shock through isolator deflection and subsequent release in a smoother form over a longer period with lower overall amplitude.

A shock pulse may contain frequency components from 0-. It is therefore not possible to avoid resonance with the isolator/mass. If however the duration of the shock pulse is less than one half period of the isolation system resonance may not be serious.

Figure 4.1
Shows the output force (into the supporting floor) v time levels from a machine produced shock wave. In case 1 the machine under consideration is connected directly to the supporting floor. Case 1 shows a high level of force over a relatively short duration. Case 2 typifies a machine installed on spring or elastomeric isolators in conjunction with a foundation block. It can be seen that the same amount of energy is transmitted in both instances. However in case 2 the energy is transmitted over a much larger time scale resulting in a substantially lower peak force. In reality the force transmitted will present itself as noise and structure borne vibration detectable by humans, it is therefore desirable in most instances to keep the peak value of transmitted force as low as possible by using on spring or elastomeric isolators between the machine and foundation or an elastically supported foundation block.

Figure 4.2
Shows a machine/structure that is rigidly connected to its foundation. The peak force into the structure is very high and of relatively short duration. Essentially all the force that occurs in the machine is transferred to the structure with the exception of that which is absorbed by the machine.

Figure 4.3
Illustrates the use of elastomeric or spring isolators between the machine and the supporting foundation. In this scenario with the correct isolator specification the peak force transmitted to the supporting foundation is significantly reduced resulting in reduced structure borne noise and transmitted vibration.

Figure 4.4
Illustrates the use of spring or elastomeric isolators supporting a foundation block. In this instance the peak force transmitted is reduced to virtually zero. The foundation block increases the system mass and reduces machine vibration and movement through mass damping.

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Theory Behind Isolated Foundations

Vibration Isolation

Vibration Isolation reduces the level of vibration transmitted to or from a machine, building or structure from another source.

The degree of isolation achieved depends on the ratio:

2: The level of isolator damping C/Cc
Referring to the diagram 3.07, the degree of isolation is given as Transmissibility (i.e. amount of vibration transmitted at a specific frequency ƒe as a fraction of the disturbing vibration at the same frequency ƒe).

Transmissibility:
> 1 = Increased transmitted vibration
= 1 = No vibration isolation
< 1 = Vibration isolation

Transmissibility T can be read from diagram 3.2 or calculated as follows:
If no damping present in isolators i.e. C/Cc = 0

fe - disturbing frequency can be determined by measurement. The isolator natural frequency fnd 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.

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

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Detailed Vibration Isolation Theory

Vibration Isolation

For effective vibration isolation the isolator natural frequency fn sholud 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.

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+

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.

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

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

Vibration Isolators Undamped

Force equation

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

Force equation

MA +CV+ Kz = F(t)

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
a) Provide efficient vibration isolation
b) 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.

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.

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
Relationships between vibration units
RMS = √ 2* (Peak)
ω = 2π ƒ
A = ωV
V = A/ω = A/2πf
V = ωD
D = V/ω

The above formula are valid for both vertical and horizontal vibrations

Vertical Axis Z
Longitudinal Axis Y
Transverse Axis X

Distribution of Load on unsymmetrical supported mass

Total Load Lt

L.A	*Lt((b-c)/b)(d/a)*
L.B	*Lt(c/b)(d/a)*
L.C	*Lt(c/b)((a-d)/a)*
L.D	*Lt((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.

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.

Stability limit

The maximum distance H of isolators below the C/g is given by an equilateral triangle connecting isolators to each other and the C/g.

Determination of undamped Vertical Natural Frequency from static vertical deflection

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

Typical Applications

Low frequency anti vibration mountings Structural bearings

Synthetic rubbers

Synthetic rubbers come in many formulations depending on application requirements.

Commonly used for anti vibration mountings requiring damping and or good oil and chemical resistance.

NBR Nitrile.

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

Typical Applications

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

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

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

Optimium 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)

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