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32.3 Vibration Isolation

Back

The purpose of vibration isolation is to “isolate” the system of interest from vibration excitations by

introducing an isolator in between them. Examples of isolators are machine mounts and vehicle

suspension systems. Two general types of isolation can be identified:

1. Force isolation (related to force transmissibility)

2. Motion isolation (related to motion transmissibility)

In force isolation, vibration forces that would be ordinarily transmitted directly from a source to a

supporting structure (isolated system) are filtered out by an isolator through its flexibility (spring) and

dissipation (damping) so that part of the force is routed through an inertial path. Clearly, the concepts of

force transmissibility are applicable here. In motion isolation, vibration motions that are applied at a

moving platform of a mechanical system (isolated system) are absorbed by an isolator through its

flexibility and dissipation so that the motion that is transmitted to the system of interest is weakened.

Motion Sickness Region

30 min. Trips

1.0

0.8

0.6

0.4

2 hr Trips

8 hr Trips

0 .2

0.1

0 .2 0.4 0 .6 0 .8 1.0

Frequency (Hz)

RMS

Acceleration

(m/s2)

FIGURE 32.3 A severe-discomfort vibration specification for ground transit vehicles.

Vibration Design and Control 32-5

The concepts of motion transmissibility are applicable in this case. The design problem in both cases is to

select applicable parameters for the isolator so that the vibrations entering the system are below specified

values within a frequency band of interest (the operating frequency range).

Let us revisit the main concepts of force transmissibility and motion transmissibility. Figure 32.4(a)

gives a schematic model of force transmissibility through an isolator. Vibration force at the source is f ðtÞ:

In view of the isolator, the source system (with impedance Zm) is made to move at the same speed as the

isolator (with impedance Zs). This is a parallel connection of impedances. Hence, the force f ðtÞ is split so

that part of it is taken up by the inertial path (broken line) of Zm: Only the remainder ð fsÞ is transmitted

through Zs to the supporting structure, which is the isolated system. Force transmissibility is

Tf ¼

f ¼

Zm þ Zs ð32:2Þ

Figure 32.4(b) gives a schematic model of motion transmissibility through an isolator. Vibration

motion vðtÞ of the source is applied through an isolator (with impedance Zs and mobility Ms) to the

isolated system (with impedance Zm and mobility Mm). The resulting force is assumed to transmit

directly from the isolator to the isolated system and hence, these two units are connected in series.

Consequently, we have the motion transmissibility:

Tm ¼

v ¼

Mm þ Ms ¼

Zs þ Zm ð32:3Þ

It can be seen that, according to these two models, we have

Tf ¼ Tm ð32:4Þ

Box 32.1

VIBRATION ENGINEERING

Vibration mitigation approaches:

* Isolation (buffers system from excitation)

* Design modification (modifies the system)

* Control (senses vibration and applies a counteracting force: passive/active)

Vibration specification:

* Peak and RMS values

* Frequency-domain specs on a nomograph

pVibration levels

pFrequency content

pExposure duration

Note:

lVelocityl ¼ v £ lDisplacementl

lAccelerationl ¼ v £ lVelocityl

Limiting specifications:

Operation (design) specifications: specify upper bounds

Testing specifications: specify lower bounds

32-6 Vibration and Shock Handbook

As a result, the concepts of force transmissibility and motion transmissibility may be studied using just

one common transmissibility function T:

Simple examples of force isolation and motion isolation are shown in Figure 32.4(c) and (d). For both

cases, the transmissibility function is given by

T ¼

k þ bjv

ðk 2 mv2 þ bjvÞ ð32:5Þ

where v is the frequency of vibration excitation. Note that the model (Equation 32.5) is not restricted to

sinusoidal vibrations. Any general vibration excitation may be represented by a Fourier spectrum, which

is a function of frequency v: Then, the response vibration spectrum is obtained by multiplying the

excitation spectrum by the transmissibility function T: The associated design problem is to select the

isolator parameters k and b to meet the specifications of isolation.

Equation 32.5 may be expressed as

T ¼

n þ 2zvnvj

ðv2

n 2 v2 þ 2zvnvjÞ ð32:6Þ

where

vn ¼

ffiffiffiffiffi

k=m p ¼ undamped natural frequency of the system

z ¼

ffiffiffiffi

km p ¼ damping ratio of the system

(a) (b)

1 =

Inertial

System

(Isolated System)

Isolator

Received

Vibration

Motion

Applied

Vibration

Motion

v(t)

Moving Platform

v(t)

k b

f (t)

ffs s

Generated

Vibration Force

f(t)

Inertial

System

(Source)

Isolator

Transmitted

Vibration Force

Fixed Supporting Structure

(Isolated System)

FIGURE 32.4 (a) Force isolation; (b) motion isolation; (c) force isolation example; (d) motion isolation example.

Vibration Design and Control 32-7

Equation 32.6 may be written in the nondimensional form:

T ¼

1 þ 2zrj

1 2 r2 þ 2zrj ð32:7Þ

where the nondimensional excitation frequency is defined as

r ¼ v=vn

The transmissibility function has a phase angle as well as magnitude. In practical applications, the level of

attenuation of the vibration excitation (rather than the phase difference between the vibration excitation

and the response) is of primary importance. Accordingly, the transmissibility magnitude

lTl ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 4z2r2

ð1 2 r2Þ2 þ 4z2r2

ð32:8Þ

is of interest. It can be shown that lTl , 1 for r .

ffiffi

2 p ; which corresponds to the isolation region. Hence,

the isolator should be designed such that the operative frequencies v are greater than

ffiffi

2 p vn:

Furthermore, a threshold value for lTl would be specified, and the parameters k and b of the isolator

should be chosen so that lTl is less than the specified threshold in the operating frequency range (which

should be given). This procedure may be illustrated using an example.

Example 32.1

A machine tool and its supporting structure are

modeled as the simple mass – spring – damper

system shown in Figure 32.5.

1. Draw a mechanical-impedance circuit for

this system in terms of the impedances of

the three elements: mass (m), spring (k),

and viscous damper (b).

2. Determine the exact value of the frequency

ratio r in terms of the damping ratio z; at

which the force transmissibility magnitude

will peak. Show that for small z; this value is

r ¼ 1:

3. Plot lTf l vs. r for the interval r ¼ ½0; 5􀀉; with

one curve for each of the five z values 0.0,

0.3, 0.7, 1.0, and 2.0, on the same plane.

Discuss the behavior of these transmissibility

curves.

4. From part (3), determine for each of the five

z values and the excitation frequency range

with respect to vn; for which the transmissibility

magnitude is:

* Less than 1.05

* Less than 0.5

5. Suppose that the device in Figure 32.5 has a primary, undamped natural frequency of 6 Hz

and a damping ratio of 0.2. It is necessary that the system has a force transmissibility magnitude

of less than 0.5 for operating frequency values greater than 12 Hz. Does the existing system

meet this requirement? If not, explain how you should modify the system to meet the

requirement.

f(t)

Flexibility and

Dissipation

Machine Tool

Inertia

Supporting

Structure

FIGURE 32.5 A simplified model of a machine tool

and its supporting structure.

32-8 Vibration and Shock Handbook

Solution

1. Here, the elements m; b, and f are in parallel

with a common velocity v across them, as

shown in Figure 32.6. In the circuit, Zm ¼ mjv;

Zb ¼ b; and Zk ¼ k=jv:

Force transmissibility

Tf ¼

F ¼

Fs=V

F=V ¼

Zs þ Z0

Zb þ Zk

Zm þ Zb þ Zk ðiÞ

Substitute the element impedances. We obtain

Tf ¼

b þ

mjv þ b þ

bjv þ k

2 v2m þ bjv þ k ¼

jvb=m þ k=m

2 v2 þ jvb=m þ k=m ðiiÞ

The last expression is obtained by dividing the numerator and the denominator by m: Now, use the

fact that

m ¼ v2

n and

m ¼ 2zvn

and divide Equation ii throughout by v2

n: We obtain

Tf ¼

n þ 2zvnjv

n 2 v2 þ 2zvnjv ¼

1 þ 2zrj

1 2 r2 þ 2zrj ðiiiÞ

The transmissibility magnitude is

lTf l ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 4z2r2

ð1 2 r2Þ2 þ 4z2r2

ð32:9Þ

where r ¼ v=vn is the normalized frequency.

2. To determine the peak point of lTf l, differentiate the expression within the square-root sign in

Equation iv and equate to zero:

½ð1 2 r2Þ2 þ 2z2r2􀀉8z2r 2 ½1 þ 4z2r2􀀉½2ð1 2 r2Þð22rÞ þ 8z2r􀀉

½ð1 2 r2Þ2 þ 4z2r2􀀉2 ¼ 0

Hence,

4r{½ð1 2 r2Þ2 þ 2z2r2􀀉2z2 þ ½1 þ 4z2r2􀀉½ð1 2 r2Þ 2 2z2􀀉} ¼ 0

which simplifies to

rð2z2r4 þ r2 2 1Þ ¼ 0

The roots are

r ¼ 0 and r2 ¼

21 ^

ffiffiffiffiffiffiffiffiffiffi

1 þ 8z2

4z2

The root r ¼ 0 corresponds to the initial stationary point at zero frequency. That does not represent a

peak. Taking only the positive root for r 2 and then its positive square root, the peak point of the

transmissibility magnitude is given by

r ¼

h ffiffiffiffiffiffiffiffiffiffi

1 þ 8z2

2 1

i1=2

2z ðivÞ

f−fs fs

f(t)

Zm Zb Zk

FIGURE 32.6 The mechanical impedance circuit of the

force isolation problem.

Vibration Design and Control 32-9

For small z; Taylor series expansion gives

ffiffiffiffiffiffiffiffiffiffi

1 þ 8z2

< 1 þ

2 £ 8z2 ¼ 1 þ 4z2

With this approximation, Equation iv equates to 1. Hence, for small damping, the transmissibility

magnitude will have a peak at r ¼ 1; and from Equation 32.9, its value is

lTf l <

ffiffiffiffiffiffiffiffiffiffi

1 þ 4z2

1 þ

2 £ 4z2

lTf l < 1

2z þ z < 1

2z ðvÞ

3. The five curves of lTf l vs. r for z ¼ 0; 0.3, 0.7, 1.0, and 2.0 are shown in Figure 32.7. Note that these

curves use the exact expression (see Equation 32.9).

From the curves, we observe the following:

1. There is always a nonzero frequency value at which the transmissibility magnitude will peak. This

is the resonance.

2. For small z; the peak transmissibility magnitude is obtained at approximately r ¼ 1: As z

increases, this peak point shifts to the left (i.e., a lower value for peak frequency).

3. The peak magnitude decreases with increasing z:

4. All the transmissibility curves pass through the magnitude value 1.0 at the same frequency r ¼

ffiffi

2 p :

5. The isolation (i.e., lTf l , 1) is given by r .

ffiffi

2 p : In this region, lTf l increases with z:

6. The transmissibility magnitude decreases for large r:

4. From the curves in Figure 32.7, we obtain:

* For lTf l , 1:05; r .

ffiffi

2 p for all z:

* For lTf l , 0:5; r . 1:73; 1.964, 2.871, 3.77, 7.075 for z ¼ 0:0; 0.3, 0.7, 1.0, and 2.0, respectively.

0.5

1.5

2.5

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

z = 0.0

z = 0.3

z = 0.7

z = 1.0

z = 2.0

Normalized Frequency r = w/wn

Transmissibility

Magnitude

|Tf |

-----

....

-.-.-.

-..-..

FIGURE 32.7 Transmissibility curves for a simple oscillator model.

32-10 Vibration and Shock Handbook

5. We need

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 4z2r2

ð1 2 r2Þ2 þ 4z2r2

1 þ 4z2r2

ð1 2 r2Þ2 þ 4z2r2 ,

4 þ 16z2r2 , ð1 2 r2Þ2 þ 4z2r2

r4 2 2r2 2 12z2r2 2 3 . 0

For z ¼ 0:2 and r ¼ 12=6 ¼ 2; the left-hand-side expression computes to

24 2 2 £ 22 2 12 £ ð0:2Þ2 £ 22 2 3 ¼ 3:08 . 0

Hence, the requirement is met. In fact, since, for r ¼ 2

LHS ¼ 24 2 2 £ 22 2 12 £ 22z2 2 3 ¼ 5 2 48z2

it follows that the requirement would be met for

5 2 48z2 . 0

z ,

ffiffiffiffiffi

¼ 0:32

If the requirement was not met (e.g., if z ¼ 0:4), the option would be to reduce damping.

32.3.1 Design Considerations

The level of isolation is defined as 1 2 T: It was noted that in the isolation region ðr .

ffiffi

2 p Þ the

transmissibility decreases (hence, the level of isolation increases) as the damping ratio z decreases. Thus,

the best conditions of isolation are given by z ¼ 0: This is not feasible in practice, but we should maintain

z as small as possible. For small z in the isolation region, Equation 32.8 may be approximated by

T ¼

ðr2 2 1Þ ð32:10Þ

Note that T is real, in this case, of z ø 0; and also is positive since r .

ffiffi

2 p : However, in general, T may

denote the magnitude of the transmissibility function. Substitute

r2 ¼ v2=v2

n ¼ v2m=k

We get

k ¼

v2mT

ð1 þ TÞ ð32:11Þ

This equation may be used to determine the design stiffness of the isolator for a specified

level of isolation ð1 2 TÞ in the operating frequency range v . v0 for a system of known mass

(including the isolator mass). Often, the static deflection ds of spring is used in design procedures and is

Vibration Design and Control 32-11

given by

ds ¼

k ð32:12Þ

Substituting Equation 32.12 into Equation 32.11, we obtain

ds ¼ ð1 þ TÞ

v2T ð32:13Þ

Since the isolation region is v .

ffiffi

2 p vn it is desirable to make vn as small as possible in order to obtain

the widest frequency range of operation. This is achieved by making the isolator as soft as possible (k as

low as possible). However, there are limits to this in terms of structural strength, stability, and availability

of springs. Then, m may be increased by adding an inertia block as the base of the system, which is then

mounted on the isolator spring (with a damping layer) or an air-filled pneumatic mount. The inertia

block will also lower the centroid of the system, thereby providing added desirable effects of stability and

reducing rocking motions and noise transmission. For improved load distribution, instead of just one

spring of design stiffness k, a set of n springs, each with stiffness k/n and uniformly distributed under the

inertia block, should be used.

Another requirement for good vibration isolation is low damping. Usually, metal springs have very low

damping (typically z less than 0.01). On the other hand, higher damping is needed to reduce resonant

vibrations that will be encountered during start-up and shutdown conditions when the excitation

frequency will vary and pass through the resonances. In addition, vibration energy has to be effectively

dissipated, even under steady operating conditions. Isolation pads made of damping material such as

cork, natural rubber, and neoprene may be used for this purpose. They can provide damping ratios of the

order of 0.01.

The basic design steps for a vibration isolator in force isolation are as follows:

1. The required level of isolation (1 to T) and the lowest frequency of operation ðv0Þ are specified.

The mass of the vibration source (m) is known.

2. Use Equation 32.11 with v ¼ v0 to compute the required stiffness k of the isolator.

3. If the component k is not satisfactory, then increase m by introducing an inertia block and

recompute k:

4. Distribute k over several springs.

5. Introduce a mounting pad of known stiffness and damping. Modify k and b accordingly and

compute T using Equation 32.8. If the specified T is exceeded, then modify the isolator parameters

as appropriate and repeat the design cycle.

Box 32.2 gives some relations that are useful in a design for vibration isolation.

Example 32.2

Consider a motor and fan unit of a building ventilation system weighing 50 kg and operating in the speed

range of 600 to 3600 rpm. Since offices are located directly underneath the motor room, a 90% vibration

isolation is desired. A set of mounting springs, each having a stiffness of 100 N/cm, is available. Design an

isolation system to mount the motor-fan unit on the room floor.

Solution

For an isolation level of 90%, the required force transmissibility is T ¼ 0:1: The lowest frequency of

operation is v ¼ ð600=60Þ2p rad=sec: First, we try four mounting points. The overall spring stiffness is

k ¼ 4 £ 100 £ 102 N=m: Substitute in Equation 32.11.

4 £ 100 £ 100 ¼ ð10 £ 2pÞ2 £ m £ 0:1

1:1

32-12 Vibration and Shock Handbook

which gives m ¼ 111:5 kg: Since the mass of the unit is 50 kg, we should use an inertia block of mass

61.5 kg or more.

32.3.2 Vibration Isolation of Flexible Systems

The simple model shown in Figure 32.4(c) and (d) may not be adequate in the design of vibration

isolators for sufficiently flexible systems. A more appropriate model for this situation is shown in

Figure 32.8. Note that the vibration isolator has an inertia block of mass m in addition to damped flexible

mounts of stiffness k and damping constant b: The vibrating system itself has a stiffness K and damping

constant B in addition to its mass M:

In the absence of K; B; and the inertia block (m) as in Figure 32.4(c), the vibrating system becomes a

simple inertia (M). Then, ya and y are the same and the equation of motion is

My€ þ by_ þ ky ¼ f ðtÞ ð32:14Þ

Box 32.2

VIBRATION ISOLATION

Transmissibility (force/force or motion/motion):

lTl ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 þ 4z2r2

ð1 2 r2Þ2 þ 4z2r2

ø 1

ðr2 2 1Þ

for r . 1 and small z

Properties:

1. Tpeak ø

ffiffiffiffiffiffiffiffiffiffi

1 þ 4z2

ø 1

for small z

2. Tpeak occurs at rpeak ¼

􀀑 ffiffiffiffiffiffiffiffiffiffi

1 þ 8z2

2 1

􀀜1=2

ø 1 for small z

3. All lTl curves coincide at r ¼

ffiffi

2 p for all z

4. Isolation region: r .

ffiffi

2 p

5. In isolation region:

lTl decreases with r (i.e., better isolation at higher frequencies)

lTl increases with z (i.e., better isolation at lower damping)

Design formulas:

Level of isolation ¼ 1 2 T

Isolator stiffness:

k ¼

v2mT

ð1 þ TÞ

where

m ¼ system mass

v ¼ operating frequency

Static deflection:

ds ¼

k ¼ ð1 þ TÞ

v2T

Vibration Design and Control 32-13

with the force transmitted to the support structure,

fs; given by

fs ¼ by_ þ ky ð32:15Þ

The force transmissibility in this case is

Tinertial ¼

f ¼

bs þ k

Ms2 þ bs þ k

with s ¼ jv

ð32:16Þ

For the flexible system and isolator shown in

Figure 32.8, the equation of motion:

My€a þ Bð y_a 2 y_Þ þ Kð ya 2 yÞ ¼ f ðtÞ ð32:17Þ

my€ þ Bð y_ 2 y_aÞ þ Kð y 2 yaÞ þ by_ þ ky ¼ 0

ð32:18Þ

Hence, in the frequency domain, we have

ðMs2 þ Bs þ KÞya 2 ðBs þ KÞy ¼ f ð32:19Þ

½ms2 þ ðB þ bÞs þ K þ k􀀉y ¼ ðBs þ KÞya

with s ¼ jv ð32:20Þ

Substitute Equation 32.20 into Equation 32.19 for

eliminating ya: We obtain

ðMs2 þ Bs þ KÞ ½ms2 þ ðB þ bÞs þ K þ k􀀉

ðBs þ KÞ

2 ðBs þ KÞ

( )

y ¼ f

which simplifies

Ms2½ms2 þ ðB þ bÞs þ K þ k􀀉 þ ðBs þ KÞðms2 þ bs þ kÞ

ðBs þ KÞ

( )

y ¼ f ð32:21Þ

The force transmitted to the supporting structure is still given by Equation 32.15. Hence, the

transmissibility with the flexible system is

Tflexible ¼ ðBs þ KÞðbs þ kÞ

{Ms2½ms2 þ ðB þ bÞs þ K þ k􀀉 þ ðBs þ KÞðms2 þ bs þ kÞ}

with s ¼ jv ð32:22Þ

From Equation 32.16 and Equation 32.22, the transmissibility magnitude ratio is

Tflexible

Tinertial ¼ ðBs þ KÞðMs2 þ bs þ kÞ

Ms2½ms2 þ ðB þ bÞs þ K þ k􀀉 þ ðBs þ KÞðms2 þ bs ¼ kÞ

􀀈 􀀈 􀀈 􀀈 􀀈

with s ¼ jv ð32:23Þ

Tflexible

Tinertial ¼ ðMs2 þ bs þ kÞ

Ms2ðms2 þ bs þ kÞ

􀀋

ðBs þ KÞ þ Ms2 þ ms2 þ bs þ k

􀀈 􀀈 􀀈 􀀈 􀀈

s ¼ jv ð32:24Þ

In the nondimensional form, we have

Tflexible

Tinertial ¼

1 2 r2 þ 2jzbr

2 r2ð1 2 rmr2 þ 2jzbrÞ

􀀋

ðr2v

þ 2jzarv rÞ þ 1 2 ð1 þ rmÞr2 þ 2jzbr

􀀈 􀀈 􀀈 􀀈 􀀈

ð32:25Þ

K B

Vibration Excitation

f(t)

k b

Transmitted

Force fs

Inertia

Block

Flexible

Vibrating

System

Vibration

Isolator

FIGURE 32.8 A model for vibration isolation of a

flexible system.

32-14 Vibration and Shock Handbook

where

r ¼

v ffiffiffiffiffi

k=M p

rm ¼

rv ¼

ffiffiffiffiffiffi

K=M p

ffiffiffiffiffi

k=M p ¼

ffiffiffiffi

za ¼

ffiffiffiffiffi

KM p

zb ¼

ffiffiffiffiffi

kM p

Again, the design problem of vibration isolation is to select the parameters rm; rv ; za; and zb so that the

required level of vibration isolation is realized for an operating frequency range of r:

A plot of Equation 32.25 for the undamped case with rm ¼ 1:0 and rv ¼ 10:0 is given in

Figure 32.9. Generally, the transmissibility ratio will be zero at r ¼ 1 (the resonance of the inertial system)

and there will be two values of r (the resonances of the flexible system), for which the ratio will

become infinity in the undamped case. The latter two neighborhoods should be avoided under steady

operating conditions.

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