Fernandes et al 2022a - Revision history

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Materiales.Compuestos: Materiales.Compuestos moved page Review 228438518289 to Fernandes et al 2022a

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Materiales.Compuestos moved page Review 228438518289 to Fernandes et al 2022a

Materiales.Compuestos: Materiales.Compuestos moved page Draft Compuestos 813470770 to Review 228438518289

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Materiales.Compuestos moved page Draft Compuestos 813470770 to Review 228438518289

Materiales.Compuestos at 16:37, 22 May 2022

2022-05-22T16:37:59Z

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 The best result obtained through the optimization process is shown in <span id='cite-_Ref8642559'></span>[[#_Ref8642559|Figure 14]] e) and has a maximum index of failure of 1.63, according to the Hashin’s failure criteria for matrix under compression, and a natural frequency of 1.22Hz. However, despite having a natural frequency higher than 1.1Hz, this solution fails to meet the damage requirements. The indexes of failure of this solution are shown in <span id='cite-_Ref7792890'></span>[[#_Ref7792890|Table 4]], while the design variables that define the geometry are shown in <span id='cite-_Ref7792903'></span>[[#_Ref7792903|Table 5]].
+[[File:Draft_Compuestos_813470770-image19.png|centre|600x600px]]
-[[Image:Draft_Compuestos_813470770-image19.png|600px]]
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;">

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 With this strategy in mind, the objective function was defined as the minimization of damage, where the damage installed in each possible design was estimated using the structural numerical model described in section <span id='cite-_Ref8116812'></span>[[#_Ref8116812|5.2]]. However, executing a structural analysis on every possible solution would lead to an exceedingly large computational cost. To avoid such situation, each candidate solution had its natural frequency estimated through the numerical model described in section <span id='cite-_Ref8116825'></span>[[#_Ref8116825|5.1]]. If the natural frequency of a possible solution was larger than 1.1Hz and lower than 1.25Hz, it would be evaluated through a structural analysis and its performance value <math display="inline">\left( Perf\right)</math>  set equal to the maximum index resulting from the application of the failure criteria included in the numerical model (listed in section <span id='cite-_Ref8116864'></span>[[#_Ref8116864|4]]). If the natural frequency was not within this range, the performance value was set equal to a penalty factor <math display="inline">\left( P\right)</math>  and the structural analysis skipped. Equation (1)  shows the formula used to determine the performance value of a solution, while <span id='cite-_Ref6320448'></span>[[#_Ref6320448|Figure 13]] shows a graphic representation of the penalty factor as a function of the natural frequency.
-<math display="inline">\left\{ \begin{matrix}Perf=15,\quad \, \, 0.0\leq f<1.0\, \vee 1.25<f\leq 1.5\, \\Perf=12,\quad \, \, 1.0\leq f<1.1\\\begin{matrix}Perf=Max\left( IF\right) ,\quad \, 1.1\leq f\leq 1.25\\Perf=20,\quad \, 1.5<f<+\infty \end{matrix}\end{matrix}\right.</math> <math display="inline">(\, SEQ\, \, Eq\, \\ast \, MERGEFORMAT\, 1)</math>
+<math display="inline">\left\{ \begin{matrix}Perf=15,\quad \, \, 0.0\leq f<1.0\, \vee 1.25<f\leq 1.5\, \\Perf=12,\quad \, \, 1.0\leq f<1.1\\\begin{matrix}Perf=Max\left( IF\right) ,\quad \, 1.1\leq f\leq 1.25\\Perf=20,\quad \, 1.5<f<+\infty \end{matrix}\end{matrix}\right.</math> <math>(1)</math>
 Where <math display="inline">Max\left( IF\right)</math>  is the largest index of failure of the failure criteria included in the structural analysis (listed in section <span id='cite-_Ref8116864'></span>[[#_Ref8116864|4]]).

@@ Line 94: / Line 94: @@
 [[File:Draft_Compuestos_813470770-image5-c.png|centre|600x600px]]
-<div id="_Ref6319941" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 <span style="text-align: center; font-size: 75%;">'''Figure 1. '''Functioning of an elastic-hinge (adapted from [5]).</span></div>
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 [[File:Draft_Compuestos_813470770-image7.png|centre|600x600px]]
-<div id="_Ref6319941" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 <span style="text-align: center; font-size: 75%;">'''Figure 3. '''Description of the deployable system considered in this research.</span></div>
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 [[File:Draft_Compuestos_813470770-image8.png|centre|600x600px]]
-<div id="_Ref6319941" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span id='_Ref6320082'></span><span style="text-align: center; font-size: 75%;">'''Figure 4. '''Highlight of the boundary conditions considered in the natural frequency model.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 4. '''Highlight of the boundary conditions considered in the natural frequency model.</span></div>
 The deployable arm was modelled with three-dimensional deformable shell elements with 4 nodes (S4R in ABAQUS<sup>®</sup>) and reduced integration. Each element had an average size of 2.5mm. The material properties introduced in the numerical model are summarized in <span id='cite-_Ref8027472'></span>[[#_Ref8027472|Table 1]].
@@ Line 168: / Line 168: @@
 [[File:Draft_Compuestos_813470770-image9.png|centre|600x600px]]
-<div id="_Ref6319941" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span id='_Ref6320118'></span><span style="text-align: center; font-size: 75%;">'''Figure 5. '''Highlight of the different regions considered in the structural model. Each region is represented with a different colour (blue, dark-grey and light-grey) and has a different mesh size.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 5. '''Highlight of the different regions considered in the structural model. Each region is represented with a different colour (blue, dark-grey and light-grey) and has a different mesh size.</span></div>
 The central region, highlighted with a light-grey colour, corresponds to the area where the slot of the elastic-hinge is located and where the highest stress concentration occurs. Therefore, the numerical model considers the existence of a composite layup feature with multiple plies in this region, allowing a more detailed analysis of the stress and strain state of each element. The elements in this region had an approximate size of 2.5mm.
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 [[File:Draft_Compuestos_813470770-image10-c.png|centre|537x537px]]
-<div id="_Ref6319941" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span id='_Ref6320163'></span><span style="text-align: center; font-size: 75%;">'''Figure 6. '''Visual representation of the three folding steps (image adapted from [5]).</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 6. '''Visual representation of the three folding steps (image adapted from [5]).</span></div>
 ==6 Validation of the Structural Model==
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 [[Image:Draft_Compuestos_813470770-image13-c.png|600px]]
-<div id="_Ref6319941" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div id="_Ref6319941" class="center" style="width: auto; margin-left: auto; margin-right: auto;"> <span id='_Ref8027518'></span>
-<span id='_Ref8027518'></span> <span style="text-align: center; font-size: 75%;">'''Figure 8. '''Geometry of the specimen used for the experimental validation of the structural model.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Figure 8. '''Geometry of the specimen used for the experimental validation of the structural model.</span></div>
 Each specimen was manufactured by  hand lay-up and cured in an autoclave according to the manufacturer specifications. Then, the slot was cut using a CNC machine.

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 |  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|MPa
 |}
 The geometry of the specimens used in the experimental validation is described by <span id='cite-_Ref8027518'></span>[[#_Ref8027518|Figure 8]]. This geometry consists on a tubular structure with two 0.18125mm thick plies oriented at +/- 45º, an internal diameter of 39mm and a cut-out, defining the slot of the elastic-hinge.
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 <div id="_Ref6319941" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span id='_Ref8027518'></span><span style="text-align: center; font-size: 75%;">'''Figure 8. '''Geometry of the specimen used for the experimental validation of the structural model.</span></div>
+<span id='_Ref8027518'></span> <span style="text-align: center; font-size: 75%;">'''Figure 8. '''Geometry of the specimen used for the experimental validation of the structural model.</span></div>
 Each specimen was manufactured by  hand lay-up and cured in an autoclave according to the manufacturer specifications. Then, the slot was cut using a CNC machine.

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 In spacecraft design, one of the major constraints is the capability of the launcher, considering both size and total mass. Designs that are based on large, rigid structures are constrained in size by the dimensions of the fairing. At the same time, developing launchers capable of sending larger payloads is an extremely difficult task. Given this scenario, adopting deployable structures, capable of changing their configuration from a packed and compact form to a large and deployed state once the payload is in orbit, is a highly attractive solution [1]. First, metallic solutions were developed and then upgraded to lighter composite systems that deploy by releasing the strain energy stored during the compaction process [2], [3]. Well-known applications of this concept are the elastic-hinges included in the monopole and dipole antennas of MARSIS (Mars Advanced Radar for Surface and Ionosphere Sounding), which allowed the folding of antennas with lengths up to 20 meters [4] (<span id='cite-_Ref6319941'></span>[[#_Ref6319941|Figure 1]] and <span id='cite-_Ref6319949'></span>[[#_Ref6319949|Figure 2]], respectively).
+[[File:Draft_Compuestos_813470770-image5-c.png|centre|600x600px]]
-[[Image:Draft_Compuestos_813470770-image5-c.png|600px]]
 <div id="_Ref6319941" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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 In this research, the deployable structure considered is a composite arm with two integrated elastic-hinges. Each end of the arm should be connected to the satellite and to an antenna, respectively. <span id='cite-_Ref6320034'></span>[[#_Ref6320034|Figure 3]] best describes the application.
+[[File:Draft_Compuestos_813470770-image7.png|centre|600x600px]]
-[[Image:Draft_Compuestos_813470770-image7.png|600px]]
 <span id='_Ref6320034'></span><span style="text-align: center; font-size: 75%;">'''Figure 3. '''Description of the deployable system considered in this research.</span>
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 The antenna is represented as a point mass of 27kg, located 1.4m away from the tip of the composite arm and connected to it through a beam multi-point constraint. The other end of the composite arm is fixed, representing its attachment to the satellite. <span id='cite-_Ref6320082'></span>[[#_Ref6320082|Figure4]] represents these boundary conditions applied to a generic antenna arm.
+[[File:Draft_Compuestos_813470770-image8.png|centre|600x600px]]
-[[Image:Draft_Compuestos_813470770-image8.png|600px]]
 <span id='_Ref6320082'></span><span style="text-align: center; font-size: 75%;">'''Figure 4. '''Highlight of the boundary conditions considered in the natural frequency model.</span>
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 The structural model considers a segment of the composite arm, corresponding to only one elastic-hinge. Similarly to the natural frequency model, three-dimensional deformable shell elements with 4 nodes (S4R in ABAQUS <sup>®</sup>) and reduced integration were used to model the elastic-hinge. Nonetheless, to reduce the computational cost of the finite element model, the elastic-hinge was divided in three regions, as shown in <span id='cite-_Ref6320118'></span>[[#_Ref6320118|Figure 5]].
+[[File:Draft_Compuestos_813470770-image9.png|centre|600x600px]]
-[[Image:Draft_Compuestos_813470770-image9.png|600px]]
 <span id='_Ref6320118'></span><span style="text-align: center; font-size: 75%;">'''Figure 5. '''Highlight of the different regions considered in the structural model. Each region is represented with a different colour (blue, dark-grey and light-grey) and has a different mesh size.</span>
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 In order to replicate the experimental testing equipment, the applied boundary conditions consider the existence of two holders, responsible for supporting the ends of the composite arm. An angular rotation is applied, forcing the arm to fold, and two plates, that pinch the tapes of the elastic hinge, force them to bend before the folding of the component starts. These boundary conditions are in accordance with [5] and represented in <span id='cite-_Ref6320163'></span>[[#_Ref6320163|Figure 6]].
+[[File:Draft_Compuestos_813470770-image10-c.png|centre|537x537px]]
-[[Image:Draft_Compuestos_813470770-image10-c.png|600px]]
 <span id='_Ref6320163'></span><span style="text-align: center; font-size: 75%;">'''Figure 6. '''Visual representation of the three folding steps (image adapted from [5]).</span>
@@ Line 191: / Line 186: @@
 The material considered in this research was AS4/8552, an unidirectional carbon/epoxy system provided by Hexcel. The mechanical properties implemented in the models are summarized in <span id='cite-_Ref8027472'></span>[[#_Ref8027472|Table 1]].
-<span id='_Ref8027472'></span><span style="text-align: center; font-size: 75%;">'''Table 1. '''Elastic and strength properties of AS4/8552, according to [19] and [20], respectively.</span>
+<span id='_Ref8027472'></span><span style="text-align: center; font-size: 75%;">'''Table 1. '''Elastic and strength properties of AS4/8552, according to [19] and [20], respectively</span>
 {| style="width: 100%;border-collapse: collapse;"
@@ Line 282: / Line 277: @@
 <span id='_Ref6396505'></span><span style="text-align: center; font-size: 75%;">'''Figure 10. '''Torque applied to an elastic-hinge as a function of the folding angle.</span>
-<span id='_Ref8642236'></span><span style="text-align: center; font-size: 75%;">'''Table 2. '''Maximum torque according to experimental and numerical results.</span>
+<span id='_Ref8642236'></span><span style="text-align: center; font-size: 75%;">'''Table 2. '''Maximum torque according to experimental and numerical results</span>
 {| style="width: 100%;border-collapse: collapse;"
@@ Line 355: / Line 350: @@
 <div id="_Ref6322674" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Table 3.''' Maximum and minimum values of each design variable.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 3.''' Maximum and minimum values of each design variable</span></div>
 {| style="width: 100%;border-collapse: collapse;"
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 <div id="_Ref7792890" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<span style="text-align: center; font-size: 75%;">'''Table 4. '''Indexes of failure observed according to each failure criterion.</span></div>
+<span style="text-align: center; font-size: 75%;">'''Table 4. '''Indexes of failure observed according to each failure criterion</span></div>
 {| style="width: 83%;margin: 1em auto 0.1em auto;border-collapse: collapse;"

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← Older revision		Revision as of 16:37, 22 May 2022
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−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	P. Fernandes<sup>a</sup>, R. Marques<sup>a</sup>, R. Pinto<sup>a</sup>, P. Mimoso<sup>a</sup>, J. Rodrigues<sup>a</sup>, A. Silva<sup>a</sup>, João Manuel R. S. Tavares<sup>a,b</sup>, G. Rodrigues<sup>c</sup>, N. Correia<sup>a</sup></div>
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−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<span style="text-align: center; font-size: 75%;"><sup>a</sup>INEGI – Institute for Science and Innovation in Mechanical and Industrial Engineering, Porto, 4200-465, Portugal</span></div>~~
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−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<span style="text-align: center; font-size: 75%;"><sup>b</sup>Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Porto, 4200-465, Portugal</span></div>~~
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−	~~<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">~~
−	~~<span style="text-align: center; font-size: 75%;"><sup>c</sup>ESTEC – European Space Research and Technology Centre, Noordwijk, 2200 AG, The Netherlands</span></div>~~
	==1 Symbols and Abbreviations==		==1 Symbols and Abbreviations==