vibration theory

4/30/12 Vibration Theory - FARRAT

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Active & Passive Shock & Vibration Isolation

Vibration Sources & Control

Shock Sources & Control

Theory Behind Isolated Foundations

Detailed Vibration Isolation Theory

Active & Passive Shock & Vibration IsolationVibration 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

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Vibration Theory

Farrat Isolevel Ltd.

Balmoral Road, Altrincham,Cheshire WA15 8HJ.

England GB

Phone: +44 (0)161 924 1600

Fax: +44 (0) 161 924 1616

email: [email protected]

www.farrat.com

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supporting structure (floor, foundation or subsoil).

Typical support natural frequencies (fn)

Structures

Natural

Frequency fn

Hz

Isolator

Frequency fni

Hz

Isolator type

Suspended

concrete floor10-15 3-5 Helical, Air Springs

Ground Floor 12-34 6-8Helical, Air Springs,

Elastomeric

Soft Clay 12 6-8Helical, Air Springs,

Elastomeric

Medium Clay 15 6-8Elastomeric

Isolators

Stiff Clay 19 8-10Elastomeric

Isolators

Loose Fill 19 8-10Elastomeric

Isolators

Very dense mixed

grain sand24 10-12

Elastomeric

Isolators

Limestone 30 10-12Elastomeric

Isolators

Hard Sandstone 34 10-12Elastomeric

Isolators

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Shock Sources & ControlReasons 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 e as a fraction of the

disturbing vibration at the same frequency

e).

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 TheoryVibration 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 e of 12 Hz and above at reasonable

cost.

To isolate frequencies below 12 Hz low frequency isolators

should be used.

Examples

Natural

Frequency

n Hz

Isolate above

Disturbing

Frequency e

Hz

Air systems 1.5+ 3+

Helical Coil spring

systems Compare2.5+ 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

e - disturbing frequency can be determined by measurement.

The isolator natural frequency nd 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

SourceDisturbing Frequency

fe Hz

Primary out of balance 1 x rpm x 0.0167

Secondary out of

balance2 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

Factorfrequency 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

= 2

Vibration Isolators with damping

Force equation

MA +CV+ Kz = F(t)

Damping is expressed as a ratio C/Cc () 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) = tan-1 (1/Q( /0 -0/) )

K = Isolator Spring Constant N/m

M = Supported Mass kg

e = 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 0.05CR Neoprene / Chloroprene

Rubbers

Good damping 0.1 NBR Nitrile Rubbers

High damping 0.2- 0.3Helical 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)

= 2

A = V

V = A/ = A/2f

V = D

D = V/

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=&= ca 0.10

Low creep

High oil and chemical resistance


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== ca 0.05

Low creep

Moderate oil and chemical resistance

Fire retardant properties


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 t (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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Farrat Isolevel Ltd.

Balmoral Road, Altrincham, Cheshire WA15 8HJ. England GB

Phone: +44 (0)161 924 1600 Fax: +44 (0) 161 924 1616

email: [email protected] w w w .farrat.com

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