shock response spectra

8/3/2019 Shock Response Spectra

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Nomenclature

SRS = Shock response spectrumDOF = Degree of freedom

SDOF = Single degree of freedom

MDOF = Multi-degree of freedom m = Mass of a SDOF system mi ith mass of a MDOF system k = Spring stiffness of a SDOF system x (t ) = Absolute displacement of mass of a SDOF system

as a function of time x i (t ) = Modal coordinate displacement for the i th mode

of vibration ui (t ) = Absolute displacement of i th mass of a MDOF

system as a function of time z(t ) = Displacement of a SDOF system mass relative to

its’ base as function of time z j (t ) = Displacement of the j th mass of a MDOF system

relative to its’ base as a function of time= Base acceleration of a SDOF or MDOF system as a

function of time w = Frequency of a one-DOF system SD = Spectral displacement SV = Spectral velocity

SA = Spectral acceleration[M ] = Mass matrix of a MDOF system[K ] = Stiffness matrix of a MDOF system

w i = Frequency of the i th mode of vibration of a MDOFsystem

= Eigen vector or mode shape for the i th mode of vibration of a MDOF system

P i = Participation factor for the i th mode of vibration of a MDOF system

= Modal mass for the i th mode of vibration of a MDOFsystem

= Absolute acceleration vector for a MDOF system= Relative acceleration vector for a MDOF system= Modal coordinate displacement vector

SAVIAC = Shock and Vibration Information Analysis Center

The intent o this paper is to provide a basic overview, or primer,o the shock response spectrum (SRS). The intended audience orthis paper is the design engineers who have need to work with theshock response spectrum, and would like to understand the under-lying detail. The shock response spectrum was frst conceived by

Dr. Maurice Biot and is described in his Ph.D. thesis published in1932. Hence, the SRS has been in existence or a long time. It has been used to characterize the requency response o shock environ-ments to estimate the maximum dynamic response o structures.The SRS is commonly used to characterize the requency contento an acceleration time-history record.

The organization o this paper is as ollows:Deinitions and examples o shock and the shock response•

spectrum.Typical events characterized by a shock response spectrum•

Examples o design spectra and how they are determined.•

How a shock response spectrum can be used in the analysis o •

linear multi-DOF (degree o reedom) systems.The naval shock design spectrum including the spectrum dip•

phenomenon.Beore addressing the details o the shock response spectrum, a

ew defnitions and examples may be helpul.Shock. Below are several defnitions o “shock” rom various

sources:At the frst Shock and Vibration Symposium in 1947, mechanical

shock was defned as “a sudden and violent change in the state o motion o the component parts or particles o a body or mediumresulting rom the sudden application o a relatively large externalorce, such as a blow or impact.”

A mechanical or physical shock is a sudden acceleration ordeceleration caused, or example, by impact, drop, earthquake, orexplosion. Shock is a transient physical excitation. (Wikipedia)

Mechanical shock may be defned as a sudden change in velocityand is a major design consideration or a wide variety o systemsand their components. (SAVIAC’s Mechanical Shock Test Tech-niques & Data Analysis course description).

Shock Response Spectrum. Similarly, several defnitions or theshock response spectrum are:

The shock response spectrum is a graphical representation o an arbitrary transient acceleration input, such as shock in termso how a single degree o reedom (SDOF) system (like a mass ona spring) responds to that input. Actually, it shows the peak ac-celeration response o an infnite number o SDOFs, each o whichhas dierent natural requencies. (Wikipedia)

Shock response spectrum analysis is the maximum responseo a series o single degree o reedom systems [having the] samedamping to a given transient signal.

Transient Shock Examples. The ollowing illustrations show

examples o transient shock. Figure 1 is the acceleration-time his-tory o the east-west component o the well known El Centro earth-quake. Figure 2 is a photograph o the damage that was done by thisearthquake. Ballistic shock to a ground combat vehicle and shockto shipboard equipment rom a near miss underwater explosion isdepicted in the photographs o Figures 3 and 4, respectively.

A shock response spectrum transorms the acceleration-timehistory rom the time domain to the maximum response o a SDOFsystem in the requency domain. Figure 5 is the shock responsespectrum o the El Centro acceleration-time history o Figure 1. Itrepresents how a series o linear single degree o reedom systemsat various requencies would respond to this shock transient.The SRS values are the maximum absolute accelerations that the

SDOF system tuned to each requency would experience over theentire transient. Notice that the spectral value at high requenciesconverges to approximately 210 cm/sec2, which corresponds to themaximum ground acceleration rom Figure 1 at approximately 22sec. The reason or this will be discussed later.

Maurice Biot. Dr. Maurice Biot (Figure 6) conceived the shockresponse spectrum as documented in his 1932 Ph.D. thesis. He de-fned the SRS as the maximum response motion rom a set o single

DOF oscillators covering the requency range. Biot showed howto pick a small number o modes which are adequate or design.For earthquake applications, he used the traditional assumptionthat the ground’s motion is not aected by the dynamic motiono the building. Later study demonstrated that this assumption isoverly conservative and leads to over design o equipment, whichis especially evident in the case o shipboard mounted equipmentsubjected to naval shock rom near-miss underwater explosions.

Figure 7 is a qualitative graphical representation on the con-struction o a shock response spectrum. Consider a transient shockacceleration-time history input applied to a base where a series o single DOF linear oscillators o dierent requencies are mounted.Also, assume that the base is sufciently massive such that themotions o the oscillators do not aect the motion o the base. Thelowest requency system (requency

1

) is on the let hand side o

the base and the highest requency system (requency 6) is on theright hand side. The transient acceleration response o each oscil-

Shock Response Spectrum – A Primer

Based on a paper presented at IMAC-XXVII, the 27th International ModalAnalysis Conerence, Society or Experimental Mechanics, Orlando, FL,February, 2009.

J. Edward Alexander, BAE Systems, US Combat Systems Minneapolis, Minneapolis, Minnesota

{ }i

f

M i

( ){ }u t

( ){ }z t

( ){ }x t

( )bu t


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lator is illustrated graphically above each oscillator.The maximum acceleration response o the oscillator 1 is G 1.

Note that the acceleration response or oscillator 1 has the lowestrequency and lower maximum acceleration amplitude than thenext ew oscillators immediately to the right. Oscillator 2 has ahigher requency and maximum amplitude G 2. Oscillator 3 has thehighest maximum response G 3 relative to the others. Progressingto the right, the requency continues to increase but the maximumacceleration response drops o. The maximum acceleration G i response o each oscillator is plotted as a unction o requency atthe top o the fgure. This is a graphical representation o the shockresponse spectrum or the system.

The SRS graph at the top o Figure 7 is used to characterize the

requency content o the transient shock input signal. Oscillator3, or example, has the greatest maximum acceleration response to

c m / s e c 2

200

100

0

–100

–200

–30010 20 30 40 50 60

Seconds

Figure 1. El Centro E-W ground acceleration.

Figure 2. Earthquake shock damage.

Figure 3. Ballistic shock.

Figure 4. Naval shock.

c m / s e c 2

1000

100

100.1 1.0 10.0

Frequency, Hz

(b)

Figure 5. Acceleration-time history o Figure 1 transormed to a shock response spectrum.

Figure 6. Maurice Biot.

Figure 7. How a shock response spectrum is developed.3



the transient input. The maximum response drops o or systemso higher and lower requencies, representing stier and sotersystems, respectively.

Figure 7 is a graphical representation o a SRS. Figures 8 and 9,along with the equations that ollow are a mathematical develop-ment o a shock response spectrum. Figure 8 depicts the applicationo an acceleration time-history to the base o a linear, non-dampedsingle degree o reedom system. Figure 9 is the shock responsespectrum o the acceleration-time history o Figure 8. The absolutecoordinate o the mass in Figure 8 is given by x (t ). The equationo motion o the SDOF system is determined by a balance o thespring and inertial orces acting on the mass:

A relative coordinate z(t ) (meaning relative to the base motion)can be defned as:

Dierentiating (2) twice and substituting into (1) yields the equa-tion o motion in relative coordinates:

From substitution o (2) into (1), it is ound that the absolute accel-eration o the mass is proportional to the relative displacement:

where w 2 is k /m. The closed orm solution to (3) is obtained romDuhamel’s Integral (5). Unless the base acceleration is a relativelysimple unction that can be integrated ater being multiplied by sinw (t –t ), (3) is typically solved by numerical integration:

Using the results rom (4), the absolute acceleration o the massis:

The shock response spectrum is defned as the maximum ( )x t or each requency:

Substituting (6) into (7), the SRS is given by (8). Since this exampledid not include damping, this is the non-damped spectrum. Whendamping is included, it can be shown that the damped SRS is

given by (9):

Maximax, Primary and Residual Shock Spectrum1

Figure 10 is an illustration o transient system amplitude as aunction o time. There are a number o dierent shock responsespectra that can be constructed rom a single transient response de-pending on which eature o the transient is selected. The primaryregion corresponds to the time that the shock excitation is appliedto the system, and the residual is the region o the response ater

the excitation has ended. Point 1 on the fgure, or example, cor-responds to three characteristics. Point 1 is the maximum absoluteresponse o the system over the entire duration (maximax), themaximum negative response o the system over the entire duration(maximum negative) and fnally the maximum negative responseduring the primary phase (primary negative). A spectrum plottermed the maximax spectrum, or example, would be a plot o themaximax points over all system requencies. Similarly, spectrumplots or the maximum negative and primary negative could bedeveloped based on the same approach.

The other points on Figure 10 are given similar terms. Point 2is maximum response during the primary phase o the response(primary positive). Points 3 and 4 occur during the residual portiono the response. Point 3 is the maximum positive response duringthe entire duration (maximum positive) and is also the maximumpositive response during the residual portion (residual positive).Point 4 is the maximum negative response during the residualportion (residual negative).

To summarize:The maximax spectrum consists o the maximum absolute re-•

sponse recorded as a unction o the system natural requency.The maximum positive spectrum contains only the maximum•

positive response.The maximum negative spectrum contains only the maximum•

negative response.The primary spectrum is made up o the maximum absolute•

responses during the excitation.The residual spectrum contains the peak response occurring•

ater the excitation has completely decayed.

Events Characterized by a Shock Response SpectrumEarthquakes, shipboard naval shock due to near-miss underwater

explosions, and pyroshock are typical events characterized by ashock response spectrum.

An earthquake results rom a sudden release o energy in theEarth’s crust that creates seismic waves. At the Earth’s surace,earthquakes maniest themselves by a shaking and sometimesdisplacement o the ground. The shaking in earthquakes can alsotrigger landslides and occasionally volcanic activity. Earthquakesare caused primarily by rupture o geological aults.

The elemental source o (naval) shock is the sudden applicationo external orces to a portion o the ship’s hull. The term “shock”is applied to the sudden, transient motion o an item o machineryas transmitted by the oundation or the mounting rom the ship’s

structure. Thus the term “shock” is employed in a relatively re-stricted sense, and does not include the destruction o the ship’s

c m / s e c 2

200

100

0

–100

–200

–300

0 10 20 30 40 50

Seconds

)(t x

2ω =

m

k

k

m

SDOF Oscillator

ub

)(t

��u t b ( ) Base Acceleration

Figure 8. Base acceleration SDOF oscillator.

c m / s e c 2

1000

100

100.1 1.0 10

Frequency, Hz

SA

ω

Figure 9. Shock response spectrum.

(1)( ) ( ) ( )( ) 0bmx t k x t u t + - =

(2)( ) ( ) ( )bz t x t u t ∫ -

(3)( ) ( ) ( )2

bz t z t u t w + = -

(4)( ) ( )2x t z t w = -

z t u t d = - -( )0

1 ( )sin ( )t b t w t t

w Ú (5)

(6)( )0

( )sin ( )t

bx t u t d w t w t t = -Ú

max( )AS x t ∫ (7)

(8)0 max

( )sin ( ) (No damping)t

A bS u t d w t w t t = -Ú

(9)( )

0 max

( ) sin ( ) (With damping)t t

A bS u e t d xw t w t w t t - -= -Ú



structure by means o direct exposure to an explosion, or the dam-age to equipment as the result o collision o a projectile or otherobject, or damage due to extreme distortion o the oundation.8

Pyroshock reers to short-duration, high-amplitude, high-requency, transient structural responses in aerospace vehicles.Pyroshock on rocket or missile systems is attributable to explosive bolts and nuts, pin pullers, separation o spent rocket boosterstages, linear cutting o the structure, and other actions that producea near-instantaneous release o strain energy. To support structuralanalysis o typical military and aerospace systems, a 20,000 Hzrequency response is always more than adequate.

Figure 11 shows the damage done to a building in the Marinadistrict o San Francisco rom the atermath o an earthquake. Soil

motion due to an earthquake is typically more horizontal thanvertical, resulting in the lateral shear-like ailures o the building

R e s p o n s e A m p l i t u d e

Excitation

Duration

Primary Residual

+

0

–

�

� Primary negativeMaximum negative

Maximax

Primary positive

Residual positive

Maximum positive

Residual negative

Figure 10. Response maxima.

Figure 11. Earthquake damage to apartment building in San Francisco’sMarina District.

Figure 12. Ship shock trials.

in Figure 11. Figures 12 and13 are photographs o the shortduration events that corre-spond to shipboard shock andpyroshock, respectively.

Earthquake Spectrum .Figure 14 is the accelerationtime-history record o the wellknown earthquake near ElCentro Caliornia in 1940. Thisparticular earthquake has beenoten cited and used or numer-ous studies. From the time-history, note that the maximum

absolute value o the ground acceleration is approximately 0.3 goccurring approximately 2 sec into the transient.

The graph o Figure 15 is the shock response spectra or thisearthquake. Note that in this case, the abscissa is not requency, but rather the system period. Hence, requency increases to thelet in this case. There are several things to note about this fgure.

The spectra are plotted on a our coordinate, tripartite grid. Usingthis plotting technique, the spectral displacement, velocity and

Figure 15. Response spectra rom El Centro quake.

Figure 16. Naval shock test fxture.

Figure 13. Pyroshock rom launcher vehicle stage separation.

Figure 14. Accelerogram rom May 18, 1940, El Centro earthquake.



acceleration are all shown on the same graph. The grid lines withthe positive 45 degree slope are measures o spectral acceleration,SA in g. The horizontal grid lines are measures o spectral velocity,SV in cm/sec. The grid lines with a negative slope o –45 degreesare a measure o spectra displacement, S D in cm. Several solidcurves are plotted on this graph where each corresponds to thespectral values or dierent amounts o damping. The percent o critical damping or each are listed by the corresponding graph.Notice also that the maximum ground motions (i.e., maximum base acceleration, velocity and displacement) are also plotted onthe graph with dotted lines. Since the spectral lines are above themaximum ground base motions in the middle region o the graph,these represent regions o amplifed response. For example, themaximum spectral acceleration is approximately 3.5 g, which is

an order o magnitude greater than the aorementioned maximumground acceleration. Observe that as the period decreases (or therequency increases) the spectral accelerations begin to convergeto the maximum ground acceleration o approximately 0.3 g. Thereason or this will be discussed later.

Naval Shock. Figures 16 through 18 show the results o a testdocumented by Regoord2 relative to shock testing associated withthe Netherlands naval shipbuilding specifcations. Figure 16 is aphotograph o the test fxture used. Figure 17 shows velocity time-history plots or three dierent test confgurations, depending onwhere the masses were mounted to the test fxture. Shock loadingwas applied to the underside o the fxture by a light weight shocktesting machine. The shock response spectra rom the three testconfgurations are plotted on Figure 18.

The intent o showing this test results is not to enlighten the

reader about the Netherlands’ shock specifcations, but rather toillustrate in general the shock levels o shipboard shock as com-

Figure 17. Velocity time histories.

Figure 18. Response spectra or multiple shocks.

pared with that o earthquakes. For the El Centro earthquake, theshock duration is on the order o 10s o seconds, where naval shockshown on Figure 17 is a much shorter duration on the order o 100so milliseconds. As expected, the shock levels or naval shock aremuch greater than or earthquakes. In the case o the earthquake,the maximum spectral acceleration is 3.5 g. The maximum ac-celeration o the naval shock response spectrum shown in Figure18, is approximately 300 m/sec2 which is about 30 g; one order o magnitude higher than earthquake shock in this example.

Pyrotechnic Shock. Figures 19 and 20 are typical durations andshock levels or pyrotechnic shock.3 From the time-history plot, theduration o the shock loading is on the order o 10s o milliseconds

and rom the shock response spectrum the spectral acceleration isapproximately 2,500 g. It is noted, however, that the SRS o Figure

Figure 19. Acceleration time history.

Figure 20. Shock response spectrum.



S h o c k R e s p o n s e

Frequency

Figure 21. Maximum SRS envelope o three dierent spectra.

20 does not correspond to the specifc time history o Figure 19.

Design SpectraEarly work done by Biot on seismology ound that the damage

potential o a single earthquake varied greatly rom one buildingto the next. He noted that requency peaks in shock spectra rom asingle earthquake are not constant within the same neighborhood.This lead to Biot recommending an envelope approach o all spec-tra or design purposes. Figure 21 is a schematic that illustrateshow a maximum envelope is constructed. For this example, threedierent spectra are superimposed and the maximum envelope isconstructed rom the maxima o the three. Similarly, a minimumenvelope can be constructed rom the minima o the three spectrashown in the fgure.

Relative to loadings on buildings, Biot ound that stresses cal-culated using the maximum envelope approach were much higher

than those observed rom an actual earthquake. Biot attributedthis to actors such as damping, plastic deormation, and possibleinteraction o nearby soil with the oundation o the building.

Housner Design Spectra. Housner4 developed the frst spectraused or seismic design o structures in the late 1950s. These wereobtained by averaging and smoothing the response spectra romeight ground motion records, two rom each o the ollowing ourearthquakes as shown in Figure 22.

El Centro (1934)•

El Centro (1940)•

Olympia (1949)•

Tekiachapi (1952)•

Newmark Design Spectra.5,6 In the 1970s Newmark developedan earthquake design spectrum approach based on amplifcationactors applied to maximum ground motions. Figures 23 and 24illustrate this technique. Figure 25 and Table 1 show the NewmarkDesign Spectrum or a specifc soil condition. The amplifcationactors in Table 1 are listed or dierent probabilities o occurrenceand also or various levels o damping o the structure. Figures 23and 24 show how a spectrum is developed rom the ground motionmaxima. The region o amplifed response is between the relativelyhigh and relatively low requency extremes o the spectrum. Notein Figure 23 at relatively high requencies, the shock spectrumlevel approaches the maximum ground acceleration. This is theaorementioned eature that was observed in the earlier El Centroearthquake example rom Figures 1 and 5.

The explanation or behavior can be aided by reerring back tothe single DOF oscillator in Figure 8. When the ground motion isapplied to the base o this oscillator, the acceleration o the mass in

general will be amplifed as shown in the middle region or Figure23. However, i the oscillator has a very high requency, as charac-terized by a relatively sti spring with respect to the mass, it willtend to respond like a rigid body relative to the base. As such, thespring does not compress or extend, and the ground motions aretransmitted directly to the mass. Hence, the mass accelerations areidentical to the base accelerations, including the maximums.

Similarly, at the low requency end o the spectrum, the oscillatorhas a relatively low requency. At this end, the spectral displace-ment converges to the maximum ground displacement. Once again,i the single DOF oscillator is considered, relative low requencyis characterized by a large mass relative to the spring stiness. Inthis case, the large mass will tend to resist motion due to its inertiawhile the base moves up and down resisted only by a relative sotspring. From equation (2) or the relative coordinate z(t ), i the

motion o the mass is zero, then the relative displacement z(t ) isidentical to the ground displacement and as such the maximums

Figure 22. Housner design velocity and acceleration spectra.

A c c e l e r a t i o n , g

V

D Region of AAmplified Response

v

d a

Ground Motion Maxima

Frequency (Log Scale)

D ≅ d

A ≅ a

Figure 23. General shape o smoothed spectrum response.

Table 1. Amplifcation actors or Newmark Spectrum

Spectral Probability Damping Ratio, %Quantity Level, % 0.5 2.0 5.0 10.0

SD 50 1.97 1.68 1.40 1.1584.1 2.99 2.51 2.04 1.62

SV 50 2.58 2.06 1.66 1.3484.1 3.81 2.98 2.32 1.81

SA 50 3.67 2.76 2.11 1.6584.1 5.12 3.65 2.67 2.01

SD = factor · d , SV = factor · v , SA = factor · a

d , v , a = maximum ground displacement, velocity, acceleration, respectivelyad /v 2 ≈ 6

are also identical.Figure 24 illustrates one example o the construction o a design

spectrum based on maximum ground motions. In this case thedesign spectrum is generated as a smoothed curve between theollowing three maximum conditions on acceleration, velocity



Maximum SRS velocity = three times maximum ground veloc-•

ity.Maximum SRS displacement = two times maximum ground•

displacement.Figure 26 is an example o the development o a design shock

response spectrum developed rom the maximum ground motionthat is quite similar to the Newmark Spectra o Figure 25. It waspublished in the 1970s by the Atomic Energy Commission or theseismic design o nuclear power plants.7

SRS Analysis of Linear MDOF Structures

Thus ar the discussion o the application o a shock responsespectrum has been only in the context o a single DOF system.Occasionally real equipment can be modeled adequately withonly one degree o reedom, but more typically real systems willrequire a multi degrees o reedom model to adequately captureoverall system dynamics accurately. When this is the case andthe system must be designed to meet a prescribed shock responsespectrum, a technique must be used to adapt the design SRS toa multi degrees o reedom structure. I the MDOF system can bemodeled as a linear system, then the maximum dynamic systemresponse can be approximated by the use o a technique calledmode superposition. This technique transorms the system dy-namics rom physical coordinates into modal coordinates in sucha way that each mode o the system behaves as i it were a singledegree o reedom.

The shock response spectrum canthen be applied to the system in themodal domain based on the moderequency and the degree o participa-tion that each mode has to the overallsystem response. The individualmodal responses are then transormed back into physical coordinates usingthe modal transormation. The modaltransormation between physical andmodal coordinates is accomplished by an Eigen value extraction o thematrix equations o motion. Theresulting Eigen values and Eigenvectors are the requencies and modeshapes or o each mode o vibrationo the system, respectively. The modeshapes, or Eigen vectors, are used totransorm the system rom physicalcoordinates to modal coordinatesand vice versa. The mathematics toaccomplish this transormation aresummarized below.

Consider the linear n-degree o reedom system shown in Figure 27.For this discussion, in the interest o keeping the mathematics relatively simple, damping is not includ-ed but the technique applies equally well to damped systems. Asnoted earlier, the system must be linear to use mode superposition.

I the system contains nonlinear elements, the resulting equationso motion are nonlinear and it is not possible to do an Eigen valueextraction o a system o nonlinear equations. This is a signifcantliability o mode superposition, since many real systems havesignifcant nonlinearities.

The system in Figure 27 has n lumped masses with displace-ments u j (t ). For this discussion, only vertical displacements areconsidered. However in general each mass could have six DOF;three translations and three rotations. As was done with the SDOFsystem rom the frst series o equations, relative coordinates aredefned as the displacement o each mass relative to the displace-ment o the base. An equation o motion similar to (1) can be writtenor each o the n masses. Similar to (2), or the j th mass a relativecoordinate is defned as:

When the relative coordinate (10) is substituted into the equa-

Figure 25. Newmark design spectra or alluvium soil (maximum ground acceleration = 1 g).

Figure 26. Atomic Energy Commission design spectra.

Figure 24. Design response constructed rom maximum ground motion.

and displacement, respectively:

Maximum SRS acceleration = our times maximum ground•

acceleration.

m n

u n (t )

m j

u j (t )

m 3

u 3 (t )

m 2

u 2 (t )

m 1

u 1(t )

Figure 27. n degree o reedomnondampened linear system.

(10)( ) ( ) ( ) j j bz t u t u t ∫ -



Recall rom (7) that the absolute value o the maximum accelera-tion o the SDOF oscillator is the spectral acceleration. Hence, orthe i th mode o the system, the ollowing holds:

Substituting (25) into (24) yields:

Since all o the modal maxima rom (24) will not occur at the sametime t , an estimate o max acceleration o the j th mass is typically

obtained using a root sum squared type approach given by (27):

This development has demonstrated that using mode superposi-tion in conjunction with the shock response spectrum, an esti-mate o the maximum dynamic response o a linear system can be approximated without solving the transient MDOF equationso motion.11

Naval Shock Design SpectraHistory. Welch8 was among the earliest to provide written guid-

ance or naval shock structural design. In the 1940s shock design based on the “Static-G” method which did not account or dier-ences in mounting/oundation requency, the location o equipment

in the ship or the ship type. The approach using envelope spectrato design shipboard equipment began about 1948.9 The maximumenvelope spectrum approach, as Biot discovered earlier, resultedin considerable over-design.

The so-called Shock Spectrum Dip was discovered in 1957 byserendipity. Strange results were observed rom reed gages dur-ing ull ship shock trials. The reeds corresponding to fxed basenatural requencies o equipment gave unexpectedly low results.The conclusion rom this observation was that the equipment inter-acted with ship structure, thereby aecting the response spectrumat the base o the equipment. Other observations rom ship shocktrials showed that heavy equipment responded with lower shocklevels than light equipment. This motivated experiments at theNaval Research Laboratory that explained the spectrum dip andthe eect o modal mass on requency response.

Shock Spectrum Dip. The Naval Research Laboratory perormedexperiments in the 1960s using the equipment in Figure 28 todemonstrate shock spectrum dip.10,11 Three double cantilever beams each o approximately the same weight, but with dier-ent stiness served as the test structures. Equipment mounted tonon-rigid oundations ed back orces into the oundations thataected the motion in such a way to result in a dip in the shockspectrum at the fxed base natural requency o the equipment asshown in Figure 29. It was observed that the spectrum values o major interest (at the equipment fxed base requency) tend to liein the region o valleys rather than in the vicinity o peaks o theshock spectrum. These experiments demonstrated that envelopespectra or design o equipment will lead to potential extreme overconservatism as shown between the dierences o the maximum

and minimum spectrum values o Figure 30. The data points romthe test are plotted on the graph, and it is evident that the actualvalues lie near the minimum spectrum envelope.

Modal Mass Considerations. The U. S. Navy’s method or shockqualifcation by analysis uses a procedure termed the DynamicDesign Analysis Method, commonly reerred to as DDAM. Thisprocedure defnes design spectrum values or each mode o thesystem as a unction o not only requency, but also the modal masso each mode. The modal mass or the i th mode o the system isgiven in Eq. 28. The modal mass is determined by the Eigen vectoror the mode o interest and the mass matrix o the system:

The modal mass represents the eective mass contribution due tothe i th mode o the structure. Figures 31 and 32 show the results

(28){ } [ ]{ }

{ } [ ]{ }

2

1T

i

i T

i i

M

M

M

j

j j

È ˘Î ˚=

tion o motion or the j th mass and all n equations o motion areassembled into one matrix equation, the system equations o mo-tion become:

[M ] and [K ] are the mass and stiness matrices o the system, re-spectively. Setting the right hand side equal to zero results in theree vibration equation:

I simple harmonic motion o the orm given by (13) is assumed:

and substituted into (12), the Eigen value problem (14) results:

The Eigen values are given by 2

i w and the Eigen vectors are given by { }

i j . The Eigen vector matrix is the assembled matrix o the

Eigen vectors:

The eigen vector matrix is used to transorm relative physicalcoordinates {z(t )} into modal coordinates {x (t )}:

Substituting [F ]{x (t )} into the equation o motion (11) and premul-

tiplying by [F ]T yields:

The matrices [ ] [ ][ ] [ ] [ ][ ]andM K F F F F are diagonal matricesdue to orthogonality properties o normal modes. Hence, the matrixequation (17) decouples into n-single DOF equations. The resultingequation or i th mode o vibration is:

The coefcient o x i (t ) in (18) is the squared circular requencyor the i th mode:

The coefcient o the base acceleration is defned as the modeparticipation actor (20). The mode participation actor determinesthe degree o “participation” the i th mode has to the overall systemresponse:

Substituting (19) and (20) into (18) yields:

Other than or the participation actor, (21) is identical to the singleDOF equation (3) used to obtain the spectral acceleration SA in

SDOF section. Dierentiating (16) twice, the transormation backto the physical coordinate accelerations become:

Ater some mathematical manipulations, the absolute physicalcoordinate accelerations can be derived in terms o the modalcoordinate accelerations given by:

I the modal coordinate accelerations are replaced with the absolutevalues o their maxima, (23) becomes (24). Notice that the equalsign has been replaced with a “less than or equal sign.” This is because the maximum modal coordinate acceleration will not oc-cur at the same time t :

(11)[ ]{ } [ ]{ } [ ]{ }( ) ( ) 1 ( )bM z t K z t M u t + = -

(12)[ ]{ } [ ]{ } { }( ) ( ) 0M z t K z t + =

(13){ } { } sin i i i z t j w =

(14)[ ]{ } [ ]{ }2

i i i M K w j j =

(15)[ ] { } { } { } { }1 2 3 n

F j j j j È ˘= ◊ ◊ ◊Î ˚

(17)[ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ] [ ]{ }1T T T

bM x K x M uF F F F F + = -

(18)i i bx t x t u t + = - ( ){ } [ ]{ }

{ } [ ]{ }( )

{ } [ ]{ }

{ } [ ]{ }( )

1T T

i i i

T T

i i i i

K M

M M

j j j

j j j j

(19){ } [ ]{ }{ } [ ]{ }

2

T

i i i T

i i

K

M

j j w

j j

=

(20){ } [ ]{ }

{ } [ ]{ }

1T

i i T

i i

M P

M

j

j j

=

(21)2( ) ( ) ( )i i i i bx t x t u t w R + = -

(22){ } { }1

( ) ( )n

i i

i

z t x t j

=

= Â

(23){ } { } { } { } { }1 1 2 2 3 31 2 3( ) ( ) ( ) ( ) ( )

n nnu t x t x t x t x t j R j R j R j R = + + + ◊ ◊

(24)1 1 2 2max max maxu x t x t x t j R j R j R £ + + ◊ ◊ ◊ + { } { } { } { }max 1 2

( ) ( ) ( )n nn

(25)i Ai x t S

max( ) =

(26){ } { }max

1

n

i Ai i

i

u P Sj

=

£ Â

(27)( )2

max

1

n

j ji i Ai i

u Sj R =

@ Â

(16){ } [ ]{ }( ) ( )z t x t F =


http://slidepdf.com/reader/full/shock-response-spectra 9/9S14 SO O /

rom O’Hara and Cuni study on mod-al mass.13 From Figure 31, note that asthe modal mass increases, the velocityand acceleration response decreases,which ollows intuition.

The U. S. Navy uses the DynamicDesign Analysis Method or shockqualifcation o shipboard and subma-rine equipment. The DDAM methoddetermines that the spectrum valuesto be used or each mode are based onthe modal mass (28) and requency.Figure 32 shows dierent design spec-tra based on the modal mass. Modes

with greater modal mass are subjectto lower maximum spectrum valuesand vice versa. DDAM also takes theaorementioned spectrum dip into

consideration or determining the maximum spectrum values to be used or each mode.

SummaryAn overview o the shock response spectrum has been presented

primarily with the intent o inorming the reader about its salienteatures, the history behind its development, and typical ways itis applied to the design o structures.

Velocity meter

Foundation beam

Velocity meter

Double cantilever beam

Double cantilever beam

Foundation beam

Shock

motion

Figure 28. Naval Research laboratory experiment.12

Figure 29. Shock spectrum or Beam A, Beam B, and Beam C.

Figure 30. Maximum and minimum envelopes o all shock spectra.

ω X , m / s

0.25

0.20

0.15

0.10

0.50

00 10 20 30 40 50 60 70 80

Modal Effective Mass, kg

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

��

��

ω 2 X / g , g

40

30

20

10

00 10 20 30 40 50 60 70 80 90 100

Modal Effective Mass, kg

Figure 31. Velocity and acceleration as unction o modal mass.

A b s o l u t e

A c c e l e r a t i o n

N , g

25

20

15

10

5

00 50 100 150 200 250 300 350 400

Frequency, Hz

M = 10 kg

20 kg

30 kg

40 kg

50 kg

2πV/g

1

Figure 32. Shock spectra as a unction o fxed base requencies and modal mass.

References1. Principles and Techniques o Shock Data Analysis, SVM-5, The Shock

and Vibration Inormation Center, United States Department o Deense,19692. R. Regoord, “Ship Shock Response o the Model Structure DSM,” S&V

Bulletin #54, 19843. Steinberg (1988), Vibration Analysis or Electronic Equipment , John

Wiley & Sons4. G. W. Housner, “Behavior o Structures During Earthquakes,” J. o Eng.

Mech. Div., ASCE, Vol. 85, No. EM4, 1959 pp. 109-1295. Newmark and Rosenblueth, Fundamentals o Earthquake Engineering ,

Prentice-Hall, 19716. A. K. Gupta, Response Spectrum Method in seismic Design and Analysis

o Structures, Blackwell Sci. Pubs., 19907. U. S. Atomic Energy Commission, Design Response Spectra or Seismic

Design o Nuclear Power Plants, Regulatory Guide, No. 1.60, 19738. Welch, W. P., 1946, “Mechanical Shock on Naval Vessels,” NAVSHIPS

250-660-26, Bureau o Ships9. Walsh, J. P. and Blake, R. E., 1948, “The Equivalent Static Accelerations

o Shock Motions,” NRL Report F-330210. O’Hara, G. J., and Sweet, A. L., 1960, “Methods or Design o Structures,”

Report o NRL Progress, pp 15-1811. O’Hara, 1961, “Eect Upon Shock Spectra o the Dynamic Reaction o

Structures,” Experimental Mechanics 1(5), pp. 145-15112. O’Hara, G. J., and Sweet, A. L., 1960, “Methods or Design o Structures,”

Report o NRL Progress, pp 15-1813. Cuni, P. F., and O’Hara, G. J., 1989, “A Procedure or Generating Shock

Design Values,” J. Sound and Vibration, 134, pp 155-164

The author can be reached at: [email protected].

shock response spectra

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