Selecting a Vibration Shock Isolator

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�� The solutions to most isolator problems begin withconsideration of the mounted system as a damped,single degree of freedom system. This allows simplecalculations of most of the parameters necessary todecide if a standard isolator will perform satisfactorilyor if a custom design is required. This approach isbased on the facts that:

1. Many isolation systems involve center-of-gravityinstallations of the equipment. That is, the center-of-gravity of the equipment coincides with the elasticcenter of the isolation system. The center-of-gravityinstallation is often recommended since it allowsperformance to be predicted more accurately and itallows the isolators to be loaded in an optimummanner. Figure 1 shows some typical center-of-gravitysystems.

2. Many equipment isolation systems are required tobe isoelastic. That is, the system translational springrates in all directions are the same.

3. Many pieces of equipment are relatively light inweight and support structures are relatively rigid incomparison to the stiffness of the isolators used tosupport and protect the equipment.

FIGURE 1TYPICAL CENTER-OF-GRAVITY INSTALLATIONS

For cases which do not fit the above conditions, orwhere more precise analysis is required, there arecomputer programs available to assist the analyst.Lord computer programs for dynamic analysis areused to determine the system response to variousdynamic disturbances. The loads, motions, andaccelerations at various points on the isolated equip-ment may be found and support structure stiffnessesmay be taken into account. Some of the more sophis-ticated programs may even accept and analyze non-linear systems. This discussion is reason to emphasizethe need for the information regarding the intendedapplication of the isolated equipment. The dynamicenvironment, the ambient environment and thephysical characteristics of the system are all importantto a proper analysis. The use of the checklist includedwith this catalog is recommended as an aid.

With the above background in mind, the aim of thistheory section will be to use the single degree-offreedom basis for the initial selection of standardisolators. This is the first step toward the design ofcustom isolators and the more complex analyses ofcritical applications.

��

Figure 2 shows the “classical” spring, mass, damperdepiction of a single degree-of-freedom dynamicsystem. Figure 3 and the related equations show thissystem as either damped or undamped. Figure 4 showsthe resulting vibration response transmissibility curvesfor the damped and undamped systems of Figure 3.

These figures and equations are well known and serveas a useful basis for beginning the analysis of anisolation problem. However, classical vibration theoryis based on one assumption that requires understand-ing in the application of the theory. That assumption isthat the properties of the elements of the systembehave in a linear, constant manner. Data to bepresented later will give an indication of the factorswhich must be considered when applying the analysisto the real world.

8

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M—Mass—Stores kinetic energyK—Spring—Stores potential energy, supports loadC—Damper—Dissipates energy, cannot support load

FIGURE 2ELEMENTS OF A VIBRATORY SYSTEM

FIGURE 3DAMPED AND UNDAMPED SINGLE DEGREE-OF-FREEDOM

BASE EXCITED VIBRATORY SYSTEMS

The equations of motion for the above model systemsare familiar to many. For review purposes, they arepresented here.

FOR THE UNDAMPED SYSTEM

The differential equation of motion is:

In which it may be seen that the forces due to thedynamic input (which varies as a function of time) arebalanced by the inertial force of the accelerating massand the spring force. From the solution of this equa-tion, comes the equation defining the natural fre-quency of an undamped spring-mass system:

Another equation which is derived from the solution ofthe basic equation of motion for the undampedvibratory system is that for transmissibility—theamount of vibration transmitted to the isolatedequipment through the mounting system depending onthe characteristics of the system and the vibrationenvironment.

Wherein, “r” is the ratio of the exciting vibrationfrequency to the system natural frquency. That is:

In a similar fashion, the damped system may beanalyzed. The equation of motion here must take intoaccount the damper which is added to the system. It is:

The equation for the natural frequency of this systemmay, for normal amounts of damping, be consideredthe same as for the undamped system. That is,

In reality, the natural frequency does vary slightly withthe amount of damping in the system. The dampingfactor is given the symbol “ζ” and is approximatelyone-half the loss factor, “η,” described in the definitionsection regarding damping in elastomers. The equa-tion for the natural frequency of a damped system,as related to that for an undamped system, is:

The damping ratio, , is defined as:

M Ý Ý X � KX � F(t)

fn �1

2�K /M

TABS �1

(1 � r2)

� � �/ 2

� � C / Cc

�

fn �1

2�K /M

r �ffn

fnd � fn 1 � �2

M Ý Ý X � C Ý X � KX � F(t)



Where, the “critical” damping level for a dampedvibratory system is defined as:

The equation for the absolute transmissibility of adamped system is written as:

The equations for the transmissibilities of the un-damped and damped systems are plotted in Figure 4.As may be seen, the addition of damping reduces theamount of transmitted vibration in the amplificationzone, around the natural frequency of the system"��0�1#. It must also be noted that the additionof damping reduces the amount of protection in theisolation region "(�� #.

FIGURE 4TYPICAL TRANSMISSIBILITY CURVES

In the real world of practical isolation systems, theelements are not linear and the actual system responsedoes not follow the above analysis rigorously. Typi-cally, elastomeric isolators are chosen for mostisolation schemes. Elastomers are sensitive to thevibration level, frequency and temperature to whichthey are exposed. The following discussion willpresent information regarding these sensitivities andprovide some guidance in the application of isolatorsfor typical installations.

��

Depending on the ambient conditions and loads,a number of elastomers may be chosen for the isolatorsin a given isolation system. As seen in the abovediscussion, the addition of damping allows more

control of the system in the region of resonance. Thecompromise which is made here though is that isola-tion is sacrificed. The higher the amount of damping,the greater the compromise. In addition, typical highlydamped elastomers exhibit poor returnability andgreater drift than elastomers which have medium orlow damping levels. The requirements of a givenapplication must be carefully weighed in order to selectthe appropriate elastomer.

Within the various families of Lord products, a numberof elastomers may be selected. Some brief descriptionsmay help to guide in their selection for a particularproblem.

$��)��— This elastomer is the baseline forcomparison of most others. It was the first elastomerand has some desirable properties, but also has somelimitations in many applications. Natural rubber hashigh strength, when compared to most syntheticelastomers. It has excellent fatigue properties and lowto medium damping which translates into efficientvibration isolation. Typically, natural rubber is not verysensitive to vibration amplitude (strain). On the limita-tion side, natural rubber is restricted to a fairly narrowtemperature range for its applications. Although itremains flexible at relatively low temperatures, it doesstiffen significantly at temperatures below 0°F. At thehigh temperature end, natural rubber is often restrictedto use below approximately 180°F.

$�� — This elastomer was originally developedas a synthetic replacement for natural rubber and hasnearly the same application range. Neoprene has moresensitivity to strain and temperature than comparablenatural rubber compounds.

�'�� — This is another synthetic elastomer whichhas been specially compounded by Lord for use inapplications requiring strength near that of naturalrubber, good low temperature flexibility and mediumdamping. The major use of SPE I elastomer has been invibration and shock mounts for the shipping containerindustry. This material has good retention of flexibilityto temperatures as low as -65°F. The high temperaturelimit for SPE I elastomer is typically +165°F.

2.)� — This elastomer is Lord’s original “BroadTemperature Range” elastomer. It is a silicone elas-tomer which was developed to have high damping anda wide span of operational temperatures. This materialhas an application range from -65°F to +300°F. Theloss factor of this material is in the range of 0.32. Thiselastomer has been widely used in isolators for MilitaryElectronics equipment for many years. It does not havethe high load carrying capability of natural rubber but

1 0

Cc � 2 KM

r � 2

TABS �1 � (2�r)2

(2�r)2 � [(1 � r2 )]2



is in the high range for materials with this broadtemperature range.

2.)� � — This material is similar in use to theBTR® elastomer except that it has a slightly morelimited temperature range and has less damping. BTRII may be used for most applications over a tempera-ture range from -40°F to + 300°F. The loss factor fortypical BTR II compounds is in the range of 0.18. Thiselastomer has better returnability, less drift, and betterstability with temperature, down to -40°F. The com-promise with BTR II elastomer is the lower damping.This means that the resonant transmissibility of asystem using BTR II elastomeric isolators will behigher than one using BTR isolators. At the same time,the high frequency isolation will be slightly better withthe BTR II. This material has found use in MilitaryElectronics isolators as well as in isolation systems foraircraft engines and shipboard equipment.

2.)�* �— This is a very highly damped elastomer.It is a silicone elastomer of the same family as theBTR elastomer described above but is speciallycompounded to have loss factors in the 0.60 to 0.70range. This would result in resonant transmissibilityreadings below 2.0 if used in a typical isolationsystem. This material is not used very often in applica-tions requiring vibration isolation. It is most often usedin products which are specifically designed fordamping, such as lead-lag dampers for helicopterrotors. If used for a vibration isolator, BTR VI willprovide excellent control of resonance but will notprovide the degree of high frequency isolation thatother elastomers will provide. The compromises hereare that this material is quite strain and temperaturesensitive, when compared to BTR and other typicalMiltronics elastomers, and that it tends to have higherdrift than the other materials.

��— This is an elastomer which has slightlyless damping than Lord’s BTR® silicone, but whichalso has less temperature and strain sensitivity. Thetypical loss factor for the MEM series of silicones is0.29, which translates into a typical resonant transmis-sibility of 3.6 at room temperature and moderate strainacross the elastomer. This material was developed byLord at a time when some electronic guidance systemsbegan to require improved performance stability ofisolation systems across a broad temperature range,down to -70°F, while maintaining a reasonable level ofdamping to control resonant response.

��/��— With miniaturization of electronic instru-mentation, equipment became slightly more ruggedand could withstand somewhat higher levels of

vibration, but still required more constant isolatorperformance over a wide temperature range. Theseindustry trends led to the development of Lord MEAsilicone. As may be seen in the material propertygraphs of Figures 5 through 8, this elastomer familyoffers significant improvement in strain and tempera-ture sensitivity over the BTR® and MEM series. Thecompromise with the MEA silicone material is that ithas less damping than the previous series. This resultsin typical loss factors in the range of 0.23 - ResonantTransmissibility of approximately 5.0. The MEAsilicone also shows less drift than the standard BTRseries elastomer.

�� — This is another specialty silicone elastomerwhich was part of the development of materials for lowtemperature service. It has excellent consistency over avery broad temperature range—even better than theMEA material described above. The compromise withthis elastomer is its low damping level. The typical lossfactor for MEE is approximately 0.11 which results inresonant transmissibility in the range of 9.0. The lowdamping does give this material the desirable feature ofproviding excellent high frequency isolation charac-teristics along with its outstanding temperature stability.

With the above background, some of the properties ofthese elastomers, as they apply to the application ofLord isolators, will be presented. As with metals,elastomers have measureable modulus properties. Thestiffness and damping characteristics of isolators aredirectly proportional to these moduli and vary as themoduli vary.

��,�.��%�� — Theengineering properties of elastomers vary with strain(the amount of deformation due to dynamic distur-bance), temperature and the frequency of the dynamicdisturbance. Of these three effects, frequency typicallyis the least and, for most isolator applications, cannormally be neglected. Strain and temperature effectsmust be considered.

�� — The general trend of dynamicmodulus with strain is that modulus decreases withincreasing strain. This same trend is true of the damp-ing modulus. The ratio of the damping modulus todynamic elastic modulus is approximately equal to theloss factor for the elastomer. The inverse of this ratiomay be equated to the expected resonant transmiss-ibility for the elastomer. This may be expressed as:

1 1

��G

��G � �

��G ��G � TR



�� ’ �� ” �� η ��

more exactly:

In general, resonant transmissibility varies onlyslightly with strain while the dynamic stiffness of anisolator may, depending on the elastomer, vary quitemarkedly with strain.

Figure 5 presents curves which depict the variation ofthe dynamic modulus of various elastomers whichmay be used in vibration isolators as related to thedynamic strain across the elastomer. These curves maybe used to approximate the change in dynamicstiffness of an isolator due to the dynamic strain acrossthe elastomer. This is based on the fact that thedynamic stiffness of an isolator is directly proportionalto the dynamic modulus of the elastomer used in it.This relationship may be written as:

�� ’ �� ’ ��

��

FIGURE 5 TYPICAL DYNAMIC ELASTIC MODULUS VALUES FOR

LORD VIBRATION ISOLATOR ELASTOMERS

This variation may be used to calculate the change in adynamic system’s natural frequency from the equation:

�� !�

�’� ��

� �� "��

As there is a change in dynamic modulus, there is avariation in damping due to the effects of strain inelastomeric materials. One indication of the amount ofdamping in a system is the resonant transmissibility ofthat system. Figure 6 shows the variation in resonanttransmissibility due to changes in vibration input forthe elastomers typically used in Lord military electron-ics isolators.

FIGURE 6TYPICAL RESONANT TRANSMISSIBILITY VALUES FOR


The data presented in Figures 5 and 6 lead to someconclusions about the application of vibration isola-tors. The following must be remembered whenanalyzing or testing an isolated system:

• It is important to specify the dynamic conditionsunder which the system will be tested.

• The performance of the isolated system will changeif the dynamic conditions (such as vibration input)change.

• The change in system performance due to changingdynamic environment may be estimated with someconfidence.

1 2

��K �A ��G

t

fn � 3.13��K T

W

TR �1 � �2

�2

Dyn

amic

Mo

du

lus

(psi

)

Single Amplitude (zero to peak) Dynamic Strain (%)

MEA

MEM

BTR

II

MEE

500 5 10 15 20 25

100

150

200

250

300

®

®

IV®

BTR

BTR

Res

on

ant T

ran

smis

sib

ility

Single Amplitude (zero to peak) Dynamic Strain (%)

MEA

MEM

BTR

II

MEE

00 5 10 15 20 25

2

4

6

8

10

12

14

16

®

BTR®

IVBTR®


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.�� — Temperature, like strain,will affect the performance of elastomers and thesystems in which elastomeric isolators are used.Figures 7 and 8 show the variations of dynamicmodulus and resonant transmissibility with tempera-ture and may be used to estimate system performancechanges as may Figures 5 and 6 in the case of strainvariation.

FIGURE 7TYPICAL TEMPERATURE CORRECTIONS FOR


FIGURE 8TRANSMISSIBILITY VS. TEMPERATURE FORLORD VIBRATION ISOLATOR ELASTOMERS

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It should be apparent from the preceding discussionthat the basic assumption of linearity in dynamicsystems must be modified when dealing with elasto-meric vibration isolators. These modifications doaffect the results of the analysis of an isolated systemand should be taken into account when writingspecifications for vibration isolators. It should also benoted that similar effects of variation with vibrationlevel have been detected with “metal mesh” isolators.Thus, care must be exercised in applying them. Theamount of variability of these isolators is somewhatdifferent than with elastomeric isolators and dependson too many factors to allow simple statements to bemade.

The following discussion will be based on the proper-ties of elastomeric isolators.

��3��*�� — Because of the strain andfrequency sensitivity of elastomers, elastomericvibration and shock isolators perform quite differentlyunder static, shock or vibration conditions.

The equation:

��#�� $�� !�

�� for elastomeric vibration/shockisolators. The static stiffness is typically less than thedynamic stiffness for these materials. To say thisanother way, the static deflection will be higher thanexpected if it were calculated, using the above for-mula, based on a vibration or shock test of the system.

Similarly, neither the static nor the vibration stiffnessof such devices is applicable to the condition of shockdisturbances of the system. It has been found empiri-cally that:

The difference in stiffness between vibration and staticconditions depends on the strain imposed by the vibra-tion on the elastomer. Figure 5 shows where the staticmodulus will lie in relation to the dynamic modulusfor some typical elastomers at various strain levels.

What this means to the packaging engineer ordynamicist is that one, single stiffness value cannot beapplied to all conditions and that the dynamic to static

dstatic �9.8fn

2

��K shock � 1.4K static

Res

on

ant T

ran

smis

sib

ility

Temperature (˚F)

0-100 -50 50 1500 100 200 250 300

2

4

6

8

10

14

12

20

18

16

MEA

MEM

BTR

II

MEE

®

BTR®

IVBTR®

Dyn

amic

Sti

ffen

ing

Rat

io -

Rel

ativ

e to

+72

˚F

Temperature (˚F)

MEA

MEM

BTR

II

MEE

0-100 -50 50 1500 100 200 250 300

0.8

1.0

1.2

1.4

1.6

1.8

2.0

®

BTR®

IVBTR®



stiffness relationship is dependent on the particularisolator being considered. What this means to theisolator designer is that each condition of use must beseparately analyzed with the correct isolator stiffnessfor each condition.

��3�� — As stated in the previousdiscussion, shock analyses for systems using elasto-meric isolators should be based on the guideline thatthe isolator stiffness will be approximately 1.4 timesthe static stiffness. In addition to this, it must beremembered that there �� be enough free deflectionin the system to allow the shock energy to be stored inthe isolators. If the system should bottom, the “g” leveltransmitted to the mounted equipment will be muchhigher than would be calculated. In short, the systemmust be allowed to oscillate freely once it has beenexposed to a shock disturbance to allow theory to beapplied appropriately. Figure 9 shows this situationschematically.

In considering the above, several items should benoted:

• Damping in the system will dissipate some of theinput energy and the peak transmitted shock will beslightly less than predicted based on a linear,undamped system.

• “τ” is the shock input pulse duration (seconds)

• “tn” is one-half of the natural period of the system(seconds)

• There �� enough free deflection allowed in thesystem to store the energy without bottoming(snubbing). If this is not considered, the transmittedshock may be significantly higher than calculatedand damage may occur in the mounted equipment.

*�� — The performance oftypical elastomeric isolators changes with changes indynamic input—the level of vibration to which thesystem is being subjected. This is definitely not whatmost textbooks on vibration would imply. The strainsensitivity of the elastomers causes the dynamiccharacteristics to change.

Figure 10 is representative of a model of a vibratorysystem proposed by Professor Snowdon of Penn StateUniversity in his book, �� This model recognizedthe changing properties of elastomers and the effectsof these changes on the typical vibration response ofan isolated system. These effects are depicted in thecomparison of a theoretically calculated transmissibil-ity response curve to one resulting from a test of anactual system using elastomeric isolators.

FIGURE 9



��!��"��

The majority of vibration and shock isolators are thoseutilizing elastomeric elements as the source of com-pliance and damping to control system responses.

FIGURE 10

��#%��&'��$

��(�′)*�↑��(�′�+)*η)

��#η$��

�↑��, ��'��′��,�� '��

�ζ��

Using this model, we may express the absolutetransmissibility of the system as:

��′��,�� '�� -�� !�.

The resulting transmissibility curve from such atreatment, compared to the classical, theoreticaltransmissibility curve, is shown in Figure 11.

FIGURE 11EFFECT OF MATERIAL SENSITIVITY ON

TRANSMISSIBILITY RESPONSE

Two important conclusions may be reached on thebasis of this comparison:

1. The “crossover” point of the transmissibility curve".��0�145# occurs at a frequency higher than6�times the natural frequency which is whatwould be expected based on classical vibrationtheory. This crossover frequency will vary depend-ing on the type of vibration input and the tempera-ture at which the test is being conducted.

2. The degree of isolation realized at high frequencies".��7�145# will be �� than calculated for anequivalent level of damping in a classical analysis.

This slower “roll-off” rate will depend,

also, on the type of elastomer, level and type ofinput and temperature.

In general, a constant amplitude sinusoidal vibrationinput will have less effect on the transmissibility curvethan a constant ‘g’ (acceleration) vibration input. Thereason is that, with increasing frequency, the strainacross the elastomer is ��! more rapidly withthe constant ‘g’ input than with a constant amplitudeinput. Remembering the fact that decreasing straincauses increasing stiffness in elastomeric isolators, thismeans that the crossover frequency will be higher andthe roll-off rate will be lower for a constant ‘g’ inputthan for a constant amplitude input. Figure 12 isrepresentative of these two types of vibration inputs asthey might appear in a test specification.

� ��G ��G � 2�

(db

octave)

TABS �1 � �2

[1 � r2 ��G

G��n ]2 � �22 � �2



FIGURE 12 COMPARISON-CONSTANT AMPLITUDE TO CONSTANT

“G” VIBRATION INPUT

No general statement of where the effects of randomvibration will lead in relationship to a sinusoidalconstant ‘g’ or constant amplitude vibration input canbe made. However, the effects will be similar to asinusoidal vibration since random vibrations typicallyproduce lower strains across isolators as frequencyincreases. There may be some exceptions to this state-ment. The section titled, “Determining NecessaryCharacteristics of Vibration/Shock Isolator” providesguidance as to how to apply the properties of elas-tomers to the various conditions which may bespecified for a typical installation requiring isolators.

��)�%��*��8��3� �� — As with any engineering activity,the selection or design of an isolator is only as good asthe information on which that selection or design isbased. Figure 13 is an example of one available Lordchecklist for isolator applications — Documentnumber SI-6106.

If the information on this checklist is provided, theselection of an appropriate isolator can be aidedgreatly, both in timeliness and suitability.

Section I provides the information about the equip-ment to be mounted (its size, weight and inertias) andthe available space for the isolation system to do itsjob. This latter item includes isolator size and availablesway space for equipment movement.

Section II tells the designer what the dynamic distur-bances are and how much of those disturbances theequipment can withstand. The difference is thefunction of the isolation system.

It is important to note here that the random vibrationmust be provided as a power spectral density versusfrequency tabulation or graph, not as an overall

�� level, in order to allow analysis of this condi-tion. Also, note that the U.S. Navy “high impact”shock test is required by specification MIL-S-901 forshipboard equipment.

Section III contains space for descriptions of anyspecial environmental exposures which the isolatorsmust withstand. Also, for critical applications, such asgyros, optics and radar isolators, the requirements forcontrol of angular motion of the isolated equipmentare requested. In such cases, particular effort should bemade to keep the elastic center of the isolation systemand the center of gravity of the equipment at the samepoint. The vibration isolators may have their dynamicproperties closely matched in order to avoid theintroduction of angular errors due to the isolationsystem itself.

All of the information listed on the checklist shown inFigure 13 is important to the selection of a propervibration isolator for a given application. As much ofthe information as possible should be supplied as earlyas possible in the design or development stage of yourequipment. Of course, any drawings or sketches of theequipment and the installation should also be madeavailable to the vibration/shock analyst who is select-ing or designing isolators.

��#� �� $��

The fragility of the equipment to be isolated istypically the determining factor in the selection ordesign of an isolator. The critical fragility level mayoccur under vibration conditions or shock conditions.Given one of these starting points, the designer canthen determine the dynamic properties required ofisolators for the application. Then, knowing theisolator required, the designer may estimate theremaining dynamic and static performance propertiesof the isolator and the mounted system.

The following sections will present a method foranalyzing the requirements for an isolation problemand for selecting an appropriate isolator.

��*�� '��— A system specification, equipment operationrequirements or a known equipment fragility spectrummay dictate what the system natural frequency must,or may, be. Figure 14 shows a fictitious fragility curvesuperimposed on a typical vibration input curve.Isolation system requirements may be derived fromthis information.

1 6



SAMPLE

1 7

�% �� A. Equipment weight ______________________________________________________________________

B. C.G. location relative to mounting points ____________________________________________________

C. Sway space ___________________________________________________________________________

D. Maximum mounting size ________________________________________________________________

E. Equipment and support structure resonance frequencies ________________________________________

F. Moment of inertia through C.G. for major axes (necessary for natural frequency and coupling calculations)

I xx _________________________ I yy _______________________ I zz __________________________

G. Fail-safe installation required? Yes No

��%�� A. Vibration requirement:

1. Sinusoidal inputs (specify sweep rate, duration and magnitude or applicable input

specification curve) ___________________________________________________________________

2. Random inputs (specify duration and magnitude (g2/Hz) applicable input specification curve)

_____________________________________________________________________________________

B. Resonant dwell (input & duration) _________________________________________________________

C. Shock requirement:

1. Pulse shape __________________ pulse period _________________ amplitude __________________

number of shocks per axis _______________________ maximum output _______________________

2. Navy hi impact required? (if yes, to what level?)____________________________________________

D. Sustained acceleration: magnitude _______________________ direction _________________________

Superimposed with vibration? Yes No

E. Vibration fragility envelope (maximum G vs. frequency preferred) or desired natural frequency and

maximum transmissibility ________________________________________________________________

F. Maximum dynamic coupling angle _________________________________________________________

matched mount required? Yes No

G . Desired returnability ____________________________________________________________________

Describe test procedure __________________________________________________________________

��%��&��A. Temperature: Operating _________________ Non-operating ________________________________

B. Salt spray per MIL ________________________ Humidity per MIL _____________________________

Sand and dust per MIL _____________________ Fungus resistance per MIL ______________________

Oil and/or gas ____________________________ Fuels _______________________________________

C. Special finishes on components ___________________________________________________________

FIGURE 13

��'��

"�� #�� $��!��%%��&��'�� '�� !��(��$�'�)��*�� '��

For Technical Assistance, Contact: Application Support,Aerospace Engineering, Lord Corporation,Mechanical Products Division, 2000 W. Grandview Blvd.,Erie, PA 16514; Phone: 814/868-0924, Ext. 6611 or 6497;FAX: 814/864-5468; E-mail: [email protected]



FIGURE 14EQUIPMENT FRAGILITY VS. VIBRATION INPUT

First, the allowable transmissibility at any frequencymay be calculated as the ratio of the allowable outputto the specified input.

The frequency at which this ratio is a maximum is onefrequency at which the system natural frequency maybe placed (assuming that it is greater than approx-imately 2.5, at some frequency). Another method ofplacing the system natural frequency is to select thatfrequency which will allow the isolation of the inputover the required frequency range. A good rule ofthumb is to select a frequency which is at least a factorof 2.0 below that frequency where the allowableresponse (output) crosses over — goes below — thespecified input curve.

Having determined an acceptable system naturalfrequency, the system stiffness (spring rate) may becalculated from the following relationship:

��′� �� -��

�� !�

� �� "��

An individual isolator spring rate may then be deter-mined by dividing this system spring rate by theallowable, or desired number of isolators to be used.The appropriate isolator may then be selected basedon the following factors:

• required dynamic spring rate

• specified vibration input at the desired natural frequency of the system

• static load supported per isolator

• allowable system transmissibility

• environmental conditions (temperature, fluid exposure, etc.)

Once a particular isolator has been selected, the prop-erties of the elastomer in the isolator may be used toestimate the performance of the isolator at other condi-tions of use, such as other vibration levels, shockinputs, steady state acceleration loading and tempera-ture extremes. The necessary elastomer property dataare found in Figures 5, 6, 7 and 8.

If the vibration input in the region of the requirednatural frequency is specified as a constant accelera-tion—constant ‘g’—it may be converted to a motioninput through the equation:

��

��

Of course, this equation may be used to convert con-stant acceleration levels to motions at any frequency. Itis necessary to know this vibratory motion input inorder to select or design an isolator. Note, that mostcatalog vibration isolators are rated for some maxi-mum vibration input level expressed in inches doubleamplitude. Also, the listed dynamic stiffnesses formany standard isolators are given for specific vibrationinputs. This information provides a starting point onFigure 5 to allow calculation of the system perfor-mance at vibration levels other than that listed for theisolator.

)��*��'��'�� — Random vibration is replacing sinusoidalvibration in specifications for much of today’s equip-ment. A good example is MIL-STD-810. Many of thevibration levels in the most recent version of thisspecification are given in the now familiar format of“power spectral density” plots. Such specifications arethe latest attempt to simulate the actual conditionsfacing sensitive equipment in various installations.

A combination of theory and experience is used in theanalysis of random vibration. As noted previously, therandom input must be specified in the units of “g2/Hz”

1 8

TABS �XoXi

orgogi

��K v �(fn )2(W)

9.8

Xi �gi

(0.051)(fn )2



in order to be analyzed and to allow proper isolatorselection. The system natural frequency may be deter-mined by a fragility versus input plot of random vibra-tion just as was done and demonstrated in Figure 14for sinusoidal vibration. Once the required naturalfrequency is known, the necessary isolator spring ratemay again be calculated from the equation:

The next steps in determining which isolator may beused are to calculate the allowable transmissibility andthe motion at which the isolated system responds at thesame natural frequency as when it is subjected to thespecified random vibration. The allowable trans-missibility, if not already specified, may be calculatedfrom the input vibration and the allowable vibration byusing the equation:

��/ ��

0� �� -�� !�0� �� -�� !�

A sinusoidal vibration input, acceleration or motion, atwhich the system will respond at approximately thesame natural frequency with the specified randomvibration may be calculated in the following manner.

��14 The analysis of random vibration is made onthe basis of probability theory. The one sigma "1σ#RMS acceleration response may be calculated from theequation:

�� σ �� 0� �� -�� 1� !�� " �� !�

��64 It has been found empirically that elastomericisolators typically respond at a 9σ vibration level. Thus,the acceleration vibration level at which the system willrespond at approximately the same natural frequency aswith the specified random level may be found to be:

��94 The above is response acceleration. To find theinput for this condition of response, we simply divideby the resonant transmissibility.

��:4 Finally, we apply the equation from a previoussection to calculate the motion input vibration equiva-ilent to this acceleration at the system natural fre-quency:

2�� 3�� .

��;4 The analysis can now follow the scheme ofprevious calculations to find the appropriate isolatorand then analyze the shock, static and temperatureperformance of the isolator.

��3�� '�� —If thefragility of the equipment in a shock environment is thecritical requirement of the application, the naturalfrequency of the system will depend on the requiredisolation of the shock input.

��14 Calculate the necessary shock transmissibility

�� -��

��64 Calculate the required �� natural frequency.This depends on the shape of the shock pulse.

1 9

��K v �(fn )2(W)

9.8

TR �SoSi

g i �g3�TR

Xi �gi

(0.051)(fn )2

TS �gogi

goRMS� (� /2)(Si )(fn)TR

g3� � 3 (� / 2)(Si)(fn)TR



The following approximate equations ��'��+��'��,�-�./

'�� .�� %��Half Sine Ts ≅ 4(fn)(to)Square Wave Ts ≅ 6(fn)(to)Triangular Ts ≅ 3.1(fn)(to)Ramp or Blast Ts ≅ 3.2(fn)(to)

��

Remember, that the system natural frequency under ashock condition will typically be different from thatunder a vibration condition for systems using elasto-meric vibration isolators.

��94 Calculate the required deflection to allow thislevel of shock protection by the equation:

�� !�

��:4 Calculate the required dynamic spring ratenecessary under the specified shock condition from theequation:

+�� <′� �� "��8��#�� 3��%�� "=-#+ ��(��"��#

��;4 Select the proper isolator from those availablein the product section, that is, one which has therequired dynamic stiffness "<′#, will support thespecified load and will allow the calculated deflection"��# without bottoming during the shock event.

��>4 Determine the dynamic stiffness "<′# of thechosen isolator, at the vibration levels specified for theapplication, by applying Figure 5 with the know-ledge that dynamic spring rate is directly proportionalto dynamic modulus "′# and by working from aknown dynamic stiffness of the isolator at a knowndynamic motion input.

��?4 Calculate system natural frequencies underspecified vibration inputs from the equation:

�� -�� !��′� ��

��-�� -�� "��

Note that the stiffness and supported weight must beconsidered on the same terms, i.e., if the stiffness is fora single mount, then the supported weight must be thatsupported on one mount. Once the system naturalfrequency is calculated, the system should be analyzedto determine what effect this resonance will have onthe operation and/or protection of the equipment.

��@4 Estimate the static stiffness of the isolatorsfrom the relationship:

�� ′� ��

Then, check the deflection of the system under the 1gload and under any steady-state (maneuver) loads fromthe equation:

�� 4��

Be sure that the chosen isolator has enough deflectioncapability to accommodate the calculated motionswithout bottoming. If the vibration isolation functionand steady state accelerations must be imposed on thesystem simultaneously, the total deflection capabilityof the isolator must be adequate to allow the deflec-tions from these two sources combined. Thus,

�"�� &� ��-��

� ��-��

�� -��

2 0

��K s �(fn )2W

9.8

fn � 3.13��K v

W

K ��K s

1.4

ds �gWK

dv �xi2

TR"��

ds �go

(0.102)(fn2 )

dtotal � dv �ds



. �� .��'�� — There area number of different types of isolators, based on con-figuration, which may be applied in supporting andprotecting various kinds of equipment. Depending onthe severity of the application and on the level of pro-tection required for the equipment, one or another ofthese mounting types may be applied.

Figures 15, 16 and 17 show some of the most common“generic” configurations of vibration isolators and thecharacteristic load versus deflection curves for thesimple shear mounting and the “buckling column”types of isolators. In general, the fully bonded orholder types of isolators are used for more criticalequipment installations because these have superiorperformance characteristics as compared to the centerbonded or unbonded configurations. The bucklingcolumn type of isolator is useful in applications wherehigh levels of shock must be reduced in order toprotect the mounted equipment. Many aerospaceequipment isolators are of the conical type becausethey are isoelastic.

In order of preference for repeatability of performancethe rank of the various isolator types is:

1. Fully Bonded

2. Holder Type

3. Center Bonded

4. Unbonded

In reviewing the standard lines of Lord isolators, theSTANDARD AVIONICS (AM), PEDESTAL (PS),PLATEFORM (100,106,150,156), HIGH DEFLEC-TION (HDM) and MINIATURE (MAA) mounts are inthe fully bonded category. The BTR (HT) mounts arethe only series in the holder type category. TheMINIATURE (MCB) series of isolators is the offeringin the center bonded type of mount. The MINIATUREGROMMETS (MGN and MGS) are in the unbondedmount category. In total, these standard offerings fromLord cover a wide range of stiffnesses and load ratingsto satisfy the requirements of many vibration andshock isolation applications.

In some instances, there may be a need to match thedynamic stiffness and damping characteristics of theisolators which are to be used on any particular pieceof equipment. Some typical applications of matchedsets of isolators are gyros, radars and optics equip-ment. For these applications, the fully bonded type ofisolator construction is highly recommended. Thedynamic performance of these mounts is much moreconsistent than other types. Dynamically matchedisolators are supplied in sets but are not standard sincematching requirements are rarely the same for anytwo applications.

FIGURE 15LOAD-DEFLECTION CURVES FOR

“SANDWICH” MOUNTS



Figure 16

Figure 17

��/��/�� — Figure 18 is a com-pleted checklist of information for a fictitious piece ofAvionics gear installed in an aircraft environment. Thefollowing section will demonstrate how the foregoing

theory and data may be applied to the selection of astandard Lord mount.

��

From the checklist, it is noted that the desired systemnatural frequency is 32 Hz with a maximum allowabletransmissibility of 4.0, or less.

��14 Determine the required dynamic spring rate:

Note that this figure is the total system spring ratesince the weight used in the calculation was the totalweight of the supported equipment. The checklistindicates that four (4) isolators will be used to supportthis unit. Thus, the required isolator is to have adynamic stiffness of:

� � ��+�� '�.�.01�� '�� 22�3�-��'� ��

��64 Make a �� +� isolator selection.

Thus far, it is known that:

1. The isolator must have a dynamic spring rate of 314lbs/in.

2. The supported static load per isolator is 3 pounds.

3. The material, or construction, of the isolator mustprovide enough damping to control resonanttransmissibility to 4.0 or less.

4. There is no special environmental resistancerequired.

Choosing a relatively small isolator available fromthose which meet the above requirements, theAM003-7, in BTR® elastomer, is selected from theproduct data section. The analysis now proceeds toconsideration of other specified conditions.

2 2

��K v �(fn )2(W)

9.8

fn � 32

W � 12 ��

=-

��8��8��K v �1254

4� 314

��8��K v �(32)2(12)

9.8� 1254



SAMPLE x

.036" D.A. 5 to 52 Hz; 5G, 52 to 500 Hz

.04 G2/Hz 10 to 300 Hz;

.036" D.A. 1/2 hr. per Axis

Half Sine 11ms 15G

3/Axis N/A

N/A (if yes, to what level?)

3G all directions

x

32 Hz with T less than 4

N.A.

N.A.

N.A.

�% �� A. Equipment weight _______________________

B. C.G. location relative to mounting points ____________________________________________________

________________________________________________________________________________________

C. Sway space ___________________________________________________________________________

D. Maximum mounting size ________________________________________________________________

E. Equipment and support structure resonance frequencies ________________________________________

F. Moment of inertia through C.G. for major axes (necessary for natural frequency and coupling

calculations)

(unknown) I xx ________________ I yy __________________ I zz_______________________

G. Fail-safe installation required? Yes No

��% �� A. Vibration requirement:

1. Sinusoidal inputs (specify sweep rate, duration and magnitude or applicable input

specification curve)

_____________________________________________________________________________________

2. Random inputs (specify duration and magnitude (g2/Hz) applicable input specification curve)

_____________________________________________________________________________________

B. Resonant dwell (input & duration) _________________________________________________________

C. Shock requirement:

1. Pulse shape _________________ pulse period _________________ amplitude __________________

number of shocks per axis _______________________ maximum output _______________________

2. Navy hi impact required? ______________________________________________________________

D. Sustained acceleration: magnitude _____________________________ direction ___________________

Superimposed with vibration? Yes No

E. Vibration fragility envelope (maximum G vs. frequency preferred) or desired natural frequency and

maximum transmissibility _______________________________________________________________

F. Maximum dynamic coupling angle ________________________________________________________

matched mount required? Yes No

G. Desired returnability ____________________________________________________________________

Describe test procedure__________________________________________________________________

��%��&��A. Temperature: Operating ________________________ Non-operating _________________________

B. Salt spray per MIL ________________________ Humidity per MIL _____________________________

Sand and dust per MIL _____________________ Fungus resistance per MIL ______________________

Oil and/or gas ____________________________ Fuels _______________________________________

C. Special finishes on components ___________________________________________________________

FIGURE 18

12 lbs.

Geometric Center

Four Mounts Desired

± 0.32"

1" High x 2” Long x 2" Wide

400 Hz

2 3

��'��

+30° to +120°F -40° to +160°F

810C 810C

810C 810C

N.A. N.A.

N.A.

"�� #�� $��!��%%��&��'�� '�� !��(��$�'�)��*�� '��

For Technical Assistance, Contact: ApplicationSupport, Aerospace Engineering, Lord Corporation,Mechanical Products Division, 2000 W. GrandviewBlvd., Erie, PA 16514; Phone: 814/868-0924,Ext. 6611 or 6497; FAX: 814/864-5468; E-mail:[email protected]



Per the previously presented material, the isolatorshould respond at a 3σ equivalent acceleration —calculated on the basis of the specified random vibra-tion at the desired natural frequency. This level willdetermine, in part, the isolator choice. The calculationis made as follows:

��(��A � 0 545:��68=-.�0 64B�"��>��2.)��

� ��#�� 0 96�=-

This is the acceleration response at the desired naturalfrequency of 32 Hz. The motion across the isolator dueto this response may be calculated as:

The ultimately selected isolator must have enough de-flection capability to allow this motion without bottom-ing (snubbing). The input acceleration is calculated as:

and the input motion as:

��!��!�(��

Step 1. Calculate a sinusoidal motion input at thedesired natural frequency with the specified randomvibration input and compare it to the specified sinevibration. Both the maximum motion and the inputmotion which would cause the isolator to respond atapproximately the same natural frequency as therandom vibration should be calculated. The maximumis calculated to check that the selected isolator willhave enough deflection capability and the resonantmotion is calculated to verify the stiffness of therequired isolator at the actual input at which it willrespond to the random vibration.

2 4

This vibration level is higher than the capability of thetentatively selected AM003-7. To remain with arelatively small isolator which will support 3 pounds,withstand the 0.047 inch double amplitude sinevibration and provide an approximate stiffness of 314lb/in per mounting point, a selection from either theAM002 or AM004 series appears to be best.

Since none of the single isolators provides enoughstiffness, a back to back (parallel) installation of apair of isolators at each mounting point is suggested.Since the AM002 is smaller than the AM004, and israted for 0.06 inch double amplitude maximum inputvibration, the selection of the AM002-8 isolator ismade. A pair of the AM002-8 isolators will providea stiffness of 346 lb/inch (two times 173 per thestiffness chart in the product section). This stiffnesswould provide a slightly higher natural frequencythan desired. However, there is a correction to bemade, based on the calculated vibration input.

The stiffnesses in the AM002 product chart are basedon an input vibration of 0.036 inch double amplitude.Figure 5 shows that the modulus of the BTR®

elastomer is sensitive to the vibration input. Themodulus is directly proportional to the stiffness of thevibration isolator. Thus, the information of Figure 5may be used to estimate the performance of anisolator at an “off spec” condition. A simple graphi-cal method may be used to estimate the performanceof an isolator at such a condition.

Knowing the geometry of the isolator, the strain atvarious conditions may be estimated. The modulusversus strain information of Figure 5 and the knowl-edge of the relationship of modulus to naturalfrequency (via the stiffness of the isolator) are used toconstruct the graph of the isolator characteristic. Theequation for calculation of the 3σ random equivalentinput at various frequencies has been shown previ-ously. The crossing point of the two lines on thegraph shown in Figure 19 is a reasonable estimate forthe response natural frequency of the selected isolatorunder the specified 0.04 g2/Hz random vibration.

The intersection of the plotted lines in Figure 19 is ata frequency of approximately 32 to 33 Hz, and at aninput vibration level of approximately 0.047 inch DA.This matches the desired system natural frequencyand confirms the selection of the AM002-8 for thisapplication. In all, eight (8) pieces of the AM002-8will be used to provide the 32 Hz system naturalfrequency, while supporting a total 12 lb unit, underthe specified random vibration of 0.04 g2/Hz. Theeight isolators will be installed in pairs at four

g o3� � 3 (� / 2)(Si)(fn )(TR )

g i 3� � 2.5g

g i 3� � go 3� / TR

xo 3� � 7.24 /(0.051)(32 2)

x o 3� � go 3� /(0.051)(fn2 )

g o3� � 7.24g

g o3� � 3 (� / 2)(0.04 )(32)(2.9)

g i 3� � 7.24 / 2.9

x i 3� � 0.048 ��

x o 3� � 0.139 ��

x i 3� � gi 3 � /(0.051 )( fn2 )

x i 3� � 2.5 /(0.051 )( 32 2 )



This makes the shock natural frequency:

Thus, the calculation for the shock transmissibilitybecomes:

FIGURE 20SINGLE DEGREE OF FREEDOM SYSTEMRESPONSE TO VARIOUS SHOCK PULSES

Since this value is above 1.0, and the equation is onlyvalid up to a value of 1.0, the information of Figure 20must be used. Use of this graph indicates that theshock transmissibility will be approximately 1.22.Thus, the shock response will be:

From this response, the next step is to calculate theexpected deflection when the selected isolator is sub-jected to the specified shock input. The equation ofinterest is:

The tentatively selected isolator, AM002-8, is capableof this much deflection without bottoming. Thus, theanalysis proceeds to another operating condition.

locations. With this portion of the analysis complete,the next operating condition - shock - is now consid-ered.

FIGURE 19

�� !�(��

The specified shock input is a 15g, 11 millisecond,half-sine pulse. From the previously presented theory,an approximation of the shock response may be foundthrough the use of the equation:

Note that the natural frequency to be used here is the�� '��#�� which may be estimated fromthe information given in Figure 5. The dynamicmodulus for the elastomer used here is approximately120 psi at a vibration level of 0.036 inch doubleamplitude and the static modulus is approximately 80psi. From this information, the static stiffness of theisolator may be estimated as follows:

As noted in previous discussion, the shock stiffness isapproximately 1.4 times the static stiffness. Thus,

2 5

Ts � 4fnto

K � (80120

)(fn

2W9.8

)

K � (80120

)( ��K )

go � Ts (gi)

fshock � 3.131170

12� 31

Ts � (4)(31)(.011) � 1.4

Go � (1.22)(15) � 18.3

ds �go

(0.102)(fn )2

K � (80120

)((32)2(12)

9.8) 0�@9>��8��

��K shock � (1.4)(836) �1170 ��

�

=-

ds �18.3

(0.102)(31)2 � 0.19 ��

AM002-8 Characteristic

Frequency (Hz)

Vib

rati

on

Inp

ut

(in

ch D

A)

0.04 g2/Hz Random 3sigma Equivalent

0.10

0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0.0020 25 30 35 40 45 50 5515



��!��A�The staticloading conditions in an isolator analysis are impor-tant from the standpoints of stress and deflection towhich the isolator will be exposed. Such conditionsare caused by the 1g load which the isolator mustsupport as well as by any maneuver and/or steady-state accelerations, which may be imposed. In thepresent example, the static �� stiffness wascalculated as being 836 lbs/in. The deflection of thesystem at any steady-state “g” loading may becalculated by using the equation:

In the example, the sustained acceleration wasspecified as being 3g. Thus, the system deflection willbe approximately:

The selected isolator, AM002-8, is able to accom-modate this deflection, even superimposed on thevibration conditions. Finally, none of the environmen-tal conditions shown on the checklist will be of anyconcern. Thus, this appears to be an appropriateisolator selection. Of course, typical testing of thisequipment, supported by the selected isolators, shouldbe conducted to prove the suitability of this system.

The isolators presented in the product portion of thiscatalog will prove appropriate for many equipmentinstallations. Should one of these products not besuitable, a custom design may be produced. Lord isparticularly well equipped to provide engineeringsupport for such opportunities. For contact informa-tion, see page 103. The following brief explanationwill provide a rough sizing method for an isolator.

�� -�A There will be occasionswhen custom designs will be required for vibrationand shock isolators. It should be remembered thatschedule and economy are in favor of the use of thestandard isolators shown in the product section here.These products should be used wherever possible.Where these will not suffice, Lord will assist byproviding the design of a special mount. The guide-lines presented here are to allow the packaging orequipment engineer to estimate the size of the isolatorso that the equipment installation can be made withthe thought in mind to allow space for the isolatorsand for the necessary deflection of the system as

supported on them. The final isolator size may beslightly larger or smaller depending on the specifica-tions being imposed.

Figure 21 shows a schematic of a conical isolator, suchas may be used for protection of avionic equipment.The two most important parameters in estimating thesize of such an isolator are the length of the elastomerwall, ��, and the available load area. For purposes ofsimplification, a conical angle of 45° is used here. Theratio of axial to radial stiffness depends on this angle.

FIGURE 21ESTIMATING AVIONICS ISOLATOR SIZE

The elastomer wall length may be estimated based onthe dynamic motion necessary for the requirements ofthe application. This length may be estimated throughthe following equation:

�� " ��&� �� -�� /��

��

From the required natural frequency, the necessarydynamic spring rate is known from:

�� ′ �� !�� "��

For a conical type isolator, the dynamic spring rate/geometry relationship is:

2 6

dstatic �(g)(W)Kstatic

dstatic �(3)(12)

836� 0.043 ��

tR �(x i)(TR )

0.30

��K �(fn)2(W)

9.8lb / in

��K �(A)( ��G )

tR



+��(��"/#��A

This area term should be determined such that thedynamic stress at resonance is kept below approx-imately 40 psi.

�

�� 5�4��-��

The combination of the elastomer wall length (tR) andload area (A), estimated from the above, and therequired attachment features will provide a goodestimate of the size of the isolator required to performthe necessary isolation functions. The proper dynamicmodulus is then selected for the isolator from anavailable range of approximately 90 to 250 psi at a0.036 inch D.A., vibration input.

)��(��A The requirement of a “resonantdwell” of isolated equipment is becoming less com-mon in today’s world. However, some projects stillhave such a requirement and it may be noted thatmany of the products described in the product sectionshave been exposed to resonant dwell conditions andhave performed very well. Isolators designed to theelastomer wall and load area guidelines given abovewill survive resonant dwell tests without significantdamage for systems with natural frequencies belowapproximately 65 Hz. Systems higher in naturalfrequency than this require special consideration andLord engineers should be consulted.

��)��A Many of the isolatorsshown in this catalog are inherently resistant to most ofthe environments (temperature, sand, dust, fungus,ozone, etc.) required by many specifications. Thesilicone elastomers are all in this category. One par-ticularly critical area is fluid resistance where specialoils, fuels or hydraulic fluids could possibly come intocontact with the elastomer. Lord engineering should becontacted for an appropriate elastomer selection.

.�� *��8��3� ��A Lord has excel-lent facilities for the testing of isolators. Electrody-namic shakers having up to eight thousand pounddynamic force capability are used to test many of theisolators designed or selected for customer use. Theseshakers are capable of sinusoidal and random vibrationtesting as well as sine-on-random and random-on-random conditions. These machines are also capable ofmany combinations of shock conditions and aresupplemented with free-fall drop test machines.Numerous isolator qualification tests have beenperformed within the test facilities at Lord.

'��

The preceding discussion presented general theorywhich is applicable to a broad class of vibration andshock problems. A special class of shock analysis isthat which involves drop tests, or specifications, suchas with protective shipping containers. This topic istreated in the following pages.

A � 1.4�(r22 �r1

2)

Pmax � (gi )(TR )W

� �PA� 40 ��


Selecting a Vibration Shock Isolator

Documents

mil humidity

mounting points

vibration

sample engineering

desired natural

inches double

dynamic spring

safe installation