Simplified vulnerability assessment procedure for a warship based on the vulnerable area approach

Transcription

1 Journal of Mechanical Science and Technology 26 (7) (2012) 2171~2181 wwwspringerlinkcom/content/ x DOI /s Simplified vulnerability assessment procedure for a warship based on the vulnerable area approach wang Sik im and Jang Hyun Lee * Department of Naval rchitecture and Ocean Engineering, Inha University, Incheon, , orea (Manuscript Received February 22, 2012; Revised March 16, 2012; ccepted pril 10, 2012) bstract The vulnerability of a warship is defined as its inability to withstand a man-made hostile environment, and can be estimated by the conditional probability of being killed by a hit We describe ship vulnerability given a penetration hit, and propose a vulnerability assessment procedure that incorporates a vulnerable area approach to naval ship survivability Measures of vulnerability indicate the killability of critical components with respect to the effectiveness of hostile weapons In the proposed methodology, a warship is considered to be an assembly of critical components representing an entire vulnerability We evaluated the vulnerability of a warship subjected to penetration effects using hypothetical models because of the paucity of available data and information, and the effectiveness of such assessment methods during initial warship design The proposed approach introduces critical component redundancy and overlap, the effects of single hits and multiple hits, and attempts to directly apply the vulnerability assessment technique to vulnerability reduction The kill tree method, Markov chain method, and oisson method are applied to multiple hits on critical components Examples show that the proposed method can provide the vulnerability parameters of a warship under the threat of being hit by a vulnerable area approach, thereby enabling an assessment of vulnerability eywords: Survivability; Vulnerability; Vulnerable area; Multiple-hit; illability; Markov chain; Warship Introduction In recent warship design, survivability has taken on a new meaning In the past, warship design has been characterized by an emphasis on armaments and performance successful design of a warship was estimated in terms of the ability to perform the mission for which it was designed, including the adaptability to rapidly defend against threat environments However, warship design has changed because the threats are increasingly lethal otential threats now carry large warheads, operate at greater distances, and employ increasing degrees of effectiveness lthough many warships have been designed with survivability in mind, it is necessary to assess survivability in terms of reduction in vulnerability Therefore, recent warship design has shifted the emphasis from performance to overall system survivability as a key measure of design In particular, the vulnerability of naval ships has become a much more common subject of discussion than in previous decades The survivability of a warship can be defined as its capability to avoid or withstand a man-made hostile environment * Corresponding author Tel: , Fax: address: jh_lee@inhaackr This paper was presented at the ICMR2011, Busan, orea, November 2011 Recommended by Guest Editor Dong-Ho Bae SME & Springer 2012 The survivability of the warship consists of three probabilities: susceptibility, vulnerability, and recoverability The inability of a warship to avoid radar, hostile weapons, and other elements that make up a hostile defense environment is referred to as susceptibility Warship susceptibility performance is measured in terms of probability of detection Recoverability refers to a warship s damage control ability and encompasses containment of damage, prevention of the loss of ship systems and equipment, and the repair and reconfiguration of critical systems so that the ship can continue to perform its mission The inability of a warship to withstand hits from a hostile environment is referred to as vulnerability Vulnerability can be measured by the conditional probability that a warship is killed given a hit Warship vulnerability assessment is an essential element of warship combat survivability discipline Vulnerability of warship given a single hit or multiple hit is usually expressed as the probability of kill or the vulnerable area in case of being given a random fragment hit on the warship Vulnerability assessment as a formal warship design discipline is a relatively new concept However, due to restricted information about warship vulnerability, a detailed vulnerability assessment model has not been introduced in the literature In contrast, a well-developed set of survivability design prin-

2 2172 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~2181 ciples has been developed for aircraft The Fundamentals of ircraft Combat Survivability nalysis and Design by Ball [1] and Weaponeering: Conventional Weapon System Effectiveness by Driels [2] comprehensively apply the fundamentals of survivability engineering to aircraft design These texts provide principles of survivability and vulnerability that are accepted and used by the aircraft design community and describe a basic taxonomy of the aircraft combat survivability discipline [3] Survivability concepts combined with design and measurement tools and techniques suggested by Ball [4] have been used in the aircraft design world for a number of years since the survivability of aircraft as a design discipline is quite well-recognized In particular, many studies [5-14] have shown that aircraft vulnerability can be successfully evaluated based on Ball s model Ball and Calvano [5] also suggested that survivability design concepts for aircraft could be applicable to naval ship design since many similarities exist We believe that the principles outlined in their study can form the basis for the structuring of survivability in naval ship design Detailed calculation methods are not mentioned in their study Therefore, it seems reasonable to explore the degree to which techniques of vulnerability assessment used by the aircraft community are applicable to naval ships In this study, we examine the applicability of Ball s aircraft combat vulnerability methods to naval ships Some of the methodology in this study is based upon Ball s method In particular, we propose a method for assessing the vulnerability of a warship to the threat of a penetrating fragment or projectile Using the vulnerable area of a vital (or critical) component, we propose a vulnerability calculation procedure Using this procedure, we analyze the vulnerability of a warship with redundancy and overlap to a single fragment hit and multiple hit n application example shows how the proposed procedure gives the probability of a kill due to a hit on a critical component 2 Basic concept about warship vulnerability 21 Characterization of warship vulnerability Surface ship combat survivability can be defined as the capability of a ship to avoid and/or withstand a manmade hostile environment while performing its mission The inability of a ship to withstand the effects of the hostile environment is called vulnerability In any mission scenario, there are many random variables that can influence vulnerability s a consequence of these uncertainties, survivability is a probability rather than a deterministic outcome The ship s vulnerability can be measured in a general way by /H, which represents the conditional probability of being killed given a hit Vulnerability has been referred to as the post-hit phase, and features that reduce vulnerability increase post-hit survivability [5] Fig 1 shows the relationship between these various measures of probability In particular, the probability of a kill is the product of the probability of a hit H (susceptibility) and the conditional probability of a kill given by a hit /H Thus, Fig 1 robability of survivability assessment = H /H (1- R ) (1) where R denotes the recoverability Hence, the probability of survival S is the complement of : S = 1- (2) Some of the features that reduce vulnerability include the ability of vital components to operate after a hit; design features that prevent or suppress the spread of damage to components not affected by the original hit; and the use of redundant systems and redundant components In this paper, by introducing the vulnerable area method used in aircraft vulnerability assessment [2] to warship vulnerability assessment, a simplified calculation model is proposed to assess warship survivability In order to provide a practical vulnerability assessment, the proposed calculation procedure considers single and multiple hits, redundancy, and the overlapping of components 22 Vulnerability assessment tasks ( /H ) ( R ) Warship vulnerability assessment of predicted threats consists of three tasks: 1) weapons systems analysis and classification of threats according to the effectiveness of weapons; 2) identification of vital or critical components and their kill modes using reliability analysis techniques including fault tree analysis (FT) and failure mode and effects analysis (FME); and 3) computation of numerical values of the vulnerability of the warship to assumed threats such as a contact-fuzed high energy (HE) warhead, a proximity-fuzed HE warhead, a torpedo, and a ballistic projectile or fragment Fig 2 summarizes the proposed vulnerability assessment procedure with respect to the threats and the features of critical components The general procedure is described as follows: (1) Weapon system analysis and identification of the threat: Identification of the threat determines the attack weapon and the weapon s effectiveness, including causing a lethal effect on the warship The main threats include non-explosive penetrators and spray fragments (2) Identification of vital or critical components: warship s components each have a level, or degree, of vulner-

3 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~ Estimates of warship vulnerability Fig 2 Flow of the vulnerability assessment methodologies Fig 3 Identification procedure for critical components ability to damage Each component s vulnerability contributes in some way to the vulnerability of the warship Vital components that are damaged or destroyed result in a defined or definable kill level The criticality of a given component is closely tied to functionality The following steps are taken to identify the critical components: (a) identify mission-essential functions; (b) identify system-essential functions and relationships; (c) conduct a failure modes and effects analysis (FME); and (d) conduct a fault tree analysis (FT) (see Fig 3) (3) Computation of vulnerability: The vulnerability of a warship to a particular threat is expressed as the probability that the warship is killed given a hit anywhere on the presented area of the warship /H, or as the single hit vulnerable area V Generally, the vulnerable area is given as V = /H (3) where and /H denote the projected area of the warship in the plane normal to the approach direction of threat and the probability that the warship is killed given a hit, respectively In this way, probability /H can be expressed as V / The computation of vulnerability is carried out at the estimates level Estimates typically use simple equations for vulnerability measures that are functions of certain major parameters of vital components The computation also requires information about the vital components and their vulnerable area The computational methodology adopted in this study is described as follows: (1) Select the threat: The threat and damage mechanics are considered according to the effectiveness of the threat since the measures of vulnerability vary with the type of threat penetrating projectile or fragment, the blast from contactfuzed HE warheads, an external blast, debris from proximityfuzed HE warheads, and other threats are considered during the computation (2) Select the vulnerability measure of vital components: One measure of vulnerability is the conditional probability that the vital component is killed by the threat nother measure of the vulnerability to impacting threats is the vulnerable area This is a theoretical non-unique area presented to the threat that would affect the whole vulnerability of the warship (3) nalyze the kill probability of critical components given the threat: If the vulnerability measures of components are determined, the vulnerable area (or killability of the i th component) is defined by a simple arithmetic equation Considering the vulnerable area Vi and the projected area i of the i th component, we define the probability of kill of the i th component given a random hit anywhere on component ki/hi as follows: = (4) vi ki / hi p i (4) nalyze the kill probability of the whole warship given the threat: The vulnerability of the warship is expressed as the probability that the warship is killed given a random hit anywhere on the presented area of the warship /H or the vulnerable area of the aircraft V Vulnerable areas provide a basis for the contribution of vital components to warship vulnerability Thus, the vulnerable area of vital components vi can provide the total vulnerable area of the warship by summing the component vulnerable areas as follows: 1 N / H vi i = 1 = (5) where and N denote the projected area of the warship and the number of vital components, respectively (The variable and subscript definitions used in this paper are summarized in Table 1)

4 2174 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~2181 Table 1 Definitions of vulnerability variables Table 3 rojected areas and vulnerable areas of critical components Definition robability of kill for i th component (i th components hit) robability of kill for i th component (j th components hit) robability of hit for i th component (random hit) robability of kill for i th component (random hit) robability of survival for i th component (i th components hit) Vulnerable area for i th component (or warship) rea of i th component (or warship) Component (i th ) Warship ki/ hi / H ki/ hj - hi/ H - ki/ H - si/ H S / H Vi i V pi vi Engine Engine Fuel oil tank Gear box Gear box ropeller ropeller Total Table 2 Combined cases of overlap and redundancy of components Overlap Non-overlap Non-redundant components Redundant components Non-redundant components Redundant components Fig 4 Examples showing the geometric description 24 Redundancy and overlap attributes In any assumed combat scenario, a warship will either be hit or not The influence of the non-redundancy and redundancy of components on /H and the vulnerable area must be considered The overlap of components is another important consideration We considered the layout of a component (redundant or non-redundant) and the condition of the location (overlap or non-overlap) as shown in Table 2 Overlap of critical components decreases the total presented area of the components When a critical component is labeled as redundant, identical or similar components duplicate some or all of its essential functions Redundancy can also decrease the degree of criticality of a critical component 3 Example of vulnerability assessment The following examples illustrate the proposed method of assessing the vulnerability of a warship under the threat being hit For comparison, consider a warship with several critical components, with or without redundancy, under a given single hit or multiple hit in the engine room and propulsion system 31 Redundancy and overlap attributes We calculated the vulnerability of a warship with several critical components in the engine room and propulsion system The most important parameter of a warship s vulnerability to weapons is /H nother parameter that can represent the /H is the vulnerable area The vulnerable area is defined as an area on a component or on the warship that, if hit, results in a kill of the component or warship The vulnerable area as a vulnerability parameter can be determined using combat damage data or live fire test and evaluation (LFT&E) data Inspection of the damage can provide a realistic value of the warship s vulnerable area with respect to the threats that caused the damage However, due to restrictions of information about real damage, the vulnerable area of each vital component was assumed This area is the product of the presented area and the probability of a kill given a hit We assumed that the vulnerable area of individual critical component was available, and we summed the vulnerable areas of all the critical components on the warship so that the vulnerable area of the warship itself could be measured To show the applicability of the proposed method, a propulsion system was considered as an example (see Fig 4) The following components of the propulsion system were defined as critical components: two main engines, two propulsion shafts, one gear box, and one fuel oil tank, without overlap The values of presented areas pi and vulnerable areas vi were assumed since the values could not be taken literally ll the critical components were assumed to be non-overlapped; therefore, the probability that a component is killed given a hit could be calculated We considered above-surface weapons such as anti-ship cruise missiles (SCMs), naval guns, and machine guns that

5 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~ Table 4 Vulnerability for a non-redundant warship model Critical components i (m²) ki/ hi have explosive or non-explosive effects on the warship, as shown in Fig However, as a simplification, we considered only the penetration effectiveness without explosive effects since the damage mechanics of explosive effects can only be estimated through extensive numerical calculations 32 Vulnerability to a single hit by a non-explosive penetrator or fragment When computing the vulnerability to a single impacting penetrator or fragment, the assumption was made that the vulnerability is expressed as the single hit vulnerable area of the warship, v 321 Warship composed of non-redundant components with non-overlap components This warship consists of N critical components whose functions are not duplicated by any other components s an example of such a model, we considered the warship shown in Fig 4(a) This warship has four critical components (N = 4): an engine, one fuel oil tank, one gear box, and one propeller Using Eqs (4) and (5), warship vulnerability can be defined in terms of the number of components killed Because of the assumption that only the component that is hit can be killed and because none of the components overlap, the union of N mutually exclusive kills can be given the following probability form: = / H k1/ H k2/ H kn / H N v1 v2 vn 1 V = = vi = i= 1 Vi (m²) ki / H Engine Fuel oil tank Gear box ropeller The remainder Total Fig 5 Weapon systems and threats (6) s a numerical illustration, Table 4 shows the assumed values for the component and warship presented areas, and the component kill criteria for a penetrator The computed values for vulnerable area and killability are indicated in the Table ssume that the propulsion system of the warship is destroyed, leading to a probability of the loss of the warship of 373% Then, the probability that the warship survives a single penetration hit is 637% 322 Warship composed of non-redundant components with overlap components Creating overlap between critical components is a means of decreasing the total presented area of the components The model described in the previous section can be extended by allowing two or more non-redundant critical components to overlap in an arbitrary manner hit in the overlap area can kill one or more of the components intersected by the shotline The layout of the warship, and the aspect from which the damage mechanism approaches, determine overlap region O ssume there are M non-redundant critical components along all shotlines within the overlap area This overlap area, with its M components, can be considered to be a composite nonredundant critical component and is denoted by subscript O When assessing the vulnerability of the overlap component, the survival expression is more appropriate than the kill expression The probability that all M components survive the hit so/ho is given by the joint probability that each of the M components survives the hit If the assumption is made that the kills of the M components are independent, then ( ) = = 1 (7) so / ho s1/ ho s2/ ho s3/ ho sm / ho ki / ho i = 1 Hence, the probability that the composite component is killed is given by M = 1 = 1 1 ko / ho so / ho ki / ho i = 1 M ( ) (8) The vulnerable area of the overlap area is the product of the overlap presented area and the probability of an overlap component kill Therefore, vo poko / ho = (9) The vulnerable area of the overlap component contributes to the warship s vulnerable area in the same manner as the vulnerability areas of the non-overlapped components However, overlapping also requires that the overlap area be subtracted from the total presented area of each overlapping component contributing to the overlap For a case in which the engine overlaps the fuel oil tank as illustrated in Fig 4(a), the vulnerable areas of the four components are presented in Table 5 The overlapping area is assumed to be 10 m 2 We note that

6 2176 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~2181 Table 5 Vulnerability area with overlap Critical component i (m²) ki/ hi overlapping two of the critical components such that one overlaps the other reduces the warship s vulnerability to 350% Thus, the net effect of component overlap can be a desirable reduction in warship vulnerability 323 Warship composed of redundant components with non-overlap components Redundancy gives a reduction in vulnerability When a critical component is designed to be redundant, identical or similar components duplicate some or all of its functions non-redundant critical component is the only component that performs a particular function Non-redundancy can increase the importance of a critical component, since the loss of a nonredundant critical component means the complete loss of a particular function in the warship If the assumption is made that a single hit cannot kill both engines, then all of the component kills are mutually exclusive, and the single hit cannot kill engine 1 and engine 2 Hence, the ship is killed only if the other vital components are killed Thus, the vulnerability of ship can be expressed as follows: = / H k1/ H k2/ H kn/ H n v 1 v2 vn 1 = = i = 1 (Except redundant component) vi Vi (m²) ki / H Engine = Fuel oil tank = Gear box ropeller rea (overlap) Total (10) where n denotes the number of non-redundant components The total vulnerable area for this case is the sum of the vulnerable areas of non-redundant components We considered a case in which a second engine was added to the warship as illustrated in Fig 4(b) and the vulnerability for a single penetration hit was calculated to be 21% as shown in Table 6 Thus, if a non-explosive penetrator of a fragment hits the diesel engine, the warship has a 21% chance of being killed The probability that a propulsion system with nonredundancy can be killed by a single penetration hit was calculated to be 373%, as shown in Table 3 This comparison clearly shows that redundancy can increase the degree of survivability Table 6 illability of propulsion system with redundancy Critical component 33 Vulnerability to a multiple hit by a non-explosive penetrator or fragment nother element in vulnerability assessment is to calculate the accumulated killability of a warship subjected to a multiple hit by non-explosive penetrators or fragments We applied the Binomial, oisson, Markov chain, and tree diagram methods to calculate multiple-hit vulnerability The binomial and oisson methods are based on the assumption that under a given hit the warship kill probability /H is constant; therefore, these methods can be applied to simple estimations The Markov chain and tree diagram methods are methods that have been recommended by the authors of several studies for their high accuracy [11] The random distribution of multiple hits over the ship is assumed to be uniform, and all hits are assumed to travel along parallel shotlines from the same direction 331 Binomial approach for N hits When N weapons encounter one ship sequentially (N is a known integer), the warship survives only if it survives all of the encounters The binomial approach is based on the assumptions that there are two outcomes for each hit, (kill and survival); that the outcomes for each hit are independent from the preceding outcomes; and that the probabilities of the kill and survival outcomes ( /H and S/H, respectively) are constant for each hit In this way, the joint probability the warship is killed in N identical encounters can be computed using the binomial approach as follows: ( 1 ) N SN / H i (m²) ki/ hi = (11) ( ) N = 1 = 1 1 (12) / NH SN / H Vi (m²) ki / H Engine Engine Fuel oil tank Gear box ropeller The rest Total = 021 where SN and /NH denote the survivability and killability for N hits, respectively If a warship composed of non-redundant components with the non-overlap components presented in Table 4 encounters five hits, the vulnerability of the warship can be calculated as

7 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~ Fig 6 Random distribution of shots on a warship follows: ( ) 5 / = 1 = = 0903 (13) NH SN The assumption that the probabilities are constant for each hit is not correct if there are redundant components Nevertheless, the binomial approach is usually used to quickly assess the approximate effectiveness 332 The oisson approach for N hits The oisson approach can also be used to determine the non-redundant model for the case of multiple hits In the oisson approach, the number of hits is a random number E The oisson process is applicable to a situation in which M shots or fragments are ejected toward the ship From zero to M of these shots can hit the ship, as shown in Fig 6 M penetrators are assumed to be uniformly distributed over a spray zone of area S where S > S The penetrator spray density ρ, number of penetrators per unit spray area, is given by ρ = M / S (14) When penetrator spray hits the entire ship, the expected number of hits on the warship E is given by E = ρ = M( / ) (15) S Given the expected E hits on the warship, the expected number of times the ship is killed is E /H Thus, the probability of zero kills of the warship when E /H kills are expected is / e E H = (16) Using the single hit vulnerability defined by Eq (6), Eq (15) can be rewritten as follows: robability of zero kills of the warship when ρ V kills are expected e ρ V = (17) Therefore, the probability that the ship is killed is the complementation of the probability that it is not killed Hence, e e ρ / 1 E H V = = 1 (18) The non-redundant ship described in Table 4 has a /H Fig 7 ill tree diagram First and second hit, non-redundant model value of 0373 We assumed that the ship was subjected to a burst of two rounds with a spray zone of 90 m 2 that covered the 300 m 2 warship Using the oisson approach, the warship is expected to be hit E = 5(90/300) = 15 according to Eq (15) Thus, the ship is expected to be killed 15 x 0373 = times The probability of zero kills of the ship when kills are expected is e = 0571 Hence the probability that the ship is killed by the burst is given by E / 1 H = e = = 0439 (19) Both the binomial approach and oisson approach are simplified methods that assume all component kills are independent, and sometimes their estimations are approximate These methods are usually used to assess vulnerability quickly for the case of multiple hits 333 Tree diagram method for N hits Tree diagram methods are exact solution methods and have been recommended in several studies for their high accuracy tree diagram can be used to determine the probability of both non-redundant and redundant ship models being killed as a result of N hits kill tree is relatively self-explanatory for the non-redundant model Fig 7 shows the mutually exclusive kill probabilities for each of the non-redundant components, and the probability that no critical components are killed after the first hit on the ship The first level of the diagram accounts for all possible outcomes for the first hit The outcomes of the first hit are the mutually exclusive outcomes of survival and kill The tree diagram can be extended to include the second hit The second level of the tree represents all possible outcomes of the second hits In Fig 7, the branch down and to the left represents a ship kill, and the branch down and to the right represents no kill or survival There are three branches at this level because each branch represents a possible outcome of the second hit The probability associated with the branches, /Hnrci, is the conditional probability of a kill of the i th non-redundant component Fig 9 defines the kill probabilities for the first hit and second hit, respectively, for a redundant ship This kill tree dia-

8 2178 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~2181 Fig 8 ill tree diagram First and second hit, redundant model Fig 10 Tree diagram for redundant components ( after 2 hits) = ( after 1 hit) + N( E + F + G + ) = ( ) = (22) Therefore, the probability that the ship survives after two hits is S ( after 2 hits ) = 1 ( after 2 hits ) = = (23) Fig 9 Tree diagram for non-redundant components gram encompasses all of the redundant and non-redundant critical components in the ship The probability associated with the branches, /Hrci, is the conditional probability of a kill of the i th redundant component Fig 9 presents the tree diagram for the ship described in Table 4 This example used the probability of kill for a single hit given in Table 4 This kill tree diagram assumes that the ship s components are non-redundant critical components The kill expression for this non-redundant ship model is (Engine) OR (Fuel Tank) OR (Gear) OR (ropeller) The four mutually exclusive outcomes of each hit are as follows The kill of engine E = e/h, the kill of fuel tank F = f/h, the kill of gear G = g/h, the kill of propeller = p/h, and no critical component kill N= 1 (E+F+G+) = 1 /H The kill probability of the i th component with j th j hit can be expressed as + = N (20) j 1 j j i i The probability that the ship is killed after the first hit is ( ) after 1 hit = E + F + G + (21) = = 0373 The kill probability of the ship caused by the second hit comes from branch N, where no critical component is killed on the first hit The probability that the ship is killed after two hits is Hence, the survivability is estimated to be 3936% kill tree diagram is shown in Fig 10 for the ship described in Table 6 In the figure, the ship s components are assumed to be redundant critical components This kill tree diagram uses the pair of diesel engines as examples of redundant critical components The example uses the probability of a single hit given in Table 6 illability can be determined using the tree diagram, similar to the approach used for the non-redundant model Considering that a kill of only engine 1 or only engine 2 does not result in a ship kill because of redundancy, the probability that the ship is killed after the first hit is given by the sum of the kill probabilities for each of the non-redundant components Consequently, as illustrated by Fig 10, the probability that the ship is killed after the first hit is ( ) after 1 hit = F + G + = = 0213 (24) The kill probability of the ship caused by the second hit comes from the branch N, where no critical component is killed on the first hit The probability the ship is killed after two hits is given by ( after 2 hits ) = ( after 1 hit ) + (E1)( E2 + F + G + ) + (E2)( E1 + F + G + ) + (N)( F + G + ) = = (25)

9 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~ Table 7 illability of propulsion system with redundancy and overlap Table 8 State transition matrix [T] i (m²) ki/ hi Vi (m²) Engine Engine Fuel oil tank Gear box ropeller / 1/ 300 robability of transitioning from this state To nrc e1 e2 rc N this state nrc e e rc N The remainder Therefore, the survivability can be obtained by S Total ( after 2 hits ) = 1 ( after 2 hits ) = = 0568 (26) Hence, the survivability significantly increased, from 396% to 568%, as a result of the addition of the second engine 334 Markov chain method for N hits The tree diagram can be continued for multiple hits; however, the computations become overwhelmingly complex The Markov chain is better suited to handle multiple hits because it can simplify the computation by using a state transition matrix The dimensions of the Markov state transition matrix can consider the number of components presented because the Markov chain assumes that a sequence of independent events associated with a system can be modeled by a state transition matrix In a Markov process, the system is defined as having two or more states or conditions in which it can reside These states are contained within the state vector {S} n event occurs such as a ship is hit, and each state of the system will transition to all other possible states with a specific probability for each transition The sequential process of evaluating the probability that the system exists in each of the several possible states after events 1, 2, 3,, J is based on the possibility that the system existed in each of the possible states after events 0, 1, 2,, J-1, respectively, and is referred to as a Markov chain The transition matrix [T] transforms the {S} (j) state vector to the {S} (j+1) state vector in the following form: ( 1) ( ) { } j + j S [ T]{ S} j 0,1,2,, J 1 = = (27) If the assumption is made that the ship has the redundant components shown in Table 7, then five distinct states can be defined as follows: 1 nrc (Non-redundant component kill) 2 rc (Redundant component kill) 3 e1 (Engine 1 kill) 4 e2 (Engine 2 kill) 5 N (No components killed) States nrc and rc are called absorbing states because the ship cannot transition from these two kill states to any of the other three non-kill states Transition matrix [T], which specifies how the ship will transition from one state to another as a result of a hit on the ship, can be constructed using a table Table 8 presents the elements of the [T] matrix for the redundant ship model defined in Table 7 Each element of the matrix represents the probability of transitioning from the state defined by the column location to the new state defined by the row location Thus, the probability of transitioning from the nrc state to the nrc state is unity (300/300) because nrc is an absorbing state, as described in Eq (28) Therefore, the probability of transitioning from the e1 state to the nrc state is the sum of the conditional probabilities of kill of the non-redundant components The probability of transitioning from the e1 state to the e2 state is zero because e2 is the state in which only engine 2 is killed The elements in the remaining columns are determined in the same manner [ T ] = The state vector {S} (j) is given by the following equation: S nrc e1 = e1 rc N ( j + 1) { } (28) (29) Eq (29) consists of the probabilities that the system is in each of the five states after the j th hit ship kill is defined by those states that specify either a kill of any of the nonredundant components (nrc), or a kill of the members of the sets of redundant components (rc) such as both engines Hence, the probability the ship is killed after j hits can be expressed as

10 2180 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~2181 = nrc + rc (30) ( j) ( j) ( j) / H Eq (28) shows the transition probabilities for the redundant model rior to the first hit, the ship is entirely in the N state Therefore, according to Eqs (28) and (29), the first state can be expressed as (1) (0) nrc 0 e1 0 e1 = [ T ] 0 rc 0 N 1 (31) Carrying out the multiplication of the transition matrix gives (1) (1) nrc 0213 e1 016 e1 = 016 rc 0 N 0467 (32) Based upon the estimation given in Eq (32), the probability that the ship is killed after the first hit can be calculated using Eq (30) Thus, (1) (1) / H= nrc = 0213 (33) Substituting Eq (28) into Eq (32) and carrying out the matrix multiplication yields the next hit, given by (2) (1) (2) nrc e e1 = [ T ] 016 = rc N (34) Eq (34) reveals that after the second hit, there is a probability that the fuel oil tank, the gear box, the propeller, or all of them have been killed; a probability that engine 1 or engine 2 has been killed; and a probability that both engines have been killed Thus, the probability that the ship is killed after a second hit is given by ( j) ( j) ( j) / H= nrc + rc = = (35) This value is the same as that obtained using the tree diagram This process can be continued for as many hits as expected Fig 11 shows the ship s as a function of j for both the redundant components and non-redundant components, given in Table 4, using the Binomial approach The difference robability of kill Binomial(Non-redundant components) Markov-chain(Redundant components) Number of hits Fig 11 for the redundant and non-redundant ship components versus the number of hits between the two curves can be explained by the reduction in vulnerability caused by redundancy s the number of hits becomes large, the effect of engine redundancy on the ship s survivability is diminished because of the increased likelihood that the large number of hits has killed both engines 4 Conclusion We proposed systematic applications of ship vulnerability given a penetration hit We considered the use of a vulnerability assessment procedure previously established for aircraft design for the vulnerability assessment of naval ships component s vulnerable area approach, which can be used to calculate single hit and multi-hit vulnerability, is described Several methods have been developed for the vulnerability assessment of warships; these include single hit vulnerability calculations, the Binomial approach, the oisson method, the Tree-diagram approach, and the Markov chain method for vulnerability calculations of fragments and blast effects Four models for calculating vulnerability (non-redundant nonoverlapping, non-redundant overlapping, redundant nonoverlapping, and redundant overlapping) are suggested and several examples are given The examples show that the proposed method can provide the vulnerability parameters of a warship under the threat being hit through proposed approaches, thereby enabling a vulnerability assessment The proposed warship vulnerability assessment methodology was developed with the aim of integrating fundamental warship design into the early design stage In addition, the method could be helpful in solving the generality problem in the evaluation of multiple hits Though the proposed method is simple in analysis, it shows the procedures necessary for ship s vulnerability calculation In this study, several assumptions and simplified models were applied due to restricted data and assessment methodology Certainly, there are areas of warship survivability theory not covered by this study that should be addressed in a complete vulnerability assessment of a warship In a complete vulnerability assessment, impact, explosion, shock, flooding, and other such surface ship phenomena should be addressed

11 S im and J H Lee / Journal of Mechanical Science and Technology 26 (7) (2012) 2171~ cknowledgement This work was supported by the research fund of the Survivability Technology Defense Research Center of the gency for Defense Development of orea (No UD090090GD) and Mid-career Researcher rogram through NRF grant funded by the MEST ( ) The authors gratefully acknowledge this support Nomenclature V O /H N [T] References : resent area : Vulnerable area : Overlap area : robability of kill given hit : No kill : Transition matrix [1] R E Ball, The fundamentals of warship combat survivability analysis and design, I education series: New York, US (1985) [2] M Driels, Weaponeering: Conventional weapon system effectiveness, I education series: Virginia, US (2004) [3] J Lillis, nalysis of the applicability of aircraft vulnerability assessment and reduction techniques to small surface craft, Naval ostgraduate School (2002) [4] R E Ball, The fundamentals of warship combat survivability analysis and design-second Ed, I education series: Virginia, US (2003) [5] R E Ball and C N Calvano, Establishing the fundamentals of a surface ship survivability design discipline, Naval Engineers Journal, 106 (1) (1994) [6] Joint Technical Coordinating Group on Warship Survivability, Survivability models and simulations, erospace systems survivability handbook series-volume 5, JTCG/S-01- D007 (2001) [7] J Fielding and O Nilubol, Integration of survivability assessment into combat aircraft design for operational effectiveness, In: ICS 2004 CD-ROM roceedings, Optimage Ltd, Edinburgh (2004) [8] S im, J H Lee and S Y Hwang, Simplified vulnerability assessment procedure for the warship based on the vulnerable area approach, Journal of the Society of Naval rchitects of orea, 48 (5) (2011) [9] Yang, B Song, Q Han and B Ou, direct simulation method for calculating multiple-hit vulnerability of warship with overlapping components, Chinese Journal of eronautics, 22 (2009) [10] Yang, B Song and Q Han, Method for assessing vulnerability of aircraft to spray fragments of missile, Systems Engineering - Theory & ractice, 27 (2) (2007) [11] J Z Zhang and H C Wang, n improved Markov chain method for multiple hit vulnerability assessment, Journal of Shenyang Institute of eronautical Engineering, 18 (2003) [12] B Song and Q Han, Theory and method for computing combat aircraft vulnerability (II), Modern Defense Technology, 27 (1999) [13] Yang, B Song and Q Han, lgorithm study on geometric description of aircraft for vulnerability assessment based on finite elements model, Journal of Northwestern oytechnical University, 22 (2004) [14] Yang, B Song, ircraft vulnerable-area decomposition method in the overlapping region of components, Journal of ircraft, 43 (2006) Jang-Hyun Lee received his hd degree from Seoul National University in 1999 Then he joined the Research Institute of Marine Science Engineering at Seoul National University and XINNOS Co, Ltd rof Lee is an associate professor of Naval architecture and Ocean engineering at Inha University in Incheon, orea, where he has organized the e-manufacturing & LM Laboratory since 2005 wang-sik im is currently a hd candidate at the Inha University in Incheon, orea He received his master degree from Inha University in 2011 He has been actively involved in research areas related to survivability of warship