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Dynamic response of the non-contact underwaterexplosions on naval equipment

Zhang Aman a,b,*, Zhou Weixing a, Wang Shiping a, Feng Linhan a

a College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, Chinab Department of Mechanical Engineering, University College London, Torrington Place, London WC1E 7JE, UK

a r t i c l e i n f o

Article his tory:

Received 8 November 2010

Received in revised form 1 April 2011

Accepted 18 May 2011

Keywords:

Underwater explosion

Shipboard equipment

Hull structure

Integration

Impact response

a b s t r a c t

Shock resistance capacity of the shipboard equipment especially for

largeones, hasbeena strongconcernof naviesall overthe world for

a long time. The shipboard equipment have previously generally

been studied separate from hull structure before. In this paper the

couplingelastic effectbetweenequipmentand hullstructureis taken

intoaccount. Withthe ABAQUSsoftware, theintegratedmodelof the

equipment coupledwith thehull structureis establishedto study the

dynamic response of the shipboard equipment to the shock wave

load as well as the bubble pulsation load. In order to verify the

numerical method, the simulated results are compared to the

experimental data, which are from a specic underwater explosion

on an actual ship. On this basis, by changing the charge location,

attack angle, equipment installation location and other parameters,

the characteristics of dynamic response under different conditions

can be obtained.In addition,the results of the integrated calculation

and the non-integrated one are compared and the characteristic

parameters whichaffect theequipmentshockresponse areanalyzed.

Some curves and conclusions are obtained for engineering applica-tions, which provides some insights into the shock resistance of

shipboard equipment.

2011 Elsevier Ltd. All rights reserved.

1. Introduction

As is known, it is inevitable for a warships to encounter impact environment during his service life.

The contact explosions cause direct damages to the ship structure as well as the internal equipment,

* Corresponding author. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China.

Tel.:86 0451 8251 8296.E-mail address: [email protected](Z. Aman).

Contents lists available at ScienceDirect

Marine Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a te /

m a r s tr u c

0951-8339/$ see front matter 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.marstruc.2011.05.005

Marine Structures 24 (2011) 396411

http://www.paper.edu.cn

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while the non-contact explosions[15]will usually not cause the breakdown of the ship structure but

will cause large-scale damages to the naval equipment[68]. Therefore, the anti-shock performanceof

shipboard equipment plays an important role in service life of a warship. Besides, both the full-scale

ship explosion tests and the model ship experiments have shown that the bubble load will cause

damages not only to the general ship structure, but also to the large-scale shipboard equipment[9,10].

The shockwave load resultingfrom underwater explosionsmainlycauses localdamages to ships, while

the bubble pulsation load with low-frequency characteristic could trigger the general step displace-

ment of warships[11]. According to the experimental study on theoating impacted platform, several

researchers have found that step displacements are the main cause of damages to the shipboard

equipment with 10 Hz installation frequency [12].

Resulting from the large volume and mass of the shipboard equipment, it is difcult and expensive

to perform the impact tests for a full-scale ship.Instead the numerical calculation becomes an effectivemethod to the study of the anti-shock performance of the shipboard equipment. In previous evalua-

tions, the equipment and the ship hull were studied separately according to the relevant standards,

such as Germanic military standard BV0430-85 [13], with the coupling effect between them rarely

considered. Some relevant studies have showed that this simplied method could not precisely match

the full-scale ship shock environment for shipboard equipment. In this paper, the warship super-

charging boiler is chosen as the study object, Based on the theory of master-slave system coupling

vibration [14], a nite element model of integrated shipboard equipment and hull is created by

considering the coupled effects between them.

Based on the ABAQUS software, the Geers-Hunter theory [15] to calculate the shock wave and

bubble load in the underwater explosions, and the acoustic medium is to simulate the shock wave and

bubble propagation in water. Once the load arrives at the ship hull, the interaction between the ship s

wet surface and surrounding oweld can be calculated by acoustic-structure coupling method. Then

the damages to the shipboard equipment resulting from bubble load during underwater explosion can

be analyzed based on the integrated ship-equipment model. Furthermore, in order to study the

mechanismof thedamage to theshipboard equipmentcaused by underwater explosionload, theeffect

of different parameters on the equipment response is investigated, including the explosion depth, the

attack angle, and the position of the detonation point along the ships length. Some curves are then

shown to represent the results obtained.

2. Numerical Calculation Method

Based on the ABAQUS software, the acoustic-structure coupling method is used to calculate the

propagation of the underwater explosion pressure in water and the interaction between the ship s wet

surface and the surrounding water. Different boundary conditions in theow eld, such as thefree surface

boundary condition and the non-reectiveboundaryconditionare allconsidered in theanalysis. Thebasic

theory of the acoustic-structure coupling method can be found in references [16,17].

Further assuming that the uid is compressible [16], adiabatic and its motion is small, the

momentum equation for theuid withvelocity-dependentmomentum lossescan be expressedas [16]:

vp

vx ax; Ki _vf Dfx; Kivf 0 (1)

Where, p is the dynamic pressure in the uid (the pressure in excess of any static pressure); x is the

uid particles spatial position; _vf and vf is the uid particle velocity and acceleration separately; Dfis

uid density;a is the force per unit volume per velocity; and Kiare dependent eld variables such as

temperature, humidity, or salinity, etc.Further assuming theuid to be inviscid, linear and compressible[16], the constitutive equation of

the uid can be expressed as:

p Rfx; Kivvfvx

0 (2)

Where,vfis the uid particle displacement,Rfis the bulk modulus of the uid.

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In order to obtain the partial differential equation used in direct integration transient analysis,

divide equation(1) by Dfand derive the result with respect tox. Assuming that the analysis is transient

and neglecting the derivative ofa/Df, combine the result with the time derivatives of equation(2), and

then get the differential equation for the uid in terms of the uid pressure can be obtained as [16]:

1

Rfp

a

DfRf_p

v

vx,

1

Df

vp

vx

! 0 (3)

Introducing an arbitraryvariation eld dp, and integrating equation (3) over thewholeuid eld, an

equivalent weak form for the equation of motion can be obtained[16]:

ZVf

dp"

1Rfp

a

DfRf_p

v

vx

1Df

v

pvx!#

dV 0 (4)

Through the coupled acoustic-structural medium analysis from ABAQUS[16], we obtain the uid

eld equilibrium equation[16]:

ZSfs

dpn,vmdS

ZVf

dp

"1

Rfp

a

DfRf_p

!

1

Df

vdp

vx ,

vp

vx

#dV

ZSfi

dp

1

d1_p

1

a1p

dS

ZSfr

dp

"a

Df

1

d1p

a

Df

1

b1

1

d1

!_p

1

b1p

#dS

ZSft

dpT0dS

ZSfrs

dp

a

Dfd1p

a

Dfb1

1

d1

!_p

1

b1pn,vm

!dS (5)

The structural behavior can be derived by using the virtual work principle [16]:

ZV

dvm,tdV ZV

de : sdVZV

acpdvm, _vmdVZV

pdvm,vmdVZV

pdvm,ndV (6)

Where, dvm is a variational displacement eld, tis thedragforceof thestructure, s is thenodalstress

in structure, p is the pressure applied on the structural wet surface, n is the normal of the structure

surface, pointing inside the uid,Dis the density of the structure, acis the mass proportional damping

factor, vm, _vm and vm are the displacement, velocity and acceleration of the structure at one point

separately, and de is the virtual displacement with respect to the virtual strain.

As above, the structure equilibrium equation in the uid eld can be obtained. Then we discretize

the structure and the acoustic medium with Finite Element Method (FEM), and dene the surfaces on

which the pressure is applied. Finally, the pressure load from the underwater explosion by Geers and

Hunters model (2002) [15] is exerted on thediscretized surfaces.As a consequence,the responseof the

Fig. 1. Thenite element model.

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structure together with the pressure propagation in the uid eld can be obtained by solving dis-

cretized equation of (5) and (6) with the explicit time integration method.

3. Verication of the numerical simulation method

In order to verify the numerical method, the numerical results are compared to the experimental

data of the warship underwater explosion.The general water displacement isD, withthe ship length L,

the width 0.14 L, and the draft 0.04 L, and the interval of the frame is 0.008 L. The nite element model

and the uid eld model are shown inFig. 1andFig. 2respectively.

The origin of the coordinatesystem is the intersection pointof central longitudinal section, midship

sectionand base plane;the X,Y, Z axis pointstowardsthe starboard, thebow andupwardsrespectively.

N kg TNTchargeis placed1.1 L away from thebroadside,0.8 L away from thefree surface and0.3 L away

from themidship section near thestern. Thetime-accelerationhistory curvesof typicalpositionon the

main deck are showed inFig. 3.

It can be seen fromFig. 3that the result of our model coincides well with the experimental data. At

the shock and pulsation stages, the time-acceleration history curve of the numerical result is similar to

that of theexperimental data.The bubblepulsation beginsat 0.57 s whichcan be observed in Fig. 3. The

peak strain at typical places of the ship is shown inTable 1, with the error dened as

relative error jE Rj

E 100%

Where Edenotes the experimental value and R is the numerical value.

Fig. 2. The uid eld model.

Fig. 3. (a) Measured value in experiment (b) Numerical result in simulation.



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It can be seen fromTable 1that the numerical results of strain coincide well with the experiment

data with maximum error as 46.3%, minimum 12.4% and average error approximately 26.3%. Further

more, the comparison between the numerical and experimental results of the shell plate deformation

under the explosion load is shown in Fig. 4.

From Fig. 4 we see that the shell plate generates large plastic deformation under the load of

underwater explosion, and serious damage is caused to the hull structure and internal equipment.

4. The integrated analysis model of shipboard equipment and ship hull

4.1. Integrated analysis model

Nowadays, most calculations on the anti-shock of the shipboard equipment are based on relevant

standards,such as Germanic military standard BV0430-85 [13], Chinese military standard GJB1060 [18]

etc., it is helpful to determine the input load and to check the anti-shock safety but not sufcient toconsider thecoupling effectof equipmentand theship structure. However, theexplosion of thecharge,

theformation of shock wave andbubble pulsation, and their transmissionto theship structureand the

equipment all occur continuously and are inter-coupling and interactive. Therefore, the integrated

effect of the equipment and the ship hull should be paid enough in the analysis.

Based on the theory of master-slave system coupling vibration, the supercharging boiler is installed

on the ship for the hull-equipment integrated calculation. During the calculation, the equipment

Table 1

Comparison of the strain peak values of typical position.

Measuring position Experimental

results (me)

Numerical

results (me)

Error Average

error

Longitudinal direction of the 63# center girder of main deck of the main

engine room

617 461 25.2% 26.3%

Cross direction of sideboard of 55# stringer toward the explosion of main

deck of the main engine room

294 430 46.3%

Cross direction of sideboard of 84# stringer away the explosion of the rear

soldier compartment of the main deck

267 300 12.4%

Vertical direction of the rib of 59# plane toward the explosion of the main

engine room of the platform

541 352 35.0%

Longitudinal direction of shell plate of 57# plane toward the explosion of

the main engine room of the platform

367 450 22.6%

Vertical direction of the rib of 55# plane away the explosion of the main

engine room of the platform

360 302 16.1%

Fig. 4. Comparison of shell plate deformation (a) Deformation of ship under underwater explosion load. (b) Deformation of

simulated result.



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installation frequency and damping are considered, and the spring-damping element is adopted tosimulate the shock absorber xed between the boilerand the shiphull. Thenal installation frequency

of thesuperchargingboiler is about 10Hz. Thenite element model of theequipment andthe ship hull

is shown inFig. 5, where the red stands for equipment and the local model of the supercharging boiler

located on the ships equipment base is shown inFig. 6.

The position of the charge is shown in Fig. 7. The length, width and draft of XXX ship are denoted as

L, B and T. A charge of N kg TNT is located at the position of 0.22 L below the naval equipment s

installation position. Generally, the underwater explosion load consists of two stages, the shock wave

stage and the bubble pulsation stage. During the shock wave stage, the head of the shock wave is the

step form. Itsamplitude valuepeakssharplybeforedecays exponentially ina short time after thephase

step. After the shock wave, the gas product of the explosion (the bubble) expands and contracts in

cycles, whilst the low-frequency pressure is radiated outwards. The underwater explosion shockwave

and bubble pulsation load in this paper are obtained by Geers and Hunters model.

4.2. Response of Warships Subjected to Underwater Explosions

Subjecting to the explosion load, the integrated model of the hull and the equipment is analyzed

numerically. The shock wave and bubbles loads act on the integrated model and generate dynamicresponse. The response of the supercharging boiler depends largely on that of the equipment base on

thehull,so thelatter is analyzed rst. Fig.8 displays the velocity-time response curveof the equipment

base andFig. 9shows the corresponding displacement-time curve.

It can be seen fromFig. 8that the high frequency response appears in the rst 0.1s which resulting

from the impact of the shock wave on the ship hull and the low-frequency part shows after 0.5 s

because of the secondary pulsation pressure, and there are multiple peaks as well. When the bubble

Fig. 5. Thenite element model of the equipment and the ship hull.

Fig. 6. The local model of the supercharging boiler located on the ship s equipment base.



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collapses at about 0.55 s, the response velocity peaks around 1.3m/s, whichis only 20% of the rstpeak

velocity caused by the shock wave. As shown in Fig. 9, the displacement response of the equipment

base is caused by the long-time pulsation pressure. The equipment base reaches the max-

imumdisplacementof about 0.2 m at 0.3 s during the period of the shock wave. However, the response

velocity reaches another peak in a shorter time (about 0.2 s) with an obvious step property resulting

from the secondary pressure wave of the bubble load.

Both the analytical and experimental results show that the shock wave is in high frequency and

the bubble load is in low frequency. Compared with the high frequency of the shock wave, that of the

bubble load pulse is much lower, which is close to the overall ships vertical natural vibration

frequency andoften tends to leadto the overallvibrationof the hull. Under theact of thebubblepulse

load, the main character of low-frequency response for warships is that the warship heaves with the

expansion and contraction of explosion bubbles, usually accompanied with the whipping movement

of the entire ship[19,20]. The motion of the entire ship at different moments caused by underwater

explosion load is comprehensively studied and the motion at specied moments can be seen in

Fig. 10.

Themotion of the whole warship at differentmoments (0.1 s, 0.3 s, 0.5 s, 0.6 sand 0.8 s) is shown in

Fig. 10. For an easier description, the warship is divided into 20 stations along the longitudinaldirection, and installation position for the supercharging boiler is between the 11th and the 12th

station (indicated by the blue broken line). As is clearly shown in Fig. 10, the ship hull makes a rst-

order vibration motion in the vertical direction during the heave oscillation.

4.3. Dynamics of Equipment Responses with the Ship hull

Through thesimulated analysis forthe whole process of thehullsubjectingto underwater explosive

load, the dynamic response of the warship and the supercharging boiler under the combined load of

the shock wave and the bubble is obtained. As two important factors of the hull and equipment

Fig. 7. Charge position and equipment s install position.

Fig. 8. Velocity-time response curve of the equipment base.



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response, the dynamic interaction and structural Mises-stress of the ship and the supercharging boiler

are shown inFig. 11.

From Fig. 11 we can see that the ship whipsunder the coupled actof the shockwave and the bubble

pulsation. Intensivelycouplingeffect exists between theboiler andthe ship structure. Theshockloadis

transmitted through the uid and reaches the boiler through the hull plate, the base and connecting

pieces. The strain of the boiler changes alternately under the shock load. At the initial time t 0 s, thebubbleexpands outward rapidly with high pressure insideand hasno inuence on theship,so there is

no response; att 0.027 s, the velocity of the equipment base is in a high frequency form with highpeak values, and the response frequency of the supercharging boiler is lower resulting from the

existence of the Shock absorber which isolates the transfer of high frequency response between the

equipment base and the ship structure. In addition, the shock absorber is compressed to its minimum

length for the rst time at this moment, storing large amount of energy which is to transfer to the

device in a low-frequency form. Att 0.28 s, the vertical displacement of the equipment base reachesits maximum value. The movement amplitude of the equipment relative to the equipment base

reduces. At t

0.47 s, thewholewarshipbegins to move in the opposite directionto that att

0.027 s;

at t 0.53 s, thespeed of theequipment base increasesrapidly, andthe high-speed peak value lasts fora long period with step displacement. The shock reducer device is compressed to the shortest length

for the second time, and then the movement amplitude of the device relative to the equipment base

reaches the maximum value. Moreover, the stress on the internal and external shell of the super-

charging boiler is relevantly large. At t 0.62 s, the vertical displacement of the equipment basereaches the maximum value again, and the whole vessel moves downwards with the underwater

explosion loads.

As the expansion and the collapse of the bubble go on, the ow eld is driven to move, and the

whole warship displacement and shape vary with time. On the whole, the warship takes a whipping

movement predominated by therst-ordervertical mode of vibrationunder thebubble load; while the

equipment takes deep vibration after the impact of shock wave, which decays gradually due to

Fig. 9. Corresponding displacement-time curve.

Fig. 10. The motion of the entire ship at different moments.



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damping. The base of the supercharging boiler manifests phase-step displacements after the act of

bubble collapse, and the response vibration of the equipment increases again and is more violent thanthat of shock wave impact. To further explain this phenomenon, the response curve of vertical velocity

for typical parts of the equipment is shown inFig.12corresponded with the velocity, and the curve of

the displacement of the equipment relative to the base of ship is shown in Fig.13.

It can be seen fromFig. 12that before t 0.5 s the velocity of the supercharging boiler increasesrapidly resulting from the impact of shock wave and then decays gradually with the same trend as the

hull, andthe oscillation of thesupercharging boileris based on thenatural installationfrequency of the

Fig. 11. The dynamic response of the warship and the supercharging boiler under the combination load of the shock wave and the

bubble.



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equipment in this process. After the bubble collapsing, i.e. after t 0.53 s, the vertical speed of theequipment peaks at 3.3 m/s and exceeds the maximum speed value of 3 m/s induced by shock wave.

As shownin Fig. 13, the vertical displacementof theequipmentrelative tothe ship hull manifeststhe

vibrationof thesupercharging boilerafter being shocked.Becausethe equipmentis connected to thehull

base by theabsorber inthe model,the positive andnegative values of thevertical displacementin Fig.13

represent the tension and compression of the absorber respectively. The absorber of the equipment is

compressed with the rising of the whole hull at initial stage, and the vibration amplitude of the equip-

mentis relativelylarge in thersttensionwitha maximumvalue of33 mm,beforethe movementbegins

to decay. The absorber is compressedduring the bubble collapsing; the vibrationamplitude value of the

equipment in the rst tension reaches 44 mm after the bubble collapse, which increases by 33%

comparing to that causedby theshock wave.Fromthe analysisabove, it canbe seen that themovement

of theequipment induced bythe bubble ismuchmoresevere than that induced byshock wave asfor the

selected model in this calculation.

In order to check the safety of the equipment, the stress response curve of the supercharging boiler

at typical positions is shown in Fig. 14. The elements at the internal and external shell of the super-charging boiler have been selected for stress analyses in the process of the calculation. The impact of

shock wave on the external shell of the supercharging boiler produces high stress with a peak value up

to 60 MPa. The vibration amplitude and stress reduce resulting from damping. The stress response of

the equipment increases rapidly and the peak value reaches about 70 MPa after the bubble collapsing,

which exceeds that induced by shock wave. However, the stress response at the internal shell of the

supercharging boiler is very low under the load of shock wave, and the peak value of the stress

increases gradually after the bubble collapsing and reaches 61 MPa at 0.72 s, which exceeds the stress

response caused by shock wave.

The analysis of the equipment stress response shows that the stress caused by underwater

explosion bubble is more severe than that caused by shock wave. Therefore, it can be concluded that

the equipment installed with absorber can be severely damaged by bubble load.

Fig. 12. Response curve of vertical velocity for typical parts.

Fig. 13. Curve of displacement of equipment relative to ship base.



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vertical natural frequency of the ship hull at that water depth, which is about 1.5 Hz. The amplitude of

whipping movement increases with the emergence of resonance and the stress response of the

equipmentbecomes more severeresulting from thelargeamplitude movementof theequipment base.

As a consequence of the bubble load, plastic deformations occur on certain joint shell parts of the

equipment when the stress values are higher than the Specic Minimum Yield Stress (SMYS) of the

material. Fig. 17 shows the curve of Equivalent Plastic Strains (PEEQ) on some joint parts with

the variation of attack angle and water depths. It indicates that PEEQ is 0 when attack angle is 30 andwater depth is deeper than 25 m, and the critical water depth of the emergence of equivalent plastic

strainis 30m whentheattackangleis 90. When thewater depth H is largerthanthis criticalvalue, thematerial of the structure is in the elastic range and there is no material yielding. On the contrary, when

the explosion depth H is less than the critical value, the equivalent plastic strain increases rapidly with

the decrease of water depth and the structure of the equipment is in great danger. The equivalent

plastic strain of the equipment increases exponentially with the increase of charge weight and the

decrease of water depth.Besides, in order to analyze the impact on the equipment from different charge locations along the

hull, ve different transverse sections have been selected for the sensitivity analysis in this paper, i.e.

sections at bow, L/4frombow,midship, L/4to sternand stern, which are labeled as S.0, S.5, S.10, S.15 and

S.20. Forthe selectedcross sections, the amount of TNT, water depth andpositionof the explosion charge

are all the same.Fig. 18shows the response amplitudes of the equipment from different explosion.

According toFig.18, the dynamic response of the equipment is relatively small when the charge is

located at the bow or stern section, with the vibration amplitude around 22 mm. While the charge is

located in the middle area of ship, i.e. the area between L/4 to bow and L/4 to stern, the response is

relatively large and the amplitude of which is about 45 mm. This phenomenon further proves that the

Fig. 16. Stress amplitude curves with water angle and depth.

Fig. 17. PEEQ curve on joint parts with angle and depth.



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majority energy of the vibration comes from the bubble load during underwater explosions. Because

shock wave normally has high frequencies and can only be a threat to the equipment within a limited

range, the damages caused by them are usually in local areas. However, the bubble load has low-

frequencypropertyand cantrigger the vibration of thewholehull.In Fig.18, the vibration amplitude is

almostthe same withinthe middlehalf ship length, by which it canbe concludedthat themain energy

of the equipment vibration derives from the bubble load. Therefore, it can be seen that the bubble load

could do effective damages to the shipboard equipment which are installed around the wide ranges of

the charge location.

5. Comparison between the integrated and the non-integrated model

As mentioned in introduction, there might be a signicant difference between the anti-shock check

of non-integrated equipment (e.g. the check with the BV043/85 standard) and that of the integrated

equipmentand thehull. Now consider thefollowingexample, theexplosion is right under thehull, and

we dene the impact factor[1]asCffiffiffiffiffiffi

Wp

=R, whereWis the amount of charge and Ris the stand-off

distance. Herecis set as 0.53 andFig.19shows the numerical results of the two cases.

As is known, the equipment is weak where the strain is large. From the contour shown in Fig. 19,signicant difference exits between theresults with differentcalculatingmethods andthe strainof the

weak part obtained through the integrated calculation is larger. It means, on the other hand, that the

result of separate anti-shock check calculation is relatively dangerous in actual situations. The time-

strain and time-acceleration history curves of the equipment are also compared in Fig. 20.

InFig. 20the green curves stand for the numerical results of the integrated anti-shock analysis and

thered forthe results of thenon-integratedone. Andthe formeris largerthanthe latterin terms of the

strain response and acceleration shock response. It can also be seen that the peak strain of different

units shows the same trend, as shown inFig. 21.

FromFig. 21we can see that the horizontal and the vertical shock response share the same regu-

larity with obvious differencesin the numerical results betweenthe integrated and the non-integrated

methods. These differences may be caused by the following reasons:

Fig. 18. Response amplitudes of the equipment along ship.

Fig. 19. Mises strain contour of equipment at 2 ms



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1) There is a strongcouplingeffect betweenthe hull andthe equipmentunderthe shockload,both of

which are elastic structures, while the coupling effect is just neglected in the non-integrated

analysis;

2) The shock environments at different spatial locations of the large equipment are different, while

they are taken asthe same inthe non-integrated analysis whichignoredthe multi-point andmulti-direction input characteristics. Therefore, the non-integrated analysis will cause a large error of

more than 20%.

We analyzed the effect of the impact factor on the results of integrated analysis and non-integrated

analysis, with its value varying from 0.1 to 1.2. Simulated results show that when the impact factor

c< 0.45 (i.e. mid and far-eld underwater explosions), the value of acceleration and stress response

Fig. 20. (a) Time-strain curves at typical position. (b) Time-acceleration curves at typical position.

Fig. 21. (a) Comparisons of responses under horizontal shock. (b) Comparisons of responses under vertical shock.



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from the non-integrated analysis is relatively larger, which means that we can adopt non-integrated

analysis in the mid and far-eld underwater explosions to evaluate the anti-shock features of large

equipment. However, when the impact factor c 0.45 (i.e. the near-eld underwater explosion),smaller responses will generate for the non-integrated anti-shock analysis of equipment compared to

actual situations, which means non-integrated analysis is somewhat dangerous for engineering

application for near-eld explosion.

6. Conclusions

Based on ABAQUS software, the numerical method is veried by comparing the numerical results

with the experimental data from the warship underwater explosion. The dynamic response can be

obtained based on the integrated model of the equipment coupling with the hull structure which iscompared with that of the non-integrated calculation. The suggestions and conclusions are shown as

follows.

1) Shock wave and bubble pulsation of underwater explosion will induce intensive impact to the hull

and shipboard equipment. From the strain and displacement response it can be seen that the

amplitude of equipment caused by the bubble pulse is greater. Therefore, bubble load which

inuences the dynamic response of the equipment can not be ignored.

2) The dynamic response of the equipment changes with waterdepth and attack angle, if charges are

located at a particular water depth, where pulsation frequency of the bubble is quite close to the

natural frequency of the ship hull, system resonance and relatively large equipment stress

responses.

3) Based on the sensitivity analyses of the equipment response with different positions of the charge

in longitudinal direction, it is found that the bubble load provides most of the energy for the

vibration of the equipment. Wherever the explosion is located within the middle half of the whole

hull length, the dymamic responses of the equipment are similar. Consequently, the bubble load

could cause effective damages to the shipboard equipment installed within a wide range of the

charge location.

4) There is a strong elastic coupling effect in integrated calculation of the equipment and the hullstructure and the impact on the equipment is multi-point and non-uniformity inputs, so the non-

integrated calculation of equipment and hull structure will make a large error when applied to

study the shock resistance.

5) When the impact factor c

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