research article application of three bioinspired...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 737502 12 pageshttpdxdoiorg1011552013737502

Research ArticleApplication of Three Bioinspired Optimization Methods forthe Design of a Nonlinear Mechanical System

Romes Antonio Borges1 Fran Seacutergio Lobato2 and Valder Steffen Jr3

1 Department of Mathematics Federal University of Goias Avendia Dr Lamartine Pinto de Avelar 112075704-220 Catalao GO Brazil

2 School of Chemical Engineering Federal University of Uberlandia Avendia Joao Naves de Avila 2121Campus Santa Monica PO Box 593 38408-144 Uberlandia MG Brazil

3 School of Mechanical Engineering Federal University of Uberlandia Avendia Joao Naves de Avila 2121Campus Santa Monica PO Box 593 38408-144 Uberlandia MG Brazil

Correspondence should be addressed to Valder Steffen Jr vsteffenmecanicaufubr

Received 21 January 2013 Revised 9 June 2013 Accepted 10 June 2013

Academic Editor Chong Wu

Copyright copy 2013 Romes Antonio Borges et al 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

The present work focuses on the optimal design of nonlinear mechanical systems by using heuristic optimization methods In thiscontext the nonlinear optimization problem is devoted to a two-degree-of-freedom nonlinear damped system constituted of aprimary mass attached to the ground by a linear spring and a secondary mass attached to the primary system by a nonlinear springThis arrangement forms a nonlinear dynamic vibration absorber (nDVA) which is used in this contribution as a representativeexample of a nonlinear mechanical system The sensitivity analysis of the suppression bandwidth namely the frequency rangeover which the ratio of the main mass displacement amplitude to the amplitude of the forcing function is less than unity withrespect to the design variables that characterize the nonlinear system based on the first order finite differences is presentedFor illustration purposes the optimization problem is written as to maximize the suppression bandwidth by using three recentbioinspired optimization methods Bees Colony Algorithm Firefly Colony Algorithm and Fish Swarm Algorithm The results arecompared with other evolutionary strategies

1 Introduction

In various mechanical design contexts engineers have todeal with nonlinear systems in which the dynamic responsedepends on a number of physical parameters An exampleof such a system is the so-called dynamic vibration absorber(DVA) The DVA is used to reduce noise and vibration invarious types of engineering systems such as compressorsrobots ships power lines airplanes and helicopters Muchof the knowledge available to date is compiled in the originalpatent by Frahm [1] in the books by Hartog [2] and Koronevand Reznikov [3] and in some review works such as theone by Steffen Jr and Rade [4] In the last two decades agreat deal of effort has been devoted to the developmentof mathematical models for characterizing the mechanicalbehavior of nonlinear dynamic vibration absorbers (nDVAs)

accounting for their typical dependence on design param-eters that influence the nonlinear behavior of the systemA particular type of nDVA is the so-called viscoelasticneutralizer as studied by Espındola and Bavastri [5] Borges etal [6] proposed and determined the robust optimal design ofan nDVA combining sensitivity analysis and multiobjectiveoptimization Different techniques have been proposed todesign viscoelastic vibration absorbers [7 8] Besides thewell-known complexity of the modeling strategy involvedin nonlinear dynamics some general methodologies havebeen suggested and have been shown to be particularlysuitable to be used in combination with structural systemsdiscretization This aspect makes them very attractive forthe modeling of nonlinear dynamic vibration absorbersAmong these strategies the theoretical study proposed byNissen and coworkers [9] and Pai and Schulz [10] should

2 Mathematical Problems in Engineering

be mentioned in which techniques to improve the stabilityand efficiency of nDVAs into a frequency band of interesthave been proposed leading to refined nDVAs Also Riceand McCraith [11] and Shaw and coworkers [12] suggestedoptimization strategies to be applied to the design of nDVAsby applying an asymmetric nonlinear Duffing-type elementincorporated in the suspension for narrow-band absorptionapplications

Nowadays different approaches based on optimizationmethods have been proposed to solve mechanical designproblems Cao and Wu [13] proposed a cellular automatabased genetic algorithm applied to mechanical systemsdesign In this algorithm the individuals in the populationare mapped onto a cellular automata to realise the localityandneighbourhoodThemapping is based on the individualsrsquofitness and the Hamming distances between individualsThe selection of individuals is controlled based on thestructure of cellular automata to avoid the fast populationdiversity loss and improve the convergence performanceduring the genetic search He and coworkers He PrempainandWu [14] proposed an improved particle swarm optimizer(PSO) associated with the fly-back mechanism to maintaina feasible population This optimization strategy was appliedto mechanical systems design involving problem-specificconstraints and mixed variables such as integer discreteand continuous variables In this sense biological systemshave contributed significantly to the development of newoptimization techniques These methodologies known asBioinspired Optimization Methods (BiOMs) are based onstrategies that seek tomimic the behavior observed in speciesfound in the nature to update a population of candidatesto solve optimization problems [15 16] These systems havethe capacity to notice and modify their environment inorder to seek for diversity and convergence In additionthis capacity makes possible the communication among theagents (individuals of population) that capture the changes inthe environment generated by local interactions [17]

In this context nature-inspired algorithms have con-tributed significantly to the development of new optimizationtechniques Among the most recent bioinspired strategiesstand the Bees Colony Algorithm (BCA) [18] the FireflyColonyAlgorithm (FCA) [19] and the Fish SwarmAlgorithm(FSA) [20]TheBCA is based on the behavior of bees coloniesin their search of raw materials for honey productionAccording to Lucic and Teodorovic [21] in each hive groupsof bees (called scouts) are recruited to explore new areas insearch for pollen and nectar These bees returning to thehive share the acquired information so that new bees areindicated to explore the best regions visited in an amountproportional to the previously passed assessment Thus themost promising regions are best explored and eventuallythe worst end up being discarded Every iteration this cyclerepeats itself with new areas being visited by scoutsThe FCAmimics the patterns of short and rhythmic flashes emitted byfireflies in order to attract other individuals to their vicinitiesThe corresponding optimization algorithm is formulated byassuming that all fireflies are unisex so that one firefly will beattracted to all other fireflies Attractiveness is proportionalto their brightness and for any two fireflies the less bright

will attract (and thus move to) the brighter one Howeverthe brightness can decrease as their distance increases and ifthere are no fireflies brighter than a given firefly it will moverandomly The brightness is associated with the objectivefunction for optimization purposes [19] Finally the FSA isa random search algorithm based on the behaviour of fishswarm observed in natureThis behavior can be summarizedas follows [20] random behaviormdashin general fish looks atrandom for food and other companion searching behaviormdashwhen the fish discovers a region with more food it willgo directly and quickly to that region swarming behaviormdashwhen swimming fish will swarm naturally in order to avoiddanger chasing behaviormdashwhen a fish in the swarmdiscoversfood the others will find the food dangling after it leapingbehaviormdashwhen fish stagnates in a region a leap is requiredto look for food in other regions

In this way the present work is dedicated to presentingalternative techniques for the optimal design of a nonlineardynamic vibration absorber (nDVA) by using heuristic opti-mization methods to maximize the suppression bandwidthFor this aim this contribution focuses on the theoreticalstudy and numerical simulation of a two-degree-of-freedomnonlinear damped system constituted of a primary massattached to the ground by a linear spring and a secondarymass attached to the primary system by a nonlinear spring(nDVA) A previous contribution regarding this type ofmechanical system can be found in Borges et al [6] Thenthe three previously mentioned optimization techniques areapplied to the design of an nDVA for illustration purposeshowever they are intended to be general in the sense thatthey can be applied to design different types of nonlinearmechanical systems This work is organized as follows Themathematical formulation of the nonlinear dynamic systemand sensitivity analysis are presented in Sections 2 and 3respectively A review of the BCA the FCA and FSA are pre-sented in Section 4 The results and discussion are describedin Section 5 Finally the conclusions and suggestions forfuture work conclude the paper

2 Mathematical Modeling of the NonlinearDynamic System

Consider the two-degree-of-freedom (dof) model shown inFigure 1

This device consists of a damped primary system attachedto the ground by a suspension that includes either a linear ora nonlinear spring and a damped secondary mass coupled tothe primary systemby a springwith nonlinear characteristicsThe external force applied to the primary system is given bythe following expression

1198651= 119901 cos (120596119905) (1)

The constitutive forces of the springs are given by

119903119894(119909119894) = 119870119894119909119894+ 119896

nl1198941199093

119894119894 = 1 2 (2)

where 1199091represents the displacement of the primary system

with respect to the ground and 1199092is the displacement of

Mathematical Problems in Engineering 3

NonlinearDVA

Primarysystem

x2m2

c2knl2

K2

K1 knl1

c1

x1

F1m1

Figure 1 Two degree-of-freedom damped system [6]

the DVA with respect to the primary system In the modelabove the dampers are linear however springs have nonlin-ear characteristics where 119870

119894and 119896

nl119894represent respectively

the linear and nonlinear coefficients of the springsWith the aim of obtaining the dimensionless normalized

equations of motion for the nonlinear system the displace-ments are normalized with respect to the length of a givenvector 119909

119888[22] according to the following expression

119910119894=

119909119894

119909119888

(3)

In addition one introduces the following relations to thesystem

1205962=

119896119894

119898119894

120577119894=

119888119894

2radic119896119894119898119894

120575119894= 2120577119894120596119894 120578

119894= 1205962

119894 120576

119894=

119896nl1198941199092

119888

1198981198941205962

120588 =1205962

1205961

119875 =119901

11989811205962119909119888

Ω =120596

1205961

120583 =1198982

1198981

(4)

By applyingNewtonrsquos second law and after some algebraicmanipulations the following normalized equation of motionof the nonlinear dynamic system can be expressed under thefollowing matrix form

My (119905) + Cy (119905) + Ky (119905) = f (119905) (5)

where the normalized mass damping and stiffness matricesare expressed respectively by the following relations

M = [1 + 120583 120583

120583 120583] C = [

1205751

0

0 1205831205752

]

K = [1205781

0

0 1205782

]

(6)

The normalized displacement and force vectors are givenby

y = [1199101

1199102

] f = [119875 cos (120591) 120576

11199103

1

minus12057621199103

2120583

] (7)

It is important to emphasized that the present contribu-tion is dedicated to maximize the suppression bandwidth

21 Steady-State Response of the Nonlinear System In thispaper the Krylov-Bogoliubov method [23] is used to inte-grate the matrix equation of motion (1)This method leads toan approximate solution of nonlinear differential equationsThe process is based on the following transformation ofvariables

y = u (120591) cos (120591) + k (120591) sin (120591) (8)

where 120591 = 120596119905 is the time dependence of u = (11990611199062)119879 and

k = (V1V2)119879 is assumed to be small for high-order terms such

as the vectors u and kAfter mathematical manipulation a nonlinear algebraic

system with four equations and four variables 1199061 1199062 V1and

V2is obtained as follows

(1 + 120583 minus 1205962

1) 1199061+ 1205831199062minus 212057711205961V1

minus

31205761(1199062

1+ V21) 1199061

4+ 1205731205962

1= 0

1205831199061+ (120583 minus 120583120588

21205962

1) 1199062minus 120583(2120577

21205881205962

1V2+

31205762(1199062

2+ V22) 1199062

4)

(1205962

1minus 1 minus 120583) V

1minus 120583V2minus 2120577112059611199061+

31205761(1199062

1+ V21) V1

4= 0

120583V1+ (120583120588

21205962

1minus 120583) V

2

minus 120583(212057721205881205962

11199062minus

31205762(1199062

2+ V22) V2

4) = 0

(9)

The system represented by (9) should be numericallysolved Then the values of 119906

1 1199062 V1and V2can be calculated

and the vibration amplitudes of the primary and secondarymasses of the nonlinear DVA are obtained The amplitudevalues are given by 119883

1and 119883

2according to the following

equation

119883119894= (1199062

119894+ V2119894)05

119894 = 1 2 (10)

3 Sensitivity Analysis of Structural Responses

In amechanical system the parameters ofmass stiffness anddamping establish the dependence with respect to a set ofdesign parameters which includes physical and geometricalcharacteristics and the parameters that are associated withthe nonlinearities [24] Such functional dependence can beexpressed in a general form as follows

r = r (M (p)C (p)K (p)) (11)

where r and p designate vectors of structural responses anddesign parameters respectively


(1) Initialize population with random solutions(2) Evaluate fitness of the population(3) While (stopping criterion not met)(4) Select sites for neighborhood search(5) Recruit bees for selected sites and evaluate fitnesses(6) Select the fittest bee from each site(7) Assign remaining bees to search randomly and evaluate their fitnesses(8) End while

Algorithm 1 Basic step in the Bees Colony Algorithm [18]

The sensitivity of the structural responses with respect toa given parameter 119901

119894evaluated for a given set of values of

the design parameter 1199010 is defined as the following partial

derivative120597r120597p119894

= limΔp119894rarr0

[1198771+ 1198772] (12)

where

1198771=

r (M (p0119894+ Δp119894) C (p0

119894+ Δp119894) K (p0

119894+ Δp119894))

Δp119894

1198772=

r (M (p0119894) C (p0

119894) K (p0

119894))

Δp119894

(13)

where Δp119894is an arbitrary variation applied to the current

value of parameter p0119894 while all other parameters remain

unchangedThe sensitivity with respect to p119894can be estimated

by finite differences by computing successively the responsescorresponding to p

119894= p0119894and p

119894= p0119894+ Δp119894 Such procedure

is an estimated approach enabling to calculate the sensitivityof the dynamic system responses with respect to small modi-fications introduced in the design parameters Moreover theresults depend upon the choice of the value of the parameterincrement Δp

119894 Another strategy consists in computing the

analytical derivatives if possible of the structural responseswith respect to the parameters of interest However thisapproach is not considered in the present paper

4 Bioinspired Algorithms

Algorithms based on swarm intelligence principles are com-mon in the literature due to the ability to find global solutionin mono- and multiobjective contexts different from algo-rithms based on gradients According to Andres and Lozano[25] swarm-based intelligence is artificial intelligence tech-nique based on the study of collective behaviour in self-organizing systems composed of a population of individualswhich takes effect between each other and environmentAlthough these systems do not have any central control of theindividual behaviour interaction between individuals andsimple behaviour between them usually lead to detection ofaggregate behaviour which is typical for whole colony Thiscould by observed by ants bees birds or bacteria in thenature By inspiration of these colonies were develop algo-rithms called swarm-based intelligence and are successfully

applied for solving complicated optimization problems [26]In this contribution three recent bioinspired optimizationmethods Bees Colony Algorithm Firefly Colony Algorithmand Fish Swarm Algorithm are considered as optimizationstrategies

41 Bee Colony Algorithm This optimization algorithm isbased on the behavior of a colony of honey bees Thecolony can extend itself over long distances and in multipledirections simultaneously to exploit a large number of foodsources In addition the colony of honey bees presents ascharacteristic the capacity of memorization learning andtransmission of information thus forming the so-calledswarm intelligence [27]

In a colony the foraging process begins by scout beesbeing sent to search randomly for promising flower patchesWhen they return to the hive those scout bees that founda patch which is rated above a certain quality threshold(measured as a combination of some constituents such assugar content) deposit their nectar or pollen and go to thewaggle dance

This dance is responsible for the transmission (colonycommunication) of information regarding a particular flowerpatch the direction inwhich it will be found its distance fromthe hive and its quality rating (or fitness) [27] The waggledance enables the colony to evaluate the relative merit ofdifferent paths according to both the quality of the food theyprovide and the amount of energy needed to harvest it [28]After waggle dancing the dancer (scout bee) goes back to theflower patch with follower bees that were waiting inside thehive More follower bees are sent to more promising patchesThis allows the colony to gather food quickly and efficientlyWhile harvesting fromapatch the beesmonitor its food levelThis is necessary to decide upon the next waggle dance whenthey return to the hive [28] If the patch is still good enough asa food source then it will be advertised in the waggle danceandmore bees will be recruited to that source In this contextPham et al [18] proposed an optimization algorithm inspiredby the natural foraging behavior of honey bees (Bees ColonyAlgorithm (BCA)) as presented in Algorithm 1

The BCA requires a number of parameters to be setnamely the number of scout bees (119899) number of sites selectedfor neighborhood search (out of 119899 visited sites) (119898) numberof top-rated (elite) sites among 119898 selected sites (119890) numberof bees recruited for the best 119890 sites (nep) number of bees


recruited for the other (119898 minus 119890) selected sites neighborhoodsearch (ngh) and the stopping criterion

The BCA starts with the 119899 scout bees being placedrandomly in the search spaceThe fitnesses of the sites visitedby the scout bees are evaluated in step 2

In step 4 bees that have the highest fitnesses are chosenas selected bees and sites visited by them are chosen forneighborhood search Then in steps 5 and 6 the algorithmconducts searches in the neighborhood of the selected sitesassigningmore bees to search near to the best 119890 sitesThe beescan be chosen directly according to the fitnesses associatedwith the sites they are visiting

Alternatively the fitness values are used to determinethe probability of the bees being selected Searches in theneighborhood of the best 119890 sites which represent morepromising solutions are made more detailed by recruitingmore bees to follow them than the other selected beesTogether with scouting this differential recruitment is a keyoperation of the BCA

However in step 6 for each patch only the bee withthe highest fitness will be selected to form the next beepopulation In nature there is no such restriction Thisrestriction is introduced here to reduce the number of pointsto be explored In step 7 the remaining bees in the populationare assigned randomly around the search space scouting fornew potential solutions

In the literature various applications using this bioin-spired approach can be found such as modeling combinato-rial optimization transportation engineering problems [21]engineering system design [15 29] mathematical functionoptimization [18] transport problems [30] dynamic opti-mization [31] optimal control problems [32] and parameterestimation in control problems [16 33] (httpmferciyesedutrabc)

42 Firefly Colony Algorithm The FCA is based on thecharacteristic of the bioluminescence of fireflies insectsnotorious for their light emission According to Yang [19]biology does not have a complete knowledge to determine allthe utilities that firefly luminescence can bring to but at leastthree functions have been identified (i) as a communicationtool and appeal to potential partners in reproduction (ii)as a bait to lure potential prey for the firefly and (iii) as awarning mechanism for potential predators reminding themthat fireflies have a bitter taste

In this way the bioluminescent signals are known to serveas elements of courtship rituals (in most cases the femalesare attracted by the light emitted by the males) methods ofprey attraction social orientation or as a warning signal topredators [34]

It was idealized some of the flashing characteristics offireflies so as to develop firefly-inspired algorithms Forsimplicity the following three idealized rules are used [35]

(1) all fireflies are unisex so that one firefly will beattracted to other fireflies regardless of their sex

(2) attractiveness is proportional to their brightness thusfor any two flashing fireflies the less brighter willmove towards the brighter one The attractiveness is

proportional to the brightness and they both decreaseas their distance increases If there is no brighter onethan a particular firefly it will move randomly

(3) the brightness of a firefly is affected or determinedby the landscape of the objective function For amaximization problem the brightness can simply beproportional to the value of the objective function

According to Yang [19] in the firefly algorithm there aretwo important issues the variation of light intensity and theformulation of the attractiveness For simplicity it is alwaysassumed that the attractiveness of a firefly is determined byits brightness which in turn is associated with the encodedobjective function

This swarm intelligence optimization technique is basedon the assumption that the solution of an optimizationproblem can be perceived as agent (firefly) which glowsproportionally to its quality in a considered problem set-ting Consequently each brighter firefly attracts its partners(regardless of their sex) which makes the search space beingexplored more efficiently The algorithm makes use of asynergic local search Each member of the swarm exploresthe problem space taking into account results obtained byothers still applying its own randomized moves as well Theinfluence of other solutions is controlled by the attractivenessvalue [34]

According to Lukasik and Zak [34] the FCA is presentedin the following Consider a continuous constrained opti-mization problem where the task is to minimize the costfunction 119891(119909) as follows

119891 (119909lowast) = min119909isin119878

119891 (119909) (14)

Assume that there exists a swarm of 119898119886agents (fireflies)

solving the above mentioned problem iteratively and 119909119894

represents a solution for a firefly 119894 in algorithmrsquos iteration119896 whereas 119891(119909

119894) denotes its cost Initially all fireflies are

dislocated in 119878 (randomly or employing some deterministicstrategy) Each firefly has its distinctive attractiveness120582whichimplies how strong it attracts othermembers of the swarmAsthe firefly attractiveness one should select anymonotonicallydecreasing function of the distance 119903

119895= 119889(119909

119894 119909119895) to the

chosen firefly 119895 for example the exponential function

120582 = 1205820exp (minus120574119903

119895) (15)

where 1205820and 120574 are predetermined algorithm parameters

maximum attractiveness value and absorption coefficientrespectively Furthermore every member of the swarm ischaracterized by its light intensity 119868

119894which can be directly

expressed as an inverse of a cost function119891(119909119894) To effectively

explore considered search space 119878 it is assumed that eachfirefly 119894 is changing its position iteratively taking into accounttwo factors attractiveness of other swarm members withhigher light intensity for example 119868

119895gt 119868119894 for all 119895 =

1 119898119886 119895 = 119894 which is varying across distance and a fixed

random step vector 119906119894 It should be noted as well that if no

brighter firefly can be found only such randomized step isbeing used


Thus moving at a given time step 119905 of a firefly 119894 toward abetter firefly 119895 is defined as

119909119905

119894= 119909119905minus1

119894+ 120582 (119909

119905minus1

119895minus 119909119905minus1

119894) + 120572 (rand minus 05) (16)

where the second term on the right side of the equationinserts the factor of attractiveness 120582 while the third termgoverned by 120572 parameter governs the insertion of certainrandomness in the path followed by the firefly rand is arandom number between 0 and 1

In the literature few works using the FCA can be foundIn this context is emphasized application in parameterestimation in control problems [16] continuous constrainedoptimization task [34] multimodal optimization [36] solu-tion of singular optimal control problems [37] and economicemissions load dispatch problem [38]

43 Fish Swarm Algorithm In the development of the FSAbased on fish swarm and observed in nature the followingcharacteristics are considered [20 39] (i) each fish representsa candidate solution of optimization problem (ii) fooddensity is related to an objective function to be optimized (inan optimization problem the amount of food in a region isinversely proportional to the objective function value) (iii)the aquarium is the design space where the fish can be found

As noted earlier the fish weight at the swarm repre-sents the accumulation of food (eg the objective function)received during the evolutionary process In this case theweight is an indicator of success [20 39]

Basically the FSA presents four operators classified intotwo classes ldquofood searchrdquo and ldquomovementrdquo Details on eachof these operators are shown in the following

431 Individual Movement Operator This operator con-tributes to the movement (individual and collective) of fishesin the swarm Each fish updates its position by using

119909119894(119905 + 1) = 119909

119894(119905) + rand times 119904ind (17)

where 119909119894is the final position of fish 119894 at current generation

rand is a random generator and 119904ind is a weighted parameter

432 Food Operator The weight of each fish is a metaphorused to measure the success of food search The higher theweight of a fish themore likely this fishwill be in a potentiallyinteresting region in the design space

According toMadeiro [39] the amount of food that a fisheats depends on the improvement in its objective function inthe current generation and the greatest value considering theswarm The weight is updated according to

119882119894(119905 + 1) = 119882

119894(119905) +

Δ119891119894

max (Δ119891) (18)

where 119882119894(119905) is the fish weight 119894 at generation 119905 and Δ119891

119894is

the difference found in the objective function between thecurrent position and the new position of fish 119894 It is importantto emphasize that Δ119891

119894= 0 for the fish in the same position

433 Instinctive Collective Movement Operator This opera-tor is important for the individual movement of fish whenΔ119891119894

= 0 Thus only the fish whose individual execution ofthe movement resulted in improvement of their fitness willinfluence the direction of motion of the swarm resulting ininstinctive collective movement In this case the resultingdirection (119868

119889) calculated using the contribution of the direc-

tions taken by the fish and the new position of the 119894th fish aregiven by

119868119889(119905) =

sum119873

119894=1Δ119909119894Δ119891119894

sum119873

119894=1Δ119891119894

119909119894(119905 + 1) = 119909

119894(119905) + 119868

119889(119905)

(19)

It is worth mentioning that in the application of thisoperator the direction chosen by a fish that located the largestamount of food exerts the greatest influence on the swarmTherefore the instinctive collectivemovement operator tendsto guide the swarm in the direction of motion chosen by thefish who found the largest amount of food in its individualmovement

434 Noninstinctive CollectiveMovement Operator As notedearlier the fish weight is a good indication of success in thesearch for food In this way the swarm weight is increasingthis means that the search process is successful So theldquoradiusrdquo of the swarm should decrease to other regions to beexplored Otherwise if the swarm weight remains constantthe radius should increase to allow the exploration of newregions

For the swarm contraction the centroid concept is usedThis is obtained by means of an average position of all fishweighted with the respective fish weights according to

119861 (119905) =sum119873

119894=1119909119894119882119894(119905)

sum119873

119894=1119882119894(119905)

(20)

If the swarm weight remains constant in the currentiteration all fish update their positions by using

119909 (119905 + 1) = 119909 (119905) minus 119904vol times rand times119909 (119905) minus 119861 (119905)

119889 (119909 (119905) 119861 (119905)) (21)

where 119889 is a function that calculates the Euclidean distancebetween the centroid and the current position of fish and 119904volis the step size used to control fish displacements

In the literature fewworks using the FSA can be found Inthis context parameter estimation in control problems [16]feed forward neural networks [40] parameter estimation inengineering systems [41] combinatorial optimization prob-lem [42] augmented Lagrangian fish swarm based methodfor global optimization [43] forecasting stock indices usingradial basis function neural networks optimized [44] andhybridization of the FSA with the Particle Swarm Algorithmto solve engineering systems are examples of successful appli-cations [45]


Table 1 Nominal values of design variables

Parameters Nominal values1205761

00011205762

001120573 011205771

0011205772

001120583 005120588 10

Response (SF 12)Sensitivity

Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

Nor

mal

ized

real

par

ts

656055504540353025201510

50

minus5

Figure 2 Sensitivity ofH(120596 p) with respect to 120588

5 Results and Discussion

The following numerical example studied in [6] is presentedto illustrate the application of the proposed methodologyto obtain the optimal design of a nDVA As previouslymentioned the nDVA is used in the present contributionto represent a large class of nonlinear mechanical systemsFigure 1 depicts the test structure consisting of a primarymass attached to the ground by a nonlinear spring andcoupled with an nDVA The nominal values of the designparameters used to generate the dynamic responses of thenonlinear system are illustrated on Table 1The computationsperformed consist in obtaining the driving point frequencyresponses associated to the displacement 119909

1

51 Sensitivities of the Frequency Response with respect toStructural Parameters To illustrate the computation proce-dure for the sensitivity of dynamic responses numerical testswere performed by using the system configuration illustratedin Figure 1 As previously mentioned the computations aredevoted to obtaining the sensitivities of the driving pointfrequency responses which are given by the elements ofH(120596 p)


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

12111009080706050403020100

Nor

mal

ized

real

par

ts



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

minus05

minus10

Nor

mal

ized

real

par

ts


In this example the normalized structural parameters 1205771

1205772 1205761 1205762 120573 120583 and 120588 are considered as the design variables

in the computation of the normalized sensitivities of thefrequency responses with respect to a given parameter p119878119873

119867(120596 p) The normalized real parts of the approximated

complex sensitivity functions calculated by finite differences(according to (17)) are shown in Figures 2 3 4 and 5 forwhich a variation of 20 of the nominal values of eachdesign parameter was adopted Also in the same figures thereal parts of the frequency responses H(120596 p) multiplied by



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

Nor

mal

ized

real

par

ts


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

16

14

12

10

8

6

4

2

0

Am

plitu

de

Suppression bandwidth

Figure 6 Representation of the objective function (maximize thesuppression bandwidth)

convenient scale factors (SF) are shownThe sensitivity func-tions denoted by 119878

119873

119867(120596 p) have been normalized according

to the following scheme

119878119873

119867(120596 p) = 120597119878

119867(120596 p)120597p

10038161003816100381610038161003816100381610038161003816(120596p0)

p0

H (120596 p) (22)

Based on the amplitudes and signs of the sensitivityfunctions one can evaluate the degree of influence of thedesign variables upon the suppression bandwidth in thefrequency bandof interest In addition the sensitivity analysisenables to decide among the design parameters those thatwill be retained in the optimization process The parameters1205771 1205772 and 120576

1do not have a significant influence on the

evaluation of the suppression bandwidth [6] Consequentlythese parameters are not considered as design variables in theoptimization run However as shown in Figures 2 3 4 and 5

the degrees of influence of the parameters 1205762 120583 and 120588 on the

suppression bandwidth are significant and will be consideredas design variables in the optimization process

After having verified the influence of each design variableon the dynamic response of the nonlinear system the interestnow is to maximize the suppression bandwidth as illustratedin Figure 6 using the bioinspired algorithms

In these simulations the following ranges are consideredfor the design parameters 09 le 120588 le 12 004 le 120583 le 006009 le 120573 le 12 and 0009 le 120576

2le 0012

In order to evaluate the performance of the three tech-niques proposed above (BCA FCA and FSA) the followingparameters were used in the algorithms

(1) BCA parameters number of scout bees (10) numberof bees recruited for the best e sites (5) number ofbees recruited for the other selected sites (5) numberof sites selected for neighborhood search (5) numberof top-rated (elite) sites among 119898 selected sites (5)neighborhood search (ngh) [10minus3 10minus4 10minus6] and gen-eration number (50)

(2) FCA parameters number of fireflies (15) maximumattractiveness value (09) absorption coefficient [0709 10] and generation number (50)

(3) FSA parameters number of fish (15) weighted pa-rameter value (1) control fish displacements [10minus110minus2 10minus3] and generation number (50)

In order to examine the quality of the solution method-ology considered the Genetic Algorithm (GA) and theParticle Swarm Optimization (PSO) have been performedThe parameters used by GA and PSO are as follows

(1) GA parameters population size (15) crossover rate(08) mutation rate (001) and generation number(50)

(2) PSO parameters population size (15) inertia weight(1) cognitive and social parameters (05) constric-tion factor (08) and generation number (50)

The stopping criterion used was the maximum numberof iterations Each case study was computed 20 times beforecalculating the average values It should be emphasized that1510 objective function evaluations for each algorithm arenecessary

In Table 2 the results (best (119861) average (119860) and worst(119882)) obtained for the design of the nonlinear vibrationabsorber are presented

This table shows that the three algorithms presented goodestimates for the unknown parameters when compared withthe GA and PSO When the BCA is analyzed in terms of theneighborhood search parameter the best result is obtainedby using 10

minus3 for example a search region with smallerdistances to exploit a large number of food sources Whenthe FCA is analyzed in terms of the absorption coefficientthe best result is found by using 120574 = 10 for examplethus emphasizing the local search Finally when the FSA isanalyzed in terms of the control fish displacements the bestresult is found by using 119904vol = 10

minus1


Table 2 Results obtained by BCA FCA FSA GA and PSO to design the nonlinear vibration absorber (OF = objective function)

120588 120583 120573 1205762

OF

BCA

10minus3

B 1165 0053 0110 0017 0236A 1189 0055 0102 0015 0230W 1027 0058 0091 0011 0218

10minus4

B 1099 0056 0103 0010 0230A 1134 0058 0095 0011 0224W 0916 0045 0103 0011 0224

10minus6

B 1134 0058 0095 0011 0224A 0981 0046 0095 0010 0218W 0961 0050 0105 0011 0218

FCA

07B 1158 0054 0110 0017 0230A 1158 0054 0110 0017 0230W 1157 0053 0110 0017 0228

09B 1148 0055 0110 0018 0230A 1148 0055 0110 0018 0230W 1148 0055 0110 0018 0230

10B 1156 0056 0111 0017 0232A 1148 0055 0110 0018 0230W 1151 0057 0109 0015 0214

FSA

10minus1

B 1166 0052 0109 0016 0235A 1191 0050 0101 0016 0232W 1019 0053 0092 0010 0219

10minus2

B 1101 0054 0099 0099 0232A 1135 0060 0094 0011 0225W 0921 0048 0102 0011 0225

10minus3

B 1135 0061 0099 0010 0226A 0993 0049 0096 0010 0219W 0971 0051 0111 0010 0214

GAB 1101 0054 0099 0098 0233A 1136 0062 0092 0012 0226W 0932 0044 0110 0012 0228

PSOB 1101 0053 0099 0099 0233A 1132 0061 0095 0010 0224W 0918 0049 0103 0011 0225

6 Conclusions

In this work the Bees Colony Algorithm the Firefly ColonyAlgorithm and the Fish Swarm Algorithm were proposedas alternative techniques to obtain the optimal design of anonlinear mechanical system For illustration purposes theywere applied in the design of a nonlinear dynamic vibrationabsorber The system nonlinearity was introduced in thesprings that connect the primary mass to the ground and theabsorber to the primary mass respectively

As observed in Table 2 the algorithms led to satisfac-tory results in terms of the effectiveness of the optimalnDVA configuration and the number of objective functionevaluations when compared with GA and PSO strategiesHowever the results obtained by the algorithms need yetto be better analyzed so that more definitive conclusionscan be drawn for example new mechanisms for diversityexploration should be studied

The choice of the design variables is based on previousknowledge regarding their sensitivities with respect to thesuppression bandwidth It is worth mentioning that theseparameters are directly associatedwith the effectiveness of thenDVA

In terms of the system resolution the equations ofmotionof the nonlinear two dof system were numerically inte-grated by using the so-called average method that providesan approximate solution to nonlinear dynamic problemsThe nonlinear algebraic equations obtained were solvednumerically enabling to determine the roots of the nonlinearalgebraic equations

It is worth mentioning that the nonlinearity factor is animportant parameter to be investigated during the designprocedure of nDVAs due to its contribution to the reductionof the vibration level However care must be taken with highvalues of nonlinearity because of the instabilities introducedin the system This point motivates an important aspect


regarding the proposed methodology obtaining the optimalspring nonlinear coefficient that guarantees the best stablesolution for a given system

Finally the optimization techniques used in this papercan be successfully applied in the design of a great numberof nonlinear mechanical systems

As further work we intend to extend the algorithms tothe multiobjective context and assess the sensitivity of theparameters with respect to the effectiveness of the solution

Nomenclature

119861 Weight matrixBCA Bees Colony AlgorithmBiOM Bioinspired Optimization Methods119862 Damping matrix119888119894 Damping coefficients

119889 Euclidean distanceDVAs Dynamic vibration absorbers119890 Number of top-rated sites119891 Objective functionf Force vector1198651 Force applied to the primary system

FCA Firefly Colony AlgorithmFSA Fish Swarm AlgorithmH(120596 p) Sensitivities of the driving point frequency

responses119868 Light intensity119868119889 Fishes movement direction

119870 Stiffness matrix119870119894 Linear coefficients of the springs

119896nl119894 Nonlinear coefficients of the springs

119872 Mass matrix119898 Number of sites selected for neighborhood

search119898119886 Number of fireflies

1198981 Primary mass

1198982 Secondary mass

119899 Number of scout beesnep Number of bees recruited for the best 119890

sitesngh Neighborhood searchp Design parameters119875 Normalized reference valuer Vectors of structural responses1199031 Constitutive forces of the springs

119877119894 Auxiliary variables vector

119903119895 Distance function

119878 Design space119878119873

119867 Sensitivity function

119904ind Weighted parameter119904vol Control fish displacementsu Auxiliary variables vectork Auxiliary variables vector1199091 Displacement of the primary system with

respect to the ground1199092 Displacement of the DVA with respect to

the primary system

119909119894 Design variables

119909119888 Displacement reference value

119883119894 Vibration amplitudes

119910119894 Normalized displacements

119882 Fish weight120572 Insertion of randomness120573 Normalized force parameter120575119894 Normalized reference value

120578119894 Normalized reference value

120582 Attractiveness1205820 Maximal attractiveness

120596 Normalized reference valueΩ Normalized frequency value120585119894 Normalized coefficient of the damping

120583 Normalized mass ratio120588 Normalized frequency ratio120591 Dimensionless time120574 Absorption coefficient120576119894 Normalized nonlinear coefficient of the

springs

Acknowledgments

The authors acknowledge the financial support providedby FAPEMIG CNPq and FUNAPEUFG The third authoracknowledges the financial support provided by FAPEMIGand CNPq (INCT-EIE) This work was presented at the 21stBrazilian Congress of Mechanical Engineering October 24-28 2011 Natal RN Brazil)

References

[1] H Frahm ldquoDevice for Damping Vibrations of Bodiesrdquo USPatent 989 958 1911

[2] J P D Hartog Mechanical Vibrations McGraw-Hill BookCompany 1934

[3] B G Koronev and L M Reznikov Dynamic Vibration Absorb-ersmdashTheory and Technical Applications John Wiley amp Sons1993

[4] V Steffen Jr and D A Rade ldquoDynamic vibration absorberrdquo inEncyclopedia of Vibration pp 9ndash26 Academic Press 2001

[5] J J Espındola and C A Bavastri ldquoViscoelastic neutralisersin vibration abatement a non-linear optimization approachrdquoBrazillian Journal of Mechanical Sciences vol 19 no 2 pp 154ndash163 1997

[6] R A Borges A M G Lima and V Steffen Jr ldquoRobust optimaldesign of a nonlinear dynamic vibration absorber combiningsensitivity analysisrdquo Shock and Vibration vol 17 no 4-5 pp507ndash520 2010

[7] J J Espındola C A Bavastri and E M de Oliveira LopesldquoDesign of optimum systems of viscoelastic vibration absorbersfor a given material based on the fractional calculus modelrdquoJournal of Vibration and Control vol 14 no 9-10 pp 1607ndash16302008

[8] J J Espındola P Pereira C A Bavastri and E M LopesldquoDesign of optimum system of viscoelastic vibration absorberswith a frobenius norm objective functionrdquo Journal of the


Brazilian Society of Mechanical Sciences and Engineering vol 31no 3 pp 210ndash219 2009

[9] J C Nissen K Popp and B Schmalhorst ldquoOptimization of anon-linear dynamic vibration absorberrdquo Journal of Sound andVibration vol 99 no 1 pp 149ndash154 1985

[10] P F Pai and M J Schulz ldquoA refined nonlinear vibrationabsorberrdquo International Journal of Mechanical Sciences vol 42no 3 pp 537ndash560 2000

[11] H J Rice and J R McCraith ldquoPractical non-linear vibrationabsorber designrdquo Journal of Sound and Vibration vol 116 no 3pp 545ndash559 1987

[12] J Shaw S W Shaw and A G Haddow ldquoOn the response ofthe nonlinear vibration absorberrdquo International Journal of Non-Linear Mechanics vol 24 no 4 pp 281ndash293 1989

[13] Y J Cao and Q HWu ldquoA cellular automata based genetic algo-rithm and its application inmechanical design optimisationrdquo inProceedings of the International Conference on Control (UKACCrsquo98) pp 1593ndash1598 September 1998

[14] S He E Prempain andQHWu ldquoAn improved particle swarmoptimizer for mechanical design optimization problemsrdquo Engi-neering Optimization vol 36 no 5 pp 585ndash605 2004

[15] F S Lobato J A Sousa C E Hori and V Steffen Jr ldquoImprovedbees colony algorithm applied to chemical engineering systemdesignrdquo International Review of Chemical Engineering vol 2 no6 pp 714ndash719 2010

[16] F S Lobato D L Souza and R Gedraite ldquoA comparative studyusing bio-inspired optimization methods applied to controllerstuningrdquo in Frontiers in Advanced Control Systems G Luiz deOliveira Serra Ed 2012

[17] J Parrish S Viscido and D Grunbaum ldquoSelf-organized fishschools an examination of emergent propertiesrdquo BiologicalBulletin vol 202 no 3 pp 296ndash305 2002

[18] D T Pham E Kog A Ghanbarzadeh S Otri S Rahimand M Zaidi ldquoThe bees algorithmmdasha novel tool for complexoptimisation problemsrdquo in Proceedings of 2nd InternationalVirtual Conference on Intelligent Production Machines andSystems Elsevier Oxford UK 2006

[19] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress Cambridge Mass USA 2008

[20] X L Li Z J Shao and J X Qian ldquoOptimizing methodbased on autonomous animats fish-swarm algorithmrdquo SystemEngineeringTheory and Practice vol 22 no 11 pp 32ndash38 2002

[21] P Lucic and D Teodorovic ldquoBee system modeling combi-natorial optimization transportation engineering problems byswarm intelligencerdquo in Proceedings of TRISTAN IV TriennialSymposium on Transportation Analysis pp 441ndash445 2011

[22] S J Zhu Y F Zheng and Y M Fu ldquoAnalysis of non-lineardynamics of a two-degree-of-freedom vibration system withnon-linear damping and non-linear springrdquo Journal of Soundand Vibration vol 271 no 1-2 pp 15ndash24 2004

[23] A H Nayfeh Perturbation Methods John Wiley amp Sons 2000[24] E J Haug K K Choi and V Komkov Design Sensitivity

Analysis of Structural Systems Academic Press 1986[25] C Andres and S Lozano ldquoA particle swarm optimization

algorithm for part-machine groupingrdquo Robotics and Computer-Integrated Manufacturing vol 22 no 5-6 pp 468ndash474 2006

[26] Q He L Wang and B Liu ldquoParameter estimation for chaoticsystems by particle swarm optimizationrdquo Chaos Solitons ampFractals vol 34 no 2 pp 654ndash661 2007

[27] K von FrischBeesTheir Vision Chemical Senses and LanguageCornell University Press Ithaca NY USA 1976

[28] S Camazine J Deneubourg N R Franks J Sneyd G Ther-aulaz and E Bonabeau Self-Organization in Biological SystemsPrinceton University Press 2003

[29] X S Yang ldquoEngineering optimizations via nature-inspiredvirtual bee algorithmsrdquo in Proceedings of the 1st InternationalWork-Conference on the Interplay BetweenNatural andArtificialComputation (IWINAC rsquo05) J M Yang and J R Alvarez Edsvol 3562 of Lecture Notes in Computer Science pp 317ndash323Springer June 2005

[30] D Teodorovic and M DellrsquoOrco ldquoBee colony optimizationmdasha cooperative learning approach to complex transportationproblemsrdquo in Proceedings of the 10th EWGT Meeting and 16thMini-EURO Conference 2005

[31] H S Chang ldquoConvergingmarriage in honey-bees optimizationand application to stochastic dynamic programmingrdquo Journal ofGlobal Optimization vol 35 no 3 pp 423ndash441 2006

[32] A Afshar O B Haddad M A Marino and B J AdamsldquoHoney-bee mating optimization (HBMO) algorithm for opti-mal reservoir operationrdquo Journal of the Franklin Institute vol344 no 5 pp 452ndash462 2007

[33] M F Azeem and A M Saad ldquoModified queen bee evolutionbased genetic algorithm for tuning of scaling factors of fuzzyknowledge base controllerrdquo in Proceedings of the 1st IndiaAnnual Conference (IEEE INDICON rsquo04) pp 299ndash303 Decem-ber 2004

[34] S Lukasik and S Zak ldquoFirefly algorithm for continuous con-strained optimization taskrdquo in Proceedings of the 1st Interna-tional Conference (ICCCI rsquo09) N T Ngugen R Kowalczykand S M Chen Eds vol 5796 of Lecture Notes in ArtificialIntelligence pp 97ndash100 October 2009

[35] X S Yang ldquoFirefly algorithm levy flights and global optimiza-tionrdquo in Research and Development in Intelligent Systems XXVIM Bramer R Ellis andM Petridis Eds pp 209ndash218 SpringerLondon UK 2010

[36] X S Yang ldquoFirefly algorithms for multimodal optimizationrdquo inStochastic Algorithms Foundations and Applications vol 5792pp 169ndash178 2009

[37] A A Pfeifer and F S Lobato ldquoSoluiton of singular optimalcontrol problems using the firefly algorithmrdquo in Proceedings ofVI Congreso Argentino de Ingenierıa Quımica (CAIQ rsquo10) 2010

[38] T Apostolopoulos and A Vlachos ldquoApplication of the fireflyalgorithm for solving the economic emissions load dispatchproblemrdquo International Journal of Combinatorics vol 2011Article ID 523806 23 pages 2011

[39] S S Madeiro Modal search for swarm based on density [PhDdissertation] Universidade de Pernambuco 2010 (Portuguese)

[40] C R Wang C L Zhou and J W Ma ldquoAn improved artificialfish-swarm algorithm and its application in feed-forward neuralnetworksrdquo in Proceedings of the 4th International Conference onMachine Learning and Cybernetics (ICMLC rsquo05) pp 2890ndash2894August 2005

[41] X L Li Y C Xue F Lu and G H Tian ldquoParameter estimationmethod based on artificial fish school algorithmrdquo Journal ofShan Dong University vol 34 no 3 pp 84ndash87 2004

[42] Y Cai ldquoArtificial fish school algorithm applied in a combinato-rial optimization problemrdquo Intelligent Systems and Applicationsvol 1 pp 37ndash43 2010

[43] A M A C Rocha T F M C Martins and E M GP Fernandes ldquoAn augmented Lagrangian fish swarm basedmethod for global optimizationrdquo Journal of Computational andApplied Mathematics vol 235 no 16 pp 4611ndash4620 2011


[44] W Shen X Guo C Wu and D Wu ldquoForecasting stockindices using radial basis function neural networks optimizedby artificial fish swarm algorithmrdquo Knowledge-Based Systemsvol 24 no 3 pp 378ndash385 2011

[45] H C Tsai and Y H Lin ldquoModification of the fish swarmalgorithm with particle swarm optimization formulation andcommunication behaviorrdquo Applied Soft Computing Journal vol11 no 8 pp 5367ndash5374 2011

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


be mentioned in which techniques to improve the stabilityand efficiency of nDVAs into a frequency band of interesthave been proposed leading to refined nDVAs Also Riceand McCraith [11] and Shaw and coworkers [12] suggestedoptimization strategies to be applied to the design of nDVAsby applying an asymmetric nonlinear Duffing-type elementincorporated in the suspension for narrow-band absorptionapplications

Nowadays different approaches based on optimizationmethods have been proposed to solve mechanical designproblems Cao and Wu [13] proposed a cellular automatabased genetic algorithm applied to mechanical systemsdesign In this algorithm the individuals in the populationare mapped onto a cellular automata to realise the localityandneighbourhoodThemapping is based on the individualsrsquofitness and the Hamming distances between individualsThe selection of individuals is controlled based on thestructure of cellular automata to avoid the fast populationdiversity loss and improve the convergence performanceduring the genetic search He and coworkers He PrempainandWu [14] proposed an improved particle swarm optimizer(PSO) associated with the fly-back mechanism to maintaina feasible population This optimization strategy was appliedto mechanical systems design involving problem-specificconstraints and mixed variables such as integer discreteand continuous variables In this sense biological systemshave contributed significantly to the development of newoptimization techniques These methodologies known asBioinspired Optimization Methods (BiOMs) are based onstrategies that seek tomimic the behavior observed in speciesfound in the nature to update a population of candidatesto solve optimization problems [15 16] These systems havethe capacity to notice and modify their environment inorder to seek for diversity and convergence In additionthis capacity makes possible the communication among theagents (individuals of population) that capture the changes inthe environment generated by local interactions [17]

In this context nature-inspired algorithms have con-tributed significantly to the development of new optimizationtechniques Among the most recent bioinspired strategiesstand the Bees Colony Algorithm (BCA) [18] the FireflyColonyAlgorithm (FCA) [19] and the Fish SwarmAlgorithm(FSA) [20]TheBCA is based on the behavior of bees coloniesin their search of raw materials for honey productionAccording to Lucic and Teodorovic [21] in each hive groupsof bees (called scouts) are recruited to explore new areas insearch for pollen and nectar These bees returning to thehive share the acquired information so that new bees areindicated to explore the best regions visited in an amountproportional to the previously passed assessment Thus themost promising regions are best explored and eventuallythe worst end up being discarded Every iteration this cyclerepeats itself with new areas being visited by scoutsThe FCAmimics the patterns of short and rhythmic flashes emitted byfireflies in order to attract other individuals to their vicinitiesThe corresponding optimization algorithm is formulated byassuming that all fireflies are unisex so that one firefly will beattracted to all other fireflies Attractiveness is proportionalto their brightness and for any two fireflies the less bright

will attract (and thus move to) the brighter one Howeverthe brightness can decrease as their distance increases and ifthere are no fireflies brighter than a given firefly it will moverandomly The brightness is associated with the objectivefunction for optimization purposes [19] Finally the FSA isa random search algorithm based on the behaviour of fishswarm observed in natureThis behavior can be summarizedas follows [20] random behaviormdashin general fish looks atrandom for food and other companion searching behaviormdashwhen the fish discovers a region with more food it willgo directly and quickly to that region swarming behaviormdashwhen swimming fish will swarm naturally in order to avoiddanger chasing behaviormdashwhen a fish in the swarmdiscoversfood the others will find the food dangling after it leapingbehaviormdashwhen fish stagnates in a region a leap is requiredto look for food in other regions

In this way the present work is dedicated to presentingalternative techniques for the optimal design of a nonlineardynamic vibration absorber (nDVA) by using heuristic opti-mization methods to maximize the suppression bandwidthFor this aim this contribution focuses on the theoreticalstudy and numerical simulation of a two-degree-of-freedomnonlinear damped system constituted of a primary massattached to the ground by a linear spring and a secondarymass attached to the primary system by a nonlinear spring(nDVA) A previous contribution regarding this type ofmechanical system can be found in Borges et al [6] Thenthe three previously mentioned optimization techniques areapplied to the design of an nDVA for illustration purposeshowever they are intended to be general in the sense thatthey can be applied to design different types of nonlinearmechanical systems This work is organized as follows Themathematical formulation of the nonlinear dynamic systemand sensitivity analysis are presented in Sections 2 and 3respectively A review of the BCA the FCA and FSA are pre-sented in Section 4 The results and discussion are describedin Section 5 Finally the conclusions and suggestions forfuture work conclude the paper

2 Mathematical Modeling of the NonlinearDynamic System

Consider the two-degree-of-freedom (dof) model shown inFigure 1

This device consists of a damped primary system attachedto the ground by a suspension that includes either a linear ora nonlinear spring and a damped secondary mass coupled tothe primary systemby a springwith nonlinear characteristicsThe external force applied to the primary system is given bythe following expression

1198651= 119901 cos (120596119905) (1)

The constitutive forces of the springs are given by

119903119894(119909119894) = 119870119894119909119894+ 119896

nl1198941199093

119894119894 = 1 2 (2)

where 1199091represents the displacement of the primary system

with respect to the ground and 1199092is the displacement of


NonlinearDVA

Primarysystem

x2m2

c2knl2

K2

K1 knl1

c1

x1

F1m1



119894and 119896





119910119894=

119909119894

119909119888

(3)


1205962=

119896119894

119898119894

120577119894=

119888119894

2radic119896119894119898119894

120575119894= 2120577119894120596119894 120578

119894= 1205962

119894 120576

119894=

119896nl1198941199092

119888

1198981198941205962

120588 =1205962

1205961

119875 =119901

11989811205962119909119888

Ω =120596

1205961

120583 =1198982

1198981

(4)


My (119905) + Cy (119905) + Ky (119905) = f (119905) (5)


M = [1 + 120583 120583

120583 120583] C = [

1205751

0

0 1205831205752

]

K = [1205781

0

0 1205782

]

(6)


y = [1199101

1199102

] f = [119875 cos (120591) 120576

11199103

1

minus12057621199103

2120583

] (7)



y = u (120591) cos (120591) + k (120591) sin (120591) (8)






(1 + 120583 minus 1205962

1) 1199061+ 1205831199062minus 212057711205961V1

minus

31205761(1199062

1+ V21) 1199061

4+ 1205731205962

1= 0

1205831199061+ (120583 minus 120583120588

21205962

1) 1199062minus 120583(2120577

21205881205962

1V2+

31205762(1199062

2+ V22) 1199062

4)

(1205962


1minus 120583V2minus 2120577112059611199061+

31205761(1199062

1+ V21) V1

4= 0

120583V1+ (120583120588

21205962

1minus 120583) V

2

minus 120583(212057721205881205962

11199062minus

31205762(1199062

2+ V22) V2

4) = 0

(9)




1and 119883


equation

119883119894= (1199062

119894+ V2119894)05

119894 = 1 2 (10)



r = r (M (p)C (p)K (p)) (11)









= limΔp119894rarr0

[1198771+ 1198772] (12)

where

1198771=

r (M (p0119894+ Δp119894) C (p0

119894+ Δp119894) K (p0

119894+ Δp119894))

Δp119894

1198772=

r (M (p0119894) C (p0

119894) K (p0

119894))

Δp119894

(13)





119894= p0119894and p





























119891 (119909lowast) = min119909isin119878

119891 (119909) (14)






119895= 119889(119909

119894 119909119895) to the


120582 = 1205820exp (minus120574119903

119895) (15)






119895gt 119868119894 for all 119895 =






119909119905

119894= 119909119905minus1

119894+ 120582 (119909

119905minus1

119895minus 119909119905minus1

119894) + 120572 (rand minus 05) (16)







119909119894(119905 + 1) = 119909

119894(119905) + rand times 119904ind (17)





119882119894(119905 + 1) = 119882

119894(119905) +

Δ119891119894

max (Δ119891) (18)


119894is







119868119889(119905) =

sum119873

119894=1Δ119909119894Δ119891119894

sum119873

119894=1Δ119891119894

119909119894(119905 + 1) = 119909

119894(119905) + 119868

119889(119905)

(19)




119861 (119905) =sum119873

119894=1119909119894119882119894(119905)

sum119873

119894=1119882119894(119905)

(20)



119889 (119909 (119905) 119861 (119905)) (21)






00011205762

001120573 011205771

0011205772

001120583 005120588 10


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

Nor

mal

ized

real

par

ts

656055504540353025201510

50

minus5




1



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

12111009080706050403020100

Nor

mal

ized

real

par

ts



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

minus05

minus10

Nor

mal

ized

real

par

ts









Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

Nor

mal

ized

real

par

ts


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

16

14

12

10

8

6

4

2

0

Am

plitu

de




119873



119878119873

119867(120596 p) = 120597119878

119867(120596 p)120597p

10038161003816100381610038161003816100381610038161003816(120596p0)

p0

H (120596 p) (22)








2le 0012












minus1



120588 120583 120573 1205762

OF

BCA

10minus3

B 1165 0053 0110 0017 0236A 1189 0055 0102 0015 0230W 1027 0058 0091 0011 0218

10minus4

B 1099 0056 0103 0010 0230A 1134 0058 0095 0011 0224W 0916 0045 0103 0011 0224

10minus6

B 1134 0058 0095 0011 0224A 0981 0046 0095 0010 0218W 0961 0050 0105 0011 0218

FCA

07B 1158 0054 0110 0017 0230A 1158 0054 0110 0017 0230W 1157 0053 0110 0017 0228

09B 1148 0055 0110 0018 0230A 1148 0055 0110 0018 0230W 1148 0055 0110 0018 0230

10B 1156 0056 0111 0017 0232A 1148 0055 0110 0018 0230W 1151 0057 0109 0015 0214

FSA

10minus1

B 1166 0052 0109 0016 0235A 1191 0050 0101 0016 0232W 1019 0053 0092 0010 0219

10minus2

B 1101 0054 0099 0099 0232A 1135 0060 0094 0011 0225W 0921 0048 0102 0011 0225

10minus3

B 1135 0061 0099 0010 0226A 0993 0049 0096 0010 0219W 0971 0051 0111 0010 0214

GAB 1101 0054 0099 0098 0233A 1136 0062 0092 0012 0226W 0932 0044 0110 0012 0228

PSOB 1101 0053 0099 0099 0233A 1132 0061 0095 0010 0224W 0918 0049 0103 0011 0225

6 Conclusions










Nomenclature















119878 Design space119878119873




the primary system










springs

Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










NonlinearDVA

Primarysystem

x2m2

c2knl2

K2

K1 knl1

c1

x1

F1m1



119894and 119896





119910119894=

119909119894

119909119888

(3)


1205962=

119896119894

119898119894

120577119894=

119888119894

2radic119896119894119898119894

120575119894= 2120577119894120596119894 120578

119894= 1205962

119894 120576

119894=

119896nl1198941199092

119888

1198981198941205962

120588 =1205962

1205961

119875 =119901

11989811205962119909119888

Ω =120596

1205961

120583 =1198982

1198981

(4)


My (119905) + Cy (119905) + Ky (119905) = f (119905) (5)


M = [1 + 120583 120583

120583 120583] C = [

1205751

0

0 1205831205752

]

K = [1205781

0

0 1205782

]

(6)


y = [1199101

1199102

] f = [119875 cos (120591) 120576

11199103

1

minus12057621199103

2120583

] (7)



y = u (120591) cos (120591) + k (120591) sin (120591) (8)






(1 + 120583 minus 1205962

1) 1199061+ 1205831199062minus 212057711205961V1

minus

31205761(1199062

1+ V21) 1199061

4+ 1205731205962

1= 0

1205831199061+ (120583 minus 120583120588

21205962

1) 1199062minus 120583(2120577

21205881205962

1V2+

31205762(1199062

2+ V22) 1199062

4)

(1205962


1minus 120583V2minus 2120577112059611199061+

31205761(1199062

1+ V21) V1

4= 0

120583V1+ (120583120588

21205962

1minus 120583) V

2

minus 120583(212057721205881205962

11199062minus

31205762(1199062

2+ V22) V2

4) = 0

(9)




1and 119883


equation

119883119894= (1199062

119894+ V2119894)05

119894 = 1 2 (10)



r = r (M (p)C (p)K (p)) (11)









= limΔp119894rarr0

[1198771+ 1198772] (12)

where

1198771=

r (M (p0119894+ Δp119894) C (p0

119894+ Δp119894) K (p0

119894+ Δp119894))

Δp119894

1198772=

r (M (p0119894) C (p0

119894) K (p0

119894))

Δp119894

(13)





119894= p0119894and p





























119891 (119909lowast) = min119909isin119878

119891 (119909) (14)






119895= 119889(119909

119894 119909119895) to the


120582 = 1205820exp (minus120574119903

119895) (15)






119895gt 119868119894 for all 119895 =






119909119905

119894= 119909119905minus1

119894+ 120582 (119909

119905minus1

119895minus 119909119905minus1

119894) + 120572 (rand minus 05) (16)







119909119894(119905 + 1) = 119909

119894(119905) + rand times 119904ind (17)





119882119894(119905 + 1) = 119882

119894(119905) +

Δ119891119894

max (Δ119891) (18)


119894is







119868119889(119905) =

sum119873

119894=1Δ119909119894Δ119891119894

sum119873

119894=1Δ119891119894

119909119894(119905 + 1) = 119909

119894(119905) + 119868

119889(119905)

(19)




119861 (119905) =sum119873

119894=1119909119894119882119894(119905)

sum119873

119894=1119882119894(119905)

(20)



119889 (119909 (119905) 119861 (119905)) (21)






00011205762

001120573 011205771

0011205772

001120583 005120588 10


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

Nor

mal

ized

real

par

ts

656055504540353025201510

50

minus5




1



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

12111009080706050403020100

Nor

mal

ized

real

par

ts



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

minus05

minus10

Nor

mal

ized

real

par

ts









Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

Nor

mal

ized

real

par

ts


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

16

14

12

10

8

6

4

2

0

Am

plitu

de




119873



119878119873

119867(120596 p) = 120597119878

119867(120596 p)120597p

10038161003816100381610038161003816100381610038161003816(120596p0)

p0

H (120596 p) (22)








2le 0012












minus1



120588 120583 120573 1205762

OF

BCA

10minus3

B 1165 0053 0110 0017 0236A 1189 0055 0102 0015 0230W 1027 0058 0091 0011 0218

10minus4

B 1099 0056 0103 0010 0230A 1134 0058 0095 0011 0224W 0916 0045 0103 0011 0224

10minus6

B 1134 0058 0095 0011 0224A 0981 0046 0095 0010 0218W 0961 0050 0105 0011 0218

FCA

07B 1158 0054 0110 0017 0230A 1158 0054 0110 0017 0230W 1157 0053 0110 0017 0228

09B 1148 0055 0110 0018 0230A 1148 0055 0110 0018 0230W 1148 0055 0110 0018 0230

10B 1156 0056 0111 0017 0232A 1148 0055 0110 0018 0230W 1151 0057 0109 0015 0214

FSA

10minus1

B 1166 0052 0109 0016 0235A 1191 0050 0101 0016 0232W 1019 0053 0092 0010 0219

10minus2

B 1101 0054 0099 0099 0232A 1135 0060 0094 0011 0225W 0921 0048 0102 0011 0225

10minus3

B 1135 0061 0099 0010 0226A 0993 0049 0096 0010 0219W 0971 0051 0111 0010 0214

GAB 1101 0054 0099 0098 0233A 1136 0062 0092 0012 0226W 0932 0044 0110 0012 0228

PSOB 1101 0053 0099 0099 0233A 1132 0061 0095 0010 0224W 0918 0049 0103 0011 0225

6 Conclusions










Nomenclature















119878 Design space119878119873




the primary system










springs

Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















= limΔp119894rarr0

[1198771+ 1198772] (12)

where

1198771=

r (M (p0119894+ Δp119894) C (p0

119894+ Δp119894) K (p0

119894+ Δp119894))

Δp119894

1198772=

r (M (p0119894) C (p0

119894) K (p0

119894))

Δp119894

(13)





119894= p0119894and p





























119891 (119909lowast) = min119909isin119878

119891 (119909) (14)






119895= 119889(119909

119894 119909119895) to the


120582 = 1205820exp (minus120574119903

119895) (15)






119895gt 119868119894 for all 119895 =






119909119905

119894= 119909119905minus1

119894+ 120582 (119909

119905minus1

119895minus 119909119905minus1

119894) + 120572 (rand minus 05) (16)







119909119894(119905 + 1) = 119909

119894(119905) + rand times 119904ind (17)





119882119894(119905 + 1) = 119882

119894(119905) +

Δ119891119894

max (Δ119891) (18)


119894is







119868119889(119905) =

sum119873

119894=1Δ119909119894Δ119891119894

sum119873

119894=1Δ119891119894

119909119894(119905 + 1) = 119909

119894(119905) + 119868

119889(119905)

(19)




119861 (119905) =sum119873

119894=1119909119894119882119894(119905)

sum119873

119894=1119882119894(119905)

(20)



119889 (119909 (119905) 119861 (119905)) (21)






00011205762

001120573 011205771

0011205772

001120583 005120588 10


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

Nor

mal

ized

real

par

ts

656055504540353025201510

50

minus5




1



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

12111009080706050403020100

Nor

mal

ized

real

par

ts



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

minus05

minus10

Nor

mal

ized

real

par

ts









Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

Nor

mal

ized

real

par

ts


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

16

14

12

10

8

6

4

2

0

Am

plitu

de




119873



119878119873

119867(120596 p) = 120597119878

119867(120596 p)120597p

10038161003816100381610038161003816100381610038161003816(120596p0)

p0

H (120596 p) (22)








2le 0012












minus1



120588 120583 120573 1205762

OF

BCA

10minus3

B 1165 0053 0110 0017 0236A 1189 0055 0102 0015 0230W 1027 0058 0091 0011 0218

10minus4

B 1099 0056 0103 0010 0230A 1134 0058 0095 0011 0224W 0916 0045 0103 0011 0224

10minus6

B 1134 0058 0095 0011 0224A 0981 0046 0095 0010 0218W 0961 0050 0105 0011 0218

FCA

07B 1158 0054 0110 0017 0230A 1158 0054 0110 0017 0230W 1157 0053 0110 0017 0228

09B 1148 0055 0110 0018 0230A 1148 0055 0110 0018 0230W 1148 0055 0110 0018 0230

10B 1156 0056 0111 0017 0232A 1148 0055 0110 0018 0230W 1151 0057 0109 0015 0214

FSA

10minus1

B 1166 0052 0109 0016 0235A 1191 0050 0101 0016 0232W 1019 0053 0092 0010 0219

10minus2

B 1101 0054 0099 0099 0232A 1135 0060 0094 0011 0225W 0921 0048 0102 0011 0225

10minus3

B 1135 0061 0099 0010 0226A 0993 0049 0096 0010 0219W 0971 0051 0111 0010 0214

GAB 1101 0054 0099 0098 0233A 1136 0062 0092 0012 0226W 0932 0044 0110 0012 0228

PSOB 1101 0053 0099 0099 0233A 1132 0061 0095 0010 0224W 0918 0049 0103 0011 0225

6 Conclusions










Nomenclature















119878 Design space119878119873




the primary system










springs

Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


























119891 (119909lowast) = min119909isin119878

119891 (119909) (14)






119895= 119889(119909

119894 119909119895) to the


120582 = 1205820exp (minus120574119903

119895) (15)






119895gt 119868119894 for all 119895 =






119909119905

119894= 119909119905minus1

119894+ 120582 (119909

119905minus1

119895minus 119909119905minus1

119894) + 120572 (rand minus 05) (16)







119909119894(119905 + 1) = 119909

119894(119905) + rand times 119904ind (17)





119882119894(119905 + 1) = 119882

119894(119905) +

Δ119891119894

max (Δ119891) (18)


119894is







119868119889(119905) =

sum119873

119894=1Δ119909119894Δ119891119894

sum119873

119894=1Δ119891119894

119909119894(119905 + 1) = 119909

119894(119905) + 119868

119889(119905)

(19)




119861 (119905) =sum119873

119894=1119909119894119882119894(119905)

sum119873

119894=1119882119894(119905)

(20)



119889 (119909 (119905) 119861 (119905)) (21)






00011205762

001120573 011205771

0011205772

001120583 005120588 10


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

Nor

mal

ized

real

par

ts

656055504540353025201510

50

minus5




1



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

12111009080706050403020100

Nor

mal

ized

real

par

ts



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

minus05

minus10

Nor

mal

ized

real

par

ts









Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

Nor

mal

ized

real

par

ts


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

16

14

12

10

8

6

4

2

0

Am

plitu

de




119873



119878119873

119867(120596 p) = 120597119878

119867(120596 p)120597p

10038161003816100381610038161003816100381610038161003816(120596p0)

p0

H (120596 p) (22)








2le 0012












minus1



120588 120583 120573 1205762

OF

BCA

10minus3

B 1165 0053 0110 0017 0236A 1189 0055 0102 0015 0230W 1027 0058 0091 0011 0218

10minus4

B 1099 0056 0103 0010 0230A 1134 0058 0095 0011 0224W 0916 0045 0103 0011 0224

10minus6

B 1134 0058 0095 0011 0224A 0981 0046 0095 0010 0218W 0961 0050 0105 0011 0218

FCA

07B 1158 0054 0110 0017 0230A 1158 0054 0110 0017 0230W 1157 0053 0110 0017 0228

09B 1148 0055 0110 0018 0230A 1148 0055 0110 0018 0230W 1148 0055 0110 0018 0230

10B 1156 0056 0111 0017 0232A 1148 0055 0110 0018 0230W 1151 0057 0109 0015 0214

FSA

10minus1

B 1166 0052 0109 0016 0235A 1191 0050 0101 0016 0232W 1019 0053 0092 0010 0219

10minus2

B 1101 0054 0099 0099 0232A 1135 0060 0094 0011 0225W 0921 0048 0102 0011 0225

10minus3

B 1135 0061 0099 0010 0226A 0993 0049 0096 0010 0219W 0971 0051 0111 0010 0214

GAB 1101 0054 0099 0098 0233A 1136 0062 0092 0012 0226W 0932 0044 0110 0012 0228

PSOB 1101 0053 0099 0099 0233A 1132 0061 0095 0010 0224W 0918 0049 0103 0011 0225

6 Conclusions










Nomenclature















119878 Design space119878119873




the primary system










springs

Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119905

119894= 119909119905minus1

119894+ 120582 (119909

119905minus1

119895minus 119909119905minus1

119894) + 120572 (rand minus 05) (16)







119909119894(119905 + 1) = 119909

119894(119905) + rand times 119904ind (17)





119882119894(119905 + 1) = 119882

119894(119905) +

Δ119891119894

max (Δ119891) (18)


119894is







119868119889(119905) =

sum119873

119894=1Δ119909119894Δ119891119894

sum119873

119894=1Δ119891119894

119909119894(119905 + 1) = 119909

119894(119905) + 119868

119889(119905)

(19)




119861 (119905) =sum119873

119894=1119909119894119882119894(119905)

sum119873

119894=1119882119894(119905)

(20)



119889 (119909 (119905) 119861 (119905)) (21)






00011205762

001120573 011205771

0011205772

001120583 005120588 10


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

Nor

mal

ized

real

par

ts

656055504540353025201510

50

minus5




1



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

12111009080706050403020100

Nor

mal

ized

real

par

ts



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

minus05

minus10

Nor

mal

ized

real

par

ts









Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

Nor

mal

ized

real

par

ts


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

16

14

12

10

8

6

4

2

0

Am

plitu

de




119873



119878119873

119867(120596 p) = 120597119878

119867(120596 p)120597p

10038161003816100381610038161003816100381610038161003816(120596p0)

p0

H (120596 p) (22)








2le 0012












minus1



120588 120583 120573 1205762

OF

BCA

10minus3

B 1165 0053 0110 0017 0236A 1189 0055 0102 0015 0230W 1027 0058 0091 0011 0218

10minus4

B 1099 0056 0103 0010 0230A 1134 0058 0095 0011 0224W 0916 0045 0103 0011 0224

10minus6

B 1134 0058 0095 0011 0224A 0981 0046 0095 0010 0218W 0961 0050 0105 0011 0218

FCA

07B 1158 0054 0110 0017 0230A 1158 0054 0110 0017 0230W 1157 0053 0110 0017 0228

09B 1148 0055 0110 0018 0230A 1148 0055 0110 0018 0230W 1148 0055 0110 0018 0230

10B 1156 0056 0111 0017 0232A 1148 0055 0110 0018 0230W 1151 0057 0109 0015 0214

FSA

10minus1

B 1166 0052 0109 0016 0235A 1191 0050 0101 0016 0232W 1019 0053 0092 0010 0219

10minus2

B 1101 0054 0099 0099 0232A 1135 0060 0094 0011 0225W 0921 0048 0102 0011 0225

10minus3

B 1135 0061 0099 0010 0226A 0993 0049 0096 0010 0219W 0971 0051 0111 0010 0214

GAB 1101 0054 0099 0098 0233A 1136 0062 0092 0012 0226W 0932 0044 0110 0012 0228

PSOB 1101 0053 0099 0099 0233A 1132 0061 0095 0010 0224W 0918 0049 0103 0011 0225

6 Conclusions










Nomenclature















119878 Design space119878119873




the primary system










springs

Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












00011205762

001120573 011205771

0011205772

001120583 005120588 10


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

Nor

mal

ized

real

par

ts

656055504540353025201510

50

minus5




1



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

12111009080706050403020100

Nor

mal

ized

real

par

ts



Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

minus05

minus10

Nor

mal

ized

real

par

ts









Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

Nor

mal

ized

real

par

ts


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

16

14

12

10

8

6

4

2

0

Am

plitu

de




119873



119878119873

119867(120596 p) = 120597119878

119867(120596 p)120597p

10038161003816100381610038161003816100381610038161003816(120596p0)

p0

H (120596 p) (22)








2le 0012












minus1



120588 120583 120573 1205762

OF

BCA

10minus3

B 1165 0053 0110 0017 0236A 1189 0055 0102 0015 0230W 1027 0058 0091 0011 0218

10minus4

B 1099 0056 0103 0010 0230A 1134 0058 0095 0011 0224W 0916 0045 0103 0011 0224

10minus6

B 1134 0058 0095 0011 0224A 0981 0046 0095 0010 0218W 0961 0050 0105 0011 0218

FCA

07B 1158 0054 0110 0017 0230A 1158 0054 0110 0017 0230W 1157 0053 0110 0017 0228

09B 1148 0055 0110 0018 0230A 1148 0055 0110 0018 0230W 1148 0055 0110 0018 0230

10B 1156 0056 0111 0017 0232A 1148 0055 0110 0018 0230W 1151 0057 0109 0015 0214

FSA

10minus1

B 1166 0052 0109 0016 0235A 1191 0050 0101 0016 0232W 1019 0053 0092 0010 0219

10minus2

B 1101 0054 0099 0099 0232A 1135 0060 0094 0011 0225W 0921 0048 0102 0011 0225

10minus3

B 1135 0061 0099 0010 0226A 0993 0049 0096 0010 0219W 0971 0051 0111 0010 0214

GAB 1101 0054 0099 0098 0233A 1136 0062 0092 0012 0226W 0932 0044 0110 0012 0228

PSOB 1101 0053 0099 0099 0233A 1132 0061 0095 0010 0224W 0918 0049 0103 0011 0225

6 Conclusions










Nomenclature















119878 Design space119878119873




the primary system










springs

Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

60555045403530252015100500

Nor

mal

ized

real

par

ts


Frequency (Hz)0750 0825 0900 0975 1050 1125 1200 1275

16

14

12

10

8

6

4

2

0

Am

plitu

de




119873



119878119873

119867(120596 p) = 120597119878

119867(120596 p)120597p

10038161003816100381610038161003816100381610038161003816(120596p0)

p0

H (120596 p) (22)








2le 0012












minus1



120588 120583 120573 1205762

OF

BCA

10minus3

B 1165 0053 0110 0017 0236A 1189 0055 0102 0015 0230W 1027 0058 0091 0011 0218

10minus4

B 1099 0056 0103 0010 0230A 1134 0058 0095 0011 0224W 0916 0045 0103 0011 0224

10minus6

B 1134 0058 0095 0011 0224A 0981 0046 0095 0010 0218W 0961 0050 0105 0011 0218

FCA

07B 1158 0054 0110 0017 0230A 1158 0054 0110 0017 0230W 1157 0053 0110 0017 0228

09B 1148 0055 0110 0018 0230A 1148 0055 0110 0018 0230W 1148 0055 0110 0018 0230

10B 1156 0056 0111 0017 0232A 1148 0055 0110 0018 0230W 1151 0057 0109 0015 0214

FSA

10minus1

B 1166 0052 0109 0016 0235A 1191 0050 0101 0016 0232W 1019 0053 0092 0010 0219

10minus2

B 1101 0054 0099 0099 0232A 1135 0060 0094 0011 0225W 0921 0048 0102 0011 0225

10minus3

B 1135 0061 0099 0010 0226A 0993 0049 0096 0010 0219W 0971 0051 0111 0010 0214

GAB 1101 0054 0099 0098 0233A 1136 0062 0092 0012 0226W 0932 0044 0110 0012 0228

PSOB 1101 0053 0099 0099 0233A 1132 0061 0095 0010 0224W 0918 0049 0103 0011 0225

6 Conclusions










Nomenclature















119878 Design space119878119873




the primary system










springs

Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588 120583 120573 1205762

OF

BCA

10minus3

B 1165 0053 0110 0017 0236A 1189 0055 0102 0015 0230W 1027 0058 0091 0011 0218

10minus4

B 1099 0056 0103 0010 0230A 1134 0058 0095 0011 0224W 0916 0045 0103 0011 0224

10minus6

B 1134 0058 0095 0011 0224A 0981 0046 0095 0010 0218W 0961 0050 0105 0011 0218

FCA

07B 1158 0054 0110 0017 0230A 1158 0054 0110 0017 0230W 1157 0053 0110 0017 0228

09B 1148 0055 0110 0018 0230A 1148 0055 0110 0018 0230W 1148 0055 0110 0018 0230

10B 1156 0056 0111 0017 0232A 1148 0055 0110 0018 0230W 1151 0057 0109 0015 0214

FSA

10minus1

B 1166 0052 0109 0016 0235A 1191 0050 0101 0016 0232W 1019 0053 0092 0010 0219

10minus2

B 1101 0054 0099 0099 0232A 1135 0060 0094 0011 0225W 0921 0048 0102 0011 0225

10minus3

B 1135 0061 0099 0010 0226A 0993 0049 0096 0010 0219W 0971 0051 0111 0010 0214

GAB 1101 0054 0099 0098 0233A 1136 0062 0092 0012 0226W 0932 0044 0110 0012 0228

PSOB 1101 0053 0099 0099 0233A 1132 0061 0095 0010 0224W 0918 0049 0103 0011 0225

6 Conclusions










Nomenclature















119878 Design space119878119873




the primary system










springs

Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Nomenclature















119878 Design space119878119873




the primary system










springs

Acknowledgments


References
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article application of three bioinspired...

Documents