Research Article Comparative Analysis Of Al-Li Alloy And

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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 815257, 12 pageshttp://dx.doi.org/10.1155/2015/815257Research ArticleComparative Analysis of Al-Li Alloy and Aluminum HoneycombPanel for Aerospace Application by Structural OptimizationNaihui Yu,1 Jianzhong Shang,1 Yujun Cao,1 Dongxi Ma,2 and Qiming Liu31College of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha 410073, ChinaOrdnance Technology Institute, Ordnance Engineering College, Shijiazhuang 050003, China3College of Mechanical and Vehicle Engineering, Hunan University, Changsha 410082, China2Correspondence should be addressed to Jianzhong Shang; jz shang nudt@163.comReceived 5 July 2015; Revised 8 September 2015; Accepted 9 September 2015Academic Editor: Mohammed NouariCopyright 2015 Naihui Yu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Al-Li alloy and aluminum honeycomb panel (AHP) are both excellent materials for aeronautical structures. In this paper, a platetype aeronautical structure (PAS), which is a base mounting structure for 172 kg functional devices, is selected for comparativeanalysis with different materials. To compare system-level performance under multidisciplinary constraints, mathematical modelsfor optimization are established and then structural optimization is carried out using Altair OptiStruct. For AHP, its honeycombcore is regarded as orthotropic material and its mechanical properties are calculated by Allen’s model in order to establish finiteelement model (FEM). The heights of facing sheet and honeycomb core are selected as design variables for size optimization. ForAl-Li alloy plate, topology optimization is carried out to obtain its most efficient load path; and then a reconstruction process isexecuted for practical manufacturing consideration; to obtain its final configuration, accurate size optimization is also used forreconstructed model of Al-Li alloy plate. Finally, the optimized mass and performance of two PASs are compared. Results showthat AHP is slightly superior to Al-Li alloy.1. IntroductionWith the development of aerospace technology, the demandof high-strength-low-density materials is becoming moreand more urgent. Severe mechanical environment and aerodynamic coupling are inevitable because of high launchacceleration and high frequency vibration, so the requirements for strength and stiffness of aeronautical structuresare extremely high. Moreover, launch costs have strongrestrictions on the overall mass of spacecraft, so the massof aeronautical structures must be minimized as far aspossible. Severe contradiction between strength and massspurs extensive utilization of advanced alloy material andcomposite material in aerospace applications, such as aluminum lithium (Al-Li) alloys [1], titanium alloy [2, 3], carbonfiber/epoxy composites, and aramid fiber/epoxy composites[4, 5]. The Al-Li products offer opportunities for significant improvements in aerostructural performance throughdensity reduction, stiffness increase, increases in fracturetoughness and fatigue crack growth resistance, and enhancedcorrosion resistance [6]. It has been found that addition of1% of lithium to aluminum reduces alloy density up to 3%and increases modulus by 6% [7]. Besides, a basic trendtowards increased utilization and integration of laminatedanisotropic composites into the construction of aeronauticaland aerospace vehicles has manifested in the last decade [8].And sandwich-type honeycomb panel [9] which is a typicallaminated anisotropic composite is an important alternativeaeronautical material. Due to their high strength-to-weightratio and stiffness-to-weight ratio, the use of honeycomb panels is particularly attractive in various aeronautical structures[10, 11].Since high-strength alloy material and honeycomb panels are both excellent options for aeronautical structures,it is necessary to conduct a comparative analysis undersame boundary conditions and load case. Currently, mostresearches concentrate on the performance of separate material [12, 13], but there are very few systematic studies on

2Mathematical Problems in EngineeringStructuraloptimizationSize optimizationDetailed design stageShape optimizationPreliminary design stageTopologyoptimizationConceptual design stageFigure 1: Three levels of structural optimization.the performance comparison of AHP and Al-Li alloy. Aeronautical structures usually have multidisciplinary requirements like minimum mass, high strength and stiffness, easedmanufacturing and assembly techniques, and functionalneeds [14]. In the design of aeronautical structures, choosingwhat kind of materials is one of the key problems which mustbe considered firstly. However, it may be a very challengingtask even for sophisticated designers only based on their professional knowledge and experience. The choice of materialsdirectly determines the dimension design of structures, itsprocessing methods, and even configurations of structures,especially when the honeycomb panels are in the scope ofselection. As a laminated cellular structure, honeycomb panelis a type of structural material considering lightweight designand is a typical anisotropic material, so it cannot be directlycompared with alloy material. Performance comparison ismeaningful only when the optimal structures of two materialsare obtained under the same conditions. Therefore, afterthe functional loads and performance constraints are almostconfirmed, structural optimization technology is needed toobtain the eventual configurations.In structural optimization, design variables can be categorized into three groups as topology, shape, and sizingvariables. Topological design variables determine an initialstructural layout whereas shape and sizing parameters givethe shape and dimensions of structures, respectively [15].The three levels of structure optimization are shown inFigure 1. Topology optimization is used in the conceptualdesign stage to obtain an initial structural configuration andto optimize material layout within a given design space fora given set of loads and boundary conditions such thatthe resulting layout meets a prescribed set of performancetargets. Shape optimization belongs to preliminary designstage. By modifying the structural boundaries, for example,detailed designs for notches, holes, and fillets, concernedmechanical performances are improved during the optimization procedure [16]. Size optimization is used in adetailed design stage and to determine the ideal thicknessof a material based on the performance goals and the forcesexpected to be placed on the component. In the field ofaerospace or aeronautical industry, structural optimizationtechnology is widely adopted for many years and is utilizedin the design of most aeronautical structures, such as spacestation [17], aircraft [18], rockets [19], spacecrafts [20]. Mauteand Allen presented a topology optimization methodologyfor the conceptual design of aeroelastic structures accounting for the fluid-structure interaction and the geometricallayout of the internal structure is optimized by materialtopology optimization [21]. A topology optimization methodis proposed to minimize the resonant response of plateswith constrained layer damping treatment under specifiedbroadband harmonic excitations [22]. For honeycomb panels, Ermolaeva et al. [23] presented the application of astructural optimization system to the optimal choice of foamsas a core material for sandwiches with aluminum alloyfaces. Hansel and Becker [24] present a simple heuristicoptimization algorithm implemented by ANSYS-macros todetermine weight-minimal laminate structures. However,there are no papers about system-level performance comparison of honeycomb panel and alloy material using structuraloptimization.In the current paper, a PAS, which is a base mounting structure for 13 different functional devices, is selectedfor comparative analysis. High-strength Al-Li alloy andaluminum honeycomb panel are both ideal material forsimple PAS, so it is very important for structure designersto determine which material is more superior. The commercial finite element package HyperMesh and OptiStruct8.0 [25] are used for structural optimization. Final sizeoptimization of facing sheet and honeycomb core heightscould be directly executed for comparative analysis, as theconfiguration of honeycomb core is almost definite. The pureAl-Li alloy plate needs topology and size optimization toobtain the optimized configuration. Mathematical modelsfor optimization are established and the mass of structureis chosen as objective function. The rest of the paper isorganized as follows. Section 2 briefly introduces the platetype aeronautical structure and its performance requirementsconsidering the launch conditions. Section 3 introduces thestructure of AHP and presents its calculation method oforthotropy mechanical properties based on Allen’s model,which is important to established its finite element model;the mathematical models for structural optimization are alsopresented for AHP and Al-Li alloy. Section 4 presents theresults of structural optimization and comparative analysis.2. Plate-Type Aeronautical Structure and ItsPerformance RequirementsFigure 2 shows a typical PAS which is often used in aspacecraft. 𝑋 direction is rocket’s flight direction. Assumethat 13 different function devices which form an independentsubsystem are installed on this PAS and their total weight is172 kg. After the layout of these function devices is determinate, the first thing which must be decided by the structuraldesigners is to select materials for this PAS. According tothe requirement of installing space, the dimension of PAS isconfirmed and the width 𝑀 and the length 𝐿 are 980 mm and1075 mm, respectively. Two lengths of this rectangle are fixedwhen the rocket launches. According to the rocket launchingenvironment, performance constraints of the system aregiven and are shown in Table 1, where 𝐺 is 9.81 m/s2 .

Mathematical Problems in Engineering3Table 1: Performance constraints of the subsystem and its PAS.PerformanceStrength and stiffnessResonance frequencyHarmonic responseMassInput loads𝑋 direction: 16.2G acceleration fieldπ‘Œ direction: 12.9G acceleration field𝑍 direction: 9.5G acceleration fieldRequirementSafety factor is not lower than 1.5Maximum deformation is less than 0.3 mmNo external input load𝑋 direction: 7.5𝐺 sin(πœ”π‘‘) harmonic excitationπ‘Œ direction: 5.7𝐺 sin(πœ”π‘‘) harmonic excitation𝑍 direction: 4.2𝐺 sin(πœ”π‘‘) harmonic excitationFrequency πœ” is changing from 0 Hz to 100 HzFundamental frequency is not less than 115 HzMaximum acceleration of function devices isnot more than 22GNot more than 30% of total mass of functiondevicesAll external input loads10751075980YZX(a) Front980(b) BackFigure 2: Typical PAS and layout of function devices installed on PAS.Because the frequency range of harmonic response is0 Hz 100 Hz, mechanical resonance will not happen as thefundamental frequency of subsystem is not less than 115 Hz.Therefore, harmonic response constraints need not be considered in the procedure of structural optimization and performance verifications should be carried out after optimizedanalysis. Lightweight design is one of the most importanttasks of structure design for aerospace applications, so themass of PAS should be selected as optimization objective andstructural stress, deformation, and the first order modal asdesign constraints.AHP and Al-Li alloy are very different aeronauticalmaterials. AHP is a kind of typical structured material and itsconfiguration is almost determinate, so only size optimization is needed to obtain its optimized structure. Differentoptimization methods for AHP and Al-Li alloy are shownin Figure 3. Software types used in the paper are SolidWorksfor geometry creation, HyperMesh for meshing, OptiStructfor optimization, and HyperView for postprocessing. CADmodels of function devices and PAS created in SolidWorksare imported into the HyperMesh for preprocessing. Preprocessing of models includes creating FE model, selection ofmaterial properties, creation of load, and applying boundaryconditions on model. For AHP, HyperLaminate module isused to model this orthotropy material. For the optimization purpose, optimization criteria are selected. Accordingto criteria, the designable and nondesignable portions aregenerated. The optimization process is executed in OptiStructand the result can be viewed in HyperView.3. Finite Element Models and MathematicalModels for Structure Optimization3.1. Honeycomb Panels. Honeycomb panel, which is shownin Figure 4, is a sort of sandwich panel consisting of twoalloy or composite plates as face sheet and hexagonal honeycomb cell as core materials, bonded together by bondingadhesive or macromolecule lamination film. In the sandwichconstruction, the facing sheets are spaced to provide mostof the bending rigidity. They also resist all or nearly all theapplied edgewise loads and flatwise bending moments. Thecore material spaces the facing sheets and transmits shearbetween them so that they are effective about a commonneutral axis. The core also provides most of the shear rigidityof the construction. The core-to-facing bonding adhesivemust be adequate to transfer the stresses from the facingsheets to the core materials so that the full properties of thetwo are utilized [10].

4Mathematical Problems in EngineeringCreate CAD models of function devicesand PAS in SolidWorksAHPAl-Li alloyCreate finite elementmodel in HyperMeshCreate finite elementmodel in HyperMeshModel AHP usingHyperLaminate moduleDefine the design optimizationproblem in HyperMeshGlobal objectiveLoad casesConstraintsDefine the design optimizationproblem in HyperMeshSize optimization inOptiStructTopology optimization inOptiStructModel reconstruction inSolidWorksManufacturabilitySize optimization inOptiStructComparativeanalysisFigure 3: Different optimization methods for AHP and Al-Li alloy.Front facing sheetBonding adhesiveHoneycomb coreBonding adhesiveBottom facing sheetZYtXhHldFigure 4: Typical structure of hexagonal honeycomb panel and itsdimensions.As shown in Figure 4, 𝐻 is the height of hexagonalhoneycomb core and β„Ž is the height of facing sheet. Forhexagonal cell, 𝑑 is cell wall thickness, 𝑙 is cell wall width,and 𝑑 is the diameter of inscribed circle of hexagon. In theprocess of optimization analysis, 2A12-T4 aluminum alloy isselected as the material of facing sheet. Honeycomb core isaluminum hexagonal honeycomb 1/8-2024-0.003 of HexcelCorporation, in which 1/8 (inch) represents the diameter ofinscribed circle of hexagon, 2024 is its material grade, and0.003 (inch) represents cell wall thickness. So the cell wallTable 2: The mechanical properties of 2A12-T4 and 2024.Material Densitygrade[kg/m 3 44Yieldstrength[MPa]28075.8width 𝑙 could be simply calculated by the equation 𝑙 𝑑 tan(30) and its value is 1.83 mm after unit conversion. Themechanical properties of 2A12-T4 and 2024 are shown inTable 2.In order to establish finite element model of honeycombpanels, honeycomb core can be equivalent to orthotropicmaterial and its mechanical properties be calculated by Allen’smodel [26] which is widely used in mechanical engineering[27]:4 𝑑 3( ) 𝐸, 3 𝑙 3𝛾 𝑑 3𝐺π‘₯𝑦 ( ) 𝐸,2𝑙 3𝛾 𝑑𝐺π‘₯𝑧 𝐺,2 𝑙𝛾 𝑑𝐺𝑦𝑧 𝐺, 3 𝑙𝑒π‘₯𝑦 0.33,𝐸π‘₯ 𝐸𝑦 (1)where 𝐸 is Young’s modulus of honeycomb core, 𝐺 is shearmodulus, 𝛾 is a correction factor, and its value depending

Mathematical Problems in Engineering5Table 3: Equivalent mechanical parameters of 1/8-2024-0.003.𝐸π‘₯ /MPa11.3𝐸𝑦 /Mpa11.3𝐺π‘₯𝑦 /MPa1.7𝐺π‘₯𝑧 /MPa370.2𝐺𝑦𝑧 /MPa246.6𝑒π‘₯𝑦0.33Table 4: Lower and upper limit values of design variables.Design variableβ„Žπ»Figure 5: Finite element model of subsystem with honeycombpanel.on the manufacturing process is generally 0.4 0.6. By theequivalent calculation, mechanical parameters of aluminumhexagonal honeycomb are shown in Table 3.A finite element model of this subsystem is established using Altair OptiStruct. Honeycomb panel is meshedin HyperMesh using four-noded linear quad elements(CQUAD4) and composite properties are applied, wherethe core is orthotropic material and the sheets are isotropicmaterial. All function devices are simplified by shell elementand their mass is equivalent to the thickness of shells.Bolt connections are simplified as multipoint constraintsand modeled by the flexible unit (RBE3). Two length ofhoneycomb panel are fixed in all six degrees of freedom. Thefinite element model is shown in Figure 5.For structure optimization, performance constraintsshown in Table 1 are used and the height of hexagonalhoneycomb core 𝐻 and the height of facing sheet β„Ž areselected as the design variables. The mass of subsystem isobjective function. And mathematical models for honeycombpanel optimization could be described asmin 𝑀 (𝑑)𝑑Subject to: 𝑔1 (𝑑) πœŽπ‘ 𝑔2 (𝑑) πœ€ 0.3 mmLower limitvalue [mm]Upper limitvalue [mm]0.15105100strength of facing sheet material, 𝑆 is safety factor andthe value is selected as 1.5 in this paper, πœ€ is maximumdeformation of PAS, 𝑓1 is fundamental frequency of thissystem, 𝑑 is design variable and 𝑑 [β„Ž, 𝐻], and 𝑑min and𝑑max are, respectively, lower and upper limit values of designvariables and are shown in Table οΏ½οΏ½3 (𝑑) 𝑓1 115 Hz𝑑min 𝑑 𝑑max ,where 𝑀(𝑑) is the total system mass, 𝑔(𝑑) is performanceconstraint, πœŽπ‘ is the actual stress value while πœŽπ‘  is yield3.2. PAS of Al-Li Alloy. The performance of frame structurelargely depends on the properties of material. In this paper,Al-Li alloy 2090 is selected for comparative analysis withhoneycomb panel while all boundary conditions, input loads,and layout of function devices are identical. 2090 is a kind ofhigh performance material and its elastic modulus increasedby about 10% while the density decreased by about 10% compared with the conventional aluminum alloy. So its stiffnessto-weight ratio and strength-to-weight ratio are improvedremarkably. The mechanical parameters of 2090 versus conventional aluminum alloy 2Al12 are shown in Table 5.For a plane-frame structure of pure Al-Li alloys, topologyoptimization is firstly used for a conceptual design proposaland then fine-tuned for manufacturability. Initial thicknessof the plane-frame structure is a critical factor for topology optimization and the value of initial thickness directlydetermined the optimization result. Therefore, four values ofinitial thickness (40 mm, 50 mm, 60 mm, and 70 mm) areused, respectively, and the best thickness is determined bycomparing the optimum results. At last, size optimizationis executed to obtain a desired thickness of plane-framestructure. Similarly with honeycomb panel, Al-Li alloys plateis meshed with four-noded linear quad elements (CQUAD4)and the function devices are meshed in the same way. Thefinite element model is shown in Figure 6. In order to retaininstallation points of function devices, the meshed plate willbe divided into designable and nondesignable portions. Thegeometry for the design space is defined by inspection ofthe bounds. As shown in Figure 5, green areas are selectedas nondesignable portions and insure that all fixing points(yellow points) are included.In the process of topology optimization of Al-Li alloysplate, the initial thickness is defined and the design variable iselement density of meshed plate 𝜌. And mathematical modelcould be described as𝑀 (𝜌)min𝜌Subject to: 𝑔1 (𝜌) πœŽπ‘ 𝑔2 (𝜌) πœ€ 0.3 mm𝑔3 (𝜌) 𝑓1 115 Hz𝜌 [0, 1] .πœŽπ‘ π‘†(3)

6Mathematical Problems in EngineeringTable 5: Mechanical parameters of 2090 versus 2Al12.Material gradeDensity 𝜌 [kg m 3 ]Young’s modulus 𝐸 [MPa]Yield strength πœŽπ‘  [MPa]25902780786007060053028020902Al12πœŽπ‘  /𝜌[MPa m3 /kg]0.20.1E/𝜌 [MPa m3 /kg]30.325.41234512(a) Finite element model(b) Designable and nondesignable portionsFigure 6: Finite element model for Al-Li alloys structure of PAS.After topology optimization, size optimization should becarried out to obtain a further optimized thickness of Al-Lialloys plate, so the thickness is selected as design variableand its initial value would be the selected result of topologyoptimization. The mathematical model of size optimization isshown inmin 𝑀 (𝑑)𝑑Subject to: 𝑔1 (𝑑) πœŽπ‘ 𝑔2 (𝑑) πœ€ 0.3 mm𝑔3 (𝑑) 𝑓1 115 Hz𝑑0 10 𝑑 𝑑0 10.πœŽπ‘ π‘†(4)4. Optimization Results andComparative Analysis4.1. Optimization Results of Honeycomb Panel. After seventimes of iteration, the results of structure optimization areobtained and shown in Figure 7. The ultimate dimensions ofdesign variables β„Ž and 𝐻 are 1 mm and 99.4 mm, respectively.The ultimate mass of subsystem is 213.40 kg and the mass ofhoneycomb panel is 41.40 kg. However, bolt sockets embedded in honeycomb panel are inevitable and their mass mustbe taken into account. The number of bolt sockets used tofix function devices by M5 bolt is 103 and the mass of eachbolt socket is 0.005 kg. Similarly, the number of bolt socketsused to fix honeycomb panel by M10 bolt is 32 and the massof each bolt socket is 0.025 kg. Therefore, the total mass ofTable 6: Maximum von Mises stress and deformation under inputloads.Input loads𝑋 direction: 16.2𝐺acceleration fieldπ‘Œ direction: 12.9𝐺acceleration field𝑍 direction: 9.5𝐺acceleration fieldMaximum vonMises stress .250.1926Table 7: Natural frequencies of first eight order modes.Mode12345678orderNaturalfrequency 114.8 137 164.7 169.6 186.9 216.5 234.4 238.9[Hz]bolt sockets is 1.315 kg and the mass of honeycomb panelincluding bolt sockets is 42.72 kg, which is 24.84% of totalmass of all function devices. The maximum von Mises stressand deformation of honeycomb panel under input loads ofthree directions are shown in Table 6.Modal analysis is carried out after structure optimization,and first two order modes of subsystem are shown in Figure 8.Natural frequencies of first eight order modes are shown inTable 7. Harmonic response analysis is also conducted using

Mathematical Problems in Engineering7390Height of facing sheet (mm)Height of honeycomb core (a) Iterations of core’s height234Iterations(b) Iterations of facing sheet’s height218217Mass of subsystem c) Iterations of subsystem’s massFigure 7: Iterative process curve of honeycomb panel optimization.Contour plotEigen mode (Mag)Analysis systemSimple averageContour plotEigen mode (Mag)Analysis systemSimple average3.674E 003.266E 002.858E 002.450E 002.041E 001.633E 001.225E 008.165E 014.083E 016.343E 16No resultMax 3.674E 00Grids 7642Min 6.343E 16Grids 1150652.631E 002.339E 002.047E 001.754E 001.462E 001.169E 008.771E 015.847E 012.924E 015.918E 16No resultMax 2.631E 00Grids 7966Min 5.918E 16Grids 115158(a) First order mode5(b) Second order modeFigure 8: First and second order mode of subsystem.67

8Mathematical Problems in EngineeringContour plotElement densities(density)Simple averageContour plotElement densities(density)Simple average1.000E 008.900E 017.800E 016.700E 015.600E 014.500E 013.400E 012.300E 011.200E 011.000E 02No resultMax 1.000E 00Grids 14007Min 1.000E 02Grids 4007961.000E 008.900E 017.800E 016.700E 015.600E 014.500E 013.400E 012.300E 011.200E 011.000E 02No resultMax 1.000E 00Grids 14007Min 1.000E 02Grids 142321(a) 40 mm(b) 50 mmContour plotElement densities(density)Simple averageContour plotElement densities(density)Simple average1.000E 008.900E 017.800E 016.700E 015.600E 014.500E 013.400E 012.300E 011.200E 011.000E 02No resultMax 1.000E 00Grids 14007Min 1.000E 02Grids 1423211.000E 008.900E 017.800E 016.700E 015.600E 014.500E 013.400E 012.300E 011.200E 011.000E 02No resultMax 1.000E 00Grids 14007Min 1.000E 02Grids 142321(c) 60 mm(d) 70 mmFigure 9: Density contours of Al-Li alloy structure with different thickness.performance constraints shown in Table 1. Modal dampingis a critical parameter for harmonic response. The dampingcoefficient of composite material is commonly 5% 8% and6% is taken for honeycomb panel in this paper. 13 codesof each function device are selected as output nodes duringharmonic analysis. The maximum acceleration response of allnodes in 𝑋, π‘Œ, and 𝑍 directions is 16.0𝐺, 9.3𝐺, and 18.5𝐺,respectively. So the acceleration response cannot exceed therequirement of subsystem, namely, 22𝐺.4.2. Optimization Results of Al-Li Alloy Structure. The topology optimization result using material distribution methodis a density distribution of the finite elements in the designdomain [16]. Density contours of Al-Li alloy structure areshown in Figure 9. Different thicknesses, 40 mm, 50 mm,60 mm, and 70 mm, are chosen to implement topologyoptimization.The iterative process curves of Al-Li alloy structure ofdifferent thickness are shown in Figure 10. The ultimate massof 40 mm plate is 54.71 kg, 50 mm plate is 48.30 kg, 60 mmplate is 47.43 kg, and 70 mm plate is 48.54 kg. Therefore, thebest thickness is 60 mm considering the light weight design.And a successive size optimization is necessary to obtaina more optimized thickness. Since boundaries of topologyoptimization result are discontinuous, the optimized structure cannot satisfy practical manufacturing requirements.So model reconstruction is needed according to densitycontour. The structure of topology optimization is mostefficient load path for various constraints, so reconstructedmodel would inevitably lead to degrading the performanceor augmenting the mass, or even degrading the performancewhile augmenting its mass. The reconstructed model of Al-Lialloy plate considering manufacturing requirements is shownin Figure 11. All inclined beams or irregular structures aremodified to straight ones. In order to enhance the strengthof fixed lengths, all of the other color parts of two lengths arechanged to red parts. Other details are also adjusted to meetthe requirements of processing. The mass of reconstructedplate is 47.23 kg and slightly lower than original 47.43 kg.Fundamental frequency is 110.7 Hz, so the performance ofreconstructed plate degraded and further optimization isneeded.

Mathematical Problems in Engineering9200Mass of Al-Li alloy plate (kg)18016014012010080604005101520Iterative times253040 mm iterative process curve50 mm iterative process curve60 mm iterative process curve70 mm iterative process curveFigure 10: Iterative process curve of Al-Li alloy plate with different thickness.Figure 11: The topology optimization modal versus reconstructed model of Al-Li alloy plate.Size optimization of reconstructed model is carried outbased on the mathematical model presented in Section 3.2.After two times of iteration, the optimized thickness of AlLi alloy plate is got and the value is 62.54 mm. The mass ofAl-Li alloy plate increases to 48.39 kg and the fundamentalfrequency increases to 115.4 Hz, simultaneously. The maximum von Mises stress and deformation of Al-Li alloy plateunder input loads of three directions are shown in Table 8.Stress nephogram of subsystem in the 𝑋 direction is shownin Figure 12. The damping coefficient of Al-Li alloy plate issupposed to be 3% and its maximum acceleration response ofall function devices in 𝑋, π‘Œ, and 𝑍 directions is 17.4𝐺, 10.5𝐺,and 19.6𝐺, respectively.Modal analysis is carried out after size optimization, andfirst two order modes of subsystem are with Al-Li alloy plateTable 8: Maximum von Mises stress and deformation under inputloads.Input loads𝑋 direction: 16.2Gacceleration fieldπ‘Œ direction: 12.9Gacceleration field𝑍 direction: 9.5Gacceleration fieldMaximum vonMises stress 17.240.1687shown in Figure 13. Natural frequencies of first eight ordermodes are shown in Table 9.

10Mathematical Problems in EngineeringContour PlotElement stresses (2D and 3D)(vonMises, max)Analysis systemContour PlotElement stresses (2D and 3D)(vonMises, max)Analysis system1.130E 011.005E 018.798E 007.548E 006.298E 005.047E 003.797E 002.547E 001.297E 004.643E 02No resultY Max 1.130E 012D 5686613Min 4.643E 022D 5684313ZX1.130E 011.005E 018.798E 007.548E 006.298E 005.047E 003.797E 002.547E 001.297E 004.643E 02No resultMax 1.130E 012D 5686613Min 4.643E 022D 5684313Figure 12: Stress nephogram of subsystem in the 𝑋 direction.Contour plotEigen mode (Mag)Analysis systemContour plotEigen mode (Mag)Analysis system4.985E 004.431E 003.877E 003.324E 002.770E 002.216E 001.662E 001.108E 005.539E 000.000E 00No result1.449E 011.288E 011.127E 019.663E 008.052E 006.442E 004.831E 003.221E 001.610E 000.000E 00No resultMax 4.985E 00Grids 14804Min 0.000E 00Grids 115139Max 1.449E 01Grids 14037Min 0.000E 00Grids 115139(a) First order mode(b) Second order modeFigure 13: First and second order mode of subsystem with Al-Li alloy plate.Table 9: Natural frequencies of first eight order 0215.2251186.9279.6294.33174.3. Comparative Analysis of Two Materials. After structuraloptimization, the optimized performance of two materialsis obtained and shown in Table 10. All performance indicesmet the requirements and the eventual mass of APH isslightly lighter than Al-Li alloy. The first order modals oftwo structures are both the vibration perpendicular to theplate face (𝑍 direction) and the deformation of 𝑍 directionis maximum. The maximum von Mises stress of Al-Li alloystructure is much larger than that of honeycomb panel.Fundamental frequency and ha

aerospace or aeronautical industry, structural optimization technology is widely adopted for many years and is utilized in the design of most aeronautical structures, such as space station[ ],aircra [], rockets[ ],spacecra s[].Maute and Allen presented a topology optimization methodology for

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