Optimization of Ring Stiffener of a Missile

8/11/2019 Optimization of Ring Stiffener of a Missile

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INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME 3, ISSUE 3, MARCH 2014 ISSN 2277-8616

217IJSTR2014www.ijstr.org

Optimization Of Ring Stiffener Of A MissileSheikh Naunehal Ahamed, Mohammed Mushraffuddin

Abstract: In general thick cylindrical structures have radial, axial, circumferential and extensional modes and some of these free vibrational modesexists within the machine operating frequency range and can lead to potential resonance. One way to avoid the resonance is to shift the system naturafrequencies away from the machine operating range. In case of turbo-generators the forcing function, which is combination of various deformation

stresses that deforms the structure into an oval shape. Our research explains how to shift oval mode frequencies using topology optimization scheme inthe context of a finite element (FE) approach. The key challenge involved in FE is that one should be able to retain the mode of interest throughout thecycle of optimization. During the optimization scheme, there will be a progressive change in the geometry and material, which may causeremoval/shifting of the mode of interest. The optimization is carried out using conventional artificial boundary condition of a missile ring stiffeners ovamode.

Index Terms: circumferential nodes, hoops stress, mesh, resonance, topology optimization, von misses stress

1INTRODUCTIONA number of theories for the prediction of the naturalfrequencies of cylinders have been developed and used overthe years. as mentioned in Guan, W. Modal Analysis of aThick-Walled Cylinder. MSc. Thesis, University ofSaskatchewan, Saskatoon, Canada, 1993. Because the

solutions for the vibrational behavior of a cylinder can not beexactly obtained by the linear elastic theory, people have triedto create various theories to solve the problem in anapproximate way. The differences between these approximatemethods are due to the various assumptions for thedisplacement components which are used in the analysis.Some of the theories are capable of dealing with finite lengthfree hollow cylinders, while others are only appropriate forsolid cylinders of infinite length. Among these researchers,only a few give a complete description of the mode shapes.The most recent work on the thick-walled cylinder is describedin Singal and Williams paper. Based on the three-dimensionaltheory of elasticity, the well-known energy method was used inthe derivation of the frequency equation of the cylinder. The

frequency equation yields natural frequencies for all thecircumferential modes of vibration, including the breathing andbeam-type modes. Experimental investigations carried out onseveral models, showed very close agreement between thetheoretical and experimental values of the naturalfrequencies.The load-carrying action of a plate is similar, to a certainextent, to that of beams thus, plates can be approximated bya gridwork of an infinite number of beams or by a network ofan infinite number of cables, depending on the flexural rigidityof the structures. This two-dimensional structural action ofplates results in lighter structures, and therefore offersnumerous economic advantages. The plate, being originallyflat, develops shear forces, bending and twisting moments toresist transverse loads.

Because the loads are generally carried in both directions andbecause the twisting rigidity in isotropic plates is quitesignificant, a plate is considerably stiffer than a beam ocomparable span and thickness. So, thin plates combine lightweight and form efficiency with high load-carrying capacityeconomy, and technological effectiveness. So far, most of the

work by several authors. as mentioned in Guan, W.ModaAnalysis of a Thick-Walled Cylinder. MSc. Thesis, University oSaskatchewan, Saskatoon,Canada, 1993. was dedicated todiscovering a theoretical or numerical way of determining thefrequencies of cylinders accurately and concisely. Some of thework considered a solid bar of infinite length only, others weresuitable for finite length circular hollow cylinders with eitherthin walls or thick walls. Few authors gave a description of themode shapes of the cylinders, and none of them provided aunique way for describing all of the modes shapes. McMahongave some mode charts showing sand patterns on the planesurface of a solid cylinder and the approximate form of thevibration at a diametrical cross section. The mode shapes othin-walled cylinders were presented. Singal and Williams

gave a description for the mode shapes of thick-walled hollowcylinders. However, all of the descriptions mentioned arebased on two parameters, the number of circumferentianodes, and number of axial nodes. Such a description isinsufficient for describing the mode shapes of a threedimensional structure uniquely. as we know from McMahonG.W. Expedmental Study of the Vibrations of Solid, lsotmprcElastic Cylinders, Journal of the Acoustical Society oAmerica. Vol. 36, 1964, pp 85-92. Mode shape information is avery important vibrational characteristic of any structureWithout mode shape information, it is difficult to find a way tocontrol and eliminate the vibration and noise of the structureitself. The energy analysis has been shown to be ideallyawaited to the determination of the natural frequencies of both

thick-walled hollow cylinders and rings, however, it does notprovide much information with regards to the eigenvectorproblem. The technique provides only the magnitude of n andwhether m is an even or odd integer. The meaning of n (where2n = the number of circumferential cross points in the radiadisplacement shape) and m (the number of cross points in theradial displacement shape along any axial generatrix) areillustrated in Figures la and lb. The actual magnitudes of theinteger m were obtained experimentally in the journal. It isimpossible for several frequencies to have the exact samemode shape, therefore it must be concluded that thedescriptors n and m alone are not adequate for describingsuch a three-dimensional problem. Natural frequencies andmode shapes of finite length thick cylinders are of

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Sheikh Naunehal Ahamed, Mohammed Mushraffuddin

(Only author names, for other information use the

space provided at the bottom (left side) of first page or

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considerable engineering importance because of strongpotential applications and more demanding requirementsimposed on cylindrical structures. In general thick cylindrical

structures have radial, axial, circumferential and extensionamodes as shown in fig 1.

Pure Radial Mode Radial Motion with radial shearingmode

Extensional ModeCircumferential

Axial Bending Mode Global Mode

FIG 1.1: Different types of Mode Shapes of a Cylinder.

FIG 1.2 : Circumferential Nodes pattern.


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FIG 1.3 : Axial Nodes Pattern.Here i is the number of circumferential nodes and j is the number of axial nodes at an instance.

When some of these free vibrational modes are coincide withthe machine operating frequency range can lead to potentialresonance and causes huge damages to structures. In orderto avoid the resonance we can shift the system naturalfrequencies away from the machine operating range by usingtopology optimization technique. One can shift the systemnatural frequency by either increasing or decreasing the Eigenvalues. In case of turbo-generators the forcing function, whichis combination of various deformating stresses that deformsthe structure into an oval shape. In this paper we prepared aring stiffener using conventional dimensions of a missile ring

stiffener provided by a research organisation and on thisstiffener we applied the Topology optimization of ring stiffenerthereby trying to reduce its weight and unwanted material andalso shifting of Oval mode Shapes.

Topology optimisationis a mathematical approach thatoptimises material layout within a given design space, for agiven set of loads and boundary conditions such that theresulting layout meets a prescribed set of performance targets.Using topology optimisation, engineers can find the bestconcept design that meets the design requirements. Topologyoptimisation has been implemented through the use of finiteelement methods for the analysis, and optimisation techniquesbased on the method of moving asymptotes, genetic

algorithms, optimality criteria method, level setsand topological derivatives. Topology optimisation is used atthe concept level of the design process to arrive at aconceptual design proposal that is then fine tuned forperformance and manufacturability. This replaces timeconsuming and costly design iterations and hence reducesdesign development time and overall cost while improvingdesign performance. In some cases, proposals from atopology optimisation, although optimal, may be expensive orinfeasible to manufacture. These challenges can be overcomethrough the use of manufacturing constraints in the topologyoptimisation problem formulation. Using manufacturingconstraints, the optimisation yields engineering designs that

would satisfy practical manufacturing requirements. In somecases Additive manufacturing technologies are used tomanufacture complex optimized shapes that would otherwiseneed manufacturing constraints.

1.1 Ring stiffener of a turbo generatorThe turbo-generator structure consists of number of cylindricarings that are arranged in parallel and are covered by theenvelope plate circumferentially and thus forms a ring stiffenedcylindrical shell. There has been an extensive work done bythe researchers to predict free vibrational behavior o

cylindrical shells, and it was found that in ring stiffenedcylindrical shell increasing the stiffeners stiffness increases thecylinders natural frequency. This Ring stiffened cylindrical shelforms a cage or support to the stator part of turbo-generatorsTypically the turbo-generator structures consist number oring-stiffened cylindrical segments arranged in parallel to eachother. A typical ring-stiffened cylindrical shell is as shown

FIG 1.4 : Normal Stiffening Ring


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1.2 Objective and Goal :The project deals with the optimization of ring stiffeners ofmissile primary goals of preventing resonance between engineand ring stiffener, retain structural stability of missile body in itshighest operating frequency range appropriate location forcutouts in ring stiffeners. All this is done by simple concept ofdetermining the highest possible oval mode of vibration of ringstiffener which is a cylinder itself, under the highest possible

operating frequency and the shifting this oval mode by makingchanges in the design. Natural frequencies and mode shapesof finite length thick cylinders are of considerable engineeringimportance because of strong potential applications and moredemanding requirements imposed on cylindrical structures.Normally this cylindrical have various free vibration modes andsome of these modes may be in the range of machinesoperating frequency. The main objective of the project is toshift the system natural frequency away from machinesoperating range. Our project particularly concentrates on aturbo generator in which the forcing function tends to deformthe structure into oval shape. Our project main aim is how toshift oval mode frequencies using topology optimizationscheme in the context of a finite element (FE) approach. In

detail, this includes:a) A theoretical modal analysis using the finite element

method to obtain the natural frequencies and modeshapes.

b) An experimental modal analysis to verify the results of

the theoretical analysis. Using the post processors. Inboth the finite element program and the experimentaltest software package, to examine the mode shaperesults and understanding of the mode shapes ingeneral.

c) Shifting Oval mode frequencies using topologyoptimization scheme in the context of a finite element(FE) approach.

d) During a topological optimization scheme, there will be

a progressive change in the geometry and material,which may cause removal/shifting of the mode ofinterest.

2 LITERATURE REVIEWThe approach of R.K singhal and W.C Guan to find truedescriptors for the mode shapes of cylinders.in their journaltitled Modal Analysis of Thick Cylinder. Where the vibrationalbehavior of a thick-walled circular cylinder is of considerableengineering importance us such elements have numerousapplications in electrical machines, stiffened cylinders. andgears, to name a few. Besides the natural frequencies of astructure, mode shapes are also an important aspect and arewry important sources of information for understanding and

controlling the vibration of a structure. A theoretical modalanalysis and an experimental modal analysis were carried outusing a thick-walled cylinder model in order to obtain itsnatural frequencies and mode shapes. The theoretical modalanalysis ws done using the finite element method. The resultsfor the frequency range from 20 Hz to 20 kHz were verifiedusing experimental modal analysis.The correlation behveenthe analytical and the experimental results is very good Thelargest error for all frequencies is 4.05 %; less than 2 % formost frequencies. The intent of the paper Aircraft EngineAttachement and Vibrational Control provided the data with afundamental background to the engine vibration/noise problemin modern aircraft and present the available solutions that can

be used to treat the engine vibration problem. Additionally, adesign approach that provides technology options to theaircraft OEM throughout the design and flight test phases ofthe program is outlined. All mounting systems need toaccomplish two basic functions:

1) constrain motion,2)

provide vibration isolation and noise reduction.

Constraining Motion refers to limiting the relative motionbetween two structures created by thrust, g loads, weightand torque. Providing isolation and reducing noise involvesminimizing the transmission of vibration from one structure toanother so as to reduce the transmitted noise into the cabinarea. To provide the first basic function, the mounting systemmust be stiff to minimize relative motions. In order to minimizetransmitted vibration (or noise), the mounting system must bedynamically soft. This inherent problem sets up competingobjectives that require compromise and flexibility in the engineattachment design. This basic issue, along with the need folonger service lives and reduced costs, is the reason for newtechnology development.In rotordynamics, nonlinear couplingforces between the rotating and surrounding stationary parts

can result in unexpected significant displacements andsubsequent high stresses leading to structural failure. Morespecifically in aircraft engines, several mechanisms cancontribute to such rotor-to-stator interactions and are usuallyclassified in three main categories:

1. interacting forces due to variations of fluid pressure

without structural contact.2.

interacting forces reduced to a unique contacting pointalong the circumference of both structures.

3. interacting forces induced by multiple simultaneouscontacting points at different locations along thecircumference.

The first category comprises phenomena such as stall flutter

forced response due to aerodynamic surroundings or acousticresonance]. This category is out of the scope of the presentpaper.The second category is more or less well understoodThe related works usually analyze the vibrations of a rotatingshaft with a non-uniform cross-section supported by journabearings where different levels of nonlinearity are considered oil-film pressure field implicating nonlinear hydrodynamicequations , direct rub and friction forces, viscous dampingforces, nonconstant angular velocity to name a few. Thesestudies mainly necessitate small models with a few couplednonlinear second-order differential equations suitable for theinvestigation of a real shaft behavior. Nonlinear and chaoticbehaviors as different as dry whip, oil whip or whirling motionare highlighted. On the other hand, the third category is an

emerging field of research and is more specific to aircraftengines like the one depicted. By virtue of the need of highmachine efficiency, it became apparent that more realisticdescriptions of fully flexible structures, principally bladed disksand outer casings, within a contact mechanic framework, wasrequired. This efficiency, simply defined as the ratio of energyoutput to energy input, strongly depends on the clearancebetween the rotating and stationary components: the wider theclearance, the less efficient the machine. Higher efficiency isachieved by reducing this tip clearance in order to avoidaerodynamic losses. Unfortunately, an obvious consequenceis a significant increase in the possibility of rub between thetwo components with origins such as: gyroscopic effect unde


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certain operating conditions, maneuvering loads during take-off and landing of the aircraft, apparition of a rotor imbalancedue to design uncertainties, bird strikes or blade-off, vibrationsdue to aerodynamic excitations, outer casing distortion causedby a temperature gradient. Depending on the nature of theinduced contact, theses interactions can give rise to eithervery short and transient dynamic responses encountered incrash analysis for example or long lived phenomena

characterized by initially intermittent soft contacts that can leadto the excitation of the mode shapes of the structures ,undesirable large amplitudes and very high stress levels intothe structures. In this paper, our emphasis will be in a betterThis approach gives the shape of initial and total deflection ofplates. From this analysis, the large deflection behaviour ofplate under transverse load can be expressed as a function ofthe pre to post buckling in-plane stiffness of plate. Liew et aldeveloped the differential quadrature method and harmonicdifferential quadrature method for static analysis of threedimensional rectangular plates. This methodology can be usedto found the bending and buckling of plates, which are simplysupported and clamped boundary conditions only. It is acertification requirement of aero-engines to demonstrate blade

containment and rotor dynamic integrity in the unlikely event ofloss of a fan blade during operation. In such an event, theengine would be shut down, but the shaft would continue torotate due to the incoming airflow. Out-of-balance forcescaused by a missing fan blade provide a source of excitationfor the whole engine-wing-aircraft structural assembly. Thiscondition is referred to as `windmilling imbalance' (FAA,1999).Rotor clearances which pertain to normal operationcould now be overcome by vibrating components, thus leadingto the rotor rubbing against the stator which, in turn, canpotentially cause a rich mixture of ejects associated withcontact/impact-related phenomena. These ejects manifestthemselves in the occurrence of multiple solutions for steady-state response scenarios, including amplitude jumps during

rotor acceleration and deceleration, and vibration responses atdi_erent/multiple frequencies of the exciting unbalance force.The studies presented here form part to completeunderstanding of the contributing mechanisms of the post-fan-blade-o_ windmill dynamics and to anciently performwindmilling analysis for large-scale engine models. The FAA,in collaboration with industry, is developing a certificationprocedure requiring engine and airframe manufacturers toprove by analysis or test that an airplane under windmillingconditions is still safe to fly. It is part of the understandingrequired for the analysis to and out what loads are involvedand to what level low-frequency vibration modes of theairframe can be excited. Computer aided topology optimizationof structures is a relatively new but rapidly expanding field of

structural mechanics. Topology optimization is used in anincreasing rate by for example the car, machine andaerospace industries as well as in materials, mechanism andMicro Electro Mechanical Systems (MEMS) design. Thereason for this is that it often achieves greater savings anddesign improvements than shape optimization. The topologyoptimization problem solves the basic engineering problem ofdistributing a limited amount of material in a design space,where a certain objective function has to be optimized. In thecase where the design domain is subjected to any static loads,we speak of topology optimization of static problems. Acommon objective in static problems is to minimize thecompliance (maximum global stiffness).In dynamic topology

optimization problems, the objective is related to Eigen valueoptimization. These problems are relevant for the design ofmachines and structures which are subjected to a dynamicload. A possible motivation for this type of problems is, forexample, to keep the Eigen frequency of a structure awayfrom the driving frequency of an attached vibrating machinewith a given frequency of vibration. A common objective indynamic topology optimization is to maximize the fundamenta

Eigen frequency, for example to shift the fundamental Eigenfrequency away from certain disturbing frequencies. Moreoverstructures with a high fundamental frequency tend to bereasonable stiff for static loads . The goal of this project is toimplement two different optimization algorithms in dynamictopology optimization problems and to compare the resultsThe implemented algorithms are the optimality criteria methodand the method of moving asymptotes (MMA). In literaturesurvey embedded topics are collected from different booksand journals. Firstly clear description of the survey is given fothe geometric detail. In Janes Strategic weapon systemsWhich is edited by Duncan Lennox in Forty-seven issue otheir book (Can also found in jsws.janes.com) it is clearlydescribed about the detail explanation of missile cross section

It gives dimension aspect of explanation for the completereport in a proper manner. In this particular journal it has beenclearly explained about the previous flight vehicle crosssections which are practically proved as air worthy. With thistype of study from this particular book it made easier toconsider the dimensional values which are acceptable andalso the further applications of this type of cross section indifferent fields. Therefore it can be concluded by stating thatwith the help of the existing matter in the given book helpedout in the analysis application which made work much easierand also gave a practical example of existence.

3 THEORETICAL BACKGROUNDS

3.1 Definition of ProblemIn any structural problem, response of the structure dependson the load applied i.e., the structure parameters, whichaffects the structure adversely and some of the definitions areto be mentioned clearly. Some of the general result oresponses are mentioned

Where the maximum deflection occurs?

What is allowable stress?Failure of structure due to external load applied?

The unambiguous definition of the problem makes thestructural analyst easy. Probably the most critical step in thestructural analysis is the definition of the problem. The trueproblem is not always what is seems to be at first glance

because this first step requires such a small part of the totatime to analyse the structure, its important is often overlookedThis project deals with the FEA Analysis of the ring stiffenerwhich are mostly used in many areas like aerospaceautomotive etc In turbo-generators or electric machine statorsthe in-plane vibrational modes are of pure radial mode, whichpersists even if the transverse vibrational modes areeliminated [4]. The turbo-generator structure consists onumber of cylindrical rings that are arranged in parallel andare covered by the envelope plate circumferentially and thusforms a ring stiffened cylindrical shell. There has been anextensive work done by the researchers [9-17] to predict freevibrational behavior of cylindrical shells, and it was found that


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in ring stiffened cylindrical shell increasing the stiffenersstiffness increases the cylinders natural frequency [5]. ThisRing stiffened cylindrical shell forms a cage or support to thestator part of turbo-generators.The turbo-generators mainlyfalls into 2 categories; 2 pole and 4-pole. This paper discussesspecifically about 2-pole machines. For a two-pole turbo-

generator the electromagnetic force acting on the stator coreis as shown in Figure 2 and it deforms the stator core into ovashape which exactly matches with the mode shape of i=2 i.eis a pure radial mode. The occurance and pattern of radiamodes in a normal turbo generator is shown in the figure

FIG 3.1 : Typical Forcing Functions Flux lines.

In this project the segment is modeled with shell 63 elements(Figure 5(a)) and a symmetric boundary condition is appliedon free edges of envelope plate. Using the Ansys TM finiteelement code, a natural frequency analysis was performedusing Block Lanczos method between 0-200Hz. As the projectdeals with the FEA Analysis of the structural plate, the

previous work is reviewed and the results are taken as areference for the present project work. The project deals withthe thickness optimization rather than shape optimization. Themain problem, is that the rectangular structural plate tthickness should be optimize for its minimum weight objectivefunction and it should also be taken care that stress should notexceed allowable which are given for the materials.

3.2. METHODOLOGY

3.2.1 Finite Element AnalysisFinite element analysis (FEA) is a powerful computationaltechnique used for solving engineering problems havingcomplex geometries that are subjected to general boundary

conditions. While the analysis is being carried out, the fieldvariables are varied from point to point, thus, possessing aninfinite number of solutions in the domain. So, the problem isquite complex. To overcome this difficulty FEA is used; thesystem is discretized into a finite number of parts known aselements by expressing the unknown field variable in terms ofthe assumed approximating functions within each element. Foreach element, systematic approximate solution is constructedby applying the variation or weighted residual methods. Thesefunctions (also called interpolation functions) are, included interms of field variables at specific points referred to as nodes.Nodes are usually located along the element boundaries, andthey connect adjacent elements. Because of its flexibility in

ability to discretize the irregular domains with finite elementsthis method has been used as a practical analysis tool fosolving problems in various engineering disciplines. FEA isused in new product design, and existing product refinementBecause of its characteristics, researchers are able to verify aproposed design to the users specifications before

manufacturing or construction. In case of structural failureFEA may be used to determine the design modifications tomeet the required conditions. Structural analysis consists oflinear and non-linear models. Linear models consider simpleparameters and assume that the material is not plasticallydeformed. Non-linear models consider that the structure ispre-stressed and is plastically deformed. FEM is a numericamethod used as an effective analysis tool in various field oengineering to provide numerical solutions to engineeringproblems. Basic idea in FEM is to find the solution ocomplicated problem by replacing simpler one by findingsolution we would able to find the approximate solution. Theexisting mathematical tools will not be sufficient to find theexact solution and sometimes even an approximate solution o

most of the practical problems. Thus, in the absence of anyother convenient method to find even the approximate solutionwe have to prefer FEM. Moreover in FEM it will often bepossible to improve or refining the approximate solution byspending more in computational method

3.2.1 Need for FEM:To predict the behavior of the structure, the designer adoptsthree tools/methods. They are:

(1) Analytical Method,(2) Experimental Method &(3) Numerical Method.


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(1)

Analytical Method:

Advantages:

Used for regular section shapes of knowngeometries or primitives where componentgeometry is expressed mathematically.

Mathematical expression used for findingunknowns using knowns.

Solution is exact. It takes less time.

Disadvantages:

Imposing boundary conditions are difficult.Cannot write generalized code.Cannot model dissimilar materials.

(2) Experimental Method:

Testing Equipments, Instruments Test setupis required.

Preparation of specimen according torequirement.

High initial cost.

More time. Exact result.

(3) Numerical Method:

There are many numerical schemes such asFEA, FDM, FVM, BEM, and CVM & Hybridmethods are used to estimate theapproximate solution of acceptabletolerances.

Predicts behavior at desired accuracy of anycomplex and irregular geometry.

Least cost.

The specific need of FEM in most of the engineering problemsthe field variables (like displacement, stress, temperature &pressure) are often represented as linear differential equationor partial differential equation. Solving these differentialequation using classical analytical method is tediousprocesses and at certain conditions becomes near toimpossible in such cases the FEM helps to convert thedifferential equations in to set of algebraic equations that canbe solved easily by using various matrix manipulationtechniques.

3.2.3 CATIA :CATIA version 5 is a process-centric computer-aideddesign/computer-assisted manufacturing/computer-aided

engineering (CAD/CAM/CAE) system that fully uses nextgeneration object technologies and leading edge industrystandards. Seamlessly integrated with Dassault SystemesProduct Lifecycle Management (PLM) solutions, it enablesusers to simulate the entire range of industrial designprocesses from initial concept to product design, analysis,assembly, and maintenance. The CATIA V5 product linecovers mechanical and shape design, styling, productsynthesis, equipment and systems engineering, NCmanufacturing, analysis and simulation, and industrial plantdesign. In addition, CATIA Knowledgeware enables broadcommunities of users to easily capture and share know-how,rules, and other intellectual property (IP) assets. CATIA V5builds on powerful smart modeling and morphing concepts to

enable the capture and reuse of process specifications andintelligence. The result is an easily scaleable, Web-enabledsystem that covers all user requirements within the digitaextended enterprise, from the simplest design to the moscomplex processes. This capability allows optimization of theentire product development process while controlling changepropagation. CATIA V5 moves beyond traditional parametricor variational approaches, accelerating the design process

and helping designers, engineers, and manufacturers increasetheir speed and productivity. CATIA V5 has an innovative andintuitive user interface that unleashes the designer's creativityContext-sensitive integrated workbenches provide engineerswith the tools they need for the task at hand, and they arebeneficial for multi-discipline integration. The workbencheshave powerful keyboard-free direct object manipulators thatmaximize user productivity. CATIA V5 applications are basedon a hybrid modeling technology. These applications provideexpanded digital product definitions, process definitions, andreview functions capable of operating on projects with anydegree of design complexity. CATIA V5 has produced domainspecific applications that have addressed global digitaenterprise requirements that span the areas of mock-up

manufacturing, plant, and operations.

3.2.4 HYPERMESH:Altair Engineering is a product design and developmentengineering software and cloud computing software companyAltair was founded by Jim Scapa, George Christ, and MarkKistner in 1985. Over its history, it has had various locationsnear Detroit, Michigan, USA. It is currently headquarteredin Troy, Michigan with regional offices throughout AmericaEurope and Asia. Altair Engineering is the creator of theHyperWorks suite of CAE software products. Altair wasestablished in 1985. In 1990, Hyper Mesh was released. In1994, Altair receives Industry Weeks "Technology of the Year"award for OptiStruct.During the 2008 economy crisis, Altai

started a program to offer free training on its product forunemployed persons in Michigan. In September 2010, Altaipurchased a 136,000-square-foot (12,600 m

2) Annex Facility

in Troy, to initially house Altairs subsidiary ilumisys, Inc. Altaialso acquired SimLab in October 2011.2011 began withanother acquisition, AcuSim, with their CFD SolverAcuSolve.In September 2011, Altair Product Design unveileda hybrid hydraulic bus.

3.2.5 ANSYS RELATED WORK :ANSYS is a general purpose finite element modeling packagefor numerically solving a wide variety of mechanical problemsTo solve the problem considered initially section is modeledwith respect to the considered dimensions. And then the

section must be meshed with in the tool. Proper meshing hasto be done to get the result with out any errors. We need to getthe structural and thermal analysis values for the designedmodel. As per the requirement load must be applied on the lefmost end of the section end ring and right most end of thesection end ring must be constrained. This is how loads areapplied and constraining is done in ANSYS. The above givenprocedures are done in preprocessor phase. Now entering into the solving phase of the model procedure to be followed isas follows. Firstly mode must be changed to static such thatafter solving we get the value of stress and deformation of thestructure when load is applied. After the first run postprocessor phase has to be opened and in this phase required


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values and analysis plots are taken. Similar solving phasemust to followed for the second cycle but this run is to get theof BLF and therefore mode must be changed to eigenbuckling. After the run it has to be saved and post processingphase must be opened. In this phase values must be noteddown and plots must be taken. This is how steps are followedto get the results in ANSYS.

3.3 MATERIAL PROPERTIES:The main material considered in this project for the ringstiffener is steel since its all mechanical physical and thermalproperties are well knowned.

3.3.1 Steel properties :Steel is an alloy of iron and other elements, including carbon.When carbon is the primary alloying element, its content in the

steel is between 0.002% and 2.1% by weight. The followingelements are always present in steecarbon, manganese, phosphorus, sulfur, silicon, and tracesof oxygen, nitrogen and aluminum. Alloying elementsintentionally added to modify the characteristics of steeinclude: manganese, nickel, chromium, molybdenum, borontitanium, vanadium and niobium. Varying the amount oalloying elements and the form of their presence in the stee

(solute elements, precipitated phase) controls qualities suchas the hardness, ductility, and tensile strength of the resultingsteel. Steel with increased carbon content can be made hardeand stronger than iron, but such steel is also less ductile thaniron. Alloys with a higher than 2.1% carbon (depending onother element content and possibly on processing) are knownas iron. Today, steel is one of the most common materials inthe world, with more than 1.3 billion tons produced annually

Table 3.3.1 : Steel material properties

SL-NO MATERIAL E(Gpa)UTIMATE-TENSILESTRENGTH

DENSITY(Kg/m3 )

YIELDSTRESS

1 Steel 200 830 7800 700

3.3.2 SHELL63 Element DescriptionSHELL63 has both bending and membrane capabilities. Bothin-plane and normal loads are permitted. The element has sixdegrees of freedom at each node: translations in the nodal x,y, and z directions and rotations about the nodal x, y, and z-axes. Stress stiffening and large deflection capabilities areincluded. A consistent tangent stiffness matrix option isavailable for use in large deflection (finite rotation) analyses.See SHELL63 in the ANSYS, Inc. Theory Reference for moredetails about this element. Similar elements are SHELL43 andSHELL181 (plastic capability), and SHELL93 (midside node

capability). The ETCHG command converts SHELL57 andSHELL157 elements to SHELL63

3.4. LOADS & BOUNDRY CONDITIONS:

3.4.1 DESIGN CONFIGURATION OF STIFFENER:The variables of cross section of a Ring Stiffener are shown inthe figure, these variables define the Natural Frequency of anyring stiffeners and are the main part in designing the ringstiffener in Catia.

FIG 3.1 : Ring Stiffener variables.

Here,2= length of Envelope Plate.2= thickness of Envelope Plate.1= length of Stiffening Ring.1=thickness of Stiffening Ring.1= Inner Radius.2= Outer Radius.y = distance from Datum line.

Typically the Turbo generator structure consists of RingStiffened Cylindrical segments arranged in parallel to eachother. A typical Ring Stiffened Cylindrical Shell is as shown inthe figure 6 above.


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3.4.2 LOADS ON THE RING STIFFENER IN A MISSILE:For any Missile, loading is divided into two major categories:limit loads and ultimate loads. Limit loads are often just flightloads and are further divided into maneuvering loads and gustloads. Ultimate loads are crash loads. Maneuvering loads aredetermined based on the performance limits of the aircraftwhether imposed by the flight manual or by the actualaerodynamic performance of aircraft. Gust loads are

determined statistically are taken from guidelines orrequirements given by the applicable regulatory agency. Crashloads are loosely bounded by the ability of humans to surviveextreme accelerations and are also typically taken fromregulations. Cylindrical or spherical pressure vessels (e.g.,hydraulic cylinders, gun barrels, pipes, boilers and tanks) arecommonly used in industry to carry both liquid s and gasesunder pressure. When the pressure vessel is exposed to thispressure, the material comprising the vessel is subjected topressure loading, and hence stresses, from all directions. Thenormal stresses resulting from this pressure are functions ofthe radius of the element under consideration, the shape ofthe pressure vessel (i.e., open ended cylinder, closed endcylinder, or sphere) as well as the applied pressure. Two types

of analysis are commonly applied to pressure vessels. Themost common method is based on a simple mechanicsapproach and is applicable to thin wall pressure vesselswhich by definition have a ratio of inner radius, r, to wallthickness, t, of r/t10. The second method is based onelasticity solution and is always applicable regardless of the r/tratio and can be referred to as the solution for thick wallpressure vessels. Both types of analysis are discussed here,although for most engineering applications, the thin wallpressure vessel can be used. Other loads that may be criticalare pressure loads (for pressurized, high-altitude aircraft) andground loads. Loads on the ground can be from adversebraking or maneuvering during taxi. Finally, you cannotdiscuss aircraft loading without hearing about fatigue and

damage tolerance. Aircraft are constantly subjected to cyclicloading. These cyclic loads initiate cracks and cause them togrow. Thermal loading is rarely considered for the analysis ofthe primary structure of aircraft but it can become critical underextreme operating conditions and should be examined wherematerials of disparate coefficients of thermal expansion are

joined.hence the loads on a ring stiffener are Pressure Loads. Machine Vibration Loads. Thermal Loads.

Bending Loads. Buckling loads. Torsional Loads.

Since these types of loads will induce different stesses in thering stiffener . some of the practically knowable stresses thatcan be known are

Hoop stresses. Von misses stresses.

Hoop stressesFor the thin-walled assumption to be valid thevessel must have a wall thickness of no more than about one-tenth (often cited as one twentieth) of its radius. This allows fortreating the wall as a surface, and subsequently usingthe YoungLaplace equation for estimating the hoop stresscreated by an internal pressure on a thin wall cylindricalpressure vessel:

(for a cylinder)

(for a sphere)

where

Pis the internal pressure

tis the wall thickness

ris the inside radius of the cylinder.

is the hoop stress.

The hoop stress equation for thin shells is also approximatelyvalid for spherical vessels, including plant cells and bacteria inwhich the internal turgor pressure may reach severaatmospheres. When the vessel has closed ends the internapressure acts on them to develop a force along the axis of thecylinder. This is known as the axial stress and is usually lessthan the hoop stress.

Though this may be approximated to

Also in this situation a radial stress is developed and maybe estimated in thin walled cylinders as:

Von misses stress : The von Mises yield criterion suggeststhat the yielding of materials begins when the second

deviatoric stress invariant reaches a critical value. For this

reason, it is sometimes called the -plasticityor flowtheory. It is part of a plasticity theory that applies bestto ductile materials, such as metals. Prior to yield, materiaresponse is assumed to be elastic. In materialsscience and engineering the von Mises yield criterion can bealso formulated in terms of the von Mises

stressor equivalent tensile stress, , a scalar stressvalue that can be computed from the stress tensor. In thiscase, a material is said to start yielding when its von Mises

stress reaches a critical value known as the yield strength,. The von Mises stress is used to predict yielding of materialsunder any loading condition from results of simple uniaxiatensile tests. The von Mises stress satisfies the property thatwo stress states with equal distortion energy have equal vonMises stress. Because the von Mises yield criterion is

independent of the first stress invariant, , it is applicable fothe analysis of plastic deformation for ductile materials suchas metals, as the onset of yield for these materials does notdepend on the hydrostatic component of the stress tensor


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Although formulated by Maxwell in 1865, it is generallyattributed to Richard Edler von Mises (1913).TytusMaksymilian Huber (1904), in a paper in Polish, anticipated tosome extent this criterion. This criterion is also referred to asthe MaxwellHuberHenckyvon Mises theory.

4 DEFINING THE GEOMETRY OF STIFFENERThe geometry of the Missile stiffener is taken by considering

various missile dimensions. As the data related to the designof any missile is a very confidential data and a very strictprivacy should be maintained, so we were asked to takeconventional design data related to the original one. From asource of one of the important Research organization thefollowing pictures of the conceptual design is taken intoconsideration.

FIG 4.1 : Different configurations of Agni missiles.


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FIG 4.2 : Comparisons of Prithvi II and other Agni Missiles.


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FIG 4.3 : Typical Ring Stiffener Dimensions.


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FIG 4.4 : Conventional Measurements of Ring Stiffener.

Studying these the conventional dimensions for a RingStiffener are calculated using proportional limits. And using thephenomenon of Theory of Similarity which states that forundistorted models in which the geometry of the undistortedmodel i-e prototype is similar to the original model then thescale effects in models will not have a deviating effect on theresults , moreover the results may vary with respect toproportionality accordingly. By applying the Dynamic similarityto Specific Model concept to these Ring Stiffener andpredicting the geometry of the Ring Stiffener as follows

4.1 CONVENTIONAL GEOMETRY DETAILS :

Stiffening Ring Thickness = 2.0 inches.Envelope Plate Thickness = 0.75 inches.Stiffening Ring Inner Diameter = 110 inches.Stiffening Ring Outer Diameter = 150 inches.Envelope Plate length = 15 inches.

5 MATHEMATICAL BACKGROUND ANDFORMULATION

5.1 MATHEMATICAL MODELINGThe use of mathematics is one of the many approaches tosolving real-world problems. Others include experimentationeither with scaled physical models or with the real world

directly. Mathematical modelling is the process by which aproblem as it appears in the real world is interpreted andrepresented in terms of abstract symbols. This makesmathematical modelling challenging and at the same timedemanding since the use of mathematics and computers fosolving real-world problems is very widespread and has animpact in all branches of learning. Structures which aresubjected to dynamic loading, particularly aircraft, vibrate oroscillate in a frequently complex manner. An aircraft, forexample, possesses an infinite number of natural or normamodes of vibration. Simplifying assumptions, such as breakingdown the structure into a number of concentrated masses

connected by weightless beams (lumped mass concept), aremade but whatever method is employed the natural modesand frequencies of vibration of a structure must be knownbefore flutter speeds and frequencies can be found. We shaldiscuss flutter and other dynamic aeroelastic phenomena inT.H.G MEGSON but for the moment we shall concentrate onthe calculation of the normal modes and frequencies ofvibration of a variety of beam and mass systems. Thedetermination of natural frequencies and normal mode shapesfor beams of nonuniform section involves the solution of andfulfilment of the appropriate boundary conditions. Howeverwith the exception of a fewspecial cases, such solutions do noexist and the natural frequencies are obtained by approximatemethods such as the Rayleigh and RayleighRitz methods


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which are presented here. Rayleighs method is discussedfirst. A beam vibrating in a normal or combination of normalmodes possesses kinetic energy by virtue of its motion andstrain energy as a result of its displacement from an initialunstrained condition. From the principle of conservation ofenergy the sum of the kinetic and strain energies is constantwith time. In computing the strain energy U of the beam weassume that displacements are due to bending strains only so

that at the ends of the beam then a good approximation to thetrue natural frequency will be obtained. We have notedpreviously that if the assumed normal mode differs only slightlyfrom the actual mode then the stationary property of thenormal modes ensures that the approximate natural frequencyis only very slightly different to the true value.Furthermore, theapproximate frequency will be higher than the actual one sincethe assumption of an approximate mode implies the presenceof some constraints which force the beam to vibrate in aparticular fashion; this has the effect of increasing thefrequency. The RayleighRitz method extends and improvesthe accuracy of the Rayleigh method by assuming a finiteseries for V(z), namely where each assumed function Vs(z)satisfies the slope and displacement conditions at the ends of

the beam and the parameters Bs are arbitrary. Substitution ofV(z) then gives approximate values for the naturalfrequencies. The parameters Bs are chosen to make thesefrequencies a minimum, thereby reducing the effects of theimplied constraints. Having chosen suitable series, the methodof solution is to form a set of equations Eliminating theparameter Bs leads to an nth-order determinant in 2 whoseroots give approximate values for the first n naturalfrequencies of the beam. The above study is w ith referenceto T.H.G MEGSON , pg no : 341 to 343.

5.2 Governing Equation for Natural Frequency:Excessive vibration due to resonance occurs when thefrequency of a dynamic excitation is close to one of the natural

frequencies of a structure. Therefore, it is necessary to restrictthe fundamental or higher natural frequencies of a structure toa prescribed range in order to avoid severe vibration

5. 2.1 Equivalent Section Stiffness:The moment of inertia of each section can be calculated byusing the following formula,

= 112

3 + 2 . . . (1)In the above equation y the position of neutral axis relative toinner diameter of the stiffening ring and can be written as

= (2)From above, using thin ring theory the deflection at the unitforce for the diametrically opposed forces can be written as,

= 0.0743 . (3)

Fig5.1 Ring under diametrically opposed forces

= 1 + Where

R is radius to neutral axis and can be written as When force isapplied diametrically and u represents the displacements. Thecompliance is defined as the external work of applied forceswhich is also equalivalent to the strain energy of structuredefined as:

=1

2 . . (4)Where U represents the strain energy and K is the stiffnessmatrix of structure. The deflection at 90 degrees from the poinof application of forces (Figure 4) can be written as

= 0.0683 In Turbo-Generator the forces are rotating with respective tothe ring and thus the average deflection of any point on thering can be written as

= 0.0713

(5)From Eqn. (6), one can write the expression radial stiffness as

1 = 1 = 713 . (6)For thick rings used in generator frame, the effect of shearmust also be included. Therefore, the stiffness (Eqn. (7)) mustbe modified as,


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0 = 1 + = 1 11 + . (7)In the above, is the deflection due to shear and the ratio ofdeflections can be written as

=4

12 . 8Where1 is stiffening ring cross-sectional area E is theyoungs modulusR is the radius to neutral axis Substitute theabove equation in (6) Then we have

0 = 1 11 +

1012 . . (9)5.2.2 Equivalent Section weight calculation:Individual segment weight can be broken down as follows,

Stiffening ring

1 = 22 121Envelope circular plate

2 = 222 2In the above, g is weight density and the total segment weight,

= 5.2.3 Section oval mode frequency :The frequency of any mode of vibration for a circular ring can

be written as

= 12 421 21 + 2 . (10)

Where

is the acceleration due to gravity. is the youngs modulus. is the moment of inertia. is stiffening ring cross-sectional area.For 2-pole turbo-generators the excitation conforms to i=2 andthe ring performs fundamental mode of flexural vibration. Forpure radial mode (i=2) the Eqn. (10) can be simplified as

2=177.50HzThe stiffness of full structure is then the sum of the stiffnesssof all the individual segments. The total structure weight is thesum of the weights of all individual segments plusmiscellaneous weights such as pipes and hence the naturalfrequency of complete structure can be written as,

2=177.5 0+ HzWhere,

is weight of miscellaneous parts5.3 Numerical Solution for Natural Frequency of RingStiffener:Overall inertial and stiffness properties are computed by usingthe above Eqns, as given below,Moment of Inertia I = 2278.99 in4Stiffness 1= 3162.2 lb/ milCorrected stiffness 0 = 2808.7lb/milWeight of segment W = 6123.46lb

Oval mode frequency F =177.50=120.2Hz.


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6 FREQUENCY PREDICTIONS THROUGH FE APPROACH

6.1. Designing The Model

TABLE 6.1 : Design Configuration of the Stiffener.

Geometry details of the RingStiffener (inches)

Stiffness Ring Thickness 2.0

Envelope plate thickness 0.75

Stiffening Ring innerdiameter

110

Stiffening Ring outerdiameter

150

Envelope plate Length 15

Material steel

Using these configurations the Ring Stiffener is modeled in Catia. The following figure show the various Projections of theStiffener in Catia.

FIG 6.1 : Catia image 1.


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FIG 6.2 : Catia image 2.

FIG 6.3 : Catia image 3

Hence the optimized structure of the Ring Stiffener is created in Catia thereby ending the modeling stage of Ring Stiffener. Thencomes the stage of Modal Analysis ,this is done using the Ansys tool.

6.2 WORK ON ANSYS :

6.2.1 Initial work : Ansys Main Menu1. Preferences> Structural Preferences> Discipline Option> h.Method and then Click ok


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Fig 6.4Structural Preferences & Discipline Option

2. Pre-processor> Element Type> add/edit/delete> Shell63(shellElastic 4 Node63) click ok

FIG 6.5: Element Type & Shell63


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6.2.2 Importing the Catia IGES file

FIG 6.6 : Procedure for Importing the IGES file

FIG 6.7 : Imported Ring Stiffener in Ansys.


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FIG 6.8: Image file of Ring Stiffener in Ansys.

6.2.3 MESHING PART IN ANSYS:

3. Pre-processor> Material Properties> material models> Structural> linear> elastic> isotropic>give density & Youngsmodulus and then Click ok

FIG 6.9 :Material models


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4. Preferences> Meshing> Size Control> Mesh Size> Areas > Pick All Areas and then

Preferences> Meshing> Mesh> Areas> Mapped> 3 or sided click ok

Fig 6.10 : Meshed model of Ring Stiffener.

5. Solution> Analysis type> Modal > ok6. Solution> Define Loads> Apply> Structural> Displacement> Onlines (select 2side) > select Ux, Uy, Uz and then ok7. Solution Type> Solve click ok


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FIG 6.11: Solution Done

6.2.4 Results of Modal analysis :The Oval mode occurs at 120.9 Hz.


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FIG 6.12: oval mode (i=2) at 120.9 Hz.

7 COMPARISON OF ANALYTICAL AND FEARESULTSThe analysis has been carried out as follows and the NaturalFrequencies of the Ring Stiffener is calculated analytically andby using the FEA tools. The comparison of the naturalfrequency is done as shown in the table.

Table 7.1:comparisons of analytical and FEA results forfrequencies

AnalyticalFormulae

Finite ElementResults

% of difference

120.2 Hz 120.9 Hz 0.5

This Results shows that FEA is in poor agreement with closedform of solutions..

8 OPTIMIZATION OF RING STIFFENER

8.1 Introduction to Topology OptimizationComputer aided topology optimization of structures is arelatively new but rapidly expanding field of structuralmechanics. Topology optimization is used in an increasing rateby for example the car, machine and aerospace industries aswell as in materials, mechanism and Micro Electro MechanicalSystems (MEMS) design. The reason for this is that it oftenachieves greater savings and design improvements thanshape optimization. The goal of structural optimization is toachieve the best performance from a structure while satisfying

various constraints, such as a certain amount of material andspecific mechanical conditions. For the past several decadestopology optimization techniques such as the homogenization

method [1], the solid isotropic material with penalization(SIMP) method [2], the evolutionary structural optimization(ESO/BESO) method [3], and the level set technique [4] havebeen developed. These methods have been successfullyadopted in topology optimization for static, dynamic, andpractical engineering problems.

3.1.1 Governing Equation for Dynamic ProblemsGoverning equation for free vibration systems considered inthis study can be written as

+ = = 0(8.1.1)By using Laplace transformation Equation can be rewritten as

2 + = 0 . . (8.1.2)By substituting 2 for l into Eq. (12), the final Eigen valueproblem is defined as

2=0 . (8.1.3)Where

K and M are the global stiffness and mass matrix, respectively is the i-th eigen frequency and denotes the correspondingeigenvector depending on

In order to numerically solve Eq


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(21), K and M have to be the symmetric and positive definite(Lehoucq et. al, 1998) stiffness and mass matrices of the finiteelement-based, generalized structural eigen value.

8.2 Topology Optimization Formulation for DynamicProblemsEigen value optimization designs are profitable for mechanicalstructural systems subjected to dynamic loading conditions

like earthquakes and wind loads. The dynamic behaviors ofstructural systems can be estimated by Eigen frequency whichdescribes structural stiffness. In general maximizing first-ordereigen frequency can be an objective for dynamic topologyoptimization problems since stiffness of structures alsoincreases when eigen frequency increases. Problems oftopology optimization for maximizing natural Eigen frequenciesof vibrating elasto static structures have been considered inthe studies (Diaz et.al, 1992; Krog et. al, 1999; Pedersen,2000) Assuming that damping can be neglected, such adynamic design problem can be formulated as follows.

2 =

=

(8.1.4)

Where and are the model stiffness and mass,respectively The optimization problem for maximizing the firstEigen frequency can be written as,

= 12 = 1111 (8.1.5)Subjected to .

11 > 121 = 00

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Optimization of Ring Stiffener of a Missile

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