EngineeringQuarterly
VOL.I...No.2 WINTER2019
WELCOME! ByTERRYPILLING
Welcometoour secondissueofthe newsletteroftheUniversityofMaryEngineeringSchool.Since ourlastissueofthe EngineeringQuarterly backinNovember2018,wehavefinished theFallsemesterofcoursesandstartedthe Springsemester.Ourcurriculumiscontinuingitsdevelopmentwithnumerousnewengineeringcoursesandlabsbeingactivelytaught inthefourdegreeswearecurrentlyoffering oncampus.ThesedegreesareConstruction Management,CivilEngineering,MechanicalEngineering,andElectricalEngineering. Thissemesteralsomarksthecompletionof ourdevelopmentofthesedegreesupthrough Junioryearandwearecurrentlyconducting afacultysearchfornewprofessorsineachof theengineeringdisciplinestojoinourteam toteachanddeveloptheseniorlevelcourses startingnextfall.Weanticipatethesuccessful completionofthesearchlaterthismonthand wewillannounceournewhiresaswellasintroduceyoutotheminourspringissueofEQ inMay.Wesincerelyhopethatyouareenjoyingournewsletterandweplantocontinue topublishanissueeverythreemonthscorrespondingtoeachseason:Fall,Winter,Spring, andSummer.Ifyouhaveanysuggestions,or youhaveanyimportantnewsthatwouldbeof interesttoengineersandengineeringstudents pleasedon’thesitatetocontactme.
TerryPilling,PhD UniversityofMary
NEW FACULTY
TheSchoolofEngineeringhashireda newfacultymember fortheConstruction Managementprogram.
ProfessorMatthew Scerbakisaregistered professionalarchitect licensedintwostates, MichiganandAlaska.
Yorkandtaughtfurthercoursesandlabsin Constructionestimating,planning,DesignBuildmanagement,CommercialDetailing, andArchitecturalDesignattheStateUniversityofNewYorkandtheParsonsSchool ofDesign.MatthewjoinedtheUniversity ofMaryEngineeringSchoolasaprofessor ofConstructionManagementstartinginJanuary2019andwillbehelpingusdesignthe ConstructionManagementprogramaswellas teachingandadvisingstudentsinConstructionManagementandArchitecture. ProfessorScerbakispassionateaboutarchitectureandconstructionmanagementandhas beenworkingtobringbacktheroleofthe MasterBuilder.AccordingtoMatthew,“The mostsignificantaccomplishmentinmycareer,andmoreimportantthananydegree, wasthepioneeringofanewparadigminconstructionprojectdelivery.For13yearswith myownbusiness,Iactedasboththearchitectandconstructionmanager,concurrently, onallofmyprojects(allcommercialwork). Thiswas,andstillis,unheardof,exceptin residentialconstruction.Ididappliedresearch,inrealtime,toseeifitwaspossibleto restoretheancientMasterBuilderconceptin acontemporarycontext,assumingallofthe liabilityandriskthatthisentailed.Itwasa veryboldmove,butIwassuccessfulinthis ‘dual’role,managingupwardsof70different tradeswhiledesigningbuildings,anddeliveringprojectsatlesscostandinhalfthetime ofmycompetitionbyoverlappingdesignand constructionbyover80%.Architect,afterall, comesfromtheGreekarchi+tekton,which meansmasterbuilder.Iwasjustbringingthe conceptbackintothecontemporaryworld, somethingthatcouldonlybeexploredbyactuallydoingit.”
InhisteachingroleMatthewwantstoshow studentshowtomanagebothconstruction anddesign,“IwanttoteachmyConstruction Managementstudentswhatitmeanstowork inahighlycollaborativeenvironment,inan essentialrolethatisnotseparatefromdesign, butoverlapswithitinacontinuous,interdependent,process.Itisaskillandmindset thatwillpositionthemontheleadingedgeof industry.”
WORKFORCE DEVELOPMENT INITIATIVE ByBRIANOPP
we’vebeendoingthisinsomefashionsince ourfoundingin1959.WhatisnewistheUniversity’slaser-likefocusonincreasingourimpactinthisall-importantarea.AnewlyestablishedUniversityofMaryworkforcedevelopmentteamwillleadeffortstogrowandexpandtheUniversity’srelationshipswithemployers,toidentifythecriticalneedsofemployers,andtocollaboratewiththoseemployerstodevelopanddelivertheexacteducation andtrainingopportunitiestheyneedtotheir employees.
Alreadywe’veaddednewworkforcedevelopmentpartnershipswithMBIEnergyServices,TheNorthDakotaDepartmentofCorrectionsandRehabilitations,andMcKenzie CountyHealthcareSystem.Theseemployersjoinover50entitiesalreadypartnering withtheUniversityofMaryforworkforce developmenttraining.Partnershipswiththe UniversityofMaryprovidevariousbenefits toemployersincludingscholarshipsfortheir employees,theabilitytopursueskill-specific certificatesinasfewas15weeks,andmore.
Perhapsthegreatestvaluewecanprovideemployersandtheirworkforceisofferingtheflexibilitytomeettheirneeds.Educationandtrainingopportunitiescanbedeliveredin-person,online,orasahybrid.They canbeemployer-specificoropentopublic enrollment.Theycanbeofferedona“for credit”basisorsimplyasameanstofurther developtheskillsandknowledgeofaworkforce.Inshort,UniversityofMaryiswilling andabletotailoreducationandtrainingopportunitiestomeettheneedofanyemployer. PleasecontactBrianOpp,Universityof MaryWorkforceDevelopmentStrategist,for moreinformationandtoexplorehowtheUniversityofMary’sworkforcedevelopmentinitiativecanbenefityou.
blopp@umary.edu
(701)355-3798
Matthew
Matthewearnedhis BachelorofScienceinArchitectureatthe UniversityofMichiganin1983,hisMasterof ArchitecturewithdistinctionattheUniversity ofMichiganin1986withconcentrationin DesignTheory,andhisMasterofFineArts degreeinPhotographyandRelatedMediaat theParsonsSchoolofDesigninNewYork. Matthewhas13yearsofexperienceinconstructionandarchitectureandhetaughtatthe ImmaculateConceptionParishinFairbanks, Alaska,until2014whenhemovedtoNew
InJanuarytheUniversityofMary launchedaWorkforceDevelopmentInitiative withtheimmediateobjectiveofhelpingemployersattractnewemployees,andretaining anddevelopingcurrentworkers.Workforceis repeatedlyidentifiedasthenumberonebarriertoeconomicgrowthinNorthDakota.The topichasbeenresearchedatlength,nowisthe timetotakeactionandtheUniversityofMary iseagertomakeanimpact.
Workforcedevelopmentandtrainingare hardlynewatUniversityofMary.Infact,
TheUniversityofMaryispreparingto beginconstructionofourengineeringbuilding,whichweplantocompletebyFall 2020.Theengineeringfaculty,alongwith LukeSeidling,theDirectoroftheUniversityofMaryPhysicalPlant,TylerAtkinson theMaintenanceandProjectManager,and VaughnRhodestheCampusArchitect,have beenworkingwithRegisteredArchitectJoe LarriveefromBartlett&West,MikeVan Dyne,P.E.fromBartlett&West,ChrisWentz, P.E.fromCWStructuralEngineers,andJohn Klein,P.E.fromApexEngineeringGroupon thefloorplans,buildingmaterials,equipment purchases,classrooms,laboratoryfacilities, machineshop,woodshop,technologyneeds, etc.,forthepasteightmonths.Theplansare nownearlyreadyandconstructionisplanned tobegininMayofthisyearforagrandopeningfortheFall2020semester.
ENGINEERING SOCIAL ByKEVINALLAN
Thefirsteverengineeringsocialwasheld onThursday,Jan.31from11:00amto 12:30pmontheUniversityofMarycampuswheretheCareerServicesofficealong withtheEngineeringDepartmentFaculty welcomed14engineeringorganizationsfrom aroundthelocalareaandstatetocampusto speakwithourstudentsaboutengineering, theirorganization,adayinthelifeofanengineer,aswellasjobandinternshipopportunities.
studentsarerequiredtoworkoncrossdisciplinarydesignprojectsthatinvolveatleast onestudentfromeachofthebranchesofengineeringthatweoffer.Itisourcontention thatmostprojectsthatstudentswillencounter intheirfuturecareerswillinvolveengineers fromdifferingareasandsoitisimportantfor ourstudentstounderstandhowtoworktogetherwithadiverseteaminaccomplishing acommongoal.Theyshouldbeabletoexplaintheirideastoengineerswhohaveadifferentbackgroundfromtheirownandthey shouldalsobeabletodivideaprojectinto subprojectsthatlendthemselvestothevariousskillsets.Thisdivisionoflaborwillinvolveleadershipandcooperationanditwill alsoshowthemhowtomaximizethegroup productivitybyunderstandingthedifferent educationalbackgroundsaswellastheindividualstrengthsofeachofthemembers.
BraunIntertecwasoncampuswiththeir drillingrigandstudentsfromtheECI412
EngineeringSocial
Wehadapproximately30studentsattend theevent.Afterwards,whenvisitingwiththe employers,theywerepleasedwiththeevent andthecaliberofstudentsthatarebeingproducedinourEngineeringSchool.Theywere alsodelightedwithhowwellthestudentsengagedthem.
Thecompaniesthatattendedwere:Applied Engineering,BarrEngineering,Bartlett& West,BasinElectricPowerCooperative, BraunIntertec,CityofBismarck,KLJ,City ofMandan,Bobcat/Doosan,GreatRiver Energy,KilldeerMountainManufacturing, MDUResources,NorthDakotaDepartment ofTransportation,andWesternAreaPower Administration.
Thiswasanexcellenteventforstudentsto interactandnetworkwithemployerswhile learningmoreaboutlocalengineeringcompaniesandwewilllooktobuildonthesuccess ofthiseventandmakeitanannualoccurrence asourengineeringschoolcontinuestogrow. Pleasecontactusifyouwouldlikerepresentativesfromyourcompanytoattend.
Wewouldliketothankallwhohelpedto makethedayasuccess,startingfirstwith thefacultyintheEngineeringDepartment, catering/diningservicesstaff,facilities,and hospitalitystaff.Thesuccessoftheevent couldn’thavebeenachievedwithouttheoverallcollaborationofallgroupsinvolved.
kallan@umary.edu
SENIOR DESIGN PROJECTS
Startinginfall2019,ourseniorlevel studentswillbestartingtheirseniordesign projects.Theprojectsareseparatedintotwo parts:duringthefirstsemesteroftheirsenioryearthestudentsarerequiredtowork onadesignprojectwithintheirowndisciplineandworkwithotherengineeringstudentsfromwithinthatdiscipline.However, wehavestructuredtheprogramsothatin thesecondsemesteroftheirsenioryearthe
Weareveryinterestedinreceivingproject requestsfromcompaniessothatourstudents canworkonaprojectofrelevancetothereal world.Ifyouworkforacompanythatmight beinterestedinhavingourseniorlevelengineeringstudentshelpthemwithaproject pleasecontactus.Weplantocollectasmany projectsaswecanfromlocalengineeringrelatedcompaniesandallowourstudentsto chooseoneofthemtoworkonandform groupsaccordingtoindividualinterestinthe project.
Thetypeofprojectsthatwearelooking forareoneswhichrequireengineeringskills andknowledgetosolveandhavecurrentrelevancetoyourcompany.Thebestprojectsare oneswhichareimportantandusefultoyou andwhichwillgivethestudentstheopportunitytoimpressyouwiththeirabilities.
THE ENGINEERING CLUB ByRODRIGODACOSTAAPARECIDO
TheUniversityofMaryEngineeringStudentsClubisanorganizationthatwasstarted bythestudentsandinvolvesundergraduate engineeringstudentsfromallofthedisciplines.Theirgoalistoservethecampusand thecommunityandatthesametimetackle funandeducationalprojectsrequiringengineeringskills.
Currently,thestudentsintheEngineeringClubarewritingaproposaltotheUniversityinordertouseCADtomapanddesignthesprinklersystemlinesonthecampus. Theproposalincludesthemethods,materials,workbreakdown,resources,andproject budget.ThestudentsinvolvedincludeJunior,Sophomore,andFreshmanlevelsfrom eachofthedifferentengineeringmajors.This projectwillgivethestudentsexperienceinengineering,design,andmanagement,andisa wonderfulopportunity.Itisourhopethatthe Universitywillconsiderthesestudents’proposalandgivethemtheopportunitytoshow offtheirskills.
BRAUN INTERTECVISITSCAMPUS
BraunIntertecDrillingRig
SoilMechanicscoursewereabletoobserve thesub-surfaceexplorationsoilboring.The boringswerebeingdonefortheEngineeringSchoolexpansionandremodelthatiscurrentlyinthedesignphase.Thesoilborings willenablethefoundationdesignfortheEngineeringLabsand,inparticular,willallow thedesignofastrongfloorfortheCivilEngineeringLab’s200,000lbloadcapacityapparatusrackandoverheadcrane.Itisveryimportanttousthatourstudentsgetasmuchexposureaspossibletoactualengineeringwork suchasthis.Observingconstructionprojects, workingondesignprojectssuppliedbyengineeringcompanies,anddoingsummerinternshipsandsemesterco-op’swithengineeringrelatedcompaniesisessentialtopreparing themfortheirfuturecareersandtohelpthem overcomeanyintimidationtheymayhaveto workingonimportantprojectswithprofessionalengineers.
The28thannualNorthDakotaRegional HighSchoolScienceBowl,hostedbyWesternAreaPowerAdministration,washeldat theUniversityofMaryonSaturday,February 2,2019.Thisevent,amathandsciencecompetitionforhighschoolstudents,isanacademiccompetitionthattestsstudents’knowledgeinallareasofscienceandmathematics andissponsoredbytheU.S.Departmentof Energy.UniversityofMaryfacultyandstudentsinthenaturalsciences,mathematicsand engineeringparticipatedintheScienceBowl asjudgesandotherofficials.
Twenty-twoteamsofstudentsrepresentingschoolsfromthroughoutNorthDakota viedforfirstplaceandafreetriptothe NationalScienceBowlinWashington,D.C., April25-29.Thefast-pacedquizformat ofthecompetitionchallengedthestudents’ knowledgeofbiology,chemistry,earthscience,physics,energyandmathematics.Afteramorninground-robinformat,therewas anafternoondouble-eliminationroundtodeterminethecompetitionchampion.The2019 NorthDakotaRegionalScienceBowlChampionshipwasawardedtoateamfromMagic CityCampusHighSchoolinMinot,ND.
Theregionalandnationaleventsencouragestudentinvolvementinmathandscience, inspirethenextgenerationtopursuecareers
inscience,technology,engineeringandmath fields,andrewardsuperioracademicachievement.Additionally,studentswithstronginterestsinscience,technologyandengineeringhadtheopportunitytomeetUniversity ofMaryEngineeringstudentsandfacultyto learnmoreaboutourprogram.
THERMODYNAMICS ByDANIELBARR
SensorontheGrottoconcreteslab
EmmeldidaprojectintheirEME341ThermodynamicscourseforprofessorDanielBarr whichinvolvedtakingtemperaturemeasurementsofthehotwaterheatingsystemthat runsunderneaththeconcreteslabonthewest sideoftheLumenVitaeUniversityCenterat thelocationoftheWarfordGrotto.Theheatingsystemisdesignedtokeepsnowandice clearfromtheWarfordGrottoandisadjusted manuallybyUniversityofMaryPhysical Plantemployeesasthetemperaturechanges overthecourseoftheyear.ThePhysical Plantwantedthestudentstostudythetemperatureoftheslab,theairtemperature,the outgoingwatertemperature,andthereturn watertemperatureovertimetoseeifthings wereworkingefficiently.Thetemperature differenceswouldallowanestimateofboth theeffectivenessoftheheatingsystemaswell asthecostovertimeofkeepingtheconcrete floorintheGrottoclear.
ThestudentsusedawaterproofDS18B20 digitaltemperaturesensorconnectedtothe
Temperaturemeasurements
Thestudentsthenwroteanalysissoftware tostudythefluctuationsinthedataandused theequationsandtheoryfromtheirthermodynamicscoursetostudythenetheatflow. Thegraphtheyobtainedshowsthechanges inthetemperaturesovertimeincludingthe fourhourcycleoftheheatexchangervalve intheLumenVitaeUniversityCenter.They alsoobservedtheincreaseofthetemperature oftheair,thepipes,andtheconcreteslabduringthedayfollowedbythedecayofthetemperatureatnightfollowingNewton’slawof cooling.Theresultshowsthattheheatingsystemkeepstheconcreteslabapproximately10 degreeswarmerthantheairtemperatureatall times.
Thestudentswroteupascientificreport oftheirfindingsandalsogaveapresentation whichwasattendedbymembersoftheUniversityofMaryPhysicalPlantaswellasEngineeringprofessors.
minethecapacitorandresistorvaluesrequiredtofilterthevariousfrequencyranges. Theresultingequalizerdisplaywillbeused onfutureelectricalengineeringdisplaytables atvariouseventssothatpeoplepassingbycan tryitout.Youcanseebythecircuitdiagram thatitworksbycomparingtheinputsignal amplitudewiththepowerrailafterthevoltage hasbeenreducedthroughaseriesofstepsand ifthedividedpowersignalislowerthanthe inputsignal,theopampsaturatestoground settingtheoutputoftheopamptoground andallowingcurrenttoflowthroughthelight emittingdiodeattachedtoit.Iftheinputsignalislowerthanthedividedpowersignalthe
Sensoronthehotwaterfeed
GeneralPurposeInputOutput(GPIO)pins onseveralRaspberryPicomputersandthey wrotecontrolsystemsanddatacollection softwareinBashshellscriptswhichcollected datapointsofthetemperatureseveryfive minutesforafourdayperiod.
Thejuniorelectricalengineeringstudents builtanequalizerdisplayusingeightoperationalamplifiersandsomeLED’s.TheinputsignalcausestheLED’stolightupasthe amplitudeofthesignalincreases.Theymade sevenoftheseandthencombinedthemwith low-passandhigh-passfilterssothateachdisplaymeasurestheoutputofadifferentfrequencyband.Thefinalprojectusedacell phoneaudiojackastheinputsothatyoucan playasongonyourphoneandtheequalizerdisplaywillshowthedifferentfrequenciesandamplitudesofthesignal.
Circuitdiagramforvolumemeter opampsaturatestorailandnocurrentwill flowthroughtheattachedLED.Thestudents addedaband-passfiltertothisandcombined abunchofthemcorrespondingtoeachfrequencyrangeofthedisplay.
FLUIDSAND STRUCTURES ByERICGARCIA
StudentsinEME306:FluidMechanics andECI351:StructuralMechanicsworked
Theyalsowrote SPICE simulationsofall ofitandperformedallofthetheoreticalcircuitanalysiscalculationsrequiredtodeter-
onacollaborativeprojectwhichinvolveddesigningandbuildinganinfrastructurethat housedahydraulicsystem.Studentsfrom eachclasshadtodesignsystemstomeetspecificrequirements.
Thecivilengineeringstudents,takingECI 351:StructuralMechanics,hadtheprimary tasktodesignandbuildatrussbridgethat hadtospan10feetwithoutdeflectingmore than0.3incheswhenatleastthreeteammemberswalkedacrossit.Eachteamwasprovidedwithlessthan$60ofmaterials.$15 worthof1×2furringstrips,$36worthof0.75 thickplywood,andwoodscrews.Thedesign processincludedcalculationstomakesurethe
forcesthatthetrussmembersexperienceddid notexceedthematerialsstrengthandtomake surethetrussstructuredidnotdeflectmore thanalloweddeflection,0.3inches(lengthof thebridgedividedby400).
Inadditiontothebridge,eachgroupin ECI351wastaskedtodesignandconstruct
lationsforpipes,openchannels,pumps,and impulseturbinestoensurethattheintegration ofallpartswouldresultinastablesteadywaterflowoftheentiresystem.
ROCKET LAUNCH TELEMETRY ByTERRYPILLING
Asaninterestingprojectfortheirdata analysisclass,ourengineeringstudentsana-
therealtimetelemetrydataofthelaunch!Unfortunately,Space-Xonlyincludesthealtitudeandthespeedovertimeintheirdisplay, butfortunately,thosenumbersareenoughfor anengineertoextractmuchmoreinteresting informationandourcleverengineeringstudentsdidexactlythat.
Thenumberstheyshowarethespeed(km/h), andthealtitude(km).Thestudentsuseda pythonscripttoextractthosetelemetrydata fromtheYouTubevideoframe-by-frameand thenwroteBashscriptsandCprogramstoextractmuchmorefromthem.
Trussbridgesupportingthewholeclass anotherstructureoftheinfrastructurewhich wasusedforthewatersystemdesignedfor EME306.Thisincludedawatertower,a buildingwithmultipleelevatedwateroutlets, andanopenchannel.
Themechanicalandcivilengineeringstudents,takingEME306:FluidMechanics, weretaskedtodevelopasteady-statehydraulicsystemwhichinvolvedhavingawatertowerprovidethenecessarypressuretothe system.Thepressurefromthewatertower neededtoprovidewaterflowtooutletson multiplefloorsofthebuildingstructureand provideahighenoughvelocitytorunanimpulseturbine.Thedesignworkrequiredof thestudentsinvolvedpressureandflowcalcu-
FalconHeavyLaunch lyzedtheYouTubevideooftheSpace-XFalconHeavytestflight.Youmightalsoliketo trysometimeaswelland,ifyouhaven’tseen it,wehighlyrecommendwatchingtheentire YouTubevideohere: https://youtu. be/wbSwFU6tY1c
Lately,Space-XhasbeenincludingsomethingalittleextraontheYouTubevideosof theirrocketlaunchesinthattheyshowasmall displaywindow(seethescreenshotabove)in thetoprightcornerofthevideowhichgives
KnowingtheAltitude(i.e.the ycomponentofthedistance)overtimealong withthespeed(i.e.themagnitudeofthe velocityvector)allowedthemtoextractthe x-component,ordownrange,distance.The figureistakenfromthereportsubmittedby PhilipSpringsteenandJosefSollmannwho areJuniorlevelElectricalEngineeringstudents.
The x and y componentsofthevelocity vectorcanbeextractedfromaknowledgeof thespeedandthetime,alongwiththealtitude.Thestudentshavelabeledtheplaces wherethespeedisclearlychangingwhichindicatesthatsomethingspecialishappeningat thattime.Whencomparingwiththecommentaryofthevideotheywereabletolabel theeventsthatoccuratthosetimes.Finally, usingthevelocitycurvetheyareabletofind thederivative(numerically)ofthatcurveand therebyfindtheaccelerationcurve.Thisindicatestheforcesthatareactingonthevehicle overthecourseoftheflight.Theywerevery gratifiedtoseethatthemajorflighteventscan beclearlyseenintheaccelerationcurve.For examplethesideboostersthrottlingdown,the BoosterEngineCutoff(BECO),theMainEngineCutoff(MECO),theSecondStageEngineStart(SES),andfinallytheSideEngine Cutoff(SECO)atthetop(notlabeledontheir
diagram).Thiswasafunwaytousetheir programminganddataanalysisskillstoshow howmuchinformationonecangatherbyusingabitofphysicsanddataanalysis.Onecan extractacompletehistoryofthelaunchfrom twosimplenumbersprovidedonthescreen duringthevideo.
WatchingaSpace-Xlaunchisalotoffun foranyonetoenjoy,butasNobelPrizewinningtheoreticalphysicistRichardFeynman alwayssaid,havingscientificcuriosityand technicalabilityenablesonetodelvedeeper intoitandtherebyenjoyitevenmore.
BAT GUANO CRAZY ByTERRYPILLING
Hereisastoryaboutscientificethics, war,famine,death...andbatguano.Thesubjectcameupduringanassignmentthestudentsdidwheretheymodeledthepopulation
growthoftheUnitedStatessince1610usingthelogisticdifferentialequation: dx dt = rx 1 x K .where x isthepopulationover time, r isthegrowthratewhichdependson thingslikereproduction,disease,immigration,war,etc.,and K isthe CarryingCapacity,whichisthemaximumpopulationthatcan besupported.TheCarryingCapacitydepends onthingslikespaceandresources.Inparticular–food.Thegraphshownbelowwas providedbysophomoreengineeringstudents intheirsolutiontooneoftheproblemsinan assignmentthissemesterandshowsthemaximumCarryingCapacityoftheUnitedStates atsomewherearound450millionpeople.
Beforewegettothebatguano,weneed tofirstgobacktotheyear1798andanEnglishclericnamedThomasMalthus.Malthus wroteabookcontaininganessayonthePrincipleofPopulationwhereinMalthusobserved thatanincreaseinanation’sfoodproduction
improvedthewell-beingofthepopulace,but thattheimprovementwasatemporaryone becauseitledtopopulationgrowth,which thenrestoredtheoriginalpercapitaproductionlevel.Inotherwords,mankindhada propensitytoutilizeabundanceforpopulationgrowthratherthanformaintainingahigh standardoflivingandthatpopulationshad atendencytogrowuntilthepoorerclasses sufferedhardshipandgreatersusceptibilityto famineanddisease,aviewthatisnowreferredtoasa Malthusiancatastrophe Malthusianmodelshavetheform: x(t)= x0 ert where x0 = x(0) istheinitialpopulationsize, r isthepopulationgrowthrate,and t isthetimeparameter.Themodelcanalso beenwrittenintheformofadifferentialequa-
pacityandthatwewouldreachitverysoon. However,itdidnothappen.Thepopulation oftheplanetisatleastthreetimeslargerthan anyonepredicteditcouldbebackinMalthus’ day.Whathappened?WasMalthuswrong? Isthelogisticmodelofpopulationgrowth wrong?Toanswerthesequestionsweneed tostartwithbatguano.
Theword‘guano’originatesfromtheAndeanindigenouslanguageQuechua,which referstoanyformofdungusedasanagriculturalfertilizer.Archaeologicalevidence suggeststhatAndeanpeoplehavecollected guanofromsmallislandsandpointslocated offthedesertcoastofPeruforuseasasoil amendmentforwellover1,500years.Spanishcolonialdocumentssuggestthattherulers oftheIncaEmpireassignedgreatvalueto guano,restrictedaccesstoit,andpunished anydisturbanceoftheanimalsproducingthe guanowithdeath.
liontons(employing60,000workersandsellingatUS$45perton)in1925tojust800,000 tons,producedby14,133workers,andselling at$19pertonin1934.Theannualworldproductionofsyntheticnitrogenfertilizeriscurrentlymorethan100milliontonsandthefood baseofhalfofthecurrentworldpopulationis basedontheHaber-Boschprocess.
Haberwasawardedthe1918NobelPrize inChemistryforthiswork.
tion: dx dt = rx andisoftenreferredtoasthe exponentiallaw.Itiswidelyregardedinthe fieldofpopulationecologyasthefirstprincipleofpopulationdynamics,withMalthusas thefounder.Theexponentiallawistherefore alsosometimesreferredtoastheMalthusian Law.
Malthuswrotethatalllifeforms,includinghumans,haveapropensitytoexponential populationgrowthwhenresourcesareabundantbutthatactualgrowthislimitedbyavailableresources.
Theincreaseofpopulationisnecessarilylimitedbythemeansof subsistence.Populationdoesinvariablyincreasewhenthemeans ofsubsistenceincreases,and,the superiorpowerofpopulationis repressedbymoralrestraint,vice andmisery.Throughtheanimalandvegetablekingdoms,naturehasscatteredtheseedsof lifeabroadwiththemostprofuseandliberalhand.Thegerms ofexistencecontainedinthis spotofearth,withamplefood, andampleroomtoexpandin, wouldfillmillionsofworldsin thecourseofafewthousand years.Necessity,thatimperious allpervadinglawofnature,restrainsthemwithintheprescribed bounds.Theraceofplants,and theraceofanimalsshrinkunder thisgreatrestrictivelaw.Andthe raceofmancannot,byanyefforts ofreason,escapefromit.Among plantsandanimalsitseffectsare wasteofseed,sickness,andprematuredeath.Amongmankind, miseryandvice.
ThomasMalthus,1798
AfterMalthus,everyoneagreedthatthe EarthhadamaximumintrinsicCarryingCa-
InNovember1802,AlexandervonHumboldtfirstencounteredguanoandbeganinvestigatingitsfertilizingpropertiesatCallao inPeruandfoundthatmassivedepositsof guanoexistedonsomeislands,insomecases morethan50mdeep.Guanofromcavebat depositscontainnitratesandwasusednot onlytofertilizecropsallovertheworldbut alsotoproducegunpowder,and,in1856,the UnitedStatespassedtheGuanoIslandsAct, whichgaveU.S.citizensdiscoveringasource ofguanoonanunclaimedislandexclusive rightstothedeposits.Controloverguano wasneededaroundtheworldbothforfertilizingcropsandforproducingexplosivesand itplayedacentralroleintheChinchaIslandsWar(1864-1866)betweenSpainanda Peruvian-Chileanalliance.
Thiswasthestateoftheworldinthelate 1800’s.Apopulationexplosionfueledby batguanothatwasheadingforaMalthusian Catastrophe.Whichbringsusto1909anda chemistnamedFritzHaber.
FritzHaberwasaGermanchemistwho workedatUniversityofKarlsruhefrom1894 to1911whereHaberandhisassistantRobert LeRossignolinventedtheHaber-Boschprocess,whichisthecatalyticformationofammoniafromhydrogenandatmosphericnitrogenunderconditionsofhightemperatureand pressure.
Tofurtherdeveloptheprocessforlargescaleammoniaproduction,Haberturnedto industry.TeamingwithCarlBoschatBASF, theprocesswassuccessfullyscaled-upto producecommercialquantitiesofammonia. TheHaber-Boschprocesswasamilestone inindustrialchemistry.Theproductionof nitrogen-basedproductssuchasfertilizerand chemicalfeedstocks,previouslydependenton acquisitionofammoniafromlimitednatural depositssuchasbatguano,nowbecamepossibleusinganeasilyavailable,abundantbase atmosphericnitrogen.Theabilitytoproduce muchlargerquantitiesofnitrogen-basedfertilizersinturnsupportedmuchgreateragriculturalyieldsandpreventedbillionsofpeoplefromstarvingtodeathandeffectively postponedthepredictedMalthusianCatastrophe.
Thediscoveryofanewwayofproducing ammoniacausedthenaturallyextractednitrateproductioninChiletofallfrom2.5mil-
Nowhereistheproblem:In1914the worldbecameengulfedinWorldWarI.Haber greetedWorldWarIwithenthusiasm,joining 92otherGermanintellectualsinsigningthe ManifestooftheNinety-ThreeinOctoberof thatyear.Haberplayedamajorroleinthe developmentofthenon-ballisticuseofchemicalwarfareinWorldWarI,inspiteofthe prohibitionoftheiruseinshellsbytheHague Conventionof1907(towhichGermanywas asignatory).Hewaspromotedtotherankof captainandmadeheadoftheChemistrySectionintheMinistryofWarsoonafterthewar began.Inadditiontoleadingtheteamsdevelopingammoniaandchlorinegasalongwith otherdeadlygasesforuseintrenchwarfare, Haberwasonhandpersonallywhenitwas firstreleasedbytheGermanmilitaryatthe SecondBattleofYpres(22Aprilto25May 1915)inBelgium.
Aspecialtroopwasformedforgaswarfare(PioneerRegiments35and36),underthe commandofOttoPeterson,withHaberand
FriedrichKerschbaumasadvisors.Haberactivelyrecruitedphysicists,chemists,andother scientiststobetransferredtotheunit.Future NobellaureatesJamesFranck,GustavHertz, andOttoHahnservedasgastroopsinHaber’s unit.
GaswarfareinWorldWarIwas,ina sense,thewarofthechemists,withHaber pittedagainstFrenchNobellaureatechemist VictorGrignard.Regardingwarandpeace, Haberoncesaid,“DuringpeacetimeascientistbelongstotheWorld,butduringwartime hebelongstohiscountry.”Thiswasanexampleoftheethicaldilemmasfacingchemistsat thattime.
Haberdefendedgaswarfareagainstaccusationsthatitwasinhumane,sayingthat deathwasdeath,bywhatevermeansitwas inflicted.Duringthe1920s,scientistsworkingathisinstitutedevelopedthecyanidegas formulationZyklonA,whichwasusedasan insecticide,especiallyasafumigantingrain stores.Thenextiteration,calledZyklonB, lateremergedasthepreferredkillingtoolof NaziGermanyforuseinexterminationcamps duringtheHolocaust.
Sohereistheethicalquestionthatwe askedourstudents:wasHaberaherooravillain?Haberdevelopedtheprocessinorderto producelargequantitiesofammoniaandnitrogenwhichwereusedinfertilizerbutwere alsousedtobuildweaponsandpoisonousgas duringWorldWarI.Hisprocessresultedin thehorribledeathsofcountlesspeopleinthe trenchesofFranceandBelgium.Atthesame time,hisprocessofproducingammoniaresultedintheaversionoftheMalthusianCatastropheandhassincesavedcountlesslives,the worldover,fromstarvationanddeath1
Youarefreetomakeupyourownmind astothemorallessonsinthisstoryandhold yourownethicaljudgments.But,asfaras Haberisconcerned,oursophomoreengineeringstudentsconcludedthatscientifictruth, andtheprogressoftechnology,areindependentoftheparticularscientistwhodiscovers themandthattheHaber-Boschprocesswould haveeventuallybeenfoundwithoutHaber’s involvement.Theybelievethattheends do not justifythemeansandthathandingouta NobelPrizetoaguylikeFritzHaberisjust batguanocrazy.
FUN STUFF
IneachofourissuesofEQwedevotea sectiontoproblemsandgamesthatwebelieve ourreaderswillenjoy.Sincethisisanewsletterdesignedforengineersandscientistsour gamesareperhapsabitmoreinvolved(anda lotmorenerdy)thanonemightfindinother newsletters.However,nooffense,butwe thinkyoumightbenerdyenoughtoenjoy them.
KENKEN PUZZLE ByNANCYKELLYHereistheKenKenpuzzleforthiseditionoftheEngineeringQuarterly.Asareminderforthoseofyouwhoarenewtoour publication,KenKenandKenDokuareastyle ofarithmeticlogicpuzzleinventedin2004
Thesolutionisatthebackofthisnewsletter.
CONTROLS PROBLEM ByANTHONYGARCIAHereisaproblemfromDr.AnthonyGarcia’sEEL405ControlSystemscourse.Note that,inthediagram, e isthevoltageEMF, τ isthetorque,and θm isthedirectionofpositiverotation.Fortheelectromechanicalsys-
Ifyouoryourcompanyisinterestedin sponsoringascholarshipforengineeringstudentspleasecontactus.Atthemomentthe onlyscholarshipspecificallyavailabletoengineeringstudentsistheannualscholarship generouslyawardedbytheNorthDakotaSocietyofProfessionalEngineers.Scholarships areimportanttoboththestudentandthesponsoringentityforthefollowingreasons:
• theyhelpstudentsfinanciallyandreducetheamountofstudentloansthey needtotakeout.
• theyarevaluableadditionstothestudent’sresumewhentheystartapplying forengineeringpositionsandareoften moremeaningfulthangradesindeterminingtheabilitiesandcharacterofan applicant.
• theyareanaddedincentiveforachievingthehighestlevelofacademicexcellenceandexcellenceofcharacterand service.
• thesponsoringpersonorcompany isprominentlymentionedandpraised everyyearwhenthescholarshipis granted.
Sopleaseconsiderhelpingourengineeringstudentsbysponsoringascholarship. Scholarshipscanbeofanyamountandthey canhaveavarietyofconditionsfordeterminingwhoisselectedeachyear.Theycanbeset upasonetimeawardsortheycanbeoffered everyyear.Youcanevenendowascholarshipsothatitcontinuestofunditselfinperpetuity-aminimumof$25,000isrequiredto setupanendowmentforapermanentscholarship.PleasenotethattheUniversityofMary isanon-profituniversityandalldonationsare taxdeductible.
byJapanesemathteacherTetsuyaMiyamoto. KenKenderivesfromtheJapanesewordfor cleverness.AsinSudoku,thegoalofeach puzzleistofillagridwithdigits1through 4fora 4 × 4 grid,1through5fora 5 × 5, 1through6fora 6 × 6,etc.sothatno digitappearsmorethanonceinanyrowor anycolumn.Thegridsaredividedintoheavilyoutlinedgroupsofcellscalledcagesand thenumbersinthecellsofeachcagemust produceacertain‘target’numberwhencombinedusingthespecifiedmathematicaloperation.Digitsmayberepeatedwithinacage,as longastheyarenotinthesameroworcolumn andasingle-cellcageisafreespace.
temgiven,determinethefollowing:
(a) TheTransferFunctionforaninputvoltage va andoutputangularposition θm
(b) TheState-Spacerepresentationforaninputvoltage va andangularposition θm Thesolutionisatthebackofthisnewsletter.
ELECTROMAGNETIC FIELDS PROBLEM ByNANCYKELLYThisproblemisfromDr.NancyKelly’s EEL316ElectricandMagneticFieldscourse. Acapacitorconsistsoftwohollowthinwalledconcentricconductingtubesoflength L.Theinnertubehasaradius Si = K andthe outertubehasaradius So =3K.Chargesof +Q and Q areplacedontheinnerandouter tubes,respectively.Thechargesareuniformly distributedonthesurfaces.Wewillignorethe fringingeffectsoftheelectricfieldnearthe cylinderends.
(a) Determinetheelectricfieldforradialdistances s<Si, Si <s<So,and s>So
(b) Computetheelectricpotentialdifference betweenthetwotubes.
(c) Computethecapacitanceofthissystemof conductors.
Thesolutionisatthebackofthisnewsletter.
STATICS PROBLEM ByJIMCARRICOD
ATA PROBLEM ByTERRYPILLING
Sometimesyouneedaquickanddirty waytodosimplethingstoadatasetandyou don’tfeellikeputtingitallintoExcelorwritingaprogramtodoit.Hereisashellscript writteninBash.Canyoufigureoutwhatit doestoadatafile?
#!/bin/bash touchresult.dat
whilereadline;do echo$line>tmp catresult.dat>>tmp mvtmpresult.dat done rmtmp
HereisaproblemfromDr.JimCarrico’s ENR201Staticscourse.Itisknownthata forcewithamomentof960Nmaboutthe
point D isrequiredtostraightenthefencepost CD showninthefigure.Ifthecapacityof thewinchpuller AB is2400N,determinethe minimumvalueofthedistance d tocreatethe specifiedmomentaboutthepoint D. Thesolutionisatthebackofthisnewsletter.
OH ROMEO,ROMEO, WHEREFOREART THOU ROMEO?
ByPATRICKKELLYInoneofthelesser-knowndraftsof Romeo andJuliet,Romeo,Benvolio,andMercutio arestudentsattheVeronaInstituteofTechnology.Inthisversionofthefamousbalcony scene[ACTII,scene2],Romeovaults (inasprightlymanner)overtherailingwiththe aidofatrebuchetthatheandhisfriendshad builtasaclassproject.Ourtask,inaidofthe star-cross’dlovers,istocalculatethelaunch parameterswithinthecontextofthemodel presentedbelow.
♥ Thebalconyrailingis 8 m aboveground andalongthewall’sedge.Thetrebuchetarmreleasesitsprojectileatan initialheightof 2 m.Therearenoobstacles,andthegroundisleveladjacentto thestructure.
♥ Taketheaccelerationduetogravity, g =10 m/s 2 ,directedstraightdownward.
♥ Theeffectsoffrictionanddragarenegligible.
♥ Romeoismovinghorizontally,with speed 1 m/s,asheclearstherailing.2
(a) Determinethelaunchspeedandtheinitialanglethatsatisfythedesignconstraints,andhowfar,horizontally,away fromthewallRomeo’slaunchpoint shouldbe.
(b) Romeoisacceleratedfromresttohis launchspeedbythetrebuchetarm. Let’ssaythatthedistancethrough whichhemoveswhileacceleratingis 2 m (theheightatwhichheisreleased). Estimatetheaverageaccelerationthat Romeomustenduretobelaunchedat thespeedcomputedin(a).
Thesolutionisatthebackofthisnewsletter.
Forexampleifyoustartwith23youget 23+32=55 andthereverseof55is55 againandsoyouarefinished.Now,thereason thattheproblemiscalledthe‘196Problem’ isbecause196isthefirstnumberforwhich itisunknownwhetherthealgorithmeverproducesapalindromicnumber.
196-Problem: Startingwiththenumber196 doesthealgorithmeverproduceapalindrome?
Theprogramlisting palindrome.c givenat theendofthisarticleisaprogramwrittenin theCprogramminglanguagethattakesnumbersorlistsofnumbersandperformsthe196algorithmonthemuntiltheyproduceapalindromicnumberorreachthemaximumnumberofiterationswithouthittingapalindrome. Notethattheprogramiswrittensothatnumbersarerepresentedasstringsofdigitsand theadditionandcarryingofdigitsisdoneby hand.Thisistoavoidthelimitationsofyour computerregardingthesizeofnumbersitcan dealwith.
IntheCprogramwehaverelegatedtheactualcomputationstothefunctions reverse, iterate and add.Thefirstonetakesa stringasinputandreturnsthereverseofthe string.Thesecondonetakestheoriginalnumberandperformsthe196-algorithmonituntiliteithergetsapalindromicnumberorit reachesthemaximumnumberofiterations. Finallythelastfunctionaddstwostringsby addingeachdigitandkeepingtrackofthe carry-overs.Ithastoincreasethelengthofthe stringrepresentingthesuminthecasethata carry-overoccursinthelastadditionandthis isdonewiththefinal else statementinthe function.Wecompiletheprogramusing gcc-opalindromepalindrome.c tocreateanexecutableprogramcalled palindrome whichwecanthenrunfrom thecommandline.Hereisanexample:
echo"155"|./palindrome appliesthealgorithmtothenumberfrom155 whichresultsin
155-->4444
wegetthefollowingoutput:
189-->1881190-->45254
191-->191192-->6996 193-->233332194-->2992
195-->9339196(nosolution)
197-->881188198-->79497
whichshowsthattheprogramfoundpalindromicnumbersforallofthemexceptfor 196.Inthatcaseitreachedthespecifiedmaximumnumberofiterationsandthenjustreturnedwhatevernumberithadatthatpoint. Youcouldtrylettingtheprogramdomoreiterations,butwemustwarnyouthattheprogramisquiteamemoryanddiskspacehog.It isverylikelythatifyoutrytogotoohighthe computerwillkilltheprogram.Butyoucan definitelydomoreiterationsthan50.
AsImentionedinmypreviousarticle ontheCollatzProbleminthelastEngineeringQuarterly,oneofthefirstthingsthatyou shouldtrywhenplayingwithunsolvedproblemslikethisistosearchforpatterns.Asan exampleofwhatImean,letuslookatasort ofpatternthatIhavefoundin196-algorithm. Tofindit,youmerelyneedtolookathowoftennon-palindromicnumberslike196appear. HereissomethingIfoundafter50iterations ofthefollowingnumbers:
196(nosolution)
-->925153265399993573340628
295(nosolution)
-->925153265399993573340628
394(nosolution)
-->925153265399993573340628
493(nosolution)
-->925153265399993573340628
592(nosolution)
-->925153265399993573340628
691(nosolution)
-->925153265399993573340628
790(nosolution)
-->925153265399993573340628
Problem: Writeacomputerprogramto explorethe196-problem.Onepossiblesolutionisgivenattheendofthisarticle.
A palindromicnumber isanumberwhose reverselooksexactlythesameastheoriginalnumber.Forexamplethenumber121is apalindromicnumberbecauseifyoureverse thedigitsyouget121backagain.Let’sbegin bydefininganiterativealgorithm:
196-algorithm
• Chooseapositiveintegerwithtwodigitsormoreandreversethedigits.
• Thenaddthereversednumbertothe originalnumber.
• Nowrepeatthiswiththenewnumber andcontinueinthisfashionuntilyou reachanumberthatisthesameboth forwardandbackward.I.e.untilyou reachapalindromicnumber.
whichsaysthat,inlessthan50iterations,the startingnumber155arrivesatthepalindromic number4444.Anotherexampleifyouare runningaformofLinuxlikeUbuntuandare usingtheBashshellis:
foriin{155..160};
doecho$i|./palindrome whichoutputs
155-->4444
156-->6666
157-->8888
158-->11011
159-->1221
160-->343
Therearemanyotherwaystorunacommand lineprogramlikethiswhichisdesignedto beusedinapipeline.Foronemoreexampleifwehadafilecalled‘numbers’which containedallthenumbersfrom189to198we couldruntheprogramlikethis:
catnumbers|./palindrome
Firstnoticethatnumberslike295and592are reallythesamethingsincetheyarethereverse ofeachotherandwhenyouapplythealgorithmyoubeginbyreversingthenumberbeforeaddingthem.So,forexample,thereis noessentialdifferencebetweenstartingwith 196orstartingwith691.Alsonoticethatall theabovenumbersgivethesameresultafter50iterationsandeachofthese‘nosolution’numbersis99awayfromtheprevious one!Thesestartingpointscanallbeconsideredthesamesequencesincetheyreachthe samenumberatsomepointasyouiterate.In otherwords,solvingtheproblemfor196automaticallysolvesitfortheotherones.
Itturnsoutthattherearenumbersthat don’tseemtogivepalindromicnumberbut arealsonotthesameasthesequencegeneratedby196.Forexample,fornumbersupto 8000thereseemstobethreedifferentgenerators.Theseare196,879and1997.Allof thebig‘nosolution’numbersthatoccurup to8000areequivalenttooneofthesethree. Itturnsoutthatthe‘nosolution’numbersare calledLychrelnumbersandthethreenumbers abovearecalled‘seeds’sincetheyarethestart ofnewsequencesofequivalentLychrelnumbers.Itwouldbeinterestingtoknowifthese threeremainindependentuptoarbitraryiterations.Perhapsthe nthiterationof879will givethesameresultasthe (n + k)thiteration of196forsome k.Ifthatoccursthenthenumberswillleadtothesamepalindromicnumber (ifoneexists).
Exercises
(1) Findallofthedifferentsequencesup to11000.Arethereanynewones otherthentheonesresultingfrom196, 879and1997thatwealreadyfound? Note:goingtohigheriterationswill slowdownthecomputerveryquickly. Perhapsyoucanfindabetterwayto writetheprogramsothatitrunsfaster andtakesuplesscomputerresources?
Answer: Thenumbers7059,10563, 10553and10877allsupposedlyform newsequencesofLychrelNumbersaccordingtovariouswebsites.
(2) Compareiterationsoftheseparate‘nosolution’numberstoseeifanyofthem join.Maybeyoucouldplotthevalueat eachiterationversustheiterationnumberandseeifthelinesjoinatanypoint. Canyoufindamathematicalproofthat theywillneverjoin?Evenamathematicalplausibilityargumentbasedon theslopesofthelengthvs.iterations plotswouldbeinteresting.Perhapsyou canshowthatthemathematicsinvolved inthe196-algorithmimpliesthatthe slopeswillalwaysbedifferentandthe lineswillneverjointherebyproving thattheyareseparatenon-palindromic numbersfrom196.
(3) Whendealingwiththenaturalnumbers wehaveessentiallybeenlookingfor palindromesinasequenceofnumbers (thedigits) base10.Wecoulduse afavoritetrickofmathematiciansand lookatadifferentbasesystemforthe numbersotherthan10.Forexampleifweworkinbase2(i.e.binarynumbers)thenumber196translatesinto11000100.Performingthe 196-algorithmonthisnumberproceeds asfollows:11000100+00100011= 11100111,whichisthereforeabinary palindromeafteronly1iteration.You canspendhoursworkingoutthepalindromesinvariousbases,bothlessthan 10andalsolargerthan10.Doingthis maygiveyoumuchgreaterinsightinto theworkingsofthesenumbers.Ifyou canfindpropertiesofpalindromesthat areinvariantunderachangeinthebase, youmaybeableusethesepropertiesto determinewhether196everreachesa palindromicnumberinbase10.
Ihopeyouenjoyplayingwiththisproblem. Ifyouareinterestedinthelatestnewsonthe problemyoucandoaGooglesearchfor“JasonDoucetteWorldRecords”,also,ifyou managetosolvetheproblemandbecomefamousdon’tforgettomentionourhumblepublicationasyouroriginalinspiration;) palindrome.c
#include < stdio.h>
#include < stdlib.h>
#include < string.h>
#define MAXSTEPS50
char ∗ iterate( char ∗ number, int max);
char ∗ reverse( char ∗ number); char ∗ add( char ∗ number1, char ∗ number2);
int main( int argc, char ∗∗ argv) { char ∗ in; in=malloc(BUFSIZ); while (scanf(”%s”,in)!=EOF) { in=iterate(in,MAXSTEPS); printf(” > %s”,in); printf(” \ n”); } return 0; }
char ∗ iterate( char ∗ number, int max) { int step=0; char ∗ reversed; reversed=malloc(BUFSIZ);
printf(”%s”,number); strcpy(reversed,number); while (step++ < max) { reversed=reverse(number); if (!strcmp(number,reversed)) return (number); number=add(number,reversed) ;
}
printf(”(nosolution)”,step 1); return (number); }
char ∗ add( char ∗ number1, char ∗ number2) { int i,carry,digit; char ∗ tmp; tmp=malloc(BUFSIZ); carry=0;
i=strlen(number2); while ( i >=0) { digit=(number1[i] ’0’)+ (number2[i] ’0’)+ carry; carry=0; if (digit < 10)number2[i] =(digit+’0’); elseif (i!=0) { digit=digit 10; number2[i]=(digit+’0’); carry=1;
}
else { digit=digit 10; number2[i]=(digit+’0’); strcpy(tmp,reverse(number2) ); strcat(tmp,”1”); number2=reverse(tmp);
}
}
}
return (number2);
char ∗ reverse( char ∗ number) { char ∗ reversed; reversed=malloc(BUFSIZ);
strcpy(reversed,number); while ( ∗ ++reversed);
∗−− reversed= ∗ number; while ( ∗ ++number) ∗−− reversed= ∗ number; return (reversed);
}