Structural optimization algorithm for vehicle suspensions

STRUCTURAL OPTIMIZATION ALGORITHM FOR VEHICLE SUSPENSIONS

Marc J. Richard1, Mohamed Bouazara2, Laouhari Khadir2, Guoqiang Q. Cai31

Departement de genie mecanique, Universite Laval, Quebec, QC, Canada2

Departement des sciences appliquees, Universite du Quebec a Chicoutimi, Chicoutimi, QC, Canada3

Department of Ocean and Mechanical Engineering, Florida Atlantic University,Boca Raton, FL, USA

E-mail: [email protected]

Received March 2009, Accepted December 2010

No. 09-CSME-19, E.I.C. Accession 3105

ABSTRACT

Stringent tolerances on mechanical components have created increasingly severe demands on thequality of new mechanical designs. The mathematical models used to analyze the various types ofmechanical systems these days need to incorporate an optimization algorithm capable of minimizingthe levels of vibrations coming from varied sources. The suggested method is based on the parallelcombination of three methods; the Rayleigh-Ritz approach (to determine the first eigenfrequencies)which is incorporated into an efficient multicriterion optimization process based on the ESO(Evolutionary Structural Optimization) method and the finite element software ABAQUS. Theanalytical resolution and the numerical calculations of the mechanical component are, finally,validated by an experimental set-up which exploits a frequency analyser, acceleration sensors and anexcitation hammer. The effectiveness of this approach is also demonstrated in the analysis of anupper car suspension arm. By gradually removing material from the initial car suspension design, thefrequency of the component can be controlled in order to optimize the structural constraints.

Keywords: optimization; structures; algorithm; vehicles.

ALGORITHME D’OPTIMISATION DE STRUCTURES POUR SUSPENSIONS DEVEHICULES

RESUME

Des tolerances rigoureuses sur les composantes mecaniques ont cree une demande de plus enplus grande sur la qualite de nouvelles conceptions mecaniques. Les modeles mathematiques quianalysent les divers types de systemes mecaniques, de nos jours, doivent incorporer unalgorithme d’optimisation capable de reduire au minimum les niveaux de vibrations provenantde sources diverses. La methode suggeree est basee sur la combinaison en parallele de troismethodes; l’approche de Rayleigh-Ritz (afin de determiner les premieres frequences propres)qui est incorpore dans un processus pouvant optimiser plusieurs criteres simultanement base surla methode ESO (optimisation structurelle evolutionnaire) et le logiciel d’elements finisABAQUS. La resolution analytique et les calculs numeriques de la composante mecanique sont,finalement, valides par une installation experimentale qui exploite un analyseur de frequence,des capteurs d’acceleration ainsi qu’un marteau d’excitation. L’efficacite de cette approche estaussi demontree en analysant un bras de suspension superieur d’une automobile. En retirantprogressivement de la matiere sur la piece de suspension initiale de la voiture, la frequence de lapiece mecanique peut etre controlee afin d’optimiser les contraintes structurelles.

Mots-cles : optimisation; structures; algorithme; vehicules.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 35, No. 1, 2011 1

1. INTRODUCTION

The vibration of mechanical systems has been a major concern for scientists and engineers forseveral centuries. During this time, almost all new mechanical design required some kind ofvibration study. In particular, the vibration of vehicles caused by the irregularities of the road isa current field which interests several researchers [1–4]. The study of aluminum is used, in orderto qualify their reliability and their mechanical resistance [5–7]. In this paper, the optimizationof suspension parts of an automobile is investigated in order to detect and correct vibrationalphenomena such as resonance [8–11]. Moreover, after review of several papers, vibrations arecharacterized by several frequency bands in the field of suspension design for vehicles [12–15].The vibrations of very low frequency (0 to 2 Hz) appear during air and terrestrial displacement.On the other hand, low frequency vibrations (2 to 20 Hz) appear especially in the terrestrialvehicles and high frequency vibrations (20 to 100 Hz) appear in vibrating or rotary machines(pneumatic tools, sanding machines, etc).

The objective of this paper consists of optimizing the mechanical properties of automobilesuspension components by reducing its weight and attempting to eliminate vibrationphenomena such as resonance. A structural design methodology has been outlined bycombining the following four critical steps in the design process: (1) calculation of the firstfrequencies of the component to be optimized by implementing an analytical model based onRayleigh-Ritz method (in a MATLAB software); (2) performing numerical simulations andfinite element analysis with the software ABAQUS; (3) conceive a simple experimentalvalidation procedure and (4) implement an efficient multicriterion optimization process forstiffness and frequency using the Evolutionary Structural Optimization (ESO) method.

Structural optimization has been a topic of interest for over 100 years. The history of shapeand topology optimization of discrete structural systems can be categorized into three periods.First, the initial period during which Maxwell [16] and Mitchell [17] made their pioneeringstudies. Although the study of Mitchell was important in view of theoretical background, thetechniques developed, at that time, applied only to limited types of discrete structures.Following these initial works, the structural topology optimization fell relatively dormant formany decades.

The second period occurred during the 1960’s and 1970’s where the interest in structuraloptimization was re-kindled with the advent of new computing technologies. During thisperiod, some theoretical results for general optimization methods were presented, andconsiderable analytical work was done on component optimization. Meanwhile, some methodsfor discrete shape optimization were exercised on very small validation problems due tocomputing restrictions. Schmit and Farshi [18] were among the first to offer a comprehensivestatement of the use of programming techniques to solve the non-linear problem of designingelastic structures under a multiplicity of loading conditions. They recognized that this form ofdefining design problems was needed to find the minimum weight of structures where finiteelement (FE) analysis was exploited to calculate the needed responses. Then, Oda [19] isprobably the first published study that may be considered a fully algorithmic approach. Thestudy treated two-dimensional cases and the optimum shape was obtained after severaliterations on pre-selected FE. Oda and Yamazoki [20,21] published subsequent applications toaxis symmetric solids and problems including body forces. Umetani and Hirai [22] proposed analgorithm called the ‘‘growing-reforming procedure’’ for the shape synthesis of structural beamsunder multiple loads. The decade of the 1970’s witnessed extensive research in structuraloptimization, but very few practical applications.


The third period, during the last 30 years, has been characterized by an extremely dramaticgrowth in the development of high-speed computers. While theoretical work on structuraltopology optimization has continued, numerical techniques have been further refined andapplied to more realistic structures. During this period, novel continuum structural topologyoptimization methods were also introduced and studied quite extensively. Rodriguez and Seireg[23] developed an algorithmic FE approach where a component was moulded to its optimalshape following the philosophy of maximum utilization of the material.

Despite the significant effort directed towards structural optimization over the past threedecades, most techniques developed so far are restricted to shape optimization with fixedtopology. The search for a general method capable of performing shape and topologyoptimization has been a great challenge. An important development in this field was proposedby Bendsoe [24–26] who introduced the homogenization method, where the structure wasrepresented by a model with micro voids and the objective consisted in seeking the optimalporosity of the porous medium using an optimality criterion [27]. Essentially, this method [28–31] has proven to be successful in generating optimum topologies for continuum structures bytreating element densities as design variables.

Allaire [32,33] also proposed a numerical method of shape optimization based on the level setmethod and shape differentiation. Initially, the level set method made possible only topologychanges since it could not solve the inherent problem of ill-posedness of shape optimizationwhich manifests itself in the frequent existence of local minima. The reason was that the level setmethod could easily remove holes but could not create new holes in the middle of the structure.However, this problem has been resolved recently [34–36] which makes the resulting optimaldesign largely independent of the initial guess.

Mattheck and Burkhardt [37] introduced an optimization approach based on biologicalgrowth. This method utilizes the growth of the finite elements to reduce localized high stresses.Many more optimization methods exist based, for example, on the genetic algorithm [38],topometry [39] and penalty functions [40].

One optimization method, however, that is currently receiving some interest because of itssimplicity and effectiveness is the evolutionary structural optimisation (ESO) method [41–44].The simple concept behind ESO consists in progressively removing inefficient material from astructure such that the residual shape of the structure evolves towards an optimum. Thisapproach inspired other researchers such as Akin [45] and Kim [46] to publish enhanced ESOprocedures. Recently, Huang, Zuo and Xie [47] proposed a new bi-directional evolutionarystructural optimization (BESO) method combine with optimality criteria for topologyfrequency optimization problems. In all of these iterative methods, a FE analysis is carriedout while the stress distribution throughout the structure is found. Then, the material noteffectively used can be eliminated from the structure using some specific criterion. Anothersoftware based on finite elements for structural analysis is OptiStruct from Altair [48].OptiStruct’s topology, topography, shape and size optimization capabilities can be used todesign and optimize structures to reduce weight and tune performance of mechanical systems.

2. DYNAMIC MODEL

The initial step in the structural design of mechanical components is to establish thefrequency characteristics of the system to be studied. Key characteristics include the resonantfrequencies of the structure and the vibratory displacements of the system. These characteristicswill typically be determined through a combination of experimental testing, analytical


calculations and FEA. Additional concerns include the environmental conditions under whichthe structure must operate. Geometric constraints, which may limit the location or size of themechanical component, must also be considered. The analytical model was developed using theRayleigh-Ritz method [49] for the free embedded aluminum plate of Fig. 1 to determine the firsteigenfrequencies of this continuous system. The Rayleigh-Ritz method can be considered anextension of Rayleigh’s method [50]. It is based on the premise that a closer approximation tothe exact natural modes can be obtained by superposing a number of assumed functions than byusing a single assumed function, as in Rayleigh’s method. The Rayleigh-Ritz method is anenergy method that can be used to approximate natural frequencies, mode shapes, and forcedresponses of continuous systems. Let w1,w2,:::,wn be a set of n linearly independent functionssatisfying at least the system’s geometric boundary conditions.

For free vibrations problems an approximation to the mode shape is of the form

Y xð Þ~Xn

i~1

aiwi xð Þ with wi~ 1{cos2k{1ð Þp

2Lx

� �ð1Þ

where we will need to determine the coefficient a1,��,an and wi represents n linearly independentfunctions which satisfy all the boundary conditions.

In the Rayleigh-Ritz method, the dynamic behaviour of the structure is represented by thefollowing general eigenvalue equations:

Xn

j~1

Kij{v2i Mij

� �aj~0 i~1,2 � � � ,n ð2Þ

Equation (2) is the well-known Galerkin equation which will be exploited in our structuraloptimization algorithm for the removal of inefficient material. Hence, the eigenvalue problemfor a structure is viewed as a discrete system of n degrees of freedom:

Fig. 1. Free embedded trapezoidal aluminum plate. (b5100 mm, bo5189 mm, L5180 mm,h56.26 mm).


K½ �{v2n M½ �

� �anf g~ 0f g ð3Þ

where K½ � and M½ � are the stiffness and mass matrices of order n|n, respectively. The nth naturalfrequency is represented by vn and anf g is the eigenvector corresponding to vn. The problem thenconsists in choosing a generating function Y xð Þ judiciously as was defined previously in Eq. (1).Finally, for a complete solution, the determinant of the coefficient matrix of the system ofequations represented by Eq. (3) is set to zero, resulting in an nth-order polynomial to solve forv2, which represents the natural frequencies. The solution of this eigenvalue problem generallygives n natural frequencies v2

i , i~1,2,:::,n and n eigenvectors, each containing a set of numbers for

a1,a2,:::,an. The ith eigenvector corresponding to vi is ~AA ið Þ~ aið Þ

1 aið Þ

2 � � �n

aið Þ

n

oT

.

3. MODELING, SIMULATION AND EXPERIMENTAL VALIDATION

A trapezoidal uniform aluminum section was first analyzed to verify the thickness effect of thismaterial on the frequency response. The aluminum alloy 6061-T6-T651 had the followingproperties: ultimate tensile strength of 310 MPa; tensile yield strength of 276 MPa; 12 %lengthening on 5 cm specimen of 1.6 mm of thickness; shear strength of 207 MPa; Young’smodulus of elasticity of 68.9 GPa; Poisson’s ratio of 0.33 and density of 2700 kg/m3. The firstthree frequencies of the aluminum plate were calculated analytically by the Rayleigh-Ritz methodwith the software MATLAB. A digital simulation with ABAQUS was also performed on thistrapezoidal aluminum plate. Finally, we also had a basic experimental set-up for validationpurposes. The loop connection for the three instruments is depicted in Fig. 2.

The experimental set-up was comprised of (1) a frequency analyser EDX-2000A-32, (2) BRUEL& KJAER acceleration sensors and (3) a DYTRAN impulse excitation hammer. The results for thefirst three modes are confined in table 1. We note that the approximate analytical method is onlyvalid for the first mode. Actually, the second and third modes of this trapezoidal plate are of littleinterest since the operating frequencies of vehicles oscillate normally between 0 and 20 Hz. The largediscrepancy between the analytical method and the other two methods, for modes 2 and 3, arelargely attributed to the selection of the function wi to get a closer approximation for the first mode.

Fig. 2. Instruments connection loop.


4. BASIC OPTIMIZATION PROCESS

It has been known for years that a reliable sign of potential structural failure is excessivestress. Ideally, the stress in every part of a structure should be as uniform as possible throughoutthe component. This leads to a rejection criterion concept, based on a local stress level, wherelow stressed material is assumed to be inefficient and will be identified for removal. Bygradually removing material with lower stress, the stress level in the new optimized designbecome more and more uniform. The optimizing procedure is simple. The design domain isdivided into a fine mesh of finite elements which should be large enough to cover the area of theoptimum design. The von Mises stress has been one of the most frequently used criteria forisotropic materials which provides an average measure of the entire stress. For plane stressproblems, the well-known von Mises stress can be calculated as

sVM~ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

xzs2y{sxsyz3t2

xy

qð4Þ

where sx and sy are normal stresses in x and y directions, respectively, and txy is the shearstress. At the end of each finite element analysis, a rejection condition must be applied:

sVMel

sVMmax

ƒRRi ð5Þ

where RRi represents the current rejection ratio. The stress level of each element is determinedby comparing the von Mises stress of each element sVM

el to the maximum von Mises stress of theentire structure sVM

max. All elements which satisfy Eq. (5) are extracted from the structure. Thecycle of finite element analysis and element removal is repeated using the same value of RRi

until a steady state is reached, which means that no more elements is being deleted at thisiteration. At this stage, an evolutionary rate (ER) can be added to the rejection ratio:

RRiz1~RRizER i~0,1,2,3::: ð6Þ

Once this increased rejection ratio is calculated, the cycle of finite element analysis andelement removal takes place again until a new steady state is reached.

5. SENSITIVITY NUMBERS FOR ELEMENT REMOVAL

The ESO procedure for the multicriteria optimization is entirely based on the evaluation ofsensitivity numbers for the stiffness and the frequency constraints. This is done exploiting a

Table 1. Frequency response for the first three vibration modes.


standard weighting approach where two criteria will be optimized simultaneously, namely, themaximization of the first mode of natural frequency and the minimization of the meancompliance of a structure (which is inversely proportional to stiffness). In other methods, toidentify the best solution for structural modifications, derivatives are often needed and, usually,a sensitivity analysis for dynamic problems is complicated. However, it can be shown [44] thatthe ESO method exploits a very simple sensitivity number to measure the effect of removing anelement on the structure.

Stiffness, along with frequency, is an important factor that needs to be taken into account inthe design of structures. It is well-known that in the finite element method, the static behavior ofa structure is given by

K½ � df g~ Pf g ð7Þ

where [K] is the global stiffness matrix and {d} and {P} are the global nodal displacement andnodal load vectors, respectively. The inverse measure of the overall stiffness of a structure isknown as the mean compliance C and is defined as

C~1

2Pf gT df g ð8Þ

Minimizing the mean compliance maximizes the overall stiffness of a structure and thestiffness constraint can be respected from CƒCall where Call is the prescribed limit for C [44].From this comes the stiffness sensitivity number for problems with an overall stiffnessconstraint [44]:

ai~1

2ai� �T

Ki

ai� �

ð9Þ

where ai� �

is the displacement vector of the ith element and Ki

is the stiffness matrix of the ithelement. The quantity ai indicates the change in the strain energy as a result of removing the ithelement. To achieve the optimization objective through element removal, the element that hasthe lowest value of ai is removed so that the increase in C is minimal.

Another important criterion is the optimization of the first mode of the natural frequencysince the response of a mechanical component to dynamic loading depends mostly on the firstfew natural frequencies. It is well known that when the frequency of the dynamic loading isclose to one of the natural frequencies, excessive vibration occurs. To avoid these vibrations, itis often necessary to shift the fundamental first frequency away from the frequency range of thedynamic load.

The ESO method exploits the concept of a frequency sensitivity number to select and removethe most inefficiently used material. This frequency sensitivity number controls the materialremoval process by defining which element has the minimum contribution to the objectivefunction of the optimization problem. It has been shown in the preceeding section that thedynamic behavior of the structure is represented by the general eigenvalue problem of Eq. (3).To decide which element should be removed from the structure so that the frequency will beshifted toward a desired value, it can be shown [44] that the frequency sensitivity number can bederived from Eq. (3) as


ain~

1

mn

ain

� �Tv2

n Mi

{ Ki � �

ain

� �ð10Þ

where Ki

and Mi

are the stiffness and mass matrix of the ith element and ain

� �is the

eigenvector of the ith element. For example, if a four node plane stress element is generated bythe finite element analysis, Ki

and Mi

will be 8 6 8 matrices and ai

n

� �will be a 8 6 1 vector.

This sensitivity number is an indicator of the change in v2n as a result of the removal of the ith

element. It is noted from equation (10) that the sensitivity number ain can be easily calculated at

the element level once the eigenvalue problem is available. By comparing Eq. (10) with Eq. (3),it becomes obvious that

Pelem ai

n~0, namely, the summation of the sensitivity numbers over allthe elements is equal to zero. A maximum increase in the chosen frequency occurs as a result ofremoving elements whose frequency sensitivity number is the highest and vice-versa. Removalof those elements whose ai

n is close to zero results in little change [43].

At this point, a linear scaling is required on the frequency sensitivity number since the basison which they are optimized is different. This scaling must be done to alleviate the problem ofcombining both sensitivity numbers (with the element removal based on the lowest and highestsensitivity values). The frequency sensitivity number linear shift is obtained using the followingrelationship:

ain

� �new~{ ai

n

� �old� �

z amaxn

� �old ð11Þ

where ain

� �newand ai

n

� �oldis the new and old frequency sensitivity number, respectively, for the

ith element and the nth natural frequency, amaxn

� �oldis the old maximum frequency sensitivity

number for the nth natural frequency. The shift imposed by Eq. (11) now allows the criteria tobe combined into a single weighted criterion:

P~XN

i~1

wjRij ð12Þ

where P now represents a global multiple criteria objective function that determines element

removal for each element i, wj is the jth criteria weighting factor (with j~1, . . . ,N), Rij~

aij

.amax

j

is the ratio of the jth criteria sensitivity number aij for each element i to the maximum value to

the jth criteria sensitivity number amaxj , and N is the total number of criteria, in our case N52.

By gradually removing the low stressed fractions, the stress distribution in the component becomesincreasingly uniform. The procedure for weighted multicriterion optimization based on ESO forstiffness and natural frequency has been implemented into a methodical five step algorithm:

(1) Discretize the structure using a fine mesh of finite elements;

(2) Add load to structure, then, solve the static problem in Eq. (8) and generate the linearstatic sensitivity number ai in Eq. (9);

(3) Remove load from structure, calculate the natural frequency sensitivity number ain using

the eigenvalue problem in Eq. (10) and obtain the new natural frequency sensitivitynumber ai

n

� �newusing the linear scaling of Eq. (11);

(4) Combine all the criterion sensitivity numbers using Eq. (12) to obtain the multiple criteriaobjective function P;


(5) Remove elements that have the lowest values of P using the following rejection ratio:

PƒRR Pmaxð Þ ð13Þ

where Pmaxð Þ is the maximum P that exists in the structure. Equation (13) can be viewed asan iterative counter used to dampen or delay the element removal process. If the number ofelements that satisfy this constraint reach zero, without achieving the optimizationobjective function, then the value of RR is increased to allow the material removingprocess to proceed. Repeat steps 2 through 5 until an optimum is reached.

Great care must be applied when increasing the rejection ratio RR. To obtain the value of thesensitivity number from the information of the previous eigenvalue solution, one must assumethat the eigenvector is approximately the same before and after the removal of that element.The assumption that the mode shape does not change significantly in between iteration cycleshas been commonly used. Hence, in light of this assumption, one must limit the evolutionaryrate ER at very low increasing steps (typically 0.001).

This algorithm was exploited for the aluminum plate of the previous section with a verticalload of 1200N applied on the x axis at the end of the plate and the weight factors (wf ~0:9 forthe frequency and ws~0:1 for the stiffness). The first three vibration modes of the optimizedstructure are tabulated in Table 2 and shown in Fig. 3. Table 2 shows the difference between thefirst three frequencies and the total mass of the aluminum plate from its initial state to theoptimized state by maximizing the first frequency for a 17 % mass reduction. We noted that thefirst frequency increases by almost 6 % while the material was deleted from each side of theplate, and also removed from two rectangular holes located on the center line of the plate asshown in Fig. 3.

6. EXAMPLE: RECTANGULAR ALUMINUM PLATE

To understand the basic principle underlying this method, let us examine a simple two-dimensional structural optimization problem. Consider an elementary rectangular aluminumplate (1506100 mm) attached with fixed support at two diagonal corners subjected to twosingle horizontal loads of 100 N as depicted in Fig. 4. These are included for the linear staticstress analysis, but are removed for the frequency analysis. Four-noded linear quadrilateralelements are used in this example. The two driving criteria are the minimization of the meancompliance and the maximization of the first mode of natural frequency.

This example is based on a rectangular plate model that was used by Xie and Steven in theirstudy of ESO for dynamic problems [43]. The physical data for this example are identical to thealuminum plate analyzed earlier. The design domain in Fig. 4 is divided into 45630 four nodeplane stress elements of equal size. The thickness of the rectangular plate is 10 mm.

Table 2. First three frequencies between the initial and optimized plate.


When the material removed is fixed at 30 %, Fig. 5 shows the optimal designs for differentweighting criteria of stiffness and natural frequency of the ESO iterations where 5(a)ws~1:0, wf ~0:0; 5(b) ws~0:9, wf ~0:1; 5(c) ws~0:7, wf ~0:3; 5(d) ws~0:5, wf ~0:5; 5(e)ws~0:3, wf ~0:7; 5(f) ws~0:2, wf ~0:8. In these evolutionary figures the dark areas representthe remaining elements and the small dots represent the nodes of the initial finite element model.

Figure 5(a) shows the case where the structure has been designed completely for stiffness;hence, the compliance is at its lowest possible value and, for a 30 % material removal from theinitial structure, the natural frequency of 2666.6 Hz for this topology is at its lowest. When theweighting criteria are modified, for example, when the frequency weighting is increased and the

Fig. 3. First three mode frequencies of the optimized trapezoidal plate (material removed 17 %).


stiffness is decreased, an increase in the natural frequency and compliance can be noted whichmeans that the overall stiffness of the structure decreases.

7. AUTOMOBILE SUSPENSION ARM COMPONENT

Automotive industry requirements for quality, productivity and cost efficiency are at a levelwhere optimization of the design and manufacturing of a product must occur in the earlieststages of conception. Car suspension systems present a special case because of their geometryand the robust constraints placed on their design by the underbody of the car. Suspensionsystems are submitted to many dynamic input loads. The induced vibrations are spread alongthe axles and the forces are transmitted to the car body through the suspension. As these inputloads induce some structural vibration, it is necessary to define the best decoupling elementsthat allow a reduction of the amplitude to meet acceptable target frequency values. Hence, theoptimum design of structures with stiffness and frequency constraints is of great importance,particularly in the automotive and aeronautical industries.

Ideally, the stress in every part of a loaded structure should be uniform and, in our case, we wantto maintain the first frequency v1 as constant as possible. As defined previously, this concept leadsto a geometrical modification criterion in which low stressed fractions of a component are removedas they add mass to the structure without significant contribution to its stiffness. When material isremoved from a structure, the frequencies of the structure will almost inevitably be changed to someextent. In this case, an attempt is made to make as little change as possible for the first frequencywhile the actual weight of the structure is being reduced. The solution procedure is the same as thatof the preceding section except that, in this case, we must define a new sensitivity number,

ain

� �new~ ai

n

� � oldz amax

n

� �old ð14Þ

where the elements having the smallest absolute values ain

will be removed. The objective functionconsists in minimizing the structural mass by removing material, element by element, by imposingthe weighting criteria wf ~0:9, ws~0:1 for a 1 %, 2 % and 3 % material removal.

Fig. 4. Initial design domain of rectangular plate under loading with fixed supports.


As mentioned earlier, to avoid severe vibration, it is often necessary to shift the firstfrequency of a structure away from the frequency range of the dynamic loading. Hence, theoptimization process developed for the simple aluminum trapezoidal plate and the rectangularplate will be applied to a higher aluminum suspension arm for an automobile as shown in Fig. 6.The dynamic head of the upper arm suspension is schematized in Fig. 7. The value of thevertical load is 3500 N and no rotation is considered on the higher knee caps (ring) around thethree axes since the algorithm is optimizing the upper component for the worst case scenario.

The optimization process developed in section 5 was applied to the suspension arm of Fig. 7.With the multicriterion ESO method, the frequency of a structure can be maintained or shiftedtowards a desired value by gradually removing material from the structure. At each iteration,the eigenvalue problem of equation (3) is exploited and then a number of elements are removedfrom the structure according to their sensitivity numbers ai

n. The design domain was divided

Fig. 5. Optimal design of rectangular plate for a 30 % material removal and various weightingbetween stiffness and natural frequency.


into 3-dimensional elements for the finite element analysis. Using the procedure outlined in theprevious section after 600 iterations, the first frequency is maintained as close as possible to itsinitial value (within 3 %). As a result the evolutionary history of the first frequency is obtainedand the corresponding new design of the upper suspension arm is given in Fig. 8.

The material was removed in three different stages. The first stage at a rejection ratio RR510 %,the upper left finite element (FE) analysis of Fig. 8(a) represents the first stage of material removalat 1 %. The second stage is shown in the upper right FE analysis of Fig. 8(b) with RR515 % for a2 % material removal. Finally, the last FE analysis is shown in the bottom of Fig. 8(c) with aRR520 % where 3 % of the material was removed. Note that, if the rejection ratio is not restrictedto a maximum value of material removed, the removal procedure can continue until thecomponent collapses.

At this stage, 3 % of the total material has been eliminated and the first frequency has beendecreased from 361 to 352 Hz with a rejection ratio of 20 %. In this analysis, only the firstfrequency has been considered for optimization with a weighting factor of wf ~0:9. Note thatthere is no control over what is happening to other frequencies during the optimization process.The first frequency has been decreasing monotonically and the second and third frequencies

Fig. 6. Automobile suspension upper arm.

Fig. 7. Aluminum 6061-T6-T651 upper suspension arm.


have dropped more rapidly. Sometimes, such behaviour is undesirable, and this procedurecould be extended to include multiple frequency constraints.

8. CONCLUSION

In this paper, we have demonstrated that a multiple criterion optimization algorithm basedon a weighting method can be introduced into the ESO method which can solve a wide range ofstiffness and frequency optimization problems. The location for element removal is determinedby weighted sensitivity numbers for each element. The sensitivity number for stiffness andnatural frequency can be easily calculated from finite element analysis. By gradually removingmaterial from the initial design, the frequency of a structure can be optimized (increased,reduced or kept constant) while the stiffness of the structure is simultaneously controlled.

The aluminum alloy used to manufacture automobile parts allows a reduction of the carmass. The results of the digital simulation for the aluminum trapezoidal plate show a smallincrease in the frequencies while an 18 % total mass reduction was achieved, whereas the digitalsimulation results for the upper suspension arm show a small reduction in the natural frequencywith a 3 % total mass reduction.

A multicriterion structural optimization design methodology has been developed whicheliminates most of the costly trial-and-error testing currently required. Automobile suspensions

Fig. 8. Final process of optimization for the suspension arm. (a) Material removal 1 %, (b) Materialremoval 2 %, (c) Material removal 3 %.


are highly nonlinear components whose frequency energy depends on a large number ofparameters whose effects may be confounded. This methodology includes design guidelines toaid in the selection of proper material extraction. It is anticipated that these guidelines will berefined, and possibly additional guidelines added, as additional structural analyses andexperimental testing is performed. The design methodology was used to optimize the designof an automobile upper suspension component for a sample application consisting of analuminum structure. Ongoing research is in progress to optimize the thickness of elements byshifting material from one element to another in order to achieve a generalized shape opti-mization. With the advance in composite and smart materials, new components in automotiveand aeronautical industries have received ever increasing attention from engineers who nolonger needs to restrict themselves to traditional designs.

Acknowledgements

The authors gratefully acknowledge the involvement of the Natural Science and EngineeringResearch Council of Canada and the CQRDA (Centre quebecois de recherche et dedeveloppement de l’aluminium) for their financial support during this work. The authorswould also like to express their gratitude toward the referees for their valuable comments.

REFERENCES

1. Lin, Y. and Zhang, Y., ‘‘Suspension optimization by a frequency domain equivalent optimal

control algorithm,’’ Journal of Sound and Vibration, Vol. 133, No. 2, pp. 239–249, 1989.

2. Rakheja, S., Computer-Aided Dynamic Analysis and Optimal Design of Suspension Systems for

Off-road Tractors, Ph.D. Thesis, Concordia University, Montreal, 1985.

3. Hsiao, M.H., Haug, E.J. and Arora, J.S., ‘‘A state space method for optimal vibration

isolators,’’ ASME Journal of Mechanical Design, Vol. 101, pp. 309–314, 1979.

4. Afimiwala, K.A. and Mayne, R.W., ‘‘Optimum design of an impact absorber,’’ ASME Journal

of Engineering for Industry, Vol. 96, pp. 124–130, 1974.

5. Jiang, D.S., Lui, T.S. and Chen, L.H., ‘‘Crack propagation behaviour of Aluminum Alloy

under resonant vibration,’’ Department of materials Science and Engineering National Cheng

Kung University, Scripta Materialia, Vol. 36, No. 1, pp. 15–20, 1997.

6. Sato, S., Enomoto, M. and Kato, R., ‘‘Aluminium extrusions for suspension arms,’’ Automotive

Engineering International, Vol. 106, No. 10, pp. 96–99, 1998.

7. Cote, F. and Akiyoshi, M., ‘‘Aluminum alloys for automobile applications,’’ Proceedings of the

6th International Conference on Aluminum Alloys ICAA-6, pp. 25–32, July, 1998.

8. Stansloski, M.J., Detection and Correction of Resonance in a Sheet Aluminum Winder Using

Vibration Signature and Finite Element Analysis, Master’s Thesis, Mississippi State University,

May, 1996.

9. Malaktar, P., Nonlinear Vibrations of Beams and Plates, Ph.D. Thesis, Virginia Polytechnic

Institute and State University, July, 2003.

10. Stangier, S.D., Theoretical and Experimental Investigation of The Free Vibration of

Parallelogram Plates with Simply Supported and Clamped Boundary Conditions, Master’s

Thesis, Virginia Polytechnic Institute and State University, August, 1997.

11. James, M.L., Smith, G.M., Wolford, J.C. and Whaley, P.W., Vibration of Mechanical and

Structural Systems: with Microcomputer Applications, Harpercollins College Div., 1994.


12. Sanders, M.S. and McCormick, E.J., Human Factors in Engineering and Design, EditionsOctares Entreprises, 1993.

13. Griffin, M., Parsons, K. and Whitham, E., ‘‘Vibration and comfort, application of experimentalresults,’’ Ergonomics, Vol. 25, pp. 721–739, 1982.

14. Pinhas, B., ‘‘Magic numbers in design of suspensions for passengers cars,’’ Passenger Car

Conference, Tennessee, SAE paper No. 911921, pp. 53–88, 1991.

15. Rakheja, S., Computer-Aided Analysis and Optimal Design of Suspension Systems for Off-road

Tractors, PhD Thesis, Concordia University, Canada, 1985.

16. Maxwell, J.C., The Scientific Papers, Cambridge University Press, Cambridge, 1890.

17. Mitchell, A.G.M., ‘‘The Limits of economy of material in frame structures,’’ Philosophical

Magazine, Vol. 8, pp. 305–316, 1904.

18. Schmit, L.A. and Farshi, B., ‘‘Some approximation concepts for structural synthesis,’’ AIAA

Journal, Vol. 12, pp. 692–699, 1974.

19. Oda, J., ‘‘On a technique to obtain an optimum strength shape by the finite element method,’’

Bulletin of the Japan Society of Mechanical Engineers, Vol. 20, pp. 160–167, 1977.

20. Oda, J. and Yamasaki, K., ‘‘On a technique to obtain an optimum strength shape of an

axisymmetric body by the finite element method,’’ Bulletin of the Japan Society of Mechanical

Engineers, Vol. 20, pp. 1524–1532, 1977.

21. Oda, J. and Yamasaki, K., ‘‘On a technique to obtain an optimum strength shape by the finite

element method: application to the problems under body force,’’ Bulletin of the Japan Society of

Mechanical Engineers, Vol. 22, pp. 131–140, 1979.

22. Umetani, Y. and Hirai, S., ‘‘Shape optimization of beams subject to displacement restrictions onthe basis of the growing-reforming procedure,’’ Bulletin of the Japan Society of Mechanical

Engineers, Vol. 21, pp. 1113–1119, 1978.

23. Rodriguez, J. and Seireg, A., ‘‘Optimizing the shapes of structures via a rule-based computer

program,’’ ASME Computers in Mechanical Engineering, Vol. 4, pp. 20–29, 1985.

24. Bendsoe, M.P. and Kikuchi, N., ‘‘Generating optimal topologies in structural design using a

homogenization method,’’ Computer Methods Applied in Mechanical Engineering, Vol. 71,

pp. 197–224, 1988.

25. Bendsoe, M.P. and Kikuchi, N., ‘‘Topology and generalized layout optimization of elastic

structures,’’ in Bendsoe, M.P. (Ed.), Topology Design of Structures, Kluwer Academic publisher,Mota Soares, CA, 1993.

26. Bendsoe, M.P., Optimization of Structural Topology, Shape and Material, Springer, Heidelberg,1995.

27. Rozvany, G.I.N., Structural Design via Optimality Criteria, Kluwer Academic Publishers,Dordrecht, 1989.

28. Tenek, L.H. and Hagiwara, I., ‘‘Static and vibrational shape and topology optimization usinghomogenization and mathematical programming,’’ Computer Methods in Applied Mechanical

Engineering, Vol. 109, pp. 143–154, 1993.

29. Allaire, G., Jouve, F. and Maillot, H., ‘‘Topology optimization for minimum stress design with the

homogenization method,’’ Structural and Multidisciplinary Optimization, Vol. 28, pp. 87–98, 2004.

30. Hassani, B. and Hinton, E., Homogenization and Structural Topology Optimization; Theory,

Practice and Software, Springer, ISBN: 3540762116, 1998.

31. Kang, Z., Zhang, C. and Cheng, G., ‘‘Structural topology optimization considering mass

moment of inertia,’’ 6th World Congress of Structural and Multidisciplinary Optimization, Rio de

Janeiro, Brazil, pp. 1–8, June, 2005.

32. Allaire, G., Jouve, F. and Toader, A.M., ‘‘A level set method for shape optimization,’’ Comptes

rendus de l’Academie des sciences Paris, serie I, Vol. 334, pp. 1125–1130, 2002.


33. Allaire, G., Jouve, F. and Toader, A.M., ‘‘Structural optimization using sensitivity analysis anda level set method,’’ Journal of Computational Physics, Vol. 194, No. 1, pp. 363–393, 2004.

34. Allaire, G., Jouve, F., de Gournay, F. and Toader, A.M., ‘‘Structural optimization usingtopological and shape sensitivity via a level set method,’’ Control and Cibernetics, Vol. 34,pp. 59–80, 2005.

35. Allaire, G. and Jouve, F., ‘‘A level set method for vibration and multiple loads structuraloptimization,’’ Computer Methods in Applied Mechanical Engineering, Vol. 194, pp. 3269–3290,2005.

36. Chungang, Z., Zhenhua, X. and Han, D., ‘‘Structural shape and topology optimization basedon level-set modelling and the element propagating method,’’ Engineering Optimization, Vol. 41,No. 6, pp. 537–555, 2009.

37. Mattheck, C. and Burkhardt, S., ‘‘A new method of structural shape optimization based onbiological growth,’’ International Journal of Fatigue, Vol. 12, pp. 185–190, 1990.

38. Nakanishi, Y., ‘‘Representation of topology using homology groups and its application tostructural optimization,’’ JSME International Journal, Series A, Solid Mechanics and Material

Engineering, Vol. 43, pp. 234–241, 2000.39. Leiva, J.P., Watson, B.C. and Kosaka, I., ‘‘A comparative study of topology and topometry

structural optimization methods within the genesis software,’’ LS-DYNA Anwenderforum,Frankhenthal, DYNAmore GmbH, pp. 23–32, 2007.

40. Kotucha, G. and Hacki, K., ‘‘Numerical instabilities in structural optimization-analogy betweentopology & shape design problem,’’ GAMM Annual Meeting 2006, Vol. 6, No. 1, pp. 229–231,2006.

41. Xie, Y.M. and Steven, G.P., ‘‘A simple evolutionary procedure for structural optimization,’’Computers Structures, Vol. 49, pp. 885–896, 1993.

42. Xie, Y.M. and Steven, G.P., ‘‘Optimal design of multiple load case structures using anevolutionary procedure,’’ Engineering Computations, Vol. 11, pp. 295–302, 1994.

43. Xie, Y.M. and Steven, G.P., ‘‘Evolutionary Structural Optimization for Dynamic Problems,’’Computers Structures, Vol. 58, No. 6, pp. 1067–1073, 1996.

44. Xie, Y.M. and Steven, G.P., Evolutionary Structural Optimisation, Springer-Verlag, ISBN 3-540-76153-5, 1997.

45. Akin, J.E. and Arjona-Baez, J., ‘‘Enhancing structural topology optimization,’’ EngineeringComputations, Vol. 18, No. 3, pp. 663–675, 2001.

46. Kim, S.R., Park, J.Y., Lee, W.G., Yu, J.S. and Han, S.Y., ‘‘Reliability-based topologyoptimization based on evolutionary structural optimization,’’ World Academy of Science,Engineering and Technology, Vol. 32, pp. 168–172, 2007.

47. Huang, X., Zuo, Z.H. and Xie, Y.M., ‘‘Evolutionary topology optimization of vibratingcontinuum structures for natural frequencies,’’ Computers and Structures, Vol. 88, No. 5,pp. 357–364, 2010.

48. Nosner, R.H., ‘‘Structural Optimisation with Altair OptiStruct in Practise,’’ Second Altair UK

Users Conference, http://tecosim.de/fileadmin/user_upload/Exposes/altair_opti-struct.pdf, con-sulted March, 2010.

49. Craver, W.L. and Egle, D.M., ‘‘A method for selection of significant terms in the assumedsolution in a Rayleigh-Ritz analysis,’’ Journal of Sound and Vibration, Vol. 22, pp. 133–142,1972.

50. Temple, G. and Bickley, W.G., Rayleigh’s Principle and Its Application to Engineering, Dover,New York, 1956.


Structural optimization algorithm for vehicle suspensions

Documents

Transcript of Structural optimization algorithm for vehicle suspensions