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Research ArticleDesigning of Elastoplastic Adaptive Truss Structures withthe Use of Particle Swarm Optimization

Jacek Szklarski1 and Marcin WikBo2

1 Institute of Fundamental Technological Research Polish Academy of Sciences 02-106 Warsaw Poland2Kazimierz Pułaski University of Technology and Humanities 26-600 Radom Poland

Correspondence should be addressed to Jacek Szklarski jszklaripptpanpl

Received 13 May 2015 Accepted 11 August 2015

Academic Editor Mohamed Ichchou

Copyright copy 2015 J Szklarski and M Wikło This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In the paper we demonstrate how Particle Swarm Optimization (PSO) can be employed to solve the Adaptive Impact Absorption(AIA) problem We consider a truss structure which is subjected to impact loads Stiff bars can be replaced by elastoplastic fuseswhich control theirs dynamical response The point of optimization is to maximize or minimize a given objective function byredesigning the structureThis is realized by redistributing the initial mass finding proper fuse localizations and adjusting in real-time the elastoplastic limits Comparing to the previous results we show that PSO is capable of achieving results at least as good asgradient-based optimization having at the same timemuch larger flexibility regarding the definition of the objective functionThisgives significantly broader field of potential applications In particular we present how PSO can be used to solve the simultaneousoptimization problem mass redistribution and fuse positioning for a set of expected various impacts

1 Introduction

The idea of optimal structural remodeling including topo-logical optimization as a problemofmaterial distribution hasbeen developed for decades (eg [1]) Consequently thereexist a large number of mathematical models and numericaltools which are capable of reliable simulation of responsesof passive structures to a predefined impact scenario andimproving the structure in the sense of finding an extremeof some desired objective function The majority of practicalsolutions are devoted to crashworthiness analysis related totraffic collisions

Due to the fast developing of the so-called smart tech-nologies there has been some considerable shift towardsfocusing on research related to adaptive structures Variouskinds of relatively inexpensive technical solution exist whichmake it possible to modify relevant structural characteristicsat themoment of impact [2] Specially designed actuators canbe applied in real-life active structures which can adapt atrequired time scales (which for obvious reasons are small)The actuators can modify structures via various possibleways For example a controlled delamination process can be

applied in order to detach selected structural joints whichwill result in a particular response Other possibilities includeshock absorbers based on magnetorheological (MR) fluidspiezo-valves used to control pressure and more see forexample [3]

Consequently existence of appropriate technologies andmathematical solutionsmakes it feasible to seriously considerthe potential gain from using Adaptive Impact Absorption(AIA) systems The focus of AIA is to obtain a device whichcanmodify its structure at themoment when a collision takesplace The point of this modification is usually to achieve adesired energy dissipation profile

One of the central problems of AIA is identification of theimpact and involved masses and velocities This means thatfirstly an impact event and its location should be detectedand then the AIA system should estimate what the mass andvelocity of the impacting body are The identification shouldhappen on timescales which enable a real-time structuralmodification which leads to optimal (in a given sense)absorption This is the main difference between AIA andclassical passive structural design the ability to react tocollision conditions in real-time Impact identification is a

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 652824 14 pageshttpdxdoiorg1011552015652824

2 Mathematical Problems in Engineering

quite demanding separate engineering problem on itself [45] and will not be considered in this paper Instead we willfocus on the process of optimal design of structures whichcan adapt by using ldquostructural fusesrdquoThese are elements withelastoplastic type of response with controllable yield stressFor example in real application such fuse can be made upof MR fluid where stresses are controlled by magnetic fieldsHowever technical details regarding construction of theseelements are not relevant in this work

In the field of typical passive structural design there isa huge amount of existing numerical methods and ready-to-use computer codes However to date the number ofexisting tools for designingAIA structures is very limitedThemain reason for this is that AIA is a relatively new conceptand also it differs in many aspects regarding the optimiza-tion process Naturally the main question which must beanswered is actually the goal of optimization This dependson the problem being solved involved energies if a structureshould answer to a critical emergency impact or maybeperiodic loads and so forth As a measure of success onecan define minimization of accelerations maximization ofstructural stiffness somehow smoothing structural responsepreserving structural integrity or maximization of dissipatedenergy in some chosen time interval Generally the requiredobjective function of optimization can be a very complex one

In this paper we consider a truss-like structure withelastoplastic fuses which can adapt to impact by adjustingthe plasticity limit The structural response is calculatedby means of virtual distortion methods (VDMs) [6] andthe presented results are continuation of the previous work[7 8] where a typical gradient-based method has beenused in order to achieve the desired optimization goal Themain disadvantage of gradient methods is the mathematicalrequirements regarding the objective function (eg it hasto be differentiable) As mentioned during AIA designprocess one can take into accountmany factors impacts withvarious parameters and at different geometrical places andcomplex goal definitions for example including costs of theused elements Some problems involve solvingmultiobjectiveoptimization (ie require simultaneous optimization ofmorethan one objective function) which can be a particularlydifficult task with any classic gradient-based approach

All this makes it tempting to use other possibly nonexactbut otherwise versatile optimization techniques like the onesbased on the so-called soft computing approach usuallysome sort of evolutionary computation Using it in thestructural design is a relatively new paradigm which hasa tremendous impact on the field and a vast amount ofrecent research is devoted to the subject A survey can befound in for example [9] and a great overview regardingmetaheuristic methods in structural design can be found in[10]

Here we consider the applicability of particle swarmoptimization (PSO) method in the process of designing AIAstructures PSO is one of the techniques which belongs tothe wide class of evolutionary computing It is based on aset of potential solutions (called ldquoswarmrdquo) which is graduallyimproved until a certain criterion of acceptance is metPSO is successfully applied in a very wide field of complex

engineering and scientific optimization problems includingmechanics robotics artificial intelligence transportationbiology and many more see [11 12] A recent survey onPSO application in the field of electrical and electronic engi-neering automation control systems communication the-ory operations research mechanical engineering fuel andenergy medicine chemistry and biology is available

Generally optimization of truss-like structures falls intothree categories size shape and topology however evenmore demanding is a simultaneous optimization where thedesign variables describing these properties are integratedVarious techniques from the field of evolutionary computa-tion have been applied to address these problems from thefield of traditional passive design for example genetic algo-rithms [13] genetic programming [14] simulated annealing[15] or ant colony optimization [16] However it seems thatrecently one of the most popular method is PSO along withits various modifications

For example in [17] the authors applied PSO to opti-mize topologies of truss structures optimizing for minimumweight under stress deflection and kinematic stability con-straints They have considered a two-stage technique wherefirst topology was optimized (withmodified binary PSO) andthen size and shape (by means of ldquoattractiverdquo and ldquorepulsiverdquoPSO) It was reported that the methodology proposed bythe authors can find superior truss structures compared withthose found with classical optimization However simultane-ous optimization for all the criteria in principle should givebetter results since these design problems are not linearlyseparable

Another challenging class of structural design problemsare the ones which involve optimization with frequencyconstraints In such problems the search space is particu-larly highly nonlinear and nonconvex with numerous localoptima When using any heuristic algorithms in such cases aneed to find a balance between exploitation and explorationarises It is of vital importance that a method explores thesearch space in a way which will enable it to find satisfyingsolution while at the same time it will not get stuck in alocal minimum It has been shown that accordingly modifiedPSO algorithmwas capable of dealing with such tasks [18 19]A recent comparison of nine metaheuristic algorithms foroptimal design of truss structures with frequency constraintscan be found in [20]

As in the case of frequency constraints the classical PSOis often modified to deal with a specific problem It is alsoincorporated together with other optimization techniquesin order to solve problems in a more efficient way Forexample to optimize layout of truss structures PSO hasbeen successfully integratedwith ldquononclassicalrdquomethods likecellular automata [21] It has been demonstrated that suchcombination has led to better solutions and more effectiveconvergence than in case of using ldquopurerdquo PSO

According to our knowledge to date there have beenno results regarding usage of PSO in the AIA domain andconsequently there are no standard problems or results whichwe can refer to In this work we demonstrate how PSO canbe employed in order to find optimal mass redistribution and

Mathematical Problems in Engineering 3

1

1 1

1

1 2 2 3

34 4

5

5 6

6

7

8

9

10

Figure 1 A simple truss structure along with the used numberingscheme Here the horizontalvertical distance between nodes is 1m

localization of structural fuses in truss structures with adap-tive elastoplastic elements Contrary to the case of passivetruss structures optimization there are no established testcases in the literature due to the innovative character of theAIA research field In the present work we choose to comparePSO with optimal results found by gradient methods pub-lished previously [7 8] It is found that for the consideredsetups PSO quickly (in asymp100 steps) converges to solution notworse than the ones found bymeans of gradient optimizationIn some cases better in the sense of the value of the objectivefunction structures are found

It is shown that such PSO approach can be appliedwhen considering multi-impact scenarios In other wordsmore complex objective function is used in order to evaluatestructures which are subject to a set of possible impacts (notjust one)Moreover it is easy to incorporate solver for optimallocalization of the structural fuses (in the previous workit has been done by replacing the stiffest elements) Sincethe considered optimization problem is a relatively simpleone (onlymass redistribution and localization of elastoplasticelements are considered) and there is a lack of previous resultsregarding PSO and AIA in the current research we limitourselves to the classical PSO without any specific improve-ments

Section 2 is devoted to short description of VDM usedto evaluate the structural response in Section 22 we definethe optimization problem and then in Section 23 we showhow it can be represented as a particle swarm Section 3discusses the problem of mass redistribution in a trussstructure in order that with the same amount of materialminimizesmaximizes some given goal function (without anyactive elements) Finally in Section 4 we show how to treatthe problem if AIA elements should be incorporated in thestructure

2 The Model

As a basic model for our consideration we use a truss can-tilever structure supported on one or both ends and poten-tially equipped with structural fuses An example of such10-bar structure is depicted in Figure 1 This setup has beeninvestigated previously for various optimization goals and

constraints [22] also with PSO approach [23] The structureis exposed to impact loads applied on one or more of itsnodes This means that at the moment 119905 = 0 a massless nodeinstantly increases its mass to 119898 and its velocity from 0 tok (119898 denoting mass of the impacting body traveling withthe velocity k) It is assumed that throughout calculations thenode remains attached to the same bars as before the impact(there is no separation between masses and the truss) Undersuch circumstances some selected quantities of interest forexample averaged displacement are subject to optimizationThis structural model was chosen due to its simplicity (eachelement is associated with only one axial plastic distortionstate) and already widely available research results regardingoptimal design and active adaptation solutions However thisdoes not reduce the generality of the proposed approachwhich is applicable to other types of structures for exampleframe or plate structures [24]

The structural response is calculated by means of specifi-cally designed impulse virtual distortion and impulse virtualforce methods (IVDM IVFM) VDMs are fast reanalysismethods used to compute the structural response of a mod-ified structure in a numerically efficient way which does notrequire solving full set of modified structural equations Theformulation of VDM is flexible and can be easily adopted toinclude structural modifications like damage plastic yielding[6] breathing cracks [25] moving masses [26] and materialdamping [27] One of the advantages of VDM over otherreanalysis methods like the method of combined approx-imations [28] is that it requires nonparametric model ofthe unmodified structure (the influence matrix) With thismodel calculation of responses of the modified structure canbe performed faster than in other reanalysis approaches sinceonly one step is needed (instead of many iterations)

The details regarding VDM are beyond scope of thispaper (more on this can be found eg in [29]) since thefocus is mainly on the process of optimization Neverthelessit is important to define how the adaptive elements aremodeled in Section 21 we briefly discuss how the structuralresponse is calculated with elastoplastic constitutive modelThen we show how particle swarm optimization method canbe applied to this optimization problem

21 Modeling the Truss Structure Containing Adaptive Ele-ments All the details regarding the numerical simulationsof the structural response of the discussed model via virtualdistortion methods were presented in [7 8] and also in [29]Here we limit ourselves to describing shortly the way theproblem is addressed The notation and used symbols arethe same as those in the cited work lowercase subindices119894 119895 refer to structural elements of the truss (bars) whilecapital subindices 119872 119873 and 119870 refer to the degrees offreedom (DOF) of nodes (joints) In the two-dimensionalcase discussed here there are two DOF per node unless anode is fixed

Generally the equation of motion for elastic and elasto-plastic structures can be stated as

119874119873119872

119872 (

119905) + 119866

119879

119873119894119878119894119895[119897119895(120576119895 (119905) minus 120573

0

119895(119905))] = 119891119873 (

119905) (1)


120574E

120574E

E

120590

120590lowast

Δ120573120573

120576

120576 120590

minus120590lowast

Figure 2 Piecewise linear elastoplastic constitutive model Figurecourtesy of Jankowski [29]

where 119874 is the mass matrix 119906119872(119905) denotes displacement of

the 119872th DOF and 119891119873(119905) is the external excitation load on

the 119873th DOF at the time 119905 The total strain of the 119894th trusselement 120576

119894(119905) obeys

119897119894120576119894 (119905) = 119866119894119872

119906119872 (

119905) (2)

119866119894119872

is a transformation matrix whose elements arerelated to the angles between elements and the directions ofdegrees of freedom 119878

119894119895is a diagonal matrix 119878

119894119894= 119864119894119860119894119897119894 119864119894

is Youngrsquos modulus 119897119894is the length of the element 119894 and 119860

119894is

the cross section of the element 119894120576119894(119905) can be split into two parts a purely elastic one and a

plastic one denoted by 1205730119894(119905) so that

120590119894 (119905) = 119864119894

(120576119894 (119905) minus 120573

0

119894(119905)) (3)

In the following only bilinear isotropic hardening plasticityis considered as a relatively basic example (see Figure 2)which requires for each element a single internal hardeningvariable Ψ

119894(119905) called the total plastic strain The evolution

of Ψ119894(119905) in time is governed by the following simple strain

hardening law

Ψ119894 (119905) =

10038161003816100381610038161003816

120573119894 (119905)

10038161003816100381610038161003816

(4)

The yield functionΦ119894(120590119894 Ψ119894) is defined as

Φ119894(120590119894 Ψ119894) =

1003816100381610038161003816120590119894

1003816100381610038161003816minus (120590

⋆

119894+

120574119894119864119894

1 minus 120574119894

Ψ119894) (5)

where 120590⋆119894 120574119894 and 119864

119894are respectively the initial plastic flow

stress the hardening coefficient and Youngrsquos modulus of the119894th truss element and the plastic modulus 120574

119894119864119894(1 minus 120574

119894) is

determined based on a simple geometric analysis of Figure 2The range of admissible stresses is defined by the require-

ment that

Φ119894(120590119894 Ψ119894) le 0 (6)

The plastic flow ( 120573

0

119894= 0) can take place only if the stress

point is on the yield surface defined by Φ119894= 0 This is stated

in the form of the following conditions of complementarityand persistency

120573

0

119894(119905)Φ119894(119905) = 0 and

120573

0

119894(119905)

Φ119894(119905) = 0 where

Φ119894(119905) = Φ

119894(120590119894(119905) Ψ119894(119905)) The yield function is used to define

the set 119884119905of indices of truss elements that are instantaneously

plastic at time 119905

(119894 isin 119884119905) equiv (Φ

119894 (119905) = 0

Φ119894 (119905) = 0) (7)

In practice the response of the linear structure 120576119871119894(119905) is

known in discrete time steps every Δ119905 and the numericalsolution for the elastoplastic structure has to be advancedin the same discrete time steps The discrete strain responseof the elastoplastic structure is expressed in the followingdiscrete form [29]

120576119894 (119905) = 120576

119871

119894(119905) + sum

119895

119905

sum

120591=0

119863

120581120581

119894119895(119905 minus 120591) 120573

0

119895(120591) (8)

where

120576

119871

119894(119905) = sum

119870

119905

sum

120591=0

119863

120581119891

119894119870(119905 minus 120591) 119891119870 (

120591) (9)

while 119863120581119891119894119870(119905) and 119863120581120581

119894119895(119905) are the discrete counterparts of the

continuous impulse response functions 119863

120581120581

119894119895(119905) is the

dynamic influence matrix describing the strain evolutionin the element 119894 in the time step 119905 in response to a unitaryimpulse of virtual distortion generated in the time 120591 in theelement 119895 119863120581119891

119894119870(119905) denotes a dynamic influence matrix

containing the displacement history induced in the 119870thDOF as a response to the unitary distortion impulse appliedin the 119894th element

The discrete update rules for plastic distortions of instan-taneously plastic elements can be stated as

Ψ119894 (119905) = Ψ119894 (

119905 minus Δ119905) +

10038161003816100381610038161003816

Δ120573

0

119894(119905)

10038161003816100381610038161003816

(10)

where Δ1205730119894(119905) denotes the increment of the plastic strain and

Δ120573

0

119894(119905) = 120573

0

119894(119905) minus 120573

0

119894(119905 minus Δ119905) (11)

TheVDMsolution is based on trial steps where the plasticdistortion increments Δ1205730

119894(119905) are determined in each time

step 119905 by freezing temporarily the plastic flow and performinga purely elastic step which yields the trial strain 120576tr

119894(119905) trial

stress 120590tr119894(119905) and trial yield function Φtr

119894(119905) The trial step is

performed by assuming that 1205730tr119894(119905) = 120573

0

119894(119905 minus Δ119905) Ψtr

119894(119905) =

Ψ119894(119905 minus Δ119905) The trial strain is

120576

tr119894(119905) = 120576

119871

119894(119905) + sum

119895

119905minusΔ119905

sum

120591=0

119863

120581120581

119894119895(119905 minus 120591) 120573

0

119895(120591)

+sum

119895

119863

120581120581

119894119895(0) 120573

0

119895(119905 minus Δ119905)

(12)


the trial stress is

120590

tr119894(119905) = 119864119894

(120576

tr119894(119905) minus 120573

0tr119894(119905))

= 119864119894(120576

tr119894(119905) minus 120573

0

119894(119905 minus Δ119905))

(13)

and the corresponding trial yield function is given by

Φ

tr119894(119905) = Φ119894

(120590

tr119894(119905) Ψ

tr119894(119905))

=

10038161003816100381610038161003816

120590

tr119894(119905)

10038161003816100381610038161003816

minus (120590

⋆

119894+

120574119894119864119894

1 minus 120574119894

Ψ119894 (119905 minus Δ119905))

(14)

The actual plastic distortions 1205730119895(119905) can be found by solv-

ing

119864119894sum

119895isin119884119905

(119863

120581120581

119894119895(0) minus

120575119894119895

1 minus 120574119894

)Δ120573

0

119895(119905)

= minusΦ

tr119894(119905) sign120590tr

119894(119905)

(15)

Having the above definition the direct problem is solvedtime step by time step that is for 119905 = 0 Δ119905 119879 The initialconditions are assumed to be zero 120576

119894(0) = 0 1205730

119894= 0 Ψ

119894(0) =

0 and 1198840= 0 At each time step 119905 = Δ119905 119879 the following

computations are necessary

(1) Trial strains stresses and yield functions by (12) to(14)

(2) Temporary assumption of 119884119905= 119884119905minusΔ119905

(3) Plastic distortion increments Δ1205730

119894(119905) by (15)

(4) The corresponding strains and stresses by (3) and (8)(5) (a) For all the elements verification of the yield

condition defined in (6) and its compliance withthe assumed set 119884

119905of the instantaneously plastic

elements needs to be performed (b) For elements119894 isin 119884119905verification of the stress compliance condition

sign120590119894(119905) = sign120590tr

119894(119905) = signΔ1205730

119894(119905) needs to be

performed If required the set 119884119905should be updated

accordingly and the computations should be repeatedfrom point (3) above

(6) Plastic strain increments and total plastic strains by(10) and (11)

Notice that no iteration with respect to the state variablesis required which is a characteristic feature of the presentedapproach Points 3ndash5 are repeated only if the set 119884

119905needs

to be updated in the current time step which happens onlywhen an elastic element enters the plastic regime or aninstantaneously plastic element is unloaded the proper set isthen usually found after just a single update

The above assumptions are used to formulate and numer-ically solve a coupled problem of material redistribution(modifications of element cross sections 119860 and associatedstiffness and mass modifications) The equations are dis-cretized in time 119905 = 0 1 119879 and solved by means of New-mark scheme119879 being the final time after which the construc-tion is evaluated that is the value of an objective function is

calculated Formore details regarding119863120576119894119896119861120576119873119896 and the entire

numerical procedure the reader is encouraged to refer to [7 8]or [29]

Regarding the material properties used throughout thepaper the values are consistent with the previous work andare as follows Youngrsquos modulus 119864 = 210GPa the density120588 = 7800 kgmminus3 and the initial cross section of all elementsis 100mm2

22 The Optimization Problem Consider an initial trussstructure S

119894consisting of the 119899 bar elements with the same

cross section 119860119894= const 119894 = 1 119899 and nodes (massless)

which join the barsThe structure is exposed to impact loadsthat is a mass 119898 with given velocity k is added at a node (ormultiple different nodes in case of a multiple impact event)As a result the initial conditions for k and 119898 in the nodechange accordingly During the entire analysis the mass isassumed to be attached to the node and to move togetherwith it In the numerical model the mass is reflected by anappropriate modification of the mass matrix 119874 in (1)

Let the volume of material used forS119894be 119881 We consider

optimization problems which can be formulated as follows(divided into two classes ldquodesignrdquo and ldquoadaptationrdquo)

(1) Design find new element cross sections 1198601015840119894such that

the corresponding material volume is equal to theinitial one 119881 and given objective function is mini-mizedmaximized

(2) Design it is as (1) but localization of adaptive struc-tural elastoplastic fuses is taken into account

(3) Adaptation find optimal plastic limit 120590

lowast at themoment of impact which will minimizemaximizegiven objective function

In the previous work [7 8] optimization problems (1)and (3) were solved with the use of gradient-based methodwith typical pros and cons related to such approach Thegradient-based method requires a proper definition of theobjective function which has to be differentiable it is proneto converging in a local minimum and so forth From theperspective of design it is tempting to obtain complete free-dom regarding definition of the objective function which canlead for example to handling multiple impact optimization

23 The Particle Swarm Optimization Particle swarm opti-mization (PSO [11]) is a metaheuristic optimization methodwhich iteratively tries to improve a candidate solution Astypical methods of this sort it does not guarantee that theoptimum is found however a very large space can besearched and there are no requirements regarding the objec-tive function Therefore PSO can be used for irregular noisycoarse problems or multiobjective optimization

In the presented approach the classical version of PSOis used only in the design process (cf Section 22) As willbe discussed later during the impact phase selection of theoptimal plastic limit 120590lowast is a smooth problem with a singleminimum which can be efficiently solved by gradient meth-ods Of course in principle also at this stage PSO can be used


(1) For each particle 119894(2) Initialize x with a random value 120583min le 119909119895 le 120583max(3) Let the particlersquos best position pi will be equal to its initial one pi larr x(4) Assign to each particle its initial velocity k so that each component of the vector

is a random number V119895larr (minus|120583max minus 120583min| |120583max minus 120583min|)

(5) Calculate the objective function for particle 119894 119891119894(pi)

(6) Update the swarm best solution blarr xi if 119891119894 le 119891(b)(7) For each particle 119894(8) Pick random numbers 119903

119901 119903119887from the range [0 1]

(9) For each component 119895(10) V

119894119895larr 120596V

119894119895+ 120601119901119903119901(119901119894119895minus 119909119894119895) + 120601119887119903119887(119887119895minus 119909119894119895)

(11) xi larr xi + ki(12) If 119891(xi) lt 119891(pi) then the particlersquos best pi larr xi(13) Update the swarm best solution if 119891

119894le 119891(b) then blarr xi

(14) If a given termination criterion is not met go to (7)

Algorithm 1 The particle swarm optimization algorithm For the basic AIA design problem the particles represent consecutive elementscross sections119909

119894119895= 120583119895 If for example the localization of structural elements is taken into account x is extended accordingly (see Section 41)

as an alternative (eg if a more complex objective function isrequired)

During the design phase each candidate structure Sis represented as a vector of real numbers xi = (119909

119894119895=1 119909119894119895=2

119909119894119895=119899

)

119879 which is in the context of PSO entitled as aldquoparticlerdquo The process of relating x to a structure is usuallyreferred to as ldquocodingrdquo and regarding thematerial redistribu-tion it will correspond to the ratio of modified cross section119860119895to the original one 119860

119895of the 119895th element

119909119895= 120583119895=

119860119895

119860119895

(16)

We will impose limit on the cross sections by requiringthat each component of the vector will be in range 120583min le

119909119895le 120583max where 120583min is the minimum element cross section

here 120583min = 0 which coincides with the element removal(which is trivial to implement unlike in the case of gradientmethods) and 120583max = 3120583

0 where 120583

0is the initial cross

section ratio 1205830= 1 After any modification the particle x

is normalized in a way that the volume of the correspondingstructure is equal to the volume of the initial one 119881 so that theamount of material needed for all the structures is identical

With each particle xi there is an associated vector calledits velocity kj whose elements lie in the range [minus|120583max minus120583min| |120583max minus 120583min|] Additionally each particle 119894 keeps trackof its best position in the history of optimization pi that isa position for which 119891(pi) is extreme (minimal or maximaldepending on the desired optimization objective)

LetN be the set (the swarm) of119873 particles representing119873 structuresThePSO algorithm iterativelymoves all the par-ticles xi through the search space according to their velocitieski Each particle is attracted to its all-time best position piand to the swarm all-time best solution b (updated at eachstep)The degree of this attractiveness along with the particleldquointentionrdquo to follow its velocity is defined by the parameters120601119901 120601119887 and 120596 respectively The PSO algorithm for finding

optimal solution in the sense in the sense of minimizingsome utility function 119891(x) is presented in Algorithm 1

The definition of the minimized objective function 119891(x)of course depends on the desired optimization goal Forexample in order to maximize structural stiffness it can beequal to the deviation of node position from their initiallocation The next section describes an example of suchoptimization

One of the issues of fundamental importance when usingPSO is the proper selection of the control parameters 120596 120601

119901

and 120601119887and the population size 119873 Since PSO is intrinsically

flexible and indeterministic it is impossible to give a formalprescription for values of these parameters which wouldgive the best results in a general case Although there existssome research concerning procedures of how to adjust thesevalues [30] possibly during the optimization process [31]often they are selected by means of a trial-and-error methodIn such procedure 120596 120601

119901 120601119887 and 119873 are chosen arbitrarily

in a way which reflects the nature of underlying objectivefunction One should keep in mind that generally speaking120596 controls the tendency to explore the entire search space120601119901the tendency to explore in the vicinity of local extrema

and 120601119887the convergence rate to the best solution found so

far 119873 is responsible for the diversity of potential solutions(which is also associated with the demand for computationalresources)

It is clear that the PSO control parameters can be verydifferent for various problems and it may be difficult to findsatisfactory balance between exploration and convergence Inorder to select values for our AIA model we have checkedhow the PSObehaves for various parameter sets (asymp30 sets) forthe scenario discussed in Section 31 We have looked at theconvergence rates and the ability to find the globally optimalsolution for various random initial conditions (keeping inmind that119873 should not be too large in order to constrain thecomputational time) This procedure led us to the followingvalues which are used for all the calculations presented in thepaper 120596 = 08 120601

119901= 120601119887= 05 and119873 = 25


m1 1

(a)

m1 1

(b)

Figure 3 (a) Original structure the two leftmost nodes are fixed and the node where impacting mass1198981= 1 kg with the velocity V

1= 5ms

is marked with the arrow (b) Remodeled structure whichmaximizes the stiffness (at the impact node at 119905 = 8ms) Here and in the subsequentfigures diamonds denote fixed points filled circles represent the nodes and a square marks the node where the impact takes placeThe widthof the lines representing elements is proportional to their cross section

0

2

4

6

0 1 2 3 4 5 6 7 8

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus2

minus4

minus6

minus8

minus10

uy1 originaluy1 remodeled

(a)

1

11

12

13

14

15

16

0 5 10 15 20 25 30PSO steps

Overall bestCurrent population best

Optimal solution

The o

bjec

tive f

unct

ion

(times10

3)

(b)

Figure 4 (a) Vertical deflections on the node at which the impact takes place for the original and the PSO remodeled structure (b)Convergence of the PSO method to the optimal solution found by means of the gradient optimization (the horizontal line)

3 Remodeling without AIA Structural Fuses

31 Designing for a Given Impact Event As the first exampleof structural PSO optimization we consider the simple trussstructure depicted in Figure 3(a) where two nodes are fixed(BCs require that the velocity is always 0) and there isone impacting mass at the node labeled with a square andan arrow Identical setup was analyzed in [32] and directcomparison with the gradient method can be made

As the objective function we choose

119891 (x) = sum

119905=0119879

(1199061199101minus 1199061199101119905=0

)

2

(17)

where the analyzed time period is 119879 = 1ms after the impact1199061199101

is the vertical displacement of the node at which theimpact takes place at the discrete time 119905 and 119906

1199101119905=0= 1 is

the nodersquos initial position The result of such optimization is

presented in Figure 3(b) and deflections for the original andthe remodeled structures are shown in Figure 4(a)

From Figure 4(b) one can see that the PSO solutionquickly converges to the best solution found by the gradientmethod (the horizontal line value taken from [32]) Thedetailed values are presented in Table 1 In this relativelysimple case there is one solution to which all the tried initialPSO populations have converged This can be interpreted asthe fact that it is likely that there are no local minimums inthis case

Let us consider another example a more complex struc-ture with two impacting masses at the same moment Identi-cal setup has been analyzed in [7]The objective function willbe analogous to the previous one

119891 (x) = sum

119905=0119879

((1199061199101minus 1199061199101119905=0

)

2

+ (1199061199102minus 1199061199102119905=0

)

2

) (18)


003

004

005

006

007

008

009

01

011

0 10 20 30 40 50 60 70PSO steps

(A)(B)

(C)(G)

The o

bjec

tive f

unct

ion

(times10

3)

(a)

0

02

04

0 02 04 06 08 1

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16

uy1 originaluy2 original

uy1 PSO (A)uy2 PSO (A)

(b)

Figure 5 (a) Convergence of PSO for various initial populations (A) (B) and (C) (G) represents the optimum as found by the gradientmethod [7] (b) Response of the original structure and the one which is a result of PSO (A)

Table 1 Cross sections (mm2) of the 10-element truss structure(Figure 1) original optimized for maximum stiffness by the gra-dient method stiffest by PSO simultaneously optimized for max-imum energy dissipation with a single adaptive element compareSection 41 (lowast marks the optimal localization of the active elementin this case)

Elem Orig Grad PSO PSO-AIA1 110 2333 2339 122lowast

2 110 1036 1031 003 110 2051 2044 7094 110 00 00 30665 110 1428 1422 4596 110 957 968 18977 110 00 00 08 110 691 699 30669 110 1543 1536 010 110 870 865 0

where 1199061199101and 1199061199102denote the vertical location of the nodes at

which the impacting masses are loaded 119879 = 1msIt is understood that the exact optimization path in

PSO depends on the initial swarm Therefore various initialconditions can lead to different solutions if there is morethan one local minimum This can be clearly seen in Figures5 and 6 where PSO converges to distinct structures whichgive very similar values of the objective function It shouldbe noted that some of these solutions are actually better (iegive smaller value of 119891) than the one found by the gradientmethod in [7]

There is no denying the fact that the outcome of suchoptimization is practically unacceptable For example solu-tion (B) will fail if the velocity of the left impacting masswould have even very small nonzero horizontal component

(O)

(A)

(B)

(C)

(G)

Figure 6 Original truss-beam structure (O) with two impactingmasses119898

1= 01 kg V

1= 5ms (the leftnodemarkedwith a square)

1198982= 2 kg V

1= 5ms (the right one) (A) (B) and (C) denote

final states of PSO for various initial conditions (G) represents thesolution obtained with the gradient optimization in [7] Here thetotal width is 11m the height is 01m

The exact final result of PSO as any other optimizationmethod will strongly depend on the chosen objective func-tion including the integration time 119879 PSO starts with aswarm of completely random structures and as it happened itconverged to configurations giving smaller119891which is definedby (18) On the other hand in the previous work the startingpoint of the gradient method was always a structure withequal cross section for all the bars

The example is presented here in order to show that in theentire search space of the problem there exists better solutionthan the one which is found by the gradient method This isnot really surprising bearing in mind the complexity of the


objective function and its dependence on the large numberof variables If one would like to obtain a solution whichcan be applied in practice one would have for exampleexploited the potential problem symmetry or define a specificset of initial conditions for PSO (possibly along with anykind of constraints during the optimization process) Alsoperforming optimization for a large set of expected impactsis desired These issues are discussed in more detail in thefollowing sections

Regarding comparison of computational requirementsfor PSO and gradient methods it should be noted that inthe previous work the latter converged after asymp100 steps For119873 = 25 this is roughly equivalent to 5 PSO steps Lookingat the convergence rates in Figures 4(b) or 5(a) one can seethat there exists computational overhead when comparing tothe gradient-based approach (here by the factor asymp5) This isusually the price one has to pay when PSO is applied How-ever it is worth mentioning that PSO algorithms are trivialto parallelize and it is easy to implement them on widelyavailable multiprocessor platforms

32 Designing for Multiple Impact Events One of the limi-tations of the gradient-based optimization in the discussedstructural design process is the difficulty of definition ofa differentiable objective function which would evaluate astructure for multiple various impacts This can be easilyachieved with PSO Consider a scenario in which a structurecan be potentially loaded with an impact characterized by4-tuples 119899 119898 V and 119901 where 119899 is the node number atwhich the impacting mass 119898 with the velocity V is appliedthe probability of such event is 119901 Let I be the set of all 119873

119868

expected impacts (ie such 4-tuples) Σ119894=0119873

119868

119901119894= 1 Then

the objective functionwhich takes into account the possibilityof multiple impacts can be defined as

119891

1015840(x) = sum

119894=0119873119868

119901119894

119873119868

(

119891 (x | 119899119894 119898119894 V119894)

119891 (xo | 119899119894 119898119894 V119894))

119896

(19)

where 119891(x | 119899119894 119898119894 V119894) denotes the calculated objective

function for the structure corresponding to the particle xprovided that the impact is at the node 119899

119894with given mass

119898119894and velocity V

119894 xo is the optimal structure designed for

the impact 119899119894 119898119894 V119894 and 119896 controls the penalty for deviation

from 119891(xo) (we use 119896 = 2)The normalization with respect to xo requires precalcu-

lation of the optimal value for a single impact This assuresthat the input from any impact to the total objective function119891

1015840 is proportional only to the probability 119901119894(and not eg to

the impacting mass where smaller mass would lead to muchsmaller deflections)This procedure is particularly importantwhen expected impacts significantly differ in kinetic energy

As an example let us consider the structure presentedin Figure 7 where an impact is expected at any node in theupper part except the ones which are fixed The impactingmassvelocity pairs are drawn from the set 2 kg 5ms05 kg 10ms All the events are assumed to be equiprob-able The objective function for each single impact 119891(x |

119899119894 119898119894 V119894) is the same as that in (17) that is the deflections

(O)

(1)

(2)

(3)

(X)

Figure 7 (O) Original structure which is a subject to multiplepossible loads (1)ndash(3) A sample optimum PSO remodeling for asingle impact at the marked nodes119898 = 2 kg V = 5ms (1) and119898 =

05 kg V = 10ms (2 3) (X) A PSO remodeled structure optimizedfor multiple impact classes x1015840O The triangles in (X) represent placesof potential impacts

at the impacting node are minimized For example (1) isoptimized for a single impact with 119898 = 2 kg V = 5msat the node marked with a square Note that according tothe limits on cross section stated in Section 23 the barson the leftmost side could not exceed 120583max = 3120583

0 During

the optimization process these elements reached 120583max Theremaining elements have virtually no influence on 119891 for thisimpact and consequently the remaining mass is randomlydistributed among these bars

In case of themultiple impact optimization using1198911015840 PSOconverged to a final solution depicted in Figure 7(X) and thefinal value of 1198911015840(x1015840O) = 4032 This should be interpreted asthat the final structure performs on average two times worse(for 119896 = 2) than the one optimized for a given impact Ideally119891

1015840 would be equal to unity which would mean that thestructure is perfectly optimized for all the possible impacts(of course usually it is impossible)

4 Adaptation to Impact Loads

Up to this moment we have discussed the process of rede-signing structures in order to maximize stiffness As men-tioned in Introduction an essential part of any AIA system isresponsiveness at the moment of an impact In our scenariothis can be realized when some elements act as structuralfuses that is have properly adjusted plastic limit see (3) Itis reasonable to assume that the dissipated plastic-like energy


128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))

Figure 8 Absorbed plastic energy (in J) as a function of the impact-ing mass and velocity for a truss structure as in Figure 3 whereelement number 1 has 120590 = const = 106

119864pl in some given time 119879 = 1ms should be maximized (as in[8])

119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)

where 119894 = 1 119873119860denotes the active elements

As a basic example consider the structure fromFigure 3(a) where element number 1 connecting nodes 1and 2 is replaced by an elastoplastic one Setting the elastop-lastic limit to the constant value 120590lowast = 106 Pa leads to energyabsorption which depends on the impacting mass and itsvelocity Such dependence is plotted in Figure 8

Assuming that a structural geometry is given alongwith the location of the active elements the AIA problemat the moment of impact is twofold identification of theimpacting mass and velocity and finding the plasticity limits120590

lowast for all the active elements As stated in Introductionimpact identification with simultaneous mass and velocityidentification is a difficult task on itself and is beyond thescope of this paper For given 119898 and V the dependence ofoptimal 120590lowast can be calculated for the truss structure with 10elements (Figure 3) it is depicted in Figure 9

Naturally using active elements with real-time 120590lowast adjust-ment will give some benefit only if the dissipated energy issignificantly larger than in case of some constant 120590lowast chosenduring the design process The potential gain is shown inFigure 9(b) where one can clearly see that in the given rangeof impacting masses active response can dissipate 7-8 timesmore energy than the passive elastoplastic constructionRemarkably for the considered structure and location of theactive element the optimal plasticity limit dependsmostly onthe impacting mass and not the velocity

Using PSO at the moment of impact in order to adjust120590

lowast is in principle possible however it would be somehowextravagant at least if the objective function like the one

defined in (20) is considered Having information about theimpacting 119898 and V a gradient-based real-time method canbe applied without any problems Another practical solutionwould be to calculate the optimal plastic limits offline andthen use them in a tabulated form during a real impact

In this AIA problem PSO can bring on two other majorimprovements Firstly in the previous work the location ofstructural fuses was rather guessed than selected on someformal basisThe reason for such procedure was due to prob-lems with the proper definition of objective function whichwould take position of the active elements into accountPSO can be deployed to find optimal location of activeelements for a given structure However the second potentialgain is the possibility of solving a problem of simultaneousmaterial redistribution and structural fuses location In thenext subsection we will look at an example of the latter

41 Using PSO for Simultaneous Material Redistribution andStructural Fuses Location Let us again consider the trussstructure from Figure 3 This time the goal will be tomaximize the dissipated energy 119864pl by using AIA fuse whichshould be located in place of one of the elastic elements Fromthe PSO point of view this will be realized by modifyingrepresentation of the structure as a particle x Let the newextended representation x1015840 be x1015840 = (x y) where y isyet another vector with the length equal to number of theelements (ie equal to the length of x) The range of possiblevalues as the elements of y is ⟨0 1⟩ (instead of 120583min and120583max as in the algorithmdescription Section 23) Now let theactive AIA element be located at the position correspondingto themaximal119910

119894(or the onewith the smallest 119894 if there exists

more than one maximal 119910119894) If more than one of the active

elements is expected one can localize them in 1198941 1198942 places

so that 1199101198941

le 1199101198942

le sdot sdot sdot le 119910119894 =1198941=1198942=sdotsdotsdot

The extended particle representation x1015840 is used to cal-culate the objective function for the structure with thematerial distribution defined by x and the AIA elementslocation by y Note that this quite versatile way of codingcan be used in order to achieve any desired goals of thestructural optimization during the design process Of coursethe cost of AIA element (and consequently their total numberin any structure) can be straightforwardly incorporatedPSO provides a surprisingly efficient mechanism for findingoptimal solution of problems defined in a complex andnoncontinuous way

Assume that one of the elements of the 10-elementexamples from Figure 3 ((a)(b) ie originalstiff) is replacedby a single adaptive element By means of a brute-forcemethod we can trivially check how much plastic-like energyis dissipated for the given impact parameters (119898

1= 1 kg V

1=

5ms ie with energy 25 J) for all the possible locations of theadaptive element It turns out that the structure remodeled formaximum stiffness can give only slightly better results In thecase of original structure maximum 119864pl = 714 J is observedwhen the adaptive element is at the link number 1 For theremodeled structuremaximum119864pl = 842 J is obtainedwhenthe link number 8 is adaptive see Table 2

Using PSOmakes it possible to simultaneously search foroptimal cross section of the elements and the localization of


20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400

Optimal log10 (120590lowast (Pa))

(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600

(Epl optimal)(Epl 120590lowast = 106(Pa))

(b)

Figure 9 (a)The optimal elastoplastic limit 120590lowast leading to the largest plastic-like energy dissipation (20) as a function of the impacting massand velocity (b) The potential gain in using the optimal 120590lowast over the constant one 120590lowast = 106 Pa

Table 2 Plastic-like energy 119864pl (in J) dissipation for optimal stressyield 120590

lowast (in MPa) for various locations of the adaptive element(instead of the elastic one) for different 10-element trusses original(Figure 3(a)) remodeled for maximal stiffness (Figure 3(b)) andoptimized by PSO (Figure 10(A))119864pl(119905) is shown in Figure 11 for thecases marked with boldface

El Original Stiff PSO120590

lowast119864pl 120590

lowast119864pl 120590

lowast119864pl

1 133 714 49 717 133 14102 37 385 65 614 mdash mdash3 133 710 75 587 178 10204 75 517 931 005 34 10825 60 282 70 497 21 0576 100 258 178 781 75 9457 1 082 mdash mdash mdash mdash8 87 216 191 842 32 13499 37 225 100 376 mdash mdash10 12 238 70 476 mdash mdash

the adaptive dissipators Figure 10 depicts a sample outcomeof such optimization and one can see that the structure differssignificantly from the one optimized for stiffness only Thegain regarding the dissipated energy is significant in this casemaximal 119864pl = 14 J

We should note that in the previous work the locationof adaptive elements was chosen rather on reasonable pre-sumptions than on optimization outcomeMoreover this wasdone on structures optimized in the first stage formaximumstiffness In order to address this problem with gradient opti-mization one can simply consider brute-force calculationsfor all possible loca-tions of active elements However incase of complex structures with many structural fuses this

(A)

(B)

Figure 10 Structure from Figure 3 remodeled for maximal energydissipation 119864pl = 1410 J The location of the active element asfound by PSO is marked with the dashed line (B) An alternativesolution giving slightly smaller dissipation 119864pl = 137 J howeverwith a different absorption characteristic compare Figure 11

approach will quickly become computationally intractablewhile still the issue of simultaneous mass redistribution andfuses localization is not taken into account

As stated above initially the swarm consists of completelyrandom particles (ie structures) and it is possible thatfor larger structures PSO will evolve towards a ldquobizarrerdquoresult for example Figure 6 Such outcome formally givesnear-optimal value of the objective function however thecommon sense indicates that such solution is rather imprac-tical Figure 12 depicts two results of PSO optimization formaximal plastic-like energy absorption for the 29-element


0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)

StiffPSO (A) Adap el 1

PSO (A) Adap el 8PSO (B) Adap el 1

Plas

tic en

ergy

Epl

(J)

Figure 11 Absorption of plastic-like energy after an impact at119905 = 0 for various structures and localization of the adaptive elementcompare Figure 10 and Table 2

(A)

(B)

Figure 12 Schematic representation of the outcome of PSO opti-mization for the structure as in Figure 7 with two adaptive elements(A) An example of a nonsymmetric ldquobizarrerdquo result giving 119864pl =

199 J (B) Result under the assumption of symmetry which is takeninto account in the particle coding procedure 119864pl = 254 J

structure supported on both sides as in Figure 7 Simultane-ous optimization for mass redistribution and localization oftwo adaptive elements (with the same 120590lowast for simplicity) forthe set of 10 possible impacts was performed (2 kg 5ms or05 kg 10ms at each of the 5 nodes marked with a square)In other words the objective function is to maximize

119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)

where 119899119894numbers the possible impacts and 119864pl | 119899119894 is calcu-

lated for the impact 119899119894 1ms after the event120590lowast being optimally

adjusted at the impact moment for maximizing the energyabsorption

The structure depicted in Figure 12(A) is characterizedby the objective function 119864pl = 199 J One expects that forthe problem defined in the way described above the outcomeshould be a symmetric structure since all the nodes haveequal probability of being loaded with the same conditionsNevertheless PSO evolved from random initial condition

0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)

Figure 12(A)Figure 12(B)

Plas

tic en

ergy

Epl

(J)

Figure 13 Absorption of energy 119864pl by the two structures depictedin Figure 12 The impact takes place at the leftmost node markedwith a triangle119898 = 05 kg V = 10ms

to this clearly nonsymmetrical configuration with two fuseslocated next to each other

Generally larger problems require searching a hugeparameter space and it is not surprising that the swarm getsstuck in a local minimumwhich clearly is not a suitable solu-tion In PSO it is easy to incorporate particle coding whichexploits symmetry and any other features expected regardingthe outcome of optimizationThe benefit is twofold the resultmore suited to expectations and reduced search space Anexample of PSO result with a coding which retains structuralsymmetry is shown in Figure 12(B) in this case 119864pl = 254 JThis is a superior value to that for case (A) moreover thesymmetric solution has better absorption characteristics inthe sense that is closer to a linear one see Figure 13

Finally note that the structure presented in Figure 7(X)evolved from random swarm towards a symmetric config-uration (although not perfectly symmetric) which reflectsthe boundary conditions In principle by appropriate mod-ification of the swarm size and PSO control parameterssimilar results can be achieved for more complex structureswith fuses For evenmore demanding multimodal problemsa specific modification of PSO should be employed forexample [33]

5 Conclusions and Outlook

We have used one of the versatile metaheuristic optimiza-tion techniques the particle swarm optimization in thenumerical process of designing structures which are ableto actively adapt to impacts AIA systems form a relativelynew branch of mathematical problems in engineering withmany subtopics open to research Up to now the consideredAIA truss structures have been optimized only by classicalgradient-based approaches One of the drawbacks of suchmethods is the need of ldquomathematically properrdquo definition


of continuous objective functions This makes it difficult todesign structures which can for example adapt to a set ofpossible impacts

We have shown that in the context of AIA design PSOcan

(i) efficiently reproduce results previously obtained bymeans of the gradient optimization (Section 31)

(ii) easily provide alternative solutionswith similar valuesof the objective function (Section 31)

(iii) be used to find out optimal structures where theobjective function is constructed from multiple variousimpact responses and therefore obtain results whichare optimized for a given set of expected impacts(Section 32)

(iv) be used to search the space for location of the struc-tural fuses (unlike the gradient methods) this canbe even done simultaneously with mass redistributionand the outcome of such optimization can be com-pletely different compared with the previous resultsobtained with gradient methods (Section 41)

As mentioned in Section 1 generally such structuraloptimization should simultaneously solve for size shape andtopology Incorporating AIA adds here yet another variablenamely solving for location of the structural fuses Followingthe presented preliminary research the future work will befocused on simultaneous optimization which will includealso topology not only adaptive element localization andmass redistribution Additionally more complex impact sce-narios will be considered for example various angles ofthe velocity vector of the impacting mass will be taken intoaccount

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This project was financed from the funds of the NationalScience Centre (Poland) allocated on the basis of decision noDEC-201205BST802971

References

[1] M P Bendsoslashe and N Kikuchi ldquoGenerating optimal topologiesin structural design using a homogenization methodrdquo Com-puterMethods in AppliedMechanics and Engineering vol 71 no2 pp 197ndash224 1988

[2] J Holnicki-Szulc and L Knap ldquoAdaptive crashworthiness con-ceptrdquo International Journal of Impact Engineering vol 30 no 6pp 639ndash663 2004

[3] J Holnicki-Szulc Ed Smart Technologies for Safety EngineeringJohn Wiley amp Sons 2008

[4] K Sekuła C Graczykowski and J Holnicki-Szulc ldquoOn-lineimpact load identificationrdquo Shock and Vibration vol 20 no 1pp 123ndash141 2013

[5] Ł Jankowski ldquoOff-line identification of dynamic loadsrdquo Struc-tural andMultidisciplinary Optimization vol 37 no 6 pp 609ndash623 2009

[6] P Kołakowski M Wikło and J Holnicki-Szulc ldquoThe virtualdistortionmethodmdasha versatile reanalysis tool for structures andsystemsrdquo Structural andMultidisciplinary Optimization vol 36no 3 pp 217ndash234 2008

[7] M Wiklo and J Holnicki-Szulc ldquoOptimal design of adaptivestructures part I Remodeling for impact receptionrdquo Structuraland Multidisciplinary Optimization vol 37 no 3 pp 305ndash3182009

[8] M Wiklo and J Holnicki-Szulc ldquoOptimal design of adaptivestructures part II Adaptation to impact loadsrdquo Structural andMultidisciplinary Optimization vol 37 no 4 pp 351ndash366 2009

[9] R Kicinger T Arciszewski and K De Jong ldquoEvolutionarycomputation and structural design a survey of the state-of-the-artrdquoComputers and Structures vol 83 no 23-24 pp 1943ndash19782005

[10] A H Gandomi X-S Yang S Talatahari and A H AlaviMetaheuristic Applications in Structures and InfrastructuresElsevier Newnes Australia 2013

[11] M Clerc Particle Swarm Optimization ISTE Wiley 2010[12] X Hu Y Shi and R C Eberhart ldquoRecent advances in particle

swarmrdquo in Proceedings of the IEEE Congress on EvolutionaryComputation vol 1 p 90 Portland Ore USA June 2004

[13] K Deb and S Gulati ldquoDesign of truss-structures for minimumweight using genetic algorithmsrdquo Finite Elements in Analysisand Design vol 37 no 5 pp 447ndash465 2001

[14] Y Yang and C Kiong Soh ldquoAutomated optimum design ofstructures using genetic programmingrdquo Computers amp Struc-tures vol 80 no 18-19 pp 1537ndash1546 2002

[15] O Hasancebi and F Erbatur ldquoLayout optimisation of trussesusing simulated annealingrdquo Advances in Engineering Softwarevol 33 no 7ndash10 pp 681ndash696 2002

[16] G-C Luh and C-Y Lin ldquoOptimal design of truss structuresusing ant algorithmrdquo Structural and Multidisciplinary Opti-mization vol 36 no 4 pp 365ndash379 2008

[17] G-C Luh and C-Y Lin ldquoOptimal design of truss-structuresusing particle swarm optimizationrdquo Computers amp Structuresvol 89 no 23-24 pp 2221ndash2232 2011

[18] H M Gomes ldquoTruss optimization with dynamic constraintsusing a particle swarm algorithmrdquo Expert Systems with Appli-cations vol 38 no 1 pp 957ndash968 2011

[19] A Kaveh and A Zolghadr ldquoDemocratic PSO for truss layoutand size optimization with frequency constraintsrdquo Computersamp Structures vol 130 pp 10ndash21 2014

[20] A Kaveh and A Zolghadr ldquoComparison of nine meta-heuristicalgorithms for optimal design of truss structures with frequencyconstraintsrdquoAdvances in Engineering Software vol 76 pp 9ndash302014

[21] S Gholizadeh ldquoLayout optimization of truss structures byhybridizing cellular automata and particle swarm optimiza-tionrdquo Computers amp Structures vol 125 pp 86ndash99 2013

[22] U T Ringertz ldquoOn methods for discrete structural optimiza-tionrdquo Engineering Optimization vol 13 no 1 pp 47ndash64 1988

[23] L J Li Z B Huang and F Liu ldquoA heuristic particle swarm opti-mization method for truss structures with discrete variablesrdquoComputers amp Structures vol 87 no 7-8 pp 435ndash443 2009

[24] J Holnicki-Szulc and J T Gierlinski Structural Analysis Designand Control by the Virtual DistortionMethodWiley ChichesterUK 1995


[25] Q Zhang L Jankowski and Z-D Duan ldquoIdentification ofcoexistent load and damagerdquo Structural and MultidisciplinaryOptimization vol 41 no 2 pp 243ndash253 2010

[26] Q Zhang Ł Jankowski and Z-D Duan ldquoSimultaneous iden-tification of moving masses and structural damagerdquo Structuraland Multidisciplinary Optimization vol 42 no 6 pp 907ndash9222010

[27] M Mroz L Jankowski and J Holnicki-Szulc ldquoVDM-basedidentification of localized damage induced dampingrdquo in Pro-ceedings of the 5th European Workshop on Structural HealthMonitoring pp 988ndash993 July 2010

[28] U Kirsch ldquoReanalysis and sensitivity reanalysis by combinedapproximationsrdquo Structural and Multidisciplinary Optimiza-tion vol 40 no 1ndash6 pp 1ndash15 2010

[29] Ł JankowskiDynamic Load Identification for Structural HealthMonitoring vol 2 of IPPT Reports Institute of FundamentalTechnological Research Polish Academy of Sciences 2013

[30] Y Shi and R C Eberhart ldquoParameter selection in particleswarm optimizationrdquo in Evolutionary Programming VII V WPorto N Saravanan DWaagen and A E Eiben Eds vol 1447of Lecture Notes in Computer Science p 591 Springer BerlinGermany 1998

[31] J Wang ldquoParticle swarm optimization with adaptive parametercontrol and oppositionrdquo Journal of Computational InformationSystems vol 7 no 12 pp 4463ndash4470 2011

[32] M Lo Projektowanie ustrojow adaptacyjnych poddawanychobciazeniom udarowym [PhD thesis] IPPT PAN WarszawaPoland 2007

[33] G Zhang and Y Li ldquoParallel and cooperative particle swarmoptimizer for multimodal problemsrdquoMathematical Problems inEngineering vol 2015 Article ID 743671 10 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


quite demanding separate engineering problem on itself [45] and will not be considered in this paper Instead we willfocus on the process of optimal design of structures whichcan adapt by using ldquostructural fusesrdquoThese are elements withelastoplastic type of response with controllable yield stressFor example in real application such fuse can be made upof MR fluid where stresses are controlled by magnetic fieldsHowever technical details regarding construction of theseelements are not relevant in this work

In the field of typical passive structural design there isa huge amount of existing numerical methods and ready-to-use computer codes However to date the number ofexisting tools for designingAIA structures is very limitedThemain reason for this is that AIA is a relatively new conceptand also it differs in many aspects regarding the optimiza-tion process Naturally the main question which must beanswered is actually the goal of optimization This dependson the problem being solved involved energies if a structureshould answer to a critical emergency impact or maybeperiodic loads and so forth As a measure of success onecan define minimization of accelerations maximization ofstructural stiffness somehow smoothing structural responsepreserving structural integrity or maximization of dissipatedenergy in some chosen time interval Generally the requiredobjective function of optimization can be a very complex one

In this paper we consider a truss-like structure withelastoplastic fuses which can adapt to impact by adjustingthe plasticity limit The structural response is calculatedby means of virtual distortion methods (VDMs) [6] andthe presented results are continuation of the previous work[7 8] where a typical gradient-based method has beenused in order to achieve the desired optimization goal Themain disadvantage of gradient methods is the mathematicalrequirements regarding the objective function (eg it hasto be differentiable) As mentioned during AIA designprocess one can take into accountmany factors impacts withvarious parameters and at different geometrical places andcomplex goal definitions for example including costs of theused elements Some problems involve solvingmultiobjectiveoptimization (ie require simultaneous optimization ofmorethan one objective function) which can be a particularlydifficult task with any classic gradient-based approach

All this makes it tempting to use other possibly nonexactbut otherwise versatile optimization techniques like the onesbased on the so-called soft computing approach usuallysome sort of evolutionary computation Using it in thestructural design is a relatively new paradigm which hasa tremendous impact on the field and a vast amount ofrecent research is devoted to the subject A survey can befound in for example [9] and a great overview regardingmetaheuristic methods in structural design can be found in[10]

Here we consider the applicability of particle swarmoptimization (PSO) method in the process of designing AIAstructures PSO is one of the techniques which belongs tothe wide class of evolutionary computing It is based on aset of potential solutions (called ldquoswarmrdquo) which is graduallyimproved until a certain criterion of acceptance is metPSO is successfully applied in a very wide field of complex

engineering and scientific optimization problems includingmechanics robotics artificial intelligence transportationbiology and many more see [11 12] A recent survey onPSO application in the field of electrical and electronic engi-neering automation control systems communication the-ory operations research mechanical engineering fuel andenergy medicine chemistry and biology is available

Generally optimization of truss-like structures falls intothree categories size shape and topology however evenmore demanding is a simultaneous optimization where thedesign variables describing these properties are integratedVarious techniques from the field of evolutionary computa-tion have been applied to address these problems from thefield of traditional passive design for example genetic algo-rithms [13] genetic programming [14] simulated annealing[15] or ant colony optimization [16] However it seems thatrecently one of the most popular method is PSO along withits various modifications

For example in [17] the authors applied PSO to opti-mize topologies of truss structures optimizing for minimumweight under stress deflection and kinematic stability con-straints They have considered a two-stage technique wherefirst topology was optimized (withmodified binary PSO) andthen size and shape (by means of ldquoattractiverdquo and ldquorepulsiverdquoPSO) It was reported that the methodology proposed bythe authors can find superior truss structures compared withthose found with classical optimization However simultane-ous optimization for all the criteria in principle should givebetter results since these design problems are not linearlyseparable

Another challenging class of structural design problemsare the ones which involve optimization with frequencyconstraints In such problems the search space is particu-larly highly nonlinear and nonconvex with numerous localoptima When using any heuristic algorithms in such cases aneed to find a balance between exploitation and explorationarises It is of vital importance that a method explores thesearch space in a way which will enable it to find satisfyingsolution while at the same time it will not get stuck in alocal minimum It has been shown that accordingly modifiedPSO algorithmwas capable of dealing with such tasks [18 19]A recent comparison of nine metaheuristic algorithms foroptimal design of truss structures with frequency constraintscan be found in [20]

As in the case of frequency constraints the classical PSOis often modified to deal with a specific problem It is alsoincorporated together with other optimization techniquesin order to solve problems in a more efficient way Forexample to optimize layout of truss structures PSO hasbeen successfully integratedwith ldquononclassicalrdquomethods likecellular automata [21] It has been demonstrated that suchcombination has led to better solutions and more effectiveconvergence than in case of using ldquopurerdquo PSO

According to our knowledge to date there have beenno results regarding usage of PSO in the AIA domain andconsequently there are no standard problems or results whichwe can refer to In this work we demonstrate how PSO canbe employed in order to find optimal mass redistribution and


1

1 1

1

1 2 2 3

34 4

5

5 6

6

7

8

9

10





2 The Model







119874119873119872

119872 (

119905) + 119866

119879

119873119894119878119894119895[119897119895(120576119895 (119905) minus 120573

0

119895(119905))] = 119891119873 (

119905) (1)


120574E

120574E

E

120590

120590lowast

Δ120573120573

120576

120576 120590

minus120590lowast





119894(119905) obeys

119897119894120576119894 (119905) = 119866119894119872

119906119872 (

119905) (2)

119866119894119872



119894119894= 119864119894119860119894119897119894 119864119894


119894is



120590119894 (119905) = 119864119894

(120576119894 (119905) minus 120573

0

119894(119905)) (3)




hardening law

Ψ119894 (119905) =

10038161003816100381610038161003816

120573119894 (119905)

10038161003816100381610038161003816

(4)


Φ119894(120590119894 Ψ119894) =

1003816100381610038161003816120590119894

1003816100381610038161003816minus (120590

⋆

119894+

120574119894119864119894

1 minus 120574119894

Ψ119894) (5)

where 120590⋆119894 120574119894 and 119864



119894119864119894(1 minus 120574

119894) is


ment that

Φ119894(120590119894 Ψ119894) le 0 (6)


0




120573

0

119894(119905)Φ119894(119905) = 0 and

120573

0

119894(119905)

Φ119894(119905) = 0 where

Φ119894(119905) = Φ




(119894 isin 119884119905) equiv (Φ

119894 (119905) = 0

Φ119894 (119905) = 0) (7)



120576119894 (119905) = 120576

119871

119894(119905) + sum

119895

119905

sum

120591=0

119863

120581120581

119894119895(119905 minus 120591) 120573

0

119895(120591) (8)

where

120576

119871

119894(119905) = sum

119870

119905

sum

120591=0

119863

120581119891

119894119870(119905 minus 120591) 119891119870 (

120591) (9)

while 119863120581119891119894119870(119905) and 119863120581120581



120581120581

119894119895(119905) is the





Ψ119894 (119905) = Ψ119894 (

119905 minus Δ119905) +

10038161003816100381610038161003816

Δ120573

0

119894(119905)

10038161003816100381610038161003816

(10)


Δ120573

0

119894(119905) = 120573

0

119894(119905) minus 120573

0

119894(119905 minus Δ119905) (11)




119894(119905) trial




0

119894(119905 minus Δ119905) Ψtr

119894(119905) =


120576

tr119894(119905) = 120576

119871

119894(119905) + sum

119895

119905minusΔ119905

sum

120591=0

119863

120581120581

119894119895(119905 minus 120591) 120573

0

119895(120591)

+sum

119895

119863

120581120581

119894119895(0) 120573

0

119895(119905 minus Δ119905)

(12)


the trial stress is

120590

tr119894(119905) = 119864119894

(120576

tr119894(119905) minus 120573

0tr119894(119905))

= 119864119894(120576

tr119894(119905) minus 120573

0

119894(119905 minus Δ119905))

(13)


Φ

tr119894(119905) = Φ119894

(120590

tr119894(119905) Ψ

tr119894(119905))

=

10038161003816100381610038161003816

120590

tr119894(119905)

10038161003816100381610038161003816

minus (120590

⋆

119894+

120574119894119864119894

1 minus 120574119894

Ψ119894 (119905 minus Δ119905))

(14)


ing

119864119894sum

119895isin119884119905

(119863

120581120581

119894119895(0) minus

120575119894119895

1 minus 120574119894

)Δ120573

0

119895(119905)

= minusΦ

tr119894(119905) sign120590tr

119894(119905)

(15)


119894(0) = 0 1205730

119894= 0 Ψ

119894(0) =






119894(119905) by (15)





sign120590119894(119905) = sign120590tr

119894(119905) = signΔ1205730

119894(119905) needs to be





119905needs

























119901 119903119887from the range [0 1]


119894119895larr 120596V

119894119895+ 120601119901119903119901(119901119894119895minus 119909119894119895) + 120601119887119903119887(119887119895minus 119909119894119895)








119894119895=1 119909119894119895=2

119909119894119895=119899

)



119909119895= 120583119895=

119860119895

119860119895

(16)




0 where 120583









119901








119901= 120601119887= 05 and119873 = 25


m1 1

(a)

m1 1

(b)


1= 5ms


0

2

4

6

0 1 2 3 4 5 6 7 8

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus2

minus4

minus6

minus8

minus10


(a)

1

11

12

13

14

15

16

0 5 10 15 20 25 30PSO steps


Optimal solution

The o

bjec

tive f

unct

ion

(times10

3)

(b)





119891 (x) = sum

119905=0119879

(1199061199101minus 1199061199101119905=0

)

2

(17)



1199101119905=0= 1 is





119891 (x) = sum

119905=0119879

((1199061199101minus 1199061199101119905=0

)

2

+ (1199061199102minus 1199061199102119905=0

)

2

) (18)


003

004

005

006

007

008

009

01

011

0 10 20 30 40 50 60 70PSO steps

(A)(B)

(C)(G)

The o

bjec

tive f

unct

ion

(times10

3)

(a)

0

02

04

0 02 04 06 08 1

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16



(b)




2 110 1036 1031 003 110 2051 2044 7094 110 00 00 30665 110 1428 1422 4596 110 957 968 18977 110 00 00 08 110 691 699 30669 110 1543 1536 010 110 870 865 0





(O)

(A)

(B)

(C)

(G)


1= 01 kg V


1198982= 2 kg V









119868


119868

119901119894= 1 Then


119891

1015840(x) = sum

119894=0119873119868

119901119894

119873119868

(

119891 (x | 119899119894 119898119894 V119894)

119891 (xo | 119899119894 119898119894 V119894))

119896

(19)













(O)

(1)

(2)

(3)

(X)




0 During







128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))



119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)














so that 1199101198941

le 1199101198942




1= 1 kg V

1=




20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

1 1

1

1 2 2 3

34 4

5

5 6

6

7

8

9

10





2 The Model







119874119873119872

119872 (

119905) + 119866

119879

119873119894119878119894119895[119897119895(120576119895 (119905) minus 120573

0

119895(119905))] = 119891119873 (

119905) (1)


120574E

120574E

E

120590

120590lowast

Δ120573120573

120576

120576 120590

minus120590lowast





119894(119905) obeys

119897119894120576119894 (119905) = 119866119894119872

119906119872 (

119905) (2)

119866119894119872



119894119894= 119864119894119860119894119897119894 119864119894


119894is



120590119894 (119905) = 119864119894

(120576119894 (119905) minus 120573

0

119894(119905)) (3)




hardening law

Ψ119894 (119905) =

10038161003816100381610038161003816

120573119894 (119905)

10038161003816100381610038161003816

(4)


Φ119894(120590119894 Ψ119894) =

1003816100381610038161003816120590119894

1003816100381610038161003816minus (120590

⋆

119894+

120574119894119864119894

1 minus 120574119894

Ψ119894) (5)

where 120590⋆119894 120574119894 and 119864



119894119864119894(1 minus 120574

119894) is


ment that

Φ119894(120590119894 Ψ119894) le 0 (6)


0




120573

0

119894(119905)Φ119894(119905) = 0 and

120573

0

119894(119905)

Φ119894(119905) = 0 where

Φ119894(119905) = Φ




(119894 isin 119884119905) equiv (Φ

119894 (119905) = 0

Φ119894 (119905) = 0) (7)



120576119894 (119905) = 120576

119871

119894(119905) + sum

119895

119905

sum

120591=0

119863

120581120581

119894119895(119905 minus 120591) 120573

0

119895(120591) (8)

where

120576

119871

119894(119905) = sum

119870

119905

sum

120591=0

119863

120581119891

119894119870(119905 minus 120591) 119891119870 (

120591) (9)

while 119863120581119891119894119870(119905) and 119863120581120581



120581120581

119894119895(119905) is the





Ψ119894 (119905) = Ψ119894 (

119905 minus Δ119905) +

10038161003816100381610038161003816

Δ120573

0

119894(119905)

10038161003816100381610038161003816

(10)


Δ120573

0

119894(119905) = 120573

0

119894(119905) minus 120573

0

119894(119905 minus Δ119905) (11)




119894(119905) trial




0

119894(119905 minus Δ119905) Ψtr

119894(119905) =


120576

tr119894(119905) = 120576

119871

119894(119905) + sum

119895

119905minusΔ119905

sum

120591=0

119863

120581120581

119894119895(119905 minus 120591) 120573

0

119895(120591)

+sum

119895

119863

120581120581

119894119895(0) 120573

0

119895(119905 minus Δ119905)

(12)


the trial stress is

120590

tr119894(119905) = 119864119894

(120576

tr119894(119905) minus 120573

0tr119894(119905))

= 119864119894(120576

tr119894(119905) minus 120573

0

119894(119905 minus Δ119905))

(13)


Φ

tr119894(119905) = Φ119894

(120590

tr119894(119905) Ψ

tr119894(119905))

=

10038161003816100381610038161003816

120590

tr119894(119905)

10038161003816100381610038161003816

minus (120590

⋆

119894+

120574119894119864119894

1 minus 120574119894

Ψ119894 (119905 minus Δ119905))

(14)


ing

119864119894sum

119895isin119884119905

(119863

120581120581

119894119895(0) minus

120575119894119895

1 minus 120574119894

)Δ120573

0

119895(119905)

= minusΦ

tr119894(119905) sign120590tr

119894(119905)

(15)


119894(0) = 0 1205730

119894= 0 Ψ

119894(0) =






119894(119905) by (15)





sign120590119894(119905) = sign120590tr

119894(119905) = signΔ1205730

119894(119905) needs to be





119905needs

























119901 119903119887from the range [0 1]


119894119895larr 120596V

119894119895+ 120601119901119903119901(119901119894119895minus 119909119894119895) + 120601119887119903119887(119887119895minus 119909119894119895)








119894119895=1 119909119894119895=2

119909119894119895=119899

)



119909119895= 120583119895=

119860119895

119860119895

(16)




0 where 120583









119901








119901= 120601119887= 05 and119873 = 25


m1 1

(a)

m1 1

(b)


1= 5ms


0

2

4

6

0 1 2 3 4 5 6 7 8

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus2

minus4

minus6

minus8

minus10


(a)

1

11

12

13

14

15

16

0 5 10 15 20 25 30PSO steps


Optimal solution

The o

bjec

tive f

unct

ion

(times10

3)

(b)





119891 (x) = sum

119905=0119879

(1199061199101minus 1199061199101119905=0

)

2

(17)



1199101119905=0= 1 is





119891 (x) = sum

119905=0119879

((1199061199101minus 1199061199101119905=0

)

2

+ (1199061199102minus 1199061199102119905=0

)

2

) (18)


003

004

005

006

007

008

009

01

011

0 10 20 30 40 50 60 70PSO steps

(A)(B)

(C)(G)

The o

bjec

tive f

unct

ion

(times10

3)

(a)

0

02

04

0 02 04 06 08 1

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16



(b)




2 110 1036 1031 003 110 2051 2044 7094 110 00 00 30665 110 1428 1422 4596 110 957 968 18977 110 00 00 08 110 691 699 30669 110 1543 1536 010 110 870 865 0





(O)

(A)

(B)

(C)

(G)


1= 01 kg V


1198982= 2 kg V









119868


119868

119901119894= 1 Then


119891

1015840(x) = sum

119894=0119873119868

119901119894

119873119868

(

119891 (x | 119899119894 119898119894 V119894)

119891 (xo | 119899119894 119898119894 V119894))

119896

(19)













(O)

(1)

(2)

(3)

(X)




0 During







128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))



119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)














so that 1199101198941

le 1199101198942




1= 1 kg V

1=




20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120574E

120574E

E

120590

120590lowast

Δ120573120573

120576

120576 120590

minus120590lowast





119894(119905) obeys

119897119894120576119894 (119905) = 119866119894119872

119906119872 (

119905) (2)

119866119894119872



119894119894= 119864119894119860119894119897119894 119864119894


119894is



120590119894 (119905) = 119864119894

(120576119894 (119905) minus 120573

0

119894(119905)) (3)




hardening law

Ψ119894 (119905) =

10038161003816100381610038161003816

120573119894 (119905)

10038161003816100381610038161003816

(4)


Φ119894(120590119894 Ψ119894) =

1003816100381610038161003816120590119894

1003816100381610038161003816minus (120590

⋆

119894+

120574119894119864119894

1 minus 120574119894

Ψ119894) (5)

where 120590⋆119894 120574119894 and 119864



119894119864119894(1 minus 120574

119894) is


ment that

Φ119894(120590119894 Ψ119894) le 0 (6)


0




120573

0

119894(119905)Φ119894(119905) = 0 and

120573

0

119894(119905)

Φ119894(119905) = 0 where

Φ119894(119905) = Φ




(119894 isin 119884119905) equiv (Φ

119894 (119905) = 0

Φ119894 (119905) = 0) (7)



120576119894 (119905) = 120576

119871

119894(119905) + sum

119895

119905

sum

120591=0

119863

120581120581

119894119895(119905 minus 120591) 120573

0

119895(120591) (8)

where

120576

119871

119894(119905) = sum

119870

119905

sum

120591=0

119863

120581119891

119894119870(119905 minus 120591) 119891119870 (

120591) (9)

while 119863120581119891119894119870(119905) and 119863120581120581



120581120581

119894119895(119905) is the





Ψ119894 (119905) = Ψ119894 (

119905 minus Δ119905) +

10038161003816100381610038161003816

Δ120573

0

119894(119905)

10038161003816100381610038161003816

(10)


Δ120573

0

119894(119905) = 120573

0

119894(119905) minus 120573

0

119894(119905 minus Δ119905) (11)




119894(119905) trial




0

119894(119905 minus Δ119905) Ψtr

119894(119905) =


120576

tr119894(119905) = 120576

119871

119894(119905) + sum

119895

119905minusΔ119905

sum

120591=0

119863

120581120581

119894119895(119905 minus 120591) 120573

0

119895(120591)

+sum

119895

119863

120581120581

119894119895(0) 120573

0

119895(119905 minus Δ119905)

(12)


the trial stress is

120590

tr119894(119905) = 119864119894

(120576

tr119894(119905) minus 120573

0tr119894(119905))

= 119864119894(120576

tr119894(119905) minus 120573

0

119894(119905 minus Δ119905))

(13)


Φ

tr119894(119905) = Φ119894

(120590

tr119894(119905) Ψ

tr119894(119905))

=

10038161003816100381610038161003816

120590

tr119894(119905)

10038161003816100381610038161003816

minus (120590

⋆

119894+

120574119894119864119894

1 minus 120574119894

Ψ119894 (119905 minus Δ119905))

(14)


ing

119864119894sum

119895isin119884119905

(119863

120581120581

119894119895(0) minus

120575119894119895

1 minus 120574119894

)Δ120573

0

119895(119905)

= minusΦ

tr119894(119905) sign120590tr

119894(119905)

(15)


119894(0) = 0 1205730

119894= 0 Ψ

119894(0) =






119894(119905) by (15)





sign120590119894(119905) = sign120590tr

119894(119905) = signΔ1205730

119894(119905) needs to be





119905needs

























119901 119903119887from the range [0 1]


119894119895larr 120596V

119894119895+ 120601119901119903119901(119901119894119895minus 119909119894119895) + 120601119887119903119887(119887119895minus 119909119894119895)








119894119895=1 119909119894119895=2

119909119894119895=119899

)



119909119895= 120583119895=

119860119895

119860119895

(16)




0 where 120583









119901








119901= 120601119887= 05 and119873 = 25


m1 1

(a)

m1 1

(b)


1= 5ms


0

2

4

6

0 1 2 3 4 5 6 7 8

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus2

minus4

minus6

minus8

minus10


(a)

1

11

12

13

14

15

16

0 5 10 15 20 25 30PSO steps


Optimal solution

The o

bjec

tive f

unct

ion

(times10

3)

(b)





119891 (x) = sum

119905=0119879

(1199061199101minus 1199061199101119905=0

)

2

(17)



1199101119905=0= 1 is





119891 (x) = sum

119905=0119879

((1199061199101minus 1199061199101119905=0

)

2

+ (1199061199102minus 1199061199102119905=0

)

2

) (18)


003

004

005

006

007

008

009

01

011

0 10 20 30 40 50 60 70PSO steps

(A)(B)

(C)(G)

The o

bjec

tive f

unct

ion

(times10

3)

(a)

0

02

04

0 02 04 06 08 1

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16



(b)




2 110 1036 1031 003 110 2051 2044 7094 110 00 00 30665 110 1428 1422 4596 110 957 968 18977 110 00 00 08 110 691 699 30669 110 1543 1536 010 110 870 865 0





(O)

(A)

(B)

(C)

(G)


1= 01 kg V


1198982= 2 kg V









119868


119868

119901119894= 1 Then


119891

1015840(x) = sum

119894=0119873119868

119901119894

119873119868

(

119891 (x | 119899119894 119898119894 V119894)

119891 (xo | 119899119894 119898119894 V119894))

119896

(19)













(O)

(1)

(2)

(3)

(X)




0 During







128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))



119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)














so that 1199101198941

le 1199101198942




1= 1 kg V

1=




20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










the trial stress is

120590

tr119894(119905) = 119864119894

(120576

tr119894(119905) minus 120573

0tr119894(119905))

= 119864119894(120576

tr119894(119905) minus 120573

0

119894(119905 minus Δ119905))

(13)


Φ

tr119894(119905) = Φ119894

(120590

tr119894(119905) Ψ

tr119894(119905))

=

10038161003816100381610038161003816

120590

tr119894(119905)

10038161003816100381610038161003816

minus (120590

⋆

119894+

120574119894119864119894

1 minus 120574119894

Ψ119894 (119905 minus Δ119905))

(14)


ing

119864119894sum

119895isin119884119905

(119863

120581120581

119894119895(0) minus

120575119894119895

1 minus 120574119894

)Δ120573

0

119895(119905)

= minusΦ

tr119894(119905) sign120590tr

119894(119905)

(15)


119894(0) = 0 1205730

119894= 0 Ψ

119894(0) =






119894(119905) by (15)





sign120590119894(119905) = sign120590tr

119894(119905) = signΔ1205730

119894(119905) needs to be





119905needs

























119901 119903119887from the range [0 1]


119894119895larr 120596V

119894119895+ 120601119901119903119901(119901119894119895minus 119909119894119895) + 120601119887119903119887(119887119895minus 119909119894119895)








119894119895=1 119909119894119895=2

119909119894119895=119899

)



119909119895= 120583119895=

119860119895

119860119895

(16)




0 where 120583









119901








119901= 120601119887= 05 and119873 = 25


m1 1

(a)

m1 1

(b)


1= 5ms


0

2

4

6

0 1 2 3 4 5 6 7 8

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus2

minus4

minus6

minus8

minus10


(a)

1

11

12

13

14

15

16

0 5 10 15 20 25 30PSO steps


Optimal solution

The o

bjec

tive f

unct

ion

(times10

3)

(b)





119891 (x) = sum

119905=0119879

(1199061199101minus 1199061199101119905=0

)

2

(17)



1199101119905=0= 1 is





119891 (x) = sum

119905=0119879

((1199061199101minus 1199061199101119905=0

)

2

+ (1199061199102minus 1199061199102119905=0

)

2

) (18)


003

004

005

006

007

008

009

01

011

0 10 20 30 40 50 60 70PSO steps

(A)(B)

(C)(G)

The o

bjec

tive f

unct

ion

(times10

3)

(a)

0

02

04

0 02 04 06 08 1

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16



(b)




2 110 1036 1031 003 110 2051 2044 7094 110 00 00 30665 110 1428 1422 4596 110 957 968 18977 110 00 00 08 110 691 699 30669 110 1543 1536 010 110 870 865 0





(O)

(A)

(B)

(C)

(G)


1= 01 kg V


1198982= 2 kg V









119868


119868

119901119894= 1 Then


119891

1015840(x) = sum

119894=0119873119868

119901119894

119873119868

(

119891 (x | 119899119894 119898119894 V119894)

119891 (xo | 119899119894 119898119894 V119894))

119896

(19)













(O)

(1)

(2)

(3)

(X)




0 During







128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))



119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)














so that 1199101198941

le 1199101198942




1= 1 kg V

1=




20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119901 119903119887from the range [0 1]


119894119895larr 120596V

119894119895+ 120601119901119903119901(119901119894119895minus 119909119894119895) + 120601119887119903119887(119887119895minus 119909119894119895)








119894119895=1 119909119894119895=2

119909119894119895=119899

)



119909119895= 120583119895=

119860119895

119860119895

(16)




0 where 120583









119901








119901= 120601119887= 05 and119873 = 25


m1 1

(a)

m1 1

(b)


1= 5ms


0

2

4

6

0 1 2 3 4 5 6 7 8

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus2

minus4

minus6

minus8

minus10


(a)

1

11

12

13

14

15

16

0 5 10 15 20 25 30PSO steps


Optimal solution

The o

bjec

tive f

unct

ion

(times10

3)

(b)





119891 (x) = sum

119905=0119879

(1199061199101minus 1199061199101119905=0

)

2

(17)



1199101119905=0= 1 is





119891 (x) = sum

119905=0119879

((1199061199101minus 1199061199101119905=0

)

2

+ (1199061199102minus 1199061199102119905=0

)

2

) (18)


003

004

005

006

007

008

009

01

011

0 10 20 30 40 50 60 70PSO steps

(A)(B)

(C)(G)

The o

bjec

tive f

unct

ion

(times10

3)

(a)

0

02

04

0 02 04 06 08 1

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16



(b)




2 110 1036 1031 003 110 2051 2044 7094 110 00 00 30665 110 1428 1422 4596 110 957 968 18977 110 00 00 08 110 691 699 30669 110 1543 1536 010 110 870 865 0





(O)

(A)

(B)

(C)

(G)


1= 01 kg V


1198982= 2 kg V









119868


119868

119901119894= 1 Then


119891

1015840(x) = sum

119894=0119873119868

119901119894

119873119868

(

119891 (x | 119899119894 119898119894 V119894)

119891 (xo | 119899119894 119898119894 V119894))

119896

(19)













(O)

(1)

(2)

(3)

(X)




0 During







128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))



119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)














so that 1199101198941

le 1199101198942




1= 1 kg V

1=




20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










m1 1

(a)

m1 1

(b)


1= 5ms


0

2

4

6

0 1 2 3 4 5 6 7 8

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus2

minus4

minus6

minus8

minus10


(a)

1

11

12

13

14

15

16

0 5 10 15 20 25 30PSO steps


Optimal solution

The o

bjec

tive f

unct

ion

(times10

3)

(b)





119891 (x) = sum

119905=0119879

(1199061199101minus 1199061199101119905=0

)

2

(17)



1199101119905=0= 1 is





119891 (x) = sum

119905=0119879

((1199061199101minus 1199061199101119905=0

)

2

+ (1199061199102minus 1199061199102119905=0

)

2

) (18)


003

004

005

006

007

008

009

01

011

0 10 20 30 40 50 60 70PSO steps

(A)(B)

(C)(G)

The o

bjec

tive f

unct

ion

(times10

3)

(a)

0

02

04

0 02 04 06 08 1

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16



(b)




2 110 1036 1031 003 110 2051 2044 7094 110 00 00 30665 110 1428 1422 4596 110 957 968 18977 110 00 00 08 110 691 699 30669 110 1543 1536 010 110 870 865 0





(O)

(A)

(B)

(C)

(G)


1= 01 kg V


1198982= 2 kg V









119868


119868

119901119894= 1 Then


119891

1015840(x) = sum

119894=0119873119868

119901119894

119873119868

(

119891 (x | 119899119894 119898119894 V119894)

119891 (xo | 119899119894 119898119894 V119894))

119896

(19)













(O)

(1)

(2)

(3)

(X)




0 During







128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))



119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)














so that 1199101198941

le 1199101198942




1= 1 kg V

1=




20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










003

004

005

006

007

008

009

01

011

0 10 20 30 40 50 60 70PSO steps

(A)(B)

(C)(G)

The o

bjec

tive f

unct

ion

(times10

3)

(a)

0

02

04

0 02 04 06 08 1

Vert

ical

disp

lace

men

t (m

m)

Time (ms)

minus02

minus04

minus06

minus08

minus1

minus12

minus14

minus16



(b)




2 110 1036 1031 003 110 2051 2044 7094 110 00 00 30665 110 1428 1422 4596 110 957 968 18977 110 00 00 08 110 691 699 30669 110 1543 1536 010 110 870 865 0





(O)

(A)

(B)

(C)

(G)


1= 01 kg V


1198982= 2 kg V









119868


119868

119901119894= 1 Then


119891

1015840(x) = sum

119894=0119873119868

119901119894

119873119868

(

119891 (x | 119899119894 119898119894 V119894)

119891 (xo | 119899119894 119898119894 V119894))

119896

(19)













(O)

(1)

(2)

(3)

(X)




0 During







128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))



119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)














so that 1199101198941

le 1199101198942




1= 1 kg V

1=




20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119868


119868

119901119894= 1 Then


119891

1015840(x) = sum

119894=0119873119868

119901119894

119873119868

(

119891 (x | 119899119894 119898119894 V119894)

119891 (xo | 119899119894 119898119894 V119894))

119896

(19)













(O)

(1)

(2)

(3)

(X)




0 During







128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))



119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)














so that 1199101198941

le 1199101198942




1= 1 kg V

1=




20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










128

112

96

80

64

48

32

16

0

30

25

20

15

10

5

20 40 60 80 100

Mass m0 (kg)

Velo

city 0

(ms

)

128000

112000

104000

120000

9600080000

880007200064000

56000

4800040000

32000

24000

160008000

Epl (J) (120590 = 106 (Pa))



119864pl = sum

119905=1119879

sum

119894=1119873119860

120590119894 (119905) Δ120573

0

119894(119905) 119897119894

119860119894 (20)














so that 1199101198941

le 1199101198942




1= 1 kg V

1=




20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20 40 60 80 100

Mass m0 (kg)

96

93

90

87

84

81

78

75

72

30

25

20

15

10

5

Velo

city 0

(ms

)

9450

930091509000

8850

8700

8550

8250

8100795078007650

8400


(a)

Mass m0 (kg)

Velo

city 0

(ms

)

80

72

64

56

48

40

32

24

16

08

20 40 60 80 100

30

25

20

15

10

5

4800

5600

6400

2400

3200

40001600


(b)





lowast119864pl 120590

lowast119864pl 120590

lowast119864pl




(A)

(B)





0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

6

8

10

12

14

16

0 1 2 3 4 5 6 7 8Time (ms)



Plas

tic en

ergy

Epl

(J)


(A)

(B)




119864pl =1

10

sum

119899119894=110

119864pl | 119899119894 (21)





0

5

10

15

20

25

30

0 01 02 03 04 05 06 07 08 09 1Time (ms)


Plas

tic en

ergy

Epl

(J)

















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Acknowledgment


References










































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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