Kenji Matsuda
email: kmatsuda@sbchem.kyoto-u.ac.jp

Takashi Hirose

Soichi Yokoyama

Denis Frath

Cooperative Self-Assembly of Photochromic Diarylethenes at Liquid/Solid Interface and Highly Sensitive Photoinduced Transformation of the Ordering

Keywords: Photochromism, Diarylethene, Scanning Tunneling Microscopy, Self-Assembly, Cooperativity

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

$.1 Introduction

By the advances of technologies in fabrication and manipulation of nano-scale structure, handling of organic molecule has become possible recently. By making the devices from organic molecule, electroconductive and photophysical properties of single molecule can be utilized in the fields of molecular-scale electronics and photonics. [START_REF] Barth | Engineering atomic and molecular nanostructures at surfaces[END_REF][START_REF] Joachim | Molecular electronics: some views on transport junctions and beyond[END_REF][START_REF] Moth-Poulsen | Molecular electronics with single molecules in solidstate devices[END_REF] By the way, photochromic molecules, which show reversible photoinduced structural change, are considered as a promising candidate for molecular-scale switching device. [START_REF]Molecular switches 2nd ed[END_REF] Among several photochromic molecules, diarylethene derivative show reversible photoinduced change of electronic structure from the viewpoint of -conjugation. Therefore, they are considered to work as a molecular-scale switch in molecular electronics. [5] However, in order to use photochromic molecules practically in molecular electronics, assembly and arrangement of the molecules are very important. Therefore, it is very important to develop methodologies to align molecules on a twodimensional surface. This chapter describes our recent achievements on the photocontrol of the self-assembled ordering composed of photochromic diarylethenes at liquid/solid interface investigated by scanning tunneling microscopy (STM). This technique, originally developed by Rabe et al., [START_REF] Rabe | Commensurability and mobility in two-dimensional molecular patterns on graphite[END_REF] is attracting interests because it can visualize the process of molecular assembly at the molecular resolution. Two-dimensional crystal engineering at liquid/solid interfaces has attracted growing attention in the last decade. [START_REF] Palma | Predicting self-assembly: from empirism to determinism[END_REF][START_REF] Elemans | Molecular and supramolecular networks on surfaces: from two-dimensional crystal engineering to reactivity[END_REF][START_REF] De Feyter | Two-dimensional supramolecular self-assembly probed by scanning tunneling microscopy[END_REF][START_REF] Xue | Patterned monolayer self-assembly programmed by side chain shape: four-component gratings[END_REF][START_REF] Nath | Crystal engineering in two dimensions: an approach to molecular nanopatterning[END_REF] To efficiently design and predict the formation of thermodynamically stable molecular ordering formed through selfassembly processes, interplays of various interactions, such as molecule-substrate interactions, intermolecular interaction, [START_REF] Zhang | Molecular templates and nano-reactors: two-dimensional hydrogen bonded supramolecular networks on solid/liquid interfaces[END_REF][START_REF] Cyr | STM Investigations of organic molecules physisorbed at the liquid-solid interface[END_REF][START_REF] Tahara | Functionalized surfaceconfined pores: guest binding directed by lateral noncovalent interactions at the solid-liquid interface[END_REF][START_REF] Xu | Two-dimensional self-assembled molecular structures formed by the competition of van der Waals forces and dipole-dipole interactions[END_REF] solvent effects, [START_REF] Yang | Solvent effects on two-dimensional molecular self-assemblies investigated by using scanning tunneling microscopy[END_REF][START_REF] Destoop | Solvent-induced homochirality in surface-confined low-density nanoporous molecular networks[END_REF][START_REF] Xu | Self-assembly polymorphism: solvent-responsive twodimensional morphologies of 2,7-ditridecyloxy-9-fluorenone by scanning tunneling microscopy[END_REF][START_REF] Takami | Solvent-induced variations in surface structure of a 2,9,16,23-tetra-tert-butyl-phthalocyanine on graphite[END_REF] and the balance of enthalpy and entropy terms, [START_REF] Blunt | Temperature-induced structural phase transitions in a two-dimensional self-assembled network[END_REF][START_REF] Song | Born-Haber cycle for monolayer self-assembly at the liquid-solid interface: assessing the enthalpic driving force[END_REF][START_REF] Miyake | Entropycontrolled 2D supramolecular structures of N,N'-bis(n-alkyl)-naphthalenediimides on a HOPG surface[END_REF][START_REF] Gutzler | Reversible phase transitions in self-assembled monolayers at the liquid-solid interface: temperature-controlled opening and closing of nanopores[END_REF] must be taken into account. The stimuli-responsive interface and the control of the phase transition is also attracting interest. [START_REF] Ciesielski | Towards supramolecular engineering of functional nanomaterials: pre-programming multi-component 2D self-assembly at solidliquid interfaces[END_REF][START_REF] Zhang | Reversible phase transformation at the solid-liquid Interface: STM reveals[END_REF] In this Chapter we will focus on the use of the cooperative self-assembly to design highly sensitive stimuli-responsive assembly.

$.2 Self-Assembly and Photoinduced Transformation of Molecular Ordering of Photochromic Diarylethenes at

Liquid/Highly Oriented Pyrolytic Graphite (HOPG) Interface [START_REF] Arai | Reversible photoinduced change in molecular ordering of diarylethene derivatives at a solution-HOPG interface[END_REF] Diarylethene-pyrene diad 1 and diarylethene-pyrene-diarylethene triad 2 were synthesized to investigate the photoinduced two-dimensional ordering change at a solution/highly oriented pyrolytic graphite (HOPG) interface by STM. The long alkyl chain and pyrene were introduced into the diarylethene in order to stabilize the 2D structure. Pyrene interacts with HOPG due to extended planar πconjugation. The alkyl chains are also effective at interacting with the HOPG substrate.

The different photochromic isomers showed different orderings reflecting the differences in their molecular structures. Because the adlayer is a monomolecular layer, most of the molecules exist in the solution rather than at the interface. Therefore, most of the photochromic reac-tions take place in the solution not at the interface. When the contents of the solution phase are thereby changed, adsorption and desorption processes exchange molecules between the interface and the solution. [START_REF] Sakano | Formation of two-dimensional ordering of diarylethene annulated isomer upon in situ UV irradiation at the liquid/HOPG interface[END_REF] As a next step, we investigated diarylethene 3 carrying amide group that forms hydrogen bond. Diarylethene 3 having two amide groups formed characteristic stripe structures at the 1-octanoic acid/HOPG interface as expected (Fig. $.2), while no ordering was observed with compounds having ester group instead of amide group under the same condition. The result clearly suggests that amide group acts as an interacting group to stabilize molecular ordering via intermolecular H-bonding network.

$.3 Self-Assembly and Photoinduced Transformation of Molecular Ordering of Diarylethenes Carrying Amide Group

The width of the bright rows was 3.5 nm for 3o (ordering ) and 1.5 nm for 3c (ordering ), suggesting different ordering patterns for the two isomers. The difference in the arrangement would be attributed to the rigid core framework of the closed-ring isomer whose delocalized -conjugation system prefers to form faceto-face - stacking assembly. Although the structural transformation from the ordering  to the ordering  was reasonably induced by in situ visible light irradiation, in situ UV irradiation resulted in an entirely different third ordering  no matter whether initial ordering was  or .

In order to reveal the origin of ordering , the annulated isomer 3a was photochemically synthesized on a milligram scale by UV irradiation to a solution of 3o overnight and subsequently isolated by HPLC. The ordering of 3a was confirmed to be identical to the ordering  that was observed after in situ UV irradiation. From these results, we concluded that the ordering  is composed of the annulated isomer 3a. The reason why the ordering  was not observed upon in situ UV irradiation is the strong preferential formation of ordering , which may be attributed to the rigid and highly symmetric structure of the annulated isomer. The control over three states was possible in this photochromic molecule. $.4 Cooperative Self-Assembly of 2-Thienyl-Type Diarylethene and Photoinduced Transformation of the Ordering [28] In order to further develop photocontrollable assembly at liquid/solid interface, we have introduced cooperative model on the analysis of adsorption process of a 2-thienyl-type diarylethene derivative 4 (Fig. $. 3). The open-ring isomer 4o, having an amide group, formed uniform stripe-patterned molecular ordering. The ordering pattern was successfully reproduced by a molecular modeling study with the molecular mechanics/molecular dynamics (MM/MD) approach. It is revealed that a parallel conformer of the open-ring isomer is stabilized by intramolecular hydrogen bonding between the embedded two amide groups.

The concentration dependence of surface coverage showed abrupt increase at around 200 mM. Meanwhile, no ordering was observed for the closed-ring isomer 4c at various concentrations. Inspired by the cooperative self-assembly in solution, we developed a cooperative self-assembly model on a 2D surface on the basis of the Langmuir adsorption model incorporating two different equilibrium constants, the nucleation constant K n and the elongation constant K e (Fig. $.4). [START_REF] Martin | Comparisons of indefinite self-association models[END_REF] The elongation constant K e and the degree of cooperativity  = K n /K e were obtained as 4600 ± 20 M -1 and (5 ± 2) × 10 -4 , respectively. The high cooperativity was found to be the origin of the experimentally observed abrupt change of surface coverage over concentration in a supernatant solution. This result suggests that reversible formation/disappearance of molecular ordering is possible by means of photoisomerization between 4o and 4c. We evalu-ated the relationship between the surface coverage and conversion ratio of 4. As expected, the surface coverage showed a rapid decrease at certain conversion ratio, at which the concentration of 4o in a supernatant solution shows the abrupt change of the surface coverage. Interestingly, the critical concentration of 4o observed from the photoirradiation experiment corresponds well to the simple dilution experiment, suggesting that the presence of the closed-ring isomer 4c did not influence the ordering formation of 4o. As a consequence, highly sensitive photocontrol over the cooperative 2D ordering formation/disappearance process on the surface is possible. [START_REF] Yokoyama | Effects of alkyl chain length and hydrogen bonds on the cooperative self-assembly of 2-thienyl-type diarylethenes at a liquid/highly oriented pyrolytic graphite (HOPG) interface[END_REF] In the previous section, we have described the cooperative self-assembly of diarylethene molecules at liquid/solid interface. To investigate the relationship between molecular structure and the mechanism of cooperative ordering formation, we synthesized 2-thienyl-type diarylethenes 5o-Cn (5o-C16 is equivalent to 4o) bearing various lengths of alkyl side chains linked via an amide group. The concentration dependence of the surface coverage was analyzed using a cooperative model for a 2D surface based on two characteristic parameters, nucleation equilibrium constant (K n ) and elongation equilibrium constant (K e ) (Fig.

$.5 Effects of Alkyl Chain Length and Hydrogen Bonds on the Cooperative Self-Assembly of 2-Thienyl-Type Diarylethene

$.5).

As the length of the alkyl chain of diarylethene increased, the K e value increased exponentially, although the pattern of molecular ordering did not change. This behavior resulted from enthalpy gain with increasing alkyl chain length, which was associated with the concentration dependence of micelle formation. For diarylethenes 6o-Cn, in which amide groups at the root of the alkyl chain were replaced by ester groups, the cooperative parameter, , approached unity and the mechanism resembled an isodesmic process. This was likely to be due to the absence of hydrogen-bonding interactions for the ester derivative. In terms of the effect of core structure for the ester derivatives of 6o-C22 and 6c-C22, the  value of the closed-ring isomer 6c-C22 was significantly smaller than that of the openring isomer 6o-C22, but the K e values were similar. This suggests that K e values of diarylethenes 6o-C22 and 6c-C22 are regulated by dispersion interactions between the alkyl chain and HOPG substrate, irrespective of the core structure; on the other hand, K n was significantly affected by the core structure. The energy gain for nucleation on substrates for the closed-ring isomer 6c-C22 was lower than that of the open-ring isomer 6o-C22, which gave rise to a large difference in cooperative self-assembly on the HOPG surface.

In conclusion of this section, it is demonstrated that the precise control of selfassembly process on 2D surface becomes possible by the careful design of the molecular structure. [START_REF] Frath | Diarylethene self-assembled monolayers: cocrystallization and mixing-induced cooperativity highlighted by scanning tunneling microscopy at the liquid/solid interface[END_REF] The adsorption behavior of amide-containing diarylethene 3 was revisited by using cooperative adsorption model. The nucleation (K n ) and elongation (K e ) equilibrium constants were determined by the analysis of concentration dependence of surface coverage. It was found that the annulated isomer has a very large equilibrium constant, which explains the predominantly observed ordering of the annulated isomer (Fig.

$.6 Cocrystallization and Mixing-Induced Cooperativity

Conclusion

We have demonstrated the quantitative study of self-assembly process of photochromic diarylethene at liquid/solid interface by scanning tunneling microscopy. By the measurement of the concentration dependence of the surface coverage and the introduction of cooperative adsorption model, the degree of cooperativity of self-assembly process can be evaluated and the guiding principle for highly sensitive photoresponsive system can be obtained. These results provided a significant amount of information on the self-assembly of photochromic molecules for the application to molecular-scale electronics and photonics.

  For the diarylethene-pyrene-diarylethene triad 2, a new ordering appeared upon irradiation with UV light to the ordering of open-open isomer 2(o-o) and returned to the original ordering upon subsequent irradiation with visible light at octanoic acid/HOPG interface (Fig. $.1). The new ordering was assigned to the ordering of the closed-closed isomer 2(c-c) by comparing the image with the images of the isolated open-closed isomer 2(o-c) and the closed-closed isomer 2(c-c).

  The UV irradiation should give both the open-closed isomer 2(o-c) and the closed-closed isomer 2(c-c), but the ordering of 2(o-c) was not observed. In order to examine the relative magnitudes of the adsorption process, we measured the STM image of the mixture with the ratio of o-o:o-c:c-c = 45:33:22. The image predominantly consisted of the closedclosed isomer 2(c-c) ordering. This implies that 2(o-o) and 2(o-c) are less likely to be physisorbed on the HOPG surface than 2(c-c).The difference in the affinity to the substrate is thought to regulate the ordering behavior.

Fig. $. 1 .

 1 Fig. $.1. Sequence of STM images at the interface of octanoic acid-HOPG upon UV and visible irradiation of compound 1(o-o) (100 × 100 nm 2 , the same areas of the sample): (a) before UV irradiation, I set = 10 pA, V bias = -2.0 V); (b) after UV irradiation for 5 s (I set = 10 pA, V bias = -2.5 V); (c) after successive visible irradiation (I set = 10 pA, V bias = -2.5 V). Adapted with permission from [26]. Copyright 2008 American Chemical Society.

Fig. $. 2 .

 2 Fig. $.2. STM images of (a) the open-ring isomer 3o (the ordering , I set = 60 pA, V bias = -1.0 V, a = 5.71 nm, b = 0.89 nm,  = 88°), (b) the closed-ring isomer 3c (the ordering , I set = 20 pA, V bias = -0.9 V, a = 4.82 nm, b = 0.56 nm,  = 85°) at the 1-octanoic acid-HOPG interface. (c) The ordering after UV irradiation for 15 min (the ordering , I set = 30 pA, V bias = -1.0 V, a = 10.2 nm, b = 0.48 nm,  = 87°). (d) Enlarged molecular model of the open-ring isomer 3o and (e) that of the closed-ring isomer 3c. Hydrogen-bond network of amide groups is highlighted in green. Adapted with permission from [27]. Copyright 2013 The Chemical Society of Japan.

Fig. $. 3 .

 3 Fig. $.3. (a) STM image of the ordering of 4o at the octanoic acid/HOPG interface (c = 250 mM, I set = 30 pA, V bias = 800 mV, a = 6.3 ± 0.2 nm, b = 1.04 ± 0.04 nm,  = 88.6 ± 0.91°). (b) Concentration dependence of the fractional coverage of 4o. The solid line denotes the best-fit curve by the model simulation (K e = 4600 ± 20 M -1 ,  = (5 ± 2) × 10 -4 ) and the dashed line denotes the simulated curve with an isodesmic model ( = 1). Adapted with permission from [28]. Copyright 2014 The Royal Society of Chemistry.

Fig. $. 4 .

 4 Fig. $.4. Schematic drawing of cooperative self-assembly at a liquid/solid interface. K n denotes nucleation equilibrium constant and K e denotes elongation equilibrium constant.

Fig. $. 5 .

 5 Fig. $.5. Concentration dependence of surface coverage of the amide derivatives 5o-C14 (circle), 5o-C16 (square), 5o-C18 (upward-pointing triangle), 5o-C20 (downward-pointing triangle), and 5o-C22 (diamond) at the octanoic acid/HOPG interface. Solid lines denote the curves of best-fit by the cooperative equilibrium model. The dashed line denotes the curve based on theoretical maximum for a monolayer. Adapted with permission from [30]. Copyright 2015 Wiley-VCH Verlag GmbH & Co. KGaA.

  $.6a).The effect of the presence of 3o on the formation of ordering β is seen in Fig.$.6b, where surface coverage of ordering β is plotted against the concentration of 3c. If the formation of ordering β is independent of the presence of the open-ring isomer 3o, the same concentration dependence is expected for any ratio of isomers (3o/3c). However, at a given concentration of 3c, the surface coverage of ordering β is significantly increased by addition of 3o. This suggests that the ordering β is likely a mixed crystal composed of the two isomers.Upon mixing the closed-ring isomer 3c to ordering α that is mainly composed of the open-ring isomer 3o, a steep rise emerged in the concentration dependence of surface coverage (Fig. $.6c). The prominent increase of the cooperativity was observed. It is suggested that the presence of 3c induces cooperativity into the formation process of molecular ordering composed of 3o. A quantitative analysis of the formation process of ordering using the cooperative model has provided a new view of the formation of 2D multicomponent molecular ordering.

Fig. $. 6 .

 6 Fig. $.6. (a) Concentration dependence of the surface coverage of 3o, 3c, and 3a at the octanoic acid/HOPG interface. Concentration dependence of the surface coverage of (b) ordering  for 3c and mixtures of 3o/3c (37:63 and 51:49) and of (c) ordering  for 3o and amixture of 3o/3c (78:22). Adapted with permission from [31]. Copyright 2015 Wiley-VCH Verlag GmbH & Co. KGaA.

Acknowledgments: This research was supported by the Funding Program for Next Generation World-Leading Researchers (NEXT program, no. GR062) and a Grant-in-Aid for Scientific Research on Innovative Areas "Photosynergetics" (no. 26107008) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.


P A Chacón Santamaría

A Sierra

Stress analysis of A González-Estrada

P Chacón
email: p_chacon96@hotmail.com

O A González-Estrada

Stress analysis of a suspension control arm

Keywords: boundary conditions, control arm, finite element method, von Mises stress boundary conditions, control arm, finite element method, von Mises stress

The suspension system of a vehicle absorbs energy to cushion and soften displacements in irregular terrains, and also supports its bodywork. Some of the components that allow the correct performance of the system are the control arms. This work aims to determine the stress distribution of a control arm for the rear suspension of a buggy vehicle. The suspension geometry was modelled in order to find the loads applied on the control arm through a movement analysis. As a result, the loads were obtained as a function of time, from these values two critical times were selected to perform the analyses. The maximum von Mises stresses were evaluated as 22 MPa and 20 MPa for each time, which are low values considering the strength of the material used to manufacture these parts.

INTRODUCTION

In the research group GIEMA, we have considered fundamental challenges of engineering to define the research lines of the group, as proposed by UNESCO. We focus on sustainable development goals, and include, e.g., research on new materials [START_REF] Martínez | Evaluación de las propiedades tribológicas de materiales compuestos de matriz metálica ( MMCs ) procesados por técnicas de fabricación aditiva con haz láser ( SLM )[END_REF]- [START_REF] González-Estrada | Evaluation of Tensile Properties and Damage of Continuous Fibre Reinforced 3D-Printed Parts[END_REF], affordable and clean energy [START_REF] Molina | Modelado de la interacción fluido estructura (FSI) para el diseño de una turbina eólica HAWT[END_REF]- [START_REF] González | Large Eddy Simulation ( LES ) Aplicado a un lecho fluidizado gas -sólido . Parte I : Reactor a escala de laboratorio[END_REF], industry innovation [START_REF] Ramirez-Matheus | Estrategia de optimización para la síntesis dimensional de un robot paralelo 5R para una aplicación de mesa de corte[END_REF]- [START_REF] Jin | Error-controlled adaptive extended finite element method for 3D linear elastic crack propagation[END_REF], good health and wellbeing [START_REF] Parra | Damage Assessment of Spinal Bones due to Prostate Cancer[END_REF]. Regarding industry innovation, it is important to take advantage of novel approaches to manufacture and design, which allow the use of computational tools [START_REF] González-Estrada | Reconstrucción de tensiones para el método de elementos finitos con mallas poligonales[END_REF] to improve the design cycle of mechanical components, especially in sensitive sectors as the automotive industry.

The suspension system of a car is the set of elements that links the sprung elements of a vehicle with the unsprung ones. There are different models depending on the mechanisms used. Suspension systems are generally divided into two groups: dependent and independent [START_REF] Vivekanandan | Design , Analysis and Simulation of Double Wishbone Suspension System[END_REF]. The selected vehicle has a double A-arm system, which is independent. This means that each wheel is isolated from the others movement. A spring shock absorber controls the movement of the assembly. This suspension type is characterised by having two control arms, the upper one and the lower one. Usually, both arms are the same length but there is an arrangement frequently known as Short Long Arm suspension, for which the upper arm is usually shorter. The control arms are the elements that connect the chassis with the ball joints. These arms support the spring shock absorber and are part of the four-bar linkage that establishes the correct movement of the system. Each suspension component ensures the contact of the wheels with the ground, allowing relative movement between the wheels and the chassis [START_REF] Jazar | Vehicle Dynamics[END_REF]. Therefore, a suspension system in poor conditions, with excessive wear or wrong designed, cannot maintain the stability of the vehicle. In this sense, Finite Element Analysis (FEA) allows to design mechanical parts by estimating their mechanical response to loading conditions [START_REF] Madenci | The Finite Element Method and Applications in Engineering Using ANSYS[END_REF]. In this work, the stress distribution of the lower control arm of a small vehicle is calculated using the software Ansys. First, the methodology that was followed is described, detailing the formulation of the problem, the procedures used for calculation and the post-processing. Then, the results of the calculated loads, von Mises stress and total deformation distributions are shown. Finally, conclusions are presented.

METHODOLOGY

To perform a stress analysis, the usual procedure for solving problems using FEM was followed, which is divided into three steps: pre-processing, calculation and post-processing.

Material and geometry

The response of the suspension system of a buggy car was studied since these vehicles are frequently used in roads with irregularities. The material of the lower control arm, the component considered for the study, is steel, with properties shown in Table 1. This material is one of the most used to manufacture this mechanical part in commercial automobiles, followed by aluminium alloys. The geometry of the part is shown in Figure 1. For the accurate calculation of the stress distribution, this geometry should be as close as possible to the real one. In the case of this suspension, the spring shock absorber is not connected to the lower arm as usual, but to the upper one.

Finite element analysis

The finite element method requires the appropriate definition of boundary conditions and generation of the mesh for the optimal convergence of the solution. The variables selected for the study were von Mises equivalent stress and total deformation (for the convergence studies).

Boundary conditions

The boundary conditions are the loads applied to the control arm and its supports. To obtain the reactions in the component that was analysed, a movement analysis was performed using SolidWorks Motion complement. In this process, the following parts of the system were modelled, see Figure 2: ▪ Upper control arm. ▪ Lower control arm. ▪ Spring shock absorber. ▪ Ball joints.

The steering knuckle, the part between the ball joints, was assumed as a rectangular plate. A quarter-vehicle model was taken into account for the study since it is a simple model but widely used in the literature [START_REF] Bouazara | Improvement in the Design of Automobile Upper Suspension Control Arms Using Aluminum Alloys[END_REF]. The assembly shown in Figure 2 was used to calculate the reactions in the lower control arm. An upward vertical force of 1125 N was applied in the middle of the steering knuckle, this magnitude corresponds to a quarter of the weight of the vehicle (including passengers). For the model, the spring stiffness was 30 N/mm and the damping coefficient of the shock absorber was 0.22 Ns/mm. It was not necessary to make the geometric model of the spring since it was simulated by the software. The forces calculated were the reactions due to the contact with the ball joint. Through the movement analysis, curves that show the variation of the components of the force with respect to time were found. The results obtained for the calculation of the reaction forces in the lower control arm are shown in Figure 3. As expected, the forces in the z-component are the most significant, and the x-components are almost nil. Since the force was not constant, the stresses were calculated for two different times, thus, two analyses were carried out. The first one for the initial time and second one for the steady state. The forces used for both analyses are shown in Table 2.

The supports considered for both analyses were cylindrical supports. These supports avoid rigid body motion for the static structural analysis. Figure 4 shows the positions were the supports and the force were applied for the first analysis. These positions were the same for the second analysis. 4. Boundary conditions for the first analysis.

Mesh

The initial mesh of the first analysis had 12811 nodes and 7047 elements, and the second had 12985 nodes and 7181 elements. The element types used were SOLID187 and SURF154. Within the configuration parameters of the mesh, the option Curvature was selected as Size Function. Although, after carrying out the convergence studies (Table 3 andTable 4), the fourth mesh was selected in both cases.

Mesh independence analysis

For finite element analysis (FEA), we should perform mesh independence verification by checking the convergence of the solution and establishing the appropriate element size. For this, we varied the size of the elements, or mesh density, and observed the change in the total deformation. The total deformation was chosen to perform the convergence because when performing it with the stress, singularities could affect convergence. The convergence curves are shown in Figure 5. It can be observed that, in both cases, the result of the maximum total deformation did not have a significant modification when changing from mesh 3 to 4. Thus, we considered the fourth mesh for the study. The data of the convergence study done for the first and second analysis are shown in Table 3 andTable 4, respectively.

RESULTS

Figure 6 shows the results for the von Mises stress. The equivalent stress distribution for the first analysis had a maximum value of approximately 22 MPa (Figure 6a) and the second had a maximum of 20 MPa (Figure 6b). Both results are low taking into account the structural steel strength. In addition, the regions of the control arm where the highest stresses occur are small zones.

CONCLUSIONS

In this work, a FEA of a lower control arm was performed to obtain the equivalent stress distribution and the total deformation of the part under critical loading conditions. For this, the forces of interest acting in part were calculated. Then, the finite element model was developed, considering a convergence study of the mesh. The calculated loads of the control arm were low, within the range of 65 to 85 N and, therefore, there were also low stresses, close to 20 MPa. These results show that it would be suitable to study the possibility of optimising its shape to save material. Additionally, a possible change of material could also be evaluated since in this case the use of a material with the strength of the steel is not justified. On the other hand, further studies could investigate more on the calculation of the loads to which the part is subjected since, in this study, only the weight of the vehicle was taken into account, but there are other conditions that may also affect the part.

Figure 1 .

 1 Figure 1. Geometry of the lower control arm.

Figure 2 .

 2 Figure 2. Suspension system modelled.

Figure 3 .

 3 Figure 3. x-component, y-component and z-component of the force acting in the control arm.

Figure 5 .

 5 Figure 5. Convergence curves.

Figure 6 .

 6 Figure 6. First (a) and second analysis (b) stress distribution.

Figure 7 Figure 7 .

 77 Figure7shows the total deformation for both analyses. The maximum value for the first analysis was approximately 5e-5 m, and for the second 1e-5 m. Notice that the maximum values of the total deformation occur at different regions of the control arm, which is explained by the different direction of the force applied in each case.

Table 1 .

 1 Structural steel mechanical properties.

	Property	Value
	Tensile ultimate strength	460 MPa
	Tensile yield strength	250 MPa
	Young's modulus	2×10 11 Pa
	Poisson's ratio	0.3
	Bulk modulus	1.6667×10 11 Pa
	Shear modulus	7.6923×10 10 Pa
	Source. Ansys database.

Table 2 .

 2 Forces applied for each analysis.

	Component	First analysis	Second analysis
	x-component	-2.3 N	-2.9 N
	y-component	61 N	-10 N
	z-component	32 N	-82 N

Table 3 .

 3 First analysis convergence data.

	Solution	Total	Change (%)	Nodes	Elements
	Number	Deformation (m)
	1	4.9491e-5	-	12811	7047
	2	4.9778e-5	0.57795	29922	17402
	3	5.0197e-5	0.83888	73205	45235
	4	5.0275e-5	0.15369	200968	132658

Table 4 .

 4 Second analysis convergence data.

	Solution	Total	Change (%)	Nodes	Elements
	Number	Deformation (m)
	1	9.3485e-6	-	12985	7181
	2	1.0248e-5	9.1838	34509	20293
	3	1.0489e-5	2.321	69577	43433
	4	1.0617e-5	1.2151	133224	86979