applied sciences

Article Multi-Criteria Optimization of an Innovative Suspension System for Race Cars

Vlad T, ot, u and Cătălin Alexandru *

Product Design, Mechatronics and Environment, Transilvania University of Bra¸sov, Bulevardul Eroilor 29, 500036 Bra¸sov, Romania; [email protected] * Correspondence: [email protected]; Tel.: +40-724-575-436

Abstract: The purpose of the present work was to design, optimize, and test an innovative suspension system for race cars. The study was based on a comprehensive approach that involved conceptual design, modeling, simulation and optimization, and development and testing of the experimental model of the proposed suspension system. The optimization process was approached through multi-objective optimal design techniques, based on design of experiments (DOE) investigation strategies and regression models. At the same time, a synthesis method based on the least squares approach was developed and integrated in the optimal design algorithm. The design in the virtual environment was achieved by using the multi-body systems (MBS) software package ADAMS, more precisely ADAMS/View—for modeling and simulation, and ADAMS/Insight—for multi-objective optimization. The physical prototype of proposed suspension system was implemented and tested

with the help of BlueStreamline, the Formula Student race car of the Transilvania University of Bras, ov. The dynamic behavior of the prototype was evaluated by specific experimental tests, similar to those the single seater would have to pass through in the competitions. Both the virtual and experimental results proved the performance of the proposed suspension system, as well as the usefulness of the



 design algorithm by which the novel suspension was developed.

Citation: T, ot,u, V.; Alexandru, C. Keywords: suspension mechanism; optimal design; kinematics; dynamics; virtual prototype; experi- Multi-Criteria Optimization of an mental model Innovative Suspension System for Race Cars. Appl. Sci. 2021, 11, 4167. https://doi.org/10.3390/app11094167 1. Introduction Academic Editor: Alessandro Gasparetto While passing over irregularities on the rolling surface, the car wheel was subjected to vertical displacement. The suspension system (which includes the wheel guiding mecha- Received: 24 March 2021 nism, and the elastic and damping elements) has the role of minimizing the impact that the Accepted: 28 April 2021 Published: 2 May 2021 rolling surface irregularities has on the chassis, reducing the forces, damping the vibration, isolating the interior (cabin/cockpit) from noise and shocks, and thus improving the ride

Publisher’s Note: MDPI stays neutral and handling of the vehicle [1–8]. Optimal suspension design, including the development with regard to jurisdictional claims in of innovative solutions such as those shown in [9,10], has been a permanent concern and published maps and institutional affil- challenge. In case of passenger car guiding (suspension and steering) systems, there are iations. many theoretical and practical researches, including various analysis and optimization methods/techniques [11–21], which reflect the performances provided by the current de- signs, and in case of the single seater and lightweight formula cars, due to their special purpose and use, there are still researchable aspects insufficiently addressed so far. The present paper introduced a new suspension design that, compared with the Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. existing ones, decupled important metrics that would be preferred to be tuned separately. This article is an open access article For instance, one of the MacPherson (suspension type mainly used in stock car racing) distributed under the terms and strut suspension main tuning disadvantages is its reduced ability to tune the camber gain. conditions of the Creative Commons The main disadvantage of a double wishbone suspension system (suspension type mainly Attribution (CC BY) license (https:// used in single seater) is the fact that some of the tunables influence each other and cannot creativecommons.org/licenses/by/ be decupled; for instance, the camber gain with wheel track variation. In the case of some 4.0/). of the best-handling street cars, such as the double wishbone suspension system which

Appl. Sci. 2021, 11, 4167. https://doi.org/10.3390/app11094167 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 4167 2 of 25

cannot cope with the desired tuning, a multi-link suspension system is used, which offers a greater degree of freedom in tuning and decuples most of the metrics. In the case of the double wishbone suspension system, based on the Formula Student tuning trials, the main conflicts in this tuning are: The wheelbase variation being influenced by tuning the bump camber (same as roll camber); and the wheel track variation being influenced by the bump caster. One of the biggest constraints in the single-seater race car classes is the fact that there is not enough vertical space to insert a MacPherson strut suspension system (as it is one of the tallest suspension systems used in stock cars) or even a double wishbone strut suspension system (as there is not enough space to insert a regular spring and damper assembly). Thus, the main suspension system used in these racing events is the double wishbone with a push/pull rod and rocker. This paper dealt with a newly (innovative) design for the wheel suspension system of a single-seater car (Formula Student type), based on a comprehensive approach that included conceptual design, modeling, simulation and optimization, and development and testing of experimental model. For the proposed suspension system, the tuning process was not as straight forward as it would be for the other suspension systems. Some of the advantages of the proposed suspension system were that it has a similar spring, damper, and rocker assembly as the double wishbone (it does not require extra space), and it can cope with tuning the bump camber and bump caster without effecting the wheel track and wheelbase. It is well known that any suspension system is designed so that it creates the best dynamic behavior of the vehicle. This behavior may vary from vehicle to vehicle, as different cars have different scopes, although they use the same suspension mechanism. Any dynamic behavior can be divided into different design objectives (targets). These are divided between ride (the ride metrics are about separating the driver from disturbances occurring as a result of the vehicle operation) and handling (the handling metrics are largely concerned with the behavior of the vehicle in response to the driver demands), and most of them are directly affected by the kinematic movement of the suspension mechanism. Most of the dynamic targets can be tuned by optimizing different kinematic metrics. Although the purpose of the tuning is to minimize the variations of camber and caster angles during the jounce and rebound of the wheel, this is just to show that the suspension system can achieve this. It is known that different race cars have different targets for the metric targets that are given by the vehicles’ desired dynamic behavior (based on the type of racing, surface they race on, wheels and tires used, and so on). The methodology applied in this regard involved defining the general context, identi- fying the problem and setting the objectives, analyzing how to use the product, defining the required functions, researching the technical and technological solutions for the identified functions, evaluating existing similar products, evaluating and validating the proposed innovative concept, and thus following the modern trends in the engineering design process [22]. In order to fulfill the research objectives, there were specific mechanical engineering design techniques/tools implemented [23], the proposed suspension being a purely mechanical one. All the single-seater classes are designed to a specific rule book. In the case of the wheel suspension systems, these regulations are very strict. After a thorough analysis of several rule books from single-seater competitions, we observed that there are specific rules that were common in all rule books, and that all the suspension systems needed to achieve [24–29]. The existing literature reveals several usable/used suspension systems for single- seater race cars, focusing on identifying design requirements and solutions for improving the effective handling and cornering capability [30–36]. However, most of them are based on classical (well-known) multi-link mechanisms with guide bars interposed/arranged between the wheel carrier and chassis (i.e., four-bar mechanism). Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 25

proving the effective handling and cornering capability [30–36]. However, most of them are based on classical (well-known) multi-link mechanisms with guide bars inter- posed/arranged between the wheel carrier and chassis (i.e., four-bar mechanism). In the following sections, several representative solutions are presented and criti- cally analyzed, with the purpose to identify and establish the novelty elements and the advantages proposed to be achieved by the innovative solution developed through this Appl. Sci. 2021, 11, 4167 research. We selected some patented solutions, which present certain3 ofnovelty 25 elements, but for which there is no theoretical or practical substantiation [32,35,36]. The suspension system of a front axle for electric propulsion car presented in [35] is composedIn the following of two sections,“A”-shaped several arms representative fixed on solutions the chassis are presented by ball and joints. critically One of the biggest analyzed,disadvantages with the of purpose this kind to identify of suspension and establish is thethat novelty it cannot elements cope and with the advan-camber adjustments, tages proposed to be achieved by the innovative solution developed through this research. which can affect the ride and handling in a negative way. Another negative point is the We selected some patented solutions, which present certain novelty elements, but for which therefact isthat no theoreticalthere is a or significant practical substantiation distance between [32,35,36]. the chassis and the arms’ ball joints that imposesThe suspension a tall chassis system with of a fronta high axle center for electric of gravity propulsion and car that presented has ain negative [35] is effect on the composedvehicle handling. of two “A”-shaped arms fixed on the chassis by ball joints. One of the biggest disadvantagesThe multi-link of this kind suspension of suspension system is that itpresente cannot coped in with [32] camber was adjustments,composed of three trans- which can affect the ride and handling in a negative way. Another negative point is the factversal that linkages there is a significantbetween distancethe chassis between and thethe chassis wheel and carrier the arms’ that ball were joints designed that to dissipate imposesthe lateral a tall forces chassis from with athe high wheel center ofto gravitythe cass andis thatand has two a negative longitudinal effect on linkages the that were vehicledesigned handling. to support the wheel carrier in a traction and braking event. The main disad- vantageThe multi-link of this suspensionkind of suspension system presented systems in [32 ]is was the composed need of of an three anti-roll transversal bar that links the linkages between the chassis and the wheel carrier that were designed to dissipate the lateraltwo wheels forces from (left the to wheel right) to theto cassiseach andother. two This longitudinal means linkages that the that wheels were designed influence each other towith support negative the wheel effects carrier during in a traction cornering. and braking Another event. important The main disadvantagedisadvantage of is the fact that thisthis kind suspension of suspension system systems transmits is the need the offorces an anti-roll with barno thatreduction links the from two wheelsthe wheel straight to (leftthe tospring right) toand each damper other. This assembly. means that the wheels influence each other with negative effectsThe during suspension cornering. Anotherdesigned important for a race disadvantage car (rear is thewheel fact thatdrive this and suspension front wheel steering) system transmits the forces with no reduction from the wheel straight to the spring and damperpresented assembly. in [36] is designed from tubular components. The front axle suspension is de- signedThe from suspension a quadrilateral designed for mechanism a race car (rear with wheel un driveequal and arms front that wheel is linked steering) to the chassis by presentedball joints. in [The36] is spring designed and from damper tubular assembly components. is mounted The front axlein a suspension quasi-horizontal is position. designedThe main from disadvantage a quadrilateral in mechanism this suspension with unequal syst armsem thatis the is linkedfact that to the it chassischanges its dynamic by ball joints. The spring and damper assembly is mounted in a quasi-horizontal position. Thebehavior main disadvantage by changing in thisthe suspension camber ( systemγ) and is caster the fact ( thatβ) angles, it changes presented its dynamic in Figure 1, and behaviorwheel track by changing during the cambercornering. (γ) and This caster resulted (β) angles, in presented non-linear in Figure behavior1, and wheel and required com- trackpensation during cornering.of the steering This resulted wheel in angles non-linear during behavior cornering. and required At the compensation same time, the constant ofchange the steering of caster wheel anglesangle duringduring cornering. cornering At theresu samelted time, in thea change constant of change zero of return (straight casterahead) angle of the during steering cornering forces, resulted and in this a change induce ofd zero variation return (straight in the ahead)required of the steering force. steering forces, and this induced variation in the required steering force.

FigureFigure 1. 1.Camber Camber and and caster caster angles. angles.

Based on the above analysis of the existing solutions, the suspension system found to Based on the above analysis of the existing solutions, the suspension system found be the most commonly used on a Formula Student single seater (for both the front and the rearto be wheels/axle) the most wascommonly based on used four-bar on guiding a Formula mechanism Student (also single called SLA—short-longseater (for both the front and arm).the Comparedrear wheels/axle) with the regular was suspension based systemson four-bar used by passenger guiding cars mechanism in which the (also called springSLA—short-long and damper assembly arm). Compared is mounted vertical with the (usually regular between suspension the upper systems suspension used by passen- armger andcars the in chassis, which as the shown spring in Figure and 2dampera), in the assembly case of the single-seateris mounted race vertical cars, the (usually between spring and damper assembly was mounted horizontally (Figure2b), and this would help the upper suspension arm and the chassis, as shown in Figure 2a), in the case of the sin- Appl. Sci. 2021, 11, x FOR PEER REVIEW 4 of 25 Appl. Sci. 2021, 11, 4167 4 of 25

gle-seater race cars, the spring and damper assembly was mounted horizontally (Figure 2b),with and forcethis would distribution help with and force keep distribution the chassis as and low keep as possible. the chassis It isas important low as possible. to mention It is importantthat both suspensionto mention systemsthat both presented suspension in Figure systems2 had presented a single degreein Figure of mobility2 had a single (i.e., the degreewheel of verticalmobility travel—Y (i.e., the Kwheel). vertical travel—YK).

YK YK

(a) (b)

FigureFigure 2. Sketch 2. Sketch of the of theclassic classic four four-bar-bar suspension suspension system system (a) and (a) and the theconfiguration configuration used used on sin- on single- gle-seaterseater carscars ((bb).).

TheThe suspension suspension system system shown shown in Figure in Figure 2b 2imposedb imposed the the use use of ofa pushrod a pushrod and and rocker rocker assemblyassembly that that transferred transferred the the vertical vertical movement movement from from the the wheel toto aa horizontalhorizontal movement move- mentto theto the spring spring and and damper damper assembly. assembly. The The rocker rocker assembly assembly was was linked linked to to the the pushrod pushrod and andthe the damper damper assembly. assembly. This This allowed allowed for thefor reduction the reduction of the of force the that force the that wheel the transferred wheel transferredto the spring to the and spring damper and assemblydamper assembly by changing by changing the length the of length the rocker of the arms rocker and arms angles, andcreating angles, thecreating opportunity the opportunity to use a smaller to use a spring smaller and spring damper and assembly.damper assembly. SuchSuch a solution a solution could could confer confer the following the following benefits: benefits: It could It could decrease decrease the variation the variation of theof camber the camber angle, angle, it would it would needneed a small a small space space to be to fitted, be fitted, and and it could it could accommodate accommodate tuningtuning for forboth both road road wheel wheel toe toe angle angle and and camber camber angle. angle. On Onthe theother other hand, hand, as a as main a main disadvantage,disadvantage, it is it well is well known known that that the thefour-bar four-bar suspension suspension mechanism mechanism has has contradictory contradictory behaviorbehavior regarding regarding the thewheel wheel track track and and camber camber angle angle variations variations (decrease (decrease of one of one of these of these variationsvariations generates generates the the increase increase of the of the other other one). one). StartingStarting from from the the classic classic four-bar four-bar suspension mechanism,mechanism, a a possible possible solution solution to avoidto avoidthe the above-mentioned above-mentioned issue issue is to is transform to transform the passivethe passive (pure (pure mechanical) mechanical) suspension suspen- into sionan into active an (controlled)active (controlled) one, and one, this and could this be could achieved be achieved with the with solution the solution shown in shownFigure 3. in FigureIn this 3. regard, In this a regard, linear actuator a linear typeactuator assembly type assembly was disposed was disposed between thebetween chassis the and the suspension. In this way, a three-contour mechanism with two degrees of freedom Appl. Sci. 2021, 11, x FOR PEER REVIEWchassis and the suspension. In this way, a three-contour mechanism with two degrees of 5 of 25 freedom(the wheel (the wheel travel andtravel the and actuator the actuator movement—expansion movement—expansion or compression, or compression, by case) by was case)obtained, was obtained, which helpedwhich helped in minimizing in minimizing the wheel the trackwheel variation. track variation. Besides the wheel track variation reduction, the active system had the advantage that it could change its ride height according to the weight of the vehicle and the dynamic loads that appeared during braking, accelerating, and cornering. The active suspension system behavior was mainly based on the sensors that had to read the road ahead and prepare the damping and spring stiffness for the upcoming irregularities in the rolling surface so that the driver (and the passengers as well) would feel as little as possible in the cockpit [37,38]. At the same time, this kind of suspension offers the possibility of having an equal weight distribution on the ground.

FigureFigure 3. 3. SchemeScheme of of an an active active suspension suspension systemsystem derivatederivate from from the the four-bar four-bar mechanism. mechanism.

The main disadvantage of the active suspension system is the size and the complex (and expensive as well) electronical system. The risk of an error in the electronical system or misbehavior of the system, as well as the low-cost efficiency, mean the active suspen- sion system is not one that can be fitted in a Formula Student single seater.

2. Conceptual Design of the Single-Seater Suspension System The development of the conceptual solution for the suspension system of the sin- gle-seater race car was based on the functional particularities (including advantages and disadvantages) of the classic suspension based on the four-bar mechanism, both in the passive (Figure 2) and active version (Figure 3). The principle behind this was to define a system with a single degree of mobility (DOM = 1)—as the passive system, which could achieve adjustments specific to the mechanism with two degrees of mobility (DOM = 2)—as the active system (considering the systems in a planar configuration, as are the representations in Figures 2 and 3). The solution of the above-mentioned system was based on a five-bar mechanism that had two degrees of mobility. The initially developed suspension system, which in- tegrated the specific pushrod-rocker assembly that helped mount the spring and damper assembly in a horizontal plane (as shown in Figure 2b) is presented in Figure 4. This would make it possible to decouple the contradictory movements (wheel track and camber angle variations) from influencing each other. The control of the second degree of mobility (with the purpose to cancel, by case, the wheel track variation or the camber angle variation) could be achieved by an actuating element (resulting in an active sus- pension) or by a mechanical element (passive suspension).

Figure 4. Suspension system based on the five-bar mechanism (DOM = 2).

Thus, the active suspension could be constituted, for example, by using a linear ac- tuator, disposed between the upper connecting rod (upper control arm as per Figure 2) of the five-bar mechanism and the chassis, with the corresponding scheme shown in Figure 5. Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 25

Appl. Sci. 2021, 11, 4167 5 of 25

Besides the wheel track variation reduction, the active system had the advantage that it could change its ride height according to the weight of the vehicle and the dynamic loads thatFigure appeared 3. Scheme during of braking,an active accelerating, suspension andsystem cornering. derivate The from active the suspensionfour-bar mechanism. system behavior was mainly based on the sensors that had to read the road ahead and prepare the dampingThe and main spring disadvantage stiffness for the of upcomingthe active irregularities suspension in system the rolling is surfacethe size so and that the complex the(and driver expensive (and the as passengers well) electronical as well) would system. feel as The little risk as possible of an error in the in cockpit the electronical [37,38]. system Ator the misbehavior same time, this of the kind system, of suspension as well offers as the possibilitylow-cost ofefficiency, having an mean equal weightthe active suspen- distribution on the ground. sion system is not one that can be fitted in a Formula Student single seater. The main disadvantage of the active suspension system is the size and the complex (and expensive as well) electronical system. The risk of an error in the electronical system or2. misbehaviorConceptual of Design the system, of asthe well Single-Seater as the low-cost Suspension efficiency, mean System the active suspension systemThe is not development one that can be of fitted the inconceptual a Formula Studentsolution single for seater.the suspension system of the sin- 2.gle-seater Conceptual race Design car was of the based Single-Seater on the functional Suspension particularities System (including advantages and disadvantages) of the classic suspension based on the four-bar mechanism, both in the The development of the conceptual solution for the suspension system of the single- seaterpassive race (Figure car was based2) and on active the functional version particularities (Figure 3). The (including principle advantages behind and this disad- was to define a vantages)system with of the a classic single suspension degree of based mobility on the (D four-barOM = mechanism,1)—as the bothpassive in the system, passive which could (Figureachieve2) andadjustments active version specific (Figure to3). the The mechan principleism behind with this two was degrees to define aof system mobility (DOM = with2)—as a single the degreeactive ofsystem mobility (considering (DOM = 1)—as the the syst passiveems system,in a planar which configuration, could achieve as are the adjustmentsrepresentations specific in to Figures the mechanism 2 and 3). with two degrees of mobility (DOM = 2)—as the active system (considering the systems in a planar configuration, as are the representations in FiguresThe2 andsolution3). of the above-mentioned system was based on a five-bar mechanism thatThe had solution two degrees of the above-mentioned of mobility. The system initially was developed based on a five-barsuspension mechanism system, which in- thattegrated had two the degrees specific of pushrod-rocker mobility. The initially assembly developed that helped suspension mount system, the spring which and damper integratedassembly the in specific a horizontal pushrod-rocker plane assembly(as shown that in helped Figure mount 2b) the is springpresented and damper in Figure 4. This assemblywould inmake a horizontal it possible plane (asto showndecouple in Figure the2 b)contradictory is presented in Figuremovements4. This would(wheel track and makecamber it possible angle tovariations) decouple the from contradictory influencing movements each other. (wheel The track control and camber of the anglesecond degree of variations) from influencing each other. The control of the second degree of mobility (with themobility purpose (with to cancel, the bypurpose case, the to wheel cancel, track by variation case, the or wheel the camber track angle variation variation) or the camber couldangle be variation) achieved bycould an actuating be achieved element by (resulting an actuating in an activeelement suspension) (resulting or in by an a active sus- mechanicalpension) or element by a mechanical (passive suspension). element (passive suspension).

FigureFigure 4. 4.Suspension Suspension system system based based on the on five-bar the five-bar mechanism mechanism (DOM = 2). (DOM = 2).

Thus, the active suspension could be constituted, for example, by using a linear actuator,Thus, disposed the active between suspension the upper connecting could be rod consti (uppertuted, control for arm example, as per Figure by using2) of a linear ac- thetuator, five-bar disposed mechanism between and the the chassis, upper with connecting the corresponding rod (upper scheme control shown arm in Figure as per5. Figure 2) of the five-bar mechanism and the chassis, with the corresponding scheme shown in Figure 5. Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 25

Appl.Appl. Sci. 2021 Sci. 2021, 11,, 11x ,FOR 4167 PEER REVIEW 6 of 25 6 of 25

Figure 5. Active suspension system based on the five-bar mechanism (DOM = 2).

With the view to obtain a pure mechanical (passive) suspension able to provide the functional parameters (behavior) of the active suspension, an original design algorithm was developed, which was based on the following steps: 1. In the five-bar suspension mechanism (Figure 4), the variation of the camber angle FigureFigurewas 5.5.Active Active canceled suspension suspension by means system system basedof a based ki onnematic the on five-bar the constraint five-bar mechanism mechanism of (DOM form = Δγ 2). (DOM = γ − = γ 2).0 = 0, where γ is the current value of the camber angle, and γ0 is the initial value; 2. WithWithPerforming the the view view the to obtainto kinematic obtain a pure a pureanalysis mechanical mechanical of (passive)the suspension (passive) suspension suspensionmechanism able to provide ableby imposing to the provide the the functional parameters (behavior) of the active suspension, an original design algorithm vertical travel of the wheel with the help of a kinematic restriction; functionalwas developed, parameters which was (behavior) based on the of following the active steps: suspension, an original design algorithm was3. developed,Obtaining the which trajectory was based of a specific on the point following on the steps:upper rod of the five-bar mechanism; 1. In the five-bar suspension mechanism (Figure4), the variation of the camber angle the trajectories of several points on the upper rod could be monitored, finally choosing 1. Inwas the canceled five-bar by meanssuspension of a kinematic mechanism constraint (Fig ofure form 4),∆ theγ = γvariation− γ = 0, whereof theγ camberis angle the point that would generate a more convenient trajectory (it could0 be a semicircle for wasthe current canceled value by of means the camber of a angle, kinematic and γ0 constraintis the initial of value; form Δγ = γ − γ0 = 0, where γ is 2. Performingthe planar variant the kinematic of mechanism, analysis ofor thea hemisphere suspension for mechanism the spatial by variant); imposing the the current value of the camber angle, and γ0 is the initial value; 4. verticalReplacing travel the of kinematic the wheel withrestriction the help from of a kinematicstep “1” with restriction; one of the following variants: 2. Performing the kinematic analysis of the suspension mechanism by imposing the 3. Obtaining4.1. Geometrical the trajectory constraint of a specific roller-guide point on the type upper (Figure rod of the6), five-barwhere the mechanism; roller would be verticalthe trajectoriesmounted travel of ofon several the the wheel pointsupper onwith rod, the the upperwhile help rod the of could gua kinematicide be monitored,of which restriction; finallywould choosing attached to the 3. Obtainingthe pointchassis that the would trajectory generate have of the a a specificshape more convenient defined point on by trajectory the the upper trajectory (it rod could obtainedof be the a semicirclefive-bar in step mechanism; “3”; thefor4.2. thetrajectories Second planar variantrocker of several of mounted mechanism, points between on or the a hemisphere u thepper upper rod for could therod spatial andbe monitored, variant);the chassis finally (Figure choosing 7), 4. Replacing the kinematic restriction from step “1” with one of the following variants: the pointlinked that to would the upper generate arm ina morethe point convenient (M) that trajectory describes (it the could trajectory be a semicircle from step for 4.1. Geometrical constraint roller-guide type (Figure6), where the roller would be the planar“3” and variant to the of chassis mechanism, at the orgeometrical a hemisphere center for (M the0) spatialof the trajectory.variant); 4. Replacingmounted the kinematic on the upper restriction rod, while from the guide step of“1” which with would one of attached the following to the variants: Both purelychassis mechanical would have solu the shapetions definedobtained by in the step trajectory “4” led obtained to mono-mobile in step “3”; (DOM = 1) 4.1. Geometrical constraint roller-guide type (Figure 6), where the roller would be mechanisms4.2. Second (in the rocker planar mounted versions), between and could the upperbe used rod on and a single-seater the chassis (Figureracing7 car), suspen- sion as theymountedlinked are re toduced theon upperthe in size,upper arm lighter, in therod, point andwhile (M)are the thatnot as describesgu complexide of the which as trajectory an activewould from suspension attached step sys-to the tem. chassis“3” and would to the chassis have the at the shape geometrical defined center by the (M 0trajectory) of the trajectory. obtained in step “3”; 4.2. Second rocker mounted between the upper rod and the chassis (Figure 7), linked to the upper arm in the point (M) that describes the trajectory from step “3” and to the chassis at the geometrical center (M0) of the trajectory. Both purely mechanical solutions obtained in step “4” led to mono-mobile (DOM = 1) mechanisms (in the planar versions), and could be used on a single-seater racing car suspen- sion as they are reduced in size, lighter, and are not as complex as an active suspension sys- tem.

FigureFigure 6. 6.Suspension Suspension system system with with guiding guiding roll (DOM roll (DOM = 1). = 1).

Figure 6. Suspension system with guiding roll (DOM = 1). Appl. Sci. 2021, 11, x FOR PEER REVIEW 7 of 25 Appl. Sci. 2021, 11, 4167 7 of 25

Figure 7. 7.Suspension Suspension system system with guidingwith guiding rocker (DOM rocker = 1). (DOM = 1). Both purely mechanical solutions obtained in step “4” led to mono-mobile (DOM = 1) mechanismsIt should (in the be planar noted versions), that the and mechanism could be used onobtained a single-seater by the racing second car suspen- variant in step “4” sion(Figure as they 7) are would reduced be in size,viable lighter, if andthe aretrajectory not as complex defined as an activeby thesuspension interest system. point (M) from the upperIt should arm approximated be noted that the a mechanismsemicircle obtained or a hemisphere. by the second In variantthese terms, in step “4”the study contin- (Figure7) would be viable if the trajectory defined by the interest point (M) from the upper armued approximated with the determination a semicircle or aof hemisphere. the global In coordinates these terms, the of studythe center continued point with (M0) defined on thethe determination chassis, a subject of the global approach coordinatesed in of the“Section center point 4” of (M 0the) defined work on by the an chassis, original synthesis amethod. subject approached in “Section4” of the work by an original synthesis method. AsAs mentioned, mentioned, the above the above approach approach was specific was to planarspecific (2D) to variants planar of (2D) suspension variants of suspen- mechanisms, with a view to simplify the formulation and related schemes. Obviously, in sion mechanisms, with a view to simplify the formulation and related schemes. Obvi- practice, as well as in the study developed in the next sections of this paper, the spatial (3D) variantously, ofin the practice, guiding as mechanism well as isin addressed/implemented. the study developed in the next sections of this paper, the spatialCompared (3D) variant with the of planar the guiding variants, themechanism corresponding is addressed/implemented. spatial configurations of sus- pensionCompared systems had with one supplementary the planar degreevariants, of mobility the corresponding that was the caster spatial angle varia- configurations of tion.suspension In this case, systems in step had “1” fromone thesupplementary above presented degr designee algorithm,of mobility two that kinematic was the caster angle restrictions had to be used: The one already used for the camber angle (∆γ = γ − γ = 0), variation. In this case, in step “1” from the above presented design algorithm,0 two kine- and a similar one for the caster angle (∆β = β − β0 = 0), which at step “4” was replaced throughmatic restrictions the solutions, had as shown to be inused: Figures The6 and one7. already used for the camber angle (Δγ = γ − γ0 = 0), andUnder a similar these terms, one for for a betterthe caster understanding angle ( ofΔβ the = designβ − β0 algorithm= 0), which of the at proposed step “4” was replaced suspensionthrough the mechanism, solutions, Figure as shown8 shows in the Figures corresponding 6 and 7. optimization flowchart. In Figure8, ∆E is the wheel track variation, ∆L is the wheelbase variation, and ∆δ is the bump Under these terms, for a better understanding of the design algorithm of the pro- steer variation. posed suspension mechanism, Figure 8 shows the corresponding optimization flowchart. In Figure 8, ΔE is the wheel track variation, ΔL is the wheelbase variation, and Δδ is the bump steer variation.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 8 of 25 Appl. Sci. 2021, 11, 4167 8 of 25

FigureFigure 8. Optimization flowchart.flowchart.

3.3. The The Kinematic OptimizationOptimization of of the the Suspension Suspension Mechanism Mechanism The kinematic optimization was carried out by considering the bi-contour mechanism, The kinematic optimization was carried out by considering the bi-contour mecha- as shown in Figure9, which was obtained from the spatial variant of a five-bar suspension nism, as shown in Figure 9, which was obtained from the spatial variant of a five-bar mechanism (whose planar scheme is that shown in Figure4) to which the toe link (6) was suspensionadded for both mechanism the front and(whose rear planar axles. Thescheme resulting is that suspension shown in system Figure had4) to three which active the toe linkdegrees (6) was of mobility, added for and both another the front two passivesand rear (theaxles. rotation The resulting of links “3”suspension and “6” system around had threetheir ownactive axes). degrees The passiveof mobility, rotations and couldanother be canceledtwo passives by adding (the rotation additional of kinematiclinks “3” and “6”constraints: around theirϕ3 = 0,ownϕ6 axes).= 0. The passive rotations could be canceled by adding additional kinematicWhen constraints: the car moves, φ3 = the0, φ vertical6 = 0. travel (up–down) of the wheel determines the modification of the suspension mechanism configuration, which generates undesirable movements, namely displacements of the wheel center along the transversal and lon- gitudinal axes (corresponding to the wheel track and wheelbase variations), as well as modifications of the wheel axis orientation (corresponding to the toe angle, camber angle, and bump steer angle variations). Appl.Appl. Sci. Sci.20212021, 11, 11, x, FOR 4167 PEER REVIEW 9 of 25 9 of 25

FigureFigure 9.9. TheThe bi-contourbi-contour spatial spatial mechanism mechanism used us ined the in kinematic the kinematic optimization. optimization.

CancelingWhen the or car at leastmoves, minimizing the vertical these travel variations (up–down) are objectives of the for wh kinematiceel determines opti- the mization, given the negative effects they can have on the dynamic behavior of the car, such modification of the suspension mechanism configuration, which generates undesirable as depreciation of stability, increasing rolling resistance, additional stress, and wear in the movements,system, which namely have been displacements stated in many of works, the wh sucheel ascenter [7,13, 17along,18,20 the,30, 39transversal,40]. and longi- tudinalTherefore, axes (corresponding the following statements to the wheel were considered track and in thewheelbase kinematic variations), optimization as of well as modificationsthe suspension of mechanism the wheel shown axis orientation in Figure9: (corresponding to the toe angle, camber angle, and• bumpThe vertical steer angle movement variations). of the wheel is an independent kinematic parameter. It is controlledCanceling byor aat Y Kleast= f(t) minimi kinematiczing restriction, these variations simulating are the ob wheeljectives passing for kinematic over a opti- mization,50-mm given (±25 mm)the negative obstacle, whicheffects is they transposed can have in the on form the ofdynamic a sinusoidal behavior function, of the car, such YasK =depreciation 25·sin(t), t—time of stability, [s]; increasing rolling resistance, additional stress, and wear in• theThe system, variations which of thehave camber been (stated∆γ) and in caster many (∆ works,β) angles such were as canceled [7,13,17,18,20,30,39,40]. by applying γ β kinematicTherefore, restrictions: the following∆ = statements 0, ∆ = 0 (further, were forconsidered the purpose in ofthe physical kinematic implemen- optimization tation, these fictive kinematic restrictions would be replaced/substituted by geometric of the suspension mechanism shown in Figure 9: constraints through the guiding rocker MM0, as schematically shown in Figure7—see • “SectionThe vertical4” of themovement work); of the wheel is an independent kinematic parameter. It is • Thecontrolled goal of theby optimizationa YK = f(t) kinematic was to minimize restriction, the variations simulating of the the wheel wheel track passing (∆E), over a wheelbase50-mm (±25 (∆ L),mm) and obstacle, bump steer which angle is ( ∆transposedδ); in the form of a sinusoidal function, • The global coordinates (X, Y, Z) of the joints that link the suspension mechanism bars YK = 25·sin(t), t—time [s]; • toThe chassis variations (A, B, C,of D—seethe camber Figure (9Δγ) are) and defined caster as ( designΔβ) angles variables were for canceled optimization, by applying assuming that the joint locations to wheel carrier are established/given by constructive kinematic restrictions: Δγ = 0, Δβ = 0 (further, for the purpose of physical imple- aspects (i.e., the wheel carrier geometry). mentation, these fictive kinematic restrictions would be replaced/substituted by The kinematic optimization was treated separately on the front and rear wheel suspen- sion mechanisms,geometric constraints using quarter-car through models. the guiding The optimization rocker MM of0 the, as suspension schematically system shown in was realizedFigure 7—see with the “Section help of 4” a multi-objective of the work); parametric design process, by using the • MSC.ADAMS The goal suiteof the of optimization software, including was to ADAMS/Insight, minimize the variations a powerful optimizationof the wheel soft- track (ΔE), warewheelbase [41], and ADAMS/View, (ΔL), and bump which steer was angle used ( toΔδ develop); the multi-body system (MBS) •model The of theglobal suspension coordinates system (X, [42 ].Y, Z) of the joints that link the suspension mechanism Thebars optimization to chassis (A, study B, wasC, D—see based on Figure design 9) of experimentsare defined (DOE)as design technique, variables which for opti- providedmization, methods assuming and tools that for the efficiently joint locations planning to and wheel running carrier a series areof established/given experiments by on theconstructive suspension systemaspects design (i.e., the and wheel analyzing carrier the resultsgeometry). [41]. In what follows, a general presentation of the optimization method was made, and subsequently it was implemented for optimizingThe kinematic the guiding optimization mechanisms was of treated the front separately and rear wheels on the of thefront single-seater and rear wheel suspensionrace car. The mechanisms, parametric design using through quarter-car the DOE mo techniquedels. The was optimization performed by of following the suspension systemthe steps was indicated realized in referencewith the [ 41help]. of a multi-objective parametric design process, by us- ing the MSC.ADAMS suite of software, including ADAMS/Insight, a powerful optimi- zation software [41], and ADAMS/View, which was used to develop the multi-body system (MBS) model of the suspension system [42]. The optimization study was based on design of experiments (DOE) technique, which provided methods and tools for efficiently planning and running a series of ex- periments on the suspension system design and analyzing the results [41]. In what fol- lows, a general presentation of the optimization method was made, and subsequently it was implemented for optimizing the guiding mechanisms of the front and rear wheels of Appl. Sci. 2021, 11, 4167 10 of 25

In ADAMS/View, a design objective was defined by creating its corresponding mea- sure (wheel track, wheelbase, and bump steer variations). For each of the three design objectives, the monitored value was the root mean square (RMS) during simulation: s v2 + v2 + v2 + ··· v2 RMS = 1 2 3 n (1) n

where v1, v2, ..., vn represent the successive values of the parameter of interest, depending on the number of steps (n) that the simulation takes to reach the end. Before running the optimization algorithm, the file that contained the MBS model designed in ADAMS/View was transferred in ADAMS/Insight. The design variables (factors) and design objectives (responses) were set up after opening the saved model in ADAMS/Insight. To run the optimization process, the responses and the factors were pro- moted from the candidates in the inclusion list (“Promote to Inclusions” in ADAMS/Insight). Each design objective was analyzed based on a regression function that approximated the model in such way that the error between the predicted and the measured values were minimal. For example, if we only considered two design variables, the regression functions would have the following forms: I Linear model:

R = a1 ∗ F1 + a3 ∗ F3 + e (2) I Linear model with factors interaction:

R = a1 + a2 ∗ F1 + a3 ∗ F2 + a4 ∗ F1 ∗ F2 + e (3) I Quadratic model:

2 2 R = a1 + a2 ∗ F1 + a3 ∗ F2 + a4 ∗ F1 ∗ F2 + a5 ∗ F1 + a6 ∗ F2 + e (4) I Cubic model:

2 2 R = a1 + a2 ∗ F1 + a3 ∗ F2 + a4 ∗ F1 ∗ F2 + a5 ∗ F1 + a6 ∗ F2 + ··· 2 2 3 3 (5) ... + a7 ∗ F1 ∗ F2 + a8 ∗ F1 ∗ F2 + a9 ∗ F1 + a10 ∗ F2 + e

where F1 and F2 were the design variables (factors), ai—the terms coefficients, R—the system response, and e—the error (that would be minimized). Depending on the selected DOE investigation strategy, we obtained the design space and the workspace of the experiment. The design space was a matrix that presented, in a normalized form, the combinations between the factors. Thus, the “−1” value corre- sponded to the minimum value of the factor, and “1” to the maximum value [41], according to the previously defined variation ranges. The effective values of the factors were seen in the workspace, along the corresponding responses. The program ran a simulation in ADAMS/View for each trial specified in this matrix to determine the responses values. After the simulations were run, the results appeared automatically in the workspace, by filling in the response columns with numerical values. Based on these results, ADAMS/Insight automatically generated the corresponding regres- sion function, according to the selected type of regression model. The response analysis was done by using the evaluation methods presented in reference [43]. To begin, the kinematic optimization algorithm was implemented for the front wheels’ suspension mechanism (the selected tuning strategy would then be used for the rear wheels’ suspension optimization as well). The MBS model developed in ADAMS/View for the front axle suspension system is shown in Figure 10. The lower control arm (1) was linked to the chassis in points A and B through spherical joints, the rocker (5) was linked to the chassis in point C by a revolute joint, and point D was the location of the spherical joint Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 25

Appl. Sci. 2021, 11, 4167 11 of 25 ADAMS/View for the front axle suspension system is shown in Figure 10. The lower control arm (1) was linked to the chassis in points A and B through spherical joints, the rocker (5) was linked to the chassis in point C by a revolute joint, and point D was the that linked the toe link/steering rod (6) to the steering rack. In the kinematic model of the location of the spherical joint that linked the toe link/steering rod (6) to the steering rack. suspension system, the steering rack was fixed, so it was rigidly connected to chassis (in In the kinematic model of the suspension system, the steering rack was fixed, so it was other words, the steering wheel is not actuated). The global coordinates of the four points rigidly connected to chassis (in other words, the steering wheel is not actuated). The were defined as design variables for the kinematic optimization of the front suspension global coordinates of the four points were defined as design variables for the kinematic system, as presented in Table1. optimization of the front suspension system, as presented in Table 1. Table 1. Front axle suspension system design variables. Table 1. Front axle suspension system design variables. X (mm) Y (mm) Z (mm) X (mm) Y (mm) Z (mm) A DV_1DV_1 DV_2DV_2 DV_3DV_3 B DV_4DV_4 DV_5DV_5 DV_6DV_6 C DV_7DV_7 DV_8DV_8 DV_9DV_9 D DV_10 DV_11 DV_12 D DV_10 DV_11 DV_12

Figure 10. Quarter-car model of the front wheel suspension system. Figure 10. Quarter-car model of the front wheel suspension system. The initial values of the design variables are (in [mm]): DV_1 = −270.37, DV_2 = 38.5, DV_3The = 108.37, initial DV_4 values = − of193.5, the design DV_5 = variables 38.5, DV_6 are = −(in279.43, [mm]): DV_7 DV_1 = − = 297.64,−270.37,DV_8 DV_2 = 214.89,= 38.5, DV_9DV_3 == 38.21,108.37, DV_10 DV_4 == −−193.5,288.5, DV_5 DV_11 = =38.5, 101.5, DV_6 DV_12 = −279.43, = 135.01. DV_7 For = each −297.64, variable, DV_8 the = variation214.89, DV_9 domain = 38.21, was DV_10±20 mm = − relative288.5, DV_11 to the initial= 101.5, value. DV_12 = 135.01. For each variable, the variationWith the domain purpose was of identifying±20 mm relative the DOE to the investigation initial value. strategy that provided the optimumWith resultsthe purpose from theof identifying point of view the DOE of the investigation regression functions strategy that accuracy, provided several the investigationoptimum results strategies from the were point tested, of view namely: of the DOE regression screening functions (level 2)—linear—Plackettaccuracy, several in- Burman;vestigation DOE strategies screening were (level tested, 2)—linear—fractional namely: DOE screening factorial; (level DOE 2)—linear—Plackett response surface— Linear—LatinBurman; DOE hypercube; screening DOE (level response 2)—linear—fr surface—interactions—D-optimal;actional factorial; DOE DOEresponse screening sur- (levelface—Linear—Latin 2)—interactions—D-optimal hypercube; DOE [41 ].respon The resultsse surface—interactions—D-optimal; thus obtained highlighted that DOE the firstscreening 4 techniques (level 2)—interacti mentionedons—D-optimal above could be [41]. used The under results the thus condition obtained of refininghighlighted the regressionthat the first models, 4 techniques but the mentioned DOE screening above co (leveluld be 2)—interactions—D-optimal used under the condition of strategyrefining couldthe regression be used withoutmodels, thebut needthe DOE for regressionscreening (level refining. 2)—interactions—D-optimal Therefore, the optimization strategy study ofcould the suspensionbe used without mechanism the need was for based regression on this refining. strategy. Therefore, the optimization study of theThe suspension regression mechanism model generated was based by theon this selected strategy. strategy allowed us to consider the interactionThe regression effects between model factors,generated which by the were sele capturedcted strategy through allowed terms us in to the consider model that the consistedinteraction of effects products between of factors, factors, as which presented were captured in Equation through (3). Theterms D-Optimal in the model design that createdconsisted a modelof products that minimized of factors, the as uncertaintypresented in of Equation the coefficients, (3). The this D-Optimal being characterized design cre- byated its a flexibility,model that allowingminimized the the total uncertainty number of the runs coefficients, in an experiment this being to characterized be specified, supplementingby its flexibility, with allowing runs fromthe total other number experiments, of runs andin an indicating experiment different to be levelsspecified, for eachsupplementing factor. with runs from other experiments, and indicating different levels for each factor.Based on the fact that all the evaluation parameters fell within acceptable limits, the regression functions generated by this model were valid (accurate), without any changes. As mentioned above, the kinematic optimization purpose was to minimize the three responses, corresponding to the wheel track, wheelbase, and bump steer variations. Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 of 25 Appl. Appl. Sci. Sci. 2021 2021, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 1212 of of 25 25

Based on the fact that all the evaluation parameters fell within acceptable limits, the BasedBased onon thethe factfact thatthat allall thethe evaluationevaluation paparametersrameters fellfell withinwithin acceptableacceptable limits,limits, thethe regression functions generated by this model were valid (accurate), without any changes. Appl. Sci. 2021, 11, 4167 regressionregression functions functions generated generated by by this this model model were were valid valid (accurate), (accurate), without without any any12 changes. changes. of 25 As mentioned above, the kinematic optimization purpose was to minimize the three re- AsAs mentionedmentioned above,above, thethe kinematickinematic optimizationoptimization purposepurpose waswas toto minimizeminimize thethe threethree re-re- sponses, corresponding to the wheel track, wheelbase, and bump steer variations. sponses,sponses, corresponding corresponding to to the the wheel wheel track, track, wheelbase, wheelbase, and and bump bump steer steer variations. variations. The algorithm used in the kinematic optimization of the suspension system was TheThe algorithmalgorithm usedused inin thethe kinematickinematic optioptimizationmization ofof thethe suspensionsuspension systemsystem waswas OptDes-generalizedThe algorithm used reduced in the gradient kinematic (GRG optimization), provided of the with suspension ADAMS/Insight, system was which OptDes-generalizedOptDes-generalized reducedreduced gradientgradient (GRG(GRG),), providedprovided withwith ADAMS/Insight,ADAMS/Insight, whichwhich OptDes-generalizedrequired that factors reduced have range gradient limits, (GRG), since provided it worked with in scaled ADAMS/Insight, space [42]. whichThis algo- requiredrequired thatthat factorsfactors havehave rangerange limits,limits, sincesince itit workedworked inin scaledscaled spacespace [42].[42]. ThisThis algo-algo- requiredrithm used that the factors MinTo have function, range limits, which since constraine it workedd in the scaled response space [(goal)42]. This to algorithmbe as close to rithmrithm usedused thethe MinToMinTo function,function, whichwhich constraineconstrainedd thethe responseresponse (goal)(goal) toto bebe asas closeclose toto usedthe target the MinTo value function, as possible which (“0”). constrained the response (goal) to be as close to the target thethe target target value value as as possible possible (“0”). (“0”). valueThe as possible optimum (“0”). values of the design variables were obtained by running the tuning to TheThe optimumoptimum values valuesvalues of ofof the thethe design designdesign variables variablesvariables were werewere obtained obtainedobtained by byby running runningrunning the thethe tuning tuningtuning toto the “0” variations of the targets. These values were as follows (in [mm]): DV_1 = −230.37, thetothe the “0”“0” “0” variationsvariations variations ofof the thethe targets.targets. These TheseThese values vavalueslues were werewere as asfollowsas followsfollows (in [mm]): (in(in [mm]):[mm]):DV_1 DV_1DV_1 = −230.37, == − −230.37,230.37, DV_2 = −1.5, DV_3 = 68.37, DV_4 = −233.5, DV_5 = −1.5, DV_6 = −319.43, DV_7 = −257.64, DV_2DV_2 = == − −−1.5,1.5, DV_3DV_3 = == 68.37, 68.37,68.37, DV_4 DV_4DV_4 = − == 233.5, −−233.5,233.5, DV_5 DV_5DV_5 = − =1.5,= −−1.5,1.5, DV_6 DV_6DV_6 = − 319.43, == −−319.43,319.43,DV_7 DV_7DV_7 = − 257.64, == −−257.64,257.64, DV_8 = 254.89, DV_9 = 78.21, DV_10 = −248.5, DV_11 = 61.5, DV_12 = 175.01. DV_8DV_8 == 254.89,254.89, DV_9DV_9 DV_9 == = 78.21,78.21, 78.21, DV_10 DV_10 DV_10 = = −= − −248.5,248.5,248.5, DV_11 DV_11 DV_11 = = = 61.5, 61.5, 61.5, DV_12 DV_12 DV_12 = = 175.01.= 175.01. 175.01. With these values, through the kinematic analysis performed in ADAMS/View, the WithWith these thesethese values, values,values, through throughthrough the thethe kinematic kinematickinematic analysis analysisanalysis performed performedperformed in ADAMS/View, inin ADAMS/View,ADAMS/View, the thethe variations of the interest parameters for the optimal (after optimization) suspension mecha- variationsvariations of ofof the thethe interest interestinterest parameters parametersparameters for forfor the thethe optimal optimaloptimal (after (after(after optimization) optimization)optimization) suspension suspensionsuspension mecha- mecha-mecha- nismnism werewere obtained obtained (Figures (Figures 11 11–13).–13). To To compare, compar thee, the results results corresponding corresponding to the to initialthe initial nismnism werewere obtainedobtained (Figures(Figures 11–13).11–13). ToTo comparcompare,e, thethe resultsresults correspondingcorresponding toto thethe initialinitial mechanismmechanism (before(before optimization) are also presented, noting noting a a significant significant improvement improvement of all mechanismmechanism (before(before optimization)optimization) areare alsoalso presented,presented, notingnoting aa significantsignificant improvementimprovement ofof allall ofthe all parameters, the parameters, which which demonstrated demonstrated the viab theility/usefulness viability/usefulness of the of implemented the implemented optimiza- thethe parameters,parameters, whichwhich demonstrateddemonstrated thethe viabviability/usefulnessility/usefulness ofof thethe implementedimplemented optimiza-optimiza- optimizationtion algorithm. algorithm. tiontion algorithm. algorithm.

FigureFigure 11.11.Wheelbase Wheelbase variation variation (initial—red, (initial—red, and and optimal—blue). optimal—blue). FigureFigure 11. 11. Wheelbase Wheelbase variation variation (initial—red, (initial—red, and and optimal—blue). optimal—blue).

Figure 12. Wheel track variation (initial—red, and optimal—blue). FigureFigure 12.12. 12. WheelWheel Wheel tracktrack track variation variation variation (initial—red, (initi (initial—red,al—red, and and and optimal—blue). optimal—blue). optimal—blue).

Figure 13. Bump steer variation (initial—red, and optimal—blue). Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 25

Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 25

Appl. Sci. 2021, 11, 4167 13 of 25 Figure 13. Bump steer variation (initial—red, and optimal—blue). Figure 13. Bump steer variation (initial—red, and optimal—blue). According to the optimization flowchart shown in Figure 8, the next step after the kinematicAccordingAccording optimization toto thethe optimizationof optimization the suspension flowchartflowchart mechanism shownshow nconsisted inin FigureFigure of8 ,8, thedetermining the next next step step the after after loca- the the tionkinematickinematic of a point optimization optimization (M) on the of of the upperthe suspension suspension control mechanism ar mechanismm whose consisted trajectory consisted of determiningduringof determining the simulation the locationthe loca- (dependingoftion a point of a (M) pointon onthe the(M) imposedupper on the controlwheel upper armvertical control whose travel) ar trajectorym whose was close during trajectory to thea convenient simulationduring the circular (depending simulation or sphericalon(depending the imposed one, soon that wheelthe laterimposed vertical it would wheel travel) be verticaleasy was to close determinetravel) to a was convenient the close center to circular a(M0) convenient of or this spherical trajectory circular one, or onsospherical the that chassis later one, it (for would so notations, that be later easy itsee towould the determine schematic be easy the to planar center determine (M0)model the of in this center Figure trajectory (M0) 7), as of onpresented this the trajectory chassis in “Section(foron the notations, 4”chassis of the see(for work. thenotations, schematicIn this seeregard, the planar schematicthe modeltrajectories planar in Figure of model several7), as in points presentedFigure on 7), the inas upperpresented“Section arm4 ” in wereof“Section the monitored, work. 4” of In the finally this work. regard, choosing In this the regard, the trajectories point the trajectoriesthat of generated several of points several more on pointsconvenient the upperon the trajectory, armupper were arm whichmonitored,were had monitored, the finally following choosingfinally global choosing the coordinates point the that point generated (in thitsat initial generated more position) convenient more (in convenient trajectory,[mm]): XM which =trajectory, −432.8, had YtheMwhich = following274.8, had Z Mthe = global 38.following The coordinates trajectory global (in coordinates (Γ its) of initial this position)point (in its during initial (in [mm]): position)the kinematic XM (in= − [mm]):432.8, analysisY XMM =of − 274.8, 432.8,the frontZYMM ==wheel 38.274.8, The suspension Z trajectoryM = 38. The mechanism, (Γ trajectory) of this point which (Γ) of during wathiss controlledpoint the kinematic during by thethe analysis kinematickinematic of the restrictionanalysis front wheel of Y Kthe = suspension25·sin(time),front wheel mechanism, suspensioncan be shown which mechanism, in Figure was controlled 14.which wa bys the controlled kinematic by restriction the kinematic YK = restriction 25·sin(time), YK can= 25·sin(time), be shown in can Figure be shown 14. in Figure 14.

Figure 14. The interest point trajectory during the kinematic analysis of the front suspension. FigureFigure 14. 14.The The interest interest point point trajectory trajectory during during the the kinematic kinematic analysis analysis of of the the front front suspension. suspension. For the rear wheel suspension system, the MBS model was also designed in ADAMS/ViewForFor the the rear rear(Figure wheel wheel suspension15). suspension The optimization system, system, the MBSthe purpose MBS model model was, was also aswas designedin also the designed case in ADAMS/Viewof the in front wheel(FigureADAMS/View suspension, 15). The optimization(Figure to minimize 15). The purpose the optimization wheel was, trac as ink, purpose thewheelbase, case ofwas, the and as front bumpin wheelthe steer case suspension, variations,of the front to whileminimizewheel the suspension, camber the wheel and to track,caster minimize wheelbase,angles the variations wheel and bumptrac werek, steercanceledwheelbase, variations, in thisand stage whilebump with thesteer camberthe variations, help and of kinematiccasterwhile anglesthe restrictions. camber variations and caster were canceledangles variations in this stage were with canceled the help in this of kinematic stage with restrictions. the help of kinematic restrictions.

FigureFigure 15. 15. Quarter-carQuarter-car model model of of the the rear rear wheel wheel suspension suspension system. system. Figure 15. Quarter-car model of the rear wheel suspension system. AsAs for for the the front front suspension suspension system, system, points points A A and and B Bwere were considered considered as as the the centers centers of of two spherical joints that link the lower control arm (1) to the chassis, point C was the link two sphericalAs for thejoints front that suspension link the lower system, control points arm A (1) and to B the were chassis, considered point C as was the thecenters link of between the rocker (5) and the chassis, and point D represented the center of the spherical betweentwo spherical the rocker joints (5) thatand linkthe chassis, the lower and control point Darm represented (1) to the chassis, the center point of the C was spherical the link joint between the toe arm (6) and the chassis. The initial values of the design variables jointbetween between the therocker toe (5)arm and (6) the and chassis, the chassis. and po Theint D initial represented values theof the center design of the variables spherical were (in [mm]): DV_1 = −341.42, DV_2 = −50.0, DV_3 = −1310.74, DV_4 = −319.62, werejoint (in between [mm]): DV_1the toe = −arm341.42, (6) andDV_2 the = −chassis.50.0, DV_3 The =initial −1310.74, values DV_4 of the= − 319.62,design DV_5variables = DV_5 = −50.0, DV_6 = −1617.95, DV_7 = −344.97, DV_8 = 211.15, DV_9 = −1743.65, −50.0,were DV_6 (in [mm]): = −1617.95, DV_1 =DV_7 −341.42, = −DV_2344.97, = −DV_850.0, DV_3 = 211.15, = −1310.74, DV_9 DV_4= −1743.65, = −319.62, DV_10 DV_5 = = DV_10 = −213.76, DV_11 = 54.01, DV_12 = −1720.29 (with correlations in notations similar −50.0, DV_6 = −1617.95, DV_7 = −344.97, DV_8 = 211.15, DV_9 = −1743.65, DV_10 = to those in Table1). As mentioned, in the rear wheel suspension optimization process, the same inves- tigation strategy as for the front wheel suspension was used, namely DOE screening (level 2)—interactions—D-optimal. The optimization algorithm used was OptDes-GRG, Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 25 Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 25 Appl. Sci. 2021, 11, x FOR PEER REVIEW 14 of 25

−213.76, DV_11 = 54.01, DV_12 = −1720.29 (with correlations in notations similar to those −213.76, DV_11 = 54.01, DV_12 = −1720.29 (with correlations in notations similar to those −in213.76, Table DV_111). = 54.01, DV_12 = −1720.29 (with correlations in notations similar to those Appl. Sci. 2021, 11, 4167 in Table 1). 14 of 25 in TableAs 1).mentioned, in the rear wheel suspension optimization process, the same inves- As mentioned, in the rear wheel suspension optimization process, the same inves- tigationAs mentioned,strategy as infor the the rear front wheel wheel suspensi suspensionon optimization was used, process, namely theDOE same screening inves- tigation strategy as for the front wheel suspension was used, namely DOE screening tigation(level 2)—interactions—D-optimal. strategy as for the front wheel The optimizationsuspension wasalgorithm used, namelyused was DOE OptDes-GRG, screening (level 2)—interactions—D-optimal. The optimization algorithm used was OptDes-GRG, with(levelwith the the2)—interactions—D-optimal. MinToMinTo operator.operator. ByBy runningrunning The the theoptimization optimizationoptimization algorithm process,process, used thethe followingwasfollowing OptDes-GRG, optimaloptimal with the MinTo operator. By running the optimization process, the following optimal valueswithvalues the ofof MinTo thethe designdesign operator. variables variables By wererunning were obtained obtained the op (in timization(in [mm]): [mm]): DV_1 process,DV_1 = =− the−381.42,381.42, following DV_2 DV_2 = optimal=− −90.0,90.0, values of the design variables were obtained (in [mm]): DV_1 = −381.42, DV_2 = −90.0, DV_3valuesDV_3 ==of − 1270.74,the1270.74, design DV_4 DV_4 variables = =−279.62,−279.62, were DV_5 DV_5obtained = −10.0, = − (in10.0, DV_6 [mm]): DV_6 = − 1657.95,DV_1 = −1657.95, = DV_7−381.42, DV_7= − 384.97,DV_2 = − = DV_8384.97, −90.0, = DV_3 = −1270.74, DV_4 = −279.62, DV_5 = −10.0, DV_6 = −1657.95, DV_7 = −384.97, DV_8 = DV_8DV_3171.15, = DV_9− 171.15,1270.74, = DV_9−1783.65, DV_4 = −= DV_10−1783.65,279.62, = DV_5DV_10−173.76, = =− 10.0,DV_11−173.76, DV_6 = 14.01, DV_11 = −1657.95, DV_12 = 14.01, DV_7= − DV_121760.29. = −384.97, = −1760.29. DV_8 = 171.15, DV_9 = −1783.65, DV_10 = −173.76, DV_11 = 14.01, DV_12 = −1760.29. 171.15,ThroughThrough DV_9 = thethe −1783.65, kinematickinematic DV_10 analysisanalysis = −173.76, performedperformed DV_11 inin =ADAMS/View, ADAMS/View,14.01, DV_12 = thethe−1760.29. variationvariation diagramsdiagrams Through the kinematic analysis performed in ADAMS/View, the variation diagrams shownshownThrough inin Figures Figures the 16kinematic 16–18–18 were were analysis obtained, obtained, performed for for the th initiale ininitial ADAMS/View, (before (before optimization) optimization) the variation and and optimal diagrams optimal (af- shown in Figures 16–18 were obtained, for the initial (before optimization) and optimal tershown(after optimization) optimization) in Figures suspension 16–18 suspension were mechanisms, obtained, mechanisms, for observing th eobserving initial (as (before in (as the in case optimization) the of case the frontof the and suspension) front optimal sus- (after optimization) suspension mechanisms, observing (as in the case of the front sus- significant(afterpension) optimization) significant reductions suspensionreductions in the monitored inmechanisms, the monitored parameters observing parameters variations. (as invariations. the case of the front sus- pension) significant reductions in the monitored parameters variations. pension) significant reductions in the monitored parameters variations.

Figure 16. Wheelbase variation (initial—red, and optimal—blue). FigureFigure 16. 16.Wheelbase Wheelbase variation variation (initial—red, (initial—red, and and optimal—blue). optimal—blue). Figure 16. Wheelbase variation (initial—red, and optimal—blue).

Figure 17. Wheel track variation (initial—red, and optimal—blue). Figure 17. Wheel track variation (initial—red, and optimal—blue). Figure 17. Wheel track variation (initial—red,(initial—red, and optimal—blue).

Figure 18. Bump steer variation (initial—red, and optimal—blue). Figure 18. Bump steer variation (initial—red, and optimal—blue). Figure 18. Bump steer variation (initial—red, and optimal—blue). Further, in a similar way to those presented for the front wheel suspension, the most Further, in a similar way to those presented for the front wheel suspension, the most convenientFurther, trajectory inin a similar (Γ) waygenerated to those by presentedpresentea point (M)d forfor located the front on wheelthewheel upper suspension,suspension, arm (3) of thethe the mostmost rear convenient trajectory (Γ) generated by a point (M) located on the upper arm (3) of the rear convenientwheel suspension trajectory trajectory mechanism ( (ΓΓ)) generated generated was by selected, by a apoint point (M)as (M) shown located located in on Figure onthe the upper 19, upper wherearm arm (3) the of (3) theinterest of rear the wheel suspension mechanism was selected, as shown in Figure 19, where the interest rearwheelpoint wheel hadsuspension the suspension following mechanism mechanism global wascoordinates was selected, selected, (in as the asshown showninitial in position) in Figure Figure 19, (in19 ,where[mm]): where theXM = interest −525.9, point had the following global coordinates (in the initial position) (in [mm]): XM = −525.9, point hadhad thethe followingfollowing globalglobal coordinatescoordinates (in (in the the initial initial position) position) (in (in [mm]): [mm]): X MXM= =− −525.9, YM = 225.9, ZM = −1663.8. For the chosen trajectory, the global coordinates of the center point M0 on the chassis were determined according to the synthesis method presented in the next section of the paper. Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 25 Appl. Sci. 2021, 11, x FOR PEER REVIEW 15 of 25

YM = 225.9, ZM = −1663.8. For the chosen trajectory, the global coordinates of the center Appl. Sci. 2021, 11, 4167 YM = 225.9, ZM = −1663.8. For the chosen trajectory, the global coordinates of the 15center of 25 point M0 on the chassis were determined according to the synthesis method presented in point M0 on the chassis were determined according to the synthesis method presented in the next section of the paper. the next section of the paper.

Figure 19. The interest point trajectory during the kinematic analysis of the rear suspension. FigureFigure 19.19.The The interestinterest pointpoint trajectorytrajectory duringduring thethe kinematickinematic analysis analysis of of the the rear rear suspension. suspension. Concluding, it was observed that with the canceling of the camber and caster angle Concluding,Concluding, itit waswas observedobserved thatthat withwith thethe cancelingcanceling ofof thethe cambercamber andand caster caster angle angle variations (which in this stage were controlled by kinematic restrictions), the wheelbase, variationsvariations (which(which inin thisthis stagestage werewere controlledcontrolled bybykinematic kinematic restrictions),restrictions), thethe wheelbase,wheelbase, wheel track, and bump steer angle variations were significantly reduced, which was not wheelwheel track,track, andand bumpbump steersteer angleangle variationsvariations werewere significantlysignificantly reduced,reduced, whichwhich waswas not not possible in the case of the classic four-bar suspension mechanism. As a result of the im- possiblepossible inin thethe case of the the classic classic four-bar four-bar su suspensionspension mechanism. mechanism. As As a result a result of the of theim- plemented optimization procedure, the variations of all kinematic parameters were implementedplemented optimization optimization procedure, procedure, the the variations variations of of all kinematickinematic parametersparameters werewere within the accepted limits for the Formula Student single seater, both for the front and withinwithin the the accepted accepted limits limits for for the the Formula Formula Student Student single single seater, seater, both both for thefor frontthe front and rearand rear wheels, with a positive effect on the ride and handling performance of the car. wheels,rear wheels, with awith positive a positive effect effect on the on ride the and ride handling and handling performance performance of the of car. the car. 4. Synthesis of the Proposed Solution 4.4. SynthesisSynthesis ofof thethe ProposedProposed SolutionSolution The innovative solution proposed in this work, for both the front and rear suspen- TheThe innovativeinnovative solution solution proposed proposed in in this this work, work, for for both both the the front front and and rear rear suspension suspen- sion systems, corresponded to the guiding mechanism shown schematically in Figure 7, systems,sion systems, corresponded corresponded to the to guiding the guiding mechanism mechanism shown shown schematically schematically in Figure in7 ,Figure whose 7, whose spatial variant (Figure 20) enabled the minimization (even canceling) of the cam- spatialwhose variantspatial variant (Figure (Figure20) enabled 20) enabled the minimization the minimization (even canceling) (even canceling) of the camber of the cam- and ber and caster angles variations by using a guiding rocker (4) mounted between the up- casterber and angles caster variations angles variations by using aby guiding using a rocker guiding (4) mountedrocker (4) betweenmounted the between upper armthe up- (3) perand arm the chassis(3) and (0). the The chassis guiding (0). rocker The was guiding arranged rocker so as was to replace/substitute arranged so as theto twore- per arm (3) and the chassis (0). The guiding rocker was arranged so as to re- place/substitutefictive kinematic the restrictions two fictive (∆ γkinematic= 0, ∆β = restrictions 0) used in the(Δγ kinematic = 0, Δβ = optimization 0) used in the depicted kine- place/substitute the two fictive kinematic restrictions (Δγ = 0, Δβ = 0) used in the kine- maticin the optimization previous section depicted of the in work. the previous section of the work. matic optimization depicted in the previous section of the work.

FigureFigure 20. 20. TheThe spatial spatial scheme scheme of of the the inno innovativevative suspension suspension system. system. Figure 20. The spatial scheme of the innovative suspension system. TheThe location location of of the point/end M on thethe guidingguiding rocker rocker was was previously previously determined determined for The location of the point/end M on the guiding rocker was previously determined forthe the initial initial position position of the of suspensionthe suspension mechanism mechanism (when (when the car the was car in was rest), in along rest), with along the for the initial position of the suspension mechanism (when the car was in rest), along withtrajectory the trajectory of this point of this during point theduring kinematic the kinematic analysis analysis (see Figures (see 14Figures and 19 14). and The 19). problem The with the trajectory of this point during the kinematic analysis (see Figures 14 and 19). The problemraised by raised the proposed by the proposed solution solution was the was fact thatthe fact we neededthat we toneeded determine to determine where the where focal pointproblem (M )raised of the by guiding the proposed rocker was solution on the was chassis, the fact so thatthat iswe would needed assure to determine the cancelation where the focal point0 (M0) of the guiding rocker was on the chassis, so that is would assure the the focal point (M0) of the guiding rocker was on the chassis, so that is would assure the cancelationof the camber of the and camber caster and variations. caster variations To do this,. To in thedo this, following, in the following, we proposed we aproposed synthesis algorithmcancelation based of the on camber the method and caster of least variations squares,. through To do this, which, in the starting following, from we the proposed imposed movement of point M (the trajectory Γ described through a series of successive positions Mk), the global coordinates of point M0 on the chassis (and thus the spatial arrangement of the guiding rocker) was determined. Appl. Sci. 2021, 11, x FOR PEER REVIEW 16 of 25

Appl. Sci. 2021, 11, 4167 a synthesis algorithm based on the method of least squares, through which, starting16 from of 25 the imposed movement of point M (the trajectory Γ described through a series of succes- sive positions Mk), the global coordinates of point M0 on the chassis (and thus the spatial arrangementThe constraint of the guiding of point rocker) M on the was guiding determined. rocker consisted in the fact that it had to be The constraint of point M on the guiding rocker consisted in the fact that it had to be on a fixed surface or curve with point M0 on the chassis. In the case of guiding point M on on a fixed surface or curve with point M0 on the chassis. In the case of guiding point M on a sphere with the center in M0 (Figure 21), the guiding (constraint) equation had the form: a sphere with the center in M0 (Figure 21), the guiding (constraint) equation had the form: 2 2 2 2 (XM − XM0) + (YM − YM0) + (ZM − ZM0) − l = 0 (6) (X −X) +(Y − Y) +(Z −Z) −𝑙4 =0 (6)

wherewhere ll44 isis the the length length of of the the guiding guiding rocker. rocker.

FigureFigure 21. 21. GuidanceGuidance of of the the interest interest point point (M) (M) on on a a sphere. sphere.

TheThe point point coordinates coordinates (X M,,Y YMM, ,ZZMM) )in in the the global global reference reference frame frame (attached (attached to to chassis) chassis) werewere input data of the synthesis problem.problem. The coordinates of the chassischassis MM00 point (XM0,, YYM0M0, ,ZZM0M0) )and and the the length length of of the the guiding guiding rocker rocker ( (ll44) )were were unknown unknown and and needed needed to to be be computed.computed. EquationEquation (6) (6) can can be be rewritten rewritten under under the the following form: 2 X +2 Y +Z2 +X∗X + Y ∗ Y +Z∗Z +R= XM +YM +ZM + X ∗ XM+ Y ∗ YM+ Z ∗ ZM + R =0 0(7) (7) where: X=−2∗X,Y=−2∗Y,Z=−2∗Z,R=X2+Y +Z2 −l2 . 2 where: X = −2∗XM0,Y = −2∗YM0,Z = −2∗ZM0,R = XM0 + YM0 + ZM0 − l4. RewritingRewriting Equation Equation (7) (7) for for the the “m” “m” number number of of positions positions that could be imposed to the wheelwheel and subtractingsubtracting thethe first first equation, equation, “m “m− 1”− 1” equations equations with with 3 unknown 3 unknown (X, Y,(X, Z) Y, were Z) wereobtained: obtained:

X·[(XM)k+1 − (XM)1] + Y·[(YM)k+1 − (YM)1] + Z·[(ZM)k+1 − (ZM)1] = (r)1 − (r)k+1 (8) X·[(XM)k+1 − (XM)1] + Y·[(YM)k+1 − (YM)1] + Z·[(ZM)k+1 − (ZM)1] = (r)1 − (r)k+1 (8) 2 2 2 where (r)k =(XM +YM +ZM)k, which leads to a function of form:  2 2 2  where (r)k = XM + YM + ZM , which leads to a function of form: Fkk (X, Y, Z) = Lk, k = 1... m − 1. (9)

For m = 4 imposed positions,Fk (X, the Y, Z)system = Lk ,(4) k =was 1... a m linear− 1. one, with 3 equations and (9) 3 unknowns. However, as there had to be more than 3 imposed finite positions, we ob- tainedFor an m over-determined = 4 imposed positions, system the that system could (4) be was solved a linear by means one, with of 3the equations least squares and 3 approach.unknowns. However, as there had to be more than 3 imposed finite positions, we obtained an over-determinedIf X’, Y’, Z’ were systema set of thatvalues could that be coul solvedd be byverified means with of the approximation least squares of approach. one part of theIf Equation X’, Y’, Z’ were(9), for a setexample, of values the that solution could of be the verified system with formed approximation by the first of one3 equa- part tions,of the would Equation result: (9), for example, the solution of the system formed by the first 3 equations, would result: k k FkF(X’, (X’, Y’, Y’, Z’) Z’) = = L’ L’k, k, k = = 1... 1... m m− − 1.1. (10) (10) SubtractingSubtracting the the Equation Equation (10) (10) from from (9) (9) we we obtain:

Fk (X, Y, Z) − Fk (X’, Y’, Z’) = Lk − L’k = dk (11) Fk (X, Y, Z) − Fk (X’, Y’, Z’) = Lk − L’k = dk (11) Assuming that the differences δx = X − X’, δy = Y − Y’, δz = Z − Z’ were small enough δ − δ − δ − as in Assumingthe developed that thefunctions differences Fk basedx = on X theX’, Taylory =Y formula,Y’, z we = ZcouldZ’ neglect were small the higher-orderenough as in terms, the developed and the functionsequation Fsystemk based (11), on thewith Taylor its unknown formula, δ wex, δ couldy, δz, neglectwould be: the higher-order terms, and the equation system (11), with its unknown δx, δy, δz, would be:

ak· δx + bk· δy + ck· δz = dk (12) Appl. Sci. 2021, 11, 4167 17 of 25

where ak = (XM)k+1 − (XM)1, bk = (YM)k+1 − (YM)1, ck = (ZM)k+1 − (ZM)1. According to the method of least squares, the best solutions of the system (12) were those values δx, δy, δz for which the objective function:

m−1 2 f(δX, δY, δZ) = ∑ (ak∗δX + bk∗δY + Ck∗δZ − dk) (13) k=0

is minimal, meaning: δf δf δf = 0, = 0, = 0 (14) δδx δδy δδz Thus, a linear system of 3 equations with 3 unknowns (δx, δy, δz) is obtained:

[aa] · δx + [ab] · δy + [ac] · δz = [ad], [ba] · δx + [bb] · δy + [bc] · δz = [bd], (15) [ca] · δx + [cb] · δy + [cc] · δz = [cd],

= m−1 2 = m−1 ∗ where [aa] ∑k=1 ak, [ab] ∑k=1 ak bk, and so on. The best approximate solution of the system (15) is:

X = X’ + δx, Y = Y’ + δy, Z = Z’ + δz, (16)

X Y Z X = − ∗Y = − ∗Z = − (17) M0 2 M0 2 M0 2 The average quadratic errors with which the solutions were calculated are: r r r ∆A ∆B ∆C µ = ε ∗ , µ = ε ∗ , µ = ε ∗ , (18) X ∆ Y ∆ Z ∆ where ∆ is the determinant of the system (15); ∆A, ∆B, and ∆C are algebraic coefficients of the [aa], [bb], and [cc] elements from the main diagonal, and ε is the error with which dk was computed: q m−1 ∗ δ + ∗ δ + ∗ δ − 2 ∑k=1 (ak X bk Y ck Z dk) ε = , (19) m − 4 The sphere radius (i.e., the guiding rocker length) could be now determined from the equation: q 2 2 2 l4 = (XM − XM0) + (YM − YM0) + (ZM − ZM0) (20)

In the particular case where the point M is moving on a circle with its center in M0 on the chassis, meaning that the guiding rocker MM0 would have two spherical joints on the 0 chassis (M0 ,M0”), the geometrical place of the M point would be on a circle Γ (Figure 22), which could be determined by intersecting the sphere S, defined by the Equation (6), with the π plane, with the following equation:

(XM − XN)∗ cos α + (YM − YN)∗ cos β + (ZM − ZN)∗ cos η = 0 (21)

where N is a point belonging to the plane, and cos α, cos β, and cos η are the cosines of the axis normal to the plane (i.e., the rotation axis). Equation (21) can be rewritten as:

XM + r1·YM + r2 + ZM + r3 = 0 (22)

where r1, r2, and r3 are given from the following equations:

cos β cos η r = , r = , r = −(X + r ·Y + r ·Z ) (23) 1 cos α 2 cos α 3 N 1 N 2 N Appl. Sci. 2021, 11, x FOR PEER REVIEW 18 of 25

Appl.Appl. Sci. Sci. 20212021, 11, 11, x, 4167FOR PEER REVIEW 18 of 25 18 of 25

Figure 22. Guidance of the interest point (M) on a semicircle.

Equation (21) can be rewritten as: + ⋅ + ⋅ + = XM r1 YM r2 ZM r3 0 (22)

where r1, r2, and r3 are given from the following equations:

cos β cosη r = , r = , r = −()X + r ⋅ Y + r ⋅ Z (23) 1 cosα 2 cos α 3 N 1 N 2 N FigureFigure 22. 22.Guidance Guidance of the of interestthe interest point (M)point on (M) a semicircle. on a semicircle. For an “m” number of finite positions of the point M, by subtracting the first equa- tion (correspondingFor an “m” number to ofthe finite initial positions position) of the from point the M, others, by subtracting the Equation the first (22) equation becomes: (correspondingEquation to (21) the initialcan be position) rewritten from as: the others, the Equation (22) becomes: (𝑋) −(𝑋) +𝑟 ∗ (𝑌) −(𝑌) +𝑟 ∗ (𝑍) −(𝑍) =0,𝑘=1,…𝑚−1 (24) (X ) − (X )  + r ∗ (Y ) − (Y )  + r ∗ (Z )+ −⋅ (Z +)  =⋅ 0, k+= 1,= . . . m − 1 (24) M k+1 M 1 1 M k+1 M 1 2 XM k+r1 YMM 1r2 ZM r3 0 (22) Equation (6) was used to determine the location of M0 according to the algorithm for Equation (6) was used to determine the location of M according to the algorithm for determining the center of a spherical surface. The orientation0 of the revolute joint axis determiningwhere r1, r2 the, and center r3 are of given a spherical from surface. the following The orientation equations: of the revolute joint axis (M0′–M0”) was obtained by (24), which had a similar solution as (8). (M00 –M0”) was obtained by (24),cos whichβ hadcos a similarη solution as (8). TheThe presentedpresented algorithm algorithmr = was transposedwas, r =transposed in, ar computer= −in()X a program+computerr ⋅ Y by+ r usingprogram⋅ Z MATLAB. by using (23) 1 α 2 α 3 N 1 N 2 N MATLAB.The actual numerical The actual implementation numericalcos implementation was reflectedcos in was the global reflected coordinates in the global of the pointscoordinates of the points M0 in which the guiding rocker was connected to the chassis, along with an M0 inFor which an the “m” guiding number rocker of wasfinite connected positions to of the the chassis, point along M, by with subtracting an additional the first equa- additionalpoint M 0 by point which M the0′ by revolute which jointthe revolute axis of the joint rocker axis was of the defined, rocker as was follows: defined, as follows: tion (corresponding0 to the initial position) from the others, the Equation (22) becomes: • TheThe frontfront wheelwheel suspension suspension mechanism mechanism (Figure (Figure 23 ):23): XM0 XM0= =− −323.9,323.9, Y YM0M0 == 245.9,245.9, ZM0 = Z = 42.9; X 0 = −318.2, Y 0 = 250.4, Z 0 = −56.8 (in mm); (𝑋)42.9;M0 −(𝑋 XM0′ = )−M0318.2, +𝑟 Y∗M0(𝑌′ = 250.4,)M0 −(𝑌ZM0′ = )−56.8M0+𝑟 (in ∗ mm);(𝑍 ) −(𝑍) =0,𝑘=1,…𝑚−1 (24) • The rear wheel suspension mechanism (Figure 24): XM0 = −282.1, YM0 = 152.4, • The rear wheel suspension mechanism (Figure 24): XM0 = −282.1, YM0 = 152.4, ZM0 = ZM0Equation= −1698.3; (6) XwasM00 =used−267.5, to determine YM00 = 154.4, the ZM0 location0 = −1599.4 of M (in0 mm).according to the algorithm for −1698.3; XM0′ = −267.5, YM0′ = 154.4, ZM0′ = −1599.4 (in mm). determining the center of a spherical surface. The orientation of the revolute joint axis (M0′–M0”) was obtained by (24), which had a similar solution as (8). The presented algorithm was transposed in a computer program by using MATLAB. The actual numerical implementation was reflected in the global coordinates of the points M0 in which the guiding rocker was connected to the chassis, along with an additional point M0′ by which the revolute joint axis of the rocker was defined, as follows:

• The front wheel suspension mechanism (Figure 23): XM0 = −323.9, YM0 = 245.9, ZM0 = 42.9; XM0′ = −318.2, YM0′ = 250.4, ZM0′ = −56.8 (in mm); • The rear wheel suspension mechanism (Figure 24): XM0 = −282.1, YM0 = 152.4, ZM0 = −1698.3; XM0′ = −267.5, YM0′ = 154.4, ZM0′ = −1599.4 (in mm).

Figure 23.23.Front Front wheel wheel suspension suspension system system with with guiding guiding rocker. rocker.

Figure 23. Front wheel suspension system with guiding rocker. Appl. Sci. 2021,, 11,, x 4167 FOR PEER REVIEW 19 of 25

FigureFigure 24. 24. RearRear wheel wheel suspension suspension syst systemem with with guiding guiding rocker. rocker. By the kinematic analysis of the proposed front and rear suspension mechanisms, By the kinematic analysis of the proposed front and rear suspension mechanisms, approximately zero variations of the camber and caster angles were obtained. In addition, approximately zero variations of the camber and caster angles were obtained. In addi- the variations of the wheelbase, wheel track, and bump steer angle for the suspension tion, the variations of the wheelbase, wheel track, and bump steer angle for the suspen- mechanisms with the guiding rocker shown in Figures 23 and 24 were very close to those sion mechanisms with the guiding rocker shown in Figures 23 and 24 were very close to corresponding to the five-bar mechanism-based suspension systems (the optimal curves those corresponding to the five-bar mechanism-based suspension systems (the optimal in Figures 11–13, and, respectively, Figures 16–18). These very good correlations between curves in Figures 11–13, and, respectively, Figures 16–18). These very good correlations results demonstrate the viability of replacing the fictive kinematic restrictions (by which between results demonstrate the viability of replacing the fictive kinematic restrictions the variations of the camber and caster angles were initially canceled) by the physical (bygeometrical which the constraints variations associated of the camber with theand guiding caster angles rockers. were initially canceled) by the physicalAs mentioned,geometrical mostconstraints Formula asso Studentciated with race carsthe guiding use the rockers. double wishbone (four-bar) suspensionAs mentioned, mechanism most (schematically Formula Student represented race cars inuse Figure the double2b). This wishbone solution (four-bar) was also suspensionimplemented mechanism on the initial (schematically race car developed represente at Transilvaniad in Figure University 2b). This solution of Brasov was (Brasov, also implementedRomania, BlueStreamline), on the initial onrace both car the develope front andd at the Transilvania rear axles, withUniversity which of the Brasov single (Brasov,seater obtained Romania, good BlueStreamline), results in the on competitions both the front where and itthe participated rear axles, (onwith the which circuits the singlefrom Silverstone, seater obtained Catalunya, good results and Varano). in the competitions Prior to implementation, where it participated that double (on wishbone the cir- cuitssuspension from Silverstone, was kinematically Catalunya, optimized and Vara by anno). algorithm Prior to basedimplementation, on DOE techniques that double and wishbonelinear regression suspension functions, was kinematically as presented inopti previousmized authors’by an algorithm papers [ 31based,32]. on The DOE proposed tech- niquessuspension and systemlinear regression solution is functions, an improvement as pres ofented the classicalin previous double authors’ wishbone papers suspension [31,32]. Thesystem, proposed by being suspension able to decouple system thesolution mentioned is an contradictoryimprovement kinematic of the classical variations double (the wishboneissues in which suspension the double system, wishbone by being suspension able to mechanism decouple wasthe me deficient),ntioned thus contradictory improving kinematicthe dynamic variations behavior (the of the issues suspension in which (car, the in double general). wishbone suspension mechanism was deficient),The values thus of severalimproving important the dynamic characteristics, behavior of when the suspension the car was (car, in staticin general). rest, of the twoThe suspensionvalues of several systems important implemented/used characteristics, on when the Formulathe car was Student in static single-seater rest, of the tworace suspension car developed systems at Transilvania implemented/used University on of the Brasov Formula (the Student classical single-seater four-bar mechanism, race car developedand the new at solutionTransilvania depicted University in this work)of Brasov are shown(the classical in Table four-bar2. It is wellmechanism, known thatand thethe new static solution characteristics depicted of in any this suspension work) are shown system in are Table given 2. byIt is the well wheel known position that the in staticspace characteristics (camber angle, of toe any angle, suspension track width system at wheel are given center) by andthe bywheel the wheelposition carrier in space ball (camberjoint positions angle, toe (kingpin angle, angle, track width steering at axiswheel offset, center) scrub and radius, by the casterwheel trail).carrier The ball staticjoint positionsparameters (kingpin were carried angle, over steering from axis theinitially offset, scru tunedb radius, four-bar caster suspension trail). system,The static and pa- as rametersthe optimization were carried process over of from the new the (proposed)initially tuned solution four-bar was suspension based by moving system, the and hard as thepoints optimization that connect process the suspension of the new system (propose to thed) chassissolution and was the based wheel by carrier moving hard the points hard pointswere not that affected connect by the this, suspension the static suspension system to systemthe chassis metrics and did the not wheel change carrier relative hard to pointsthe classical were not four-bar affected suspension. by this, the static suspension system metrics did not change rel- ative to the classical four-bar suspension.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 20 of 25

Appl. Sci. 2021, 11, 4167 20 of 25 Table 2. Suspension system characteristics.

Four-Bar New Solution Four-Bar New Solution Table 2. SuspensionFront system characteristics. Rear Roll center height [mm] 250 246 280 290 Caster center height [mm] Four-Bar 150 New Solution 147 Four-Bar 190 New Solution 188 Anti-dive/squat [%] 50 Front 49 65 Rear 67 RollCamber center height [deg] [mm] 250 1.5 246 280 1 290 CasterToe center [deg] height [mm] 150 0.7 147 190 0.5 188 KingpinAnti-dive/squat angle [deg] [%] 50−3.9 49 65−4.07 67 SteeringCamber axis offset [deg] [mm] 1.5 3.1 1 0 Scrub Toeradius [deg] [mm] 0.7 30.75 0.5 2 Kingpin angle [deg] −3.9 −4.07 SteeringCaster axis trail offset [mm] [mm] 3.1 17.25 0 8 Track widthScrub at radiuswheel [mm]center [mm] 30.75 1210 2 1150 Caster trail [mm] 17.25 8 Track width at wheel center [mm]5. Development and Testing 1210 of the Physical (Experimental) 1150 Prototype The process of modeling, simulation, and optimization in a virtual environment, the results5. Development of which and are Testing presented of the in Physical Figure (Experimental)25 in the form Prototype of the full-vehicle virtual proto- type,The preceded process ofthe modeling, implementation simulation, of andthe optimizationphysical prototype, in a virtual a environment,subject addressed the from hereresults on. of whichIn order are presentedto develop in Figure the physical 25 in the formprototype, of the full-vehicle the following virtual steps prototype, were taken: preceded the implementation of the physical prototype, a subject addressed from here on. Embodiment design of the full-vehicle suspension system, starting from which the exe- In order to develop the physical prototype, the following steps were taken: Embodiment cutiondesign ofand the assembly full-vehicle drawings suspension were system, generated; starting from executing which the or executionacquisitioning and assem- the compo- nents;bly drawings assembling were generated; the components; executing oracquisitioning acquisitioning theand components; mounting assemblingthe data monitoring the systemcomponents; (sensors). acquisitioning and mounting the data monitoring system (sensors).

Figure 25.25.Full-vehicle Full-vehicle virtual virtual prototype prototype of the of Formulathe Formula Student Student single seater.single seater. The physical prototype of the proposed suspension system was implemented and The physical prototype of the proposed suspension system was implemented and tested with the help of BlueStreamline, the Formula Student team of the Transilvania tested with the help of BlueStreamline, the Formula Student team of the Transilvania University of Bras, ov (Figure 26). The data acquisition system used to measure damper Universitytravel consisted of Bra of twoșov short (Figure distance 26). sensors The data (sharp), acquisitio an Arduinon system MEGA used 2560 developmentto measure damper travelboard, aconsisted breadboard, of two and ashort SD memory distance card sens socket.ors (sharp), The programming an Arduino language MEGA used2560 devel- opmentby the microcontroller board, a breadboard, was C#. Theand dataa SD acquisition memory card system socket. used The to measure programming the forces language usedtransmitted by the in microcontroller the chassis consists was of an C#. AQ-1 The data data logger, acquisition which had system a built-in used force to sensor measure the forcesthat was transmitted mounted on in the the chassis chassis near consists its center of ofan mass.AQ-1 The data data logger, thus collectedwhich had were a built-in forcetransferred sensor for that postprocessing was mounted into EXCEL.on the chassis near its center of mass. The data thus col- lected were transferred for postprocessing into EXCEL. Appl. Sci. 2021, 11, x FOR PEER REVIEW 21 of 25

Appl. Sci. 2021, 11, 4167 21 of 25 Appl. Sci. 2021, 11, x FOR PEER REVIEW 21 of 25

Appl. Sci. 2021, 11, x FOR PEER REVIEW 21 of 25

Figure 26. The Formula Student single seater equipped with the proposed suspension.

FigureFigureIn 26. 26. Theorder The Formula Formula to evaluate Student Student single single the seater seaterdynamic equipped equipped behavior with with the the proposed proposed of the suspension. suspension.single-seater prototype, several

tests were carried out, which were similar to those the single seater would have to pass FigureInIn 26. order order The Formula to to evaluate evaluate Student the the dynamicsingle dynamic seater behavior eq behavioruipped of with the of single-seaterthethe proposed single-seater suspension. prototype, prototype, several several tests werethroughtests carriedwere incarried out, the which out,competitions. werewhich similar were tosimilarFurther, those to the thosethe single resultsthe seater single wouldobtained seater have would in to pass thehave throughcase to pass of the left–right incorneringthrough theIn competitions. order in theand to evaluatecompetitions. passing Further, the over the dynamic Further, results vibrators, obtainedbehavior the results as inof well the theobtained case single-seateras ofa in theskid the left–right pad caseprototype, typeof cornering the event, severalleft–right and were presented. testspassingThecornering weretests over carried andwere vibrators, passing out,performed which asover well wevibrators, as reon asimilar skidthe as padkarting towell those type as a event,circuitthe skid single pad were from seatertype presented. Prejmer event, would were Thehave (Figure testspresented. to pass were 27). For the over- throughperformedpassThe tests vibrators’ in were onthe the competitions. performed karting test, all circuit onthe Further, the fromcorresponding karting Prejmerthe resultscircuit (Figure obtainedfrom vibr 27 Prejmer).ators For in thethe of (Figure overpass caseall cornersof 27). the vibrators’ For left–right mustthe over- test, be reached. This corneringallpass the vibrators’ corresponding and passing test, all vibrators over the vibrators,corresponding of all corners as well vibr must as atorsa beskid reached.of pad all cornerstype This event, test must aimedwere be reached.presented. to check This the test aimed to check the speed response (reaction) of the suspension system (Figure 28), Thespeedtest tests aimed response were to checkperformed (reaction) the speed on of thethe response suspensionkarting circuit(react systemion) from of (Figure Prejmer the suspension 28 (Figure), and the 27).system way For itthe(Figure supports over- 28), passlateralandand vibrators’thethe forces way way (Figureit test, supportsit supports all 29 the). lateral corresponding lateral forces forces (Figure vibr (Figureators 29). of all 29). corners must be reached. This test aimed to check the speed response (reaction) of the suspension system (Figure 28), and the way it supports lateral forces (Figure 29).

Figure 27. Map of the karting circuit from Prejmer (2012 www.prejmercircuit.ro).

FigureFigure 27. 27. MapMap Map of of the theof karting kartingthe karting circuit circuit fromcircuit from Prejmer Prejmer from (2012 (2012Prejmer www.prejmercircuit.ro). www.prejmercircuit.ro (2012 www.prejmercircuit.ro). 24 March 2021).

Figure 28. Front spring and damper movement—circuit lap. Figure 28. FrontFront spring spring and and damper damper movement—circuit movement—circuit lap. lap.

According to the diagrams shown in Figure 29, it is possible to observe the shift of loadingFigure 28. forces Front that spring act on and the cardamper from onmovement—circuit one side to the other lap. (left–right) at the time of turning, in both wide and tight corners. At the same time, a linear behavior was observed in the suspension system, which induced a good handling of the car and reduced the necessary corrections in the steering wheel while turning. Appl. Sci. 2021, 11, x FOR PEER REVIEW 22 of 25

Appl.Appl. Sci. Sci. 20212021,, 1111,, x 4167 FOR PEER REVIEW 2222 of of 25 25

Figure 29. Lateral force—circuit lap.

According to the diagrams shown in Figure 29, it is possible to observe the shift of loading forces that act on the car from on one side to the other (left–right) at the time of turning, in both wide and tight corners. At the same time, a linear behavior was observed

in the suspension system, which induced a good handling of the car and reduced the Figure 29. Lateral force—circuit lap. necessaryFigure 29. correctionsLateral force—circuit in the steering lap. wheel while turning. The skid pad type event was carried out in an especially arranged track that mim- AccordingThe skid pad to typethe diagrams event was shown carried in out Figure in an 29, especially it is possible arranged to observe track that the mimicked shift of icked the Formula Student layout shown in Figure 30. The diagrams in Figure 31 show loadingthe Formula forces Student that act layouton the showncar from in on Figure one side30. to The the diagrams other (left–right) in Figure at 31 the show time theof the movement of the front axle spring and damper assembly during constant cornering turning,movement in both of the wide front and axle tight spring corners. and At damper the same assembly time, a during linear behavior constant corneringwas observed (left (left and right). By analyzing the results from Figure 31 (maximum damper travel− of inand the right). suspension By analyzing system, the which results induced from Figure a good 31 (maximum handling damperof the car travel and of reduced +/ 20 mm)the +/−20 mm) and Figure 28 (maximum damper− travel of +/−30 mm), it is clearly seen that necessaryand Figure corrections 28 (maximum in the damper steering travel wheel of +/ while30 mm),turning. it is clearly seen that the suspension thedid suspension not reach itsdid end not stops reach during its end the stops skid duri padng event. the skid On the pad other event. hand, On bythe correlating other hand, the The skid pad type event was carried out in an especially arranged track that mim- bylateral correlating force diagram the lateral presented force diagram in Figure presen32 withted the in circuit Figure layout 32 with shown the in circuitFigure layout 30, it is icked the Formula Student layout shown in Figure 30. The diagrams in Figure 31 show shownclear thatin Figure the suspension 30, it is clear had that a linear the suspension behavior that had was a linear easy behavior to anticipate that fromwas easy a driver to the movement of the front axle spring and damper assembly during constant cornering anticipatepoint of view.from a driver point of view. (left and right). By analyzing the results from Figure 31 (maximum damper travel of +/−20 mm) and Figure 28 (maximum damper travel of +/−30 mm), it is clearly seen that the suspension did not reach its end stops during the skid pad event. On the other hand, by correlating the lateral force diagram presented in Figure 32 with the circuit layout shown in Figure 30, it is clear that the suspension had a linear behavior that was easy to anticipate from a driver point of view.

FigureFigure 30. 30. SkidSkid pad pad track track according according to tothe the Formula Formula Student Student rules. rules.

Furthermore,Furthermore, as as the the bump bump steer steer variation variation was was reduced reduced to to zero zero it itwas was easy easy to to deduce deduce thatthat the the only only toe toe angle angle that that the the front front wheels wheels had had was was the the input input given given by by the the driver driver from from ◦ thethe steering steering wheel, wheel, and and the the rear rear toe toe angle angle was was eq equalual to to the the static static toe toe angle angle (0.5°). (0.5 ).As As the the

bumpbump camber camber and and bump bump caster caster variations variations were were reduced reduced to to zero zero as as well, well, the the camber camber and and Figurecaster 30. angles Skid duringpad track cornering according would to the Formula be equal Student to the rules. static ones (1.5◦ front and 1◦ on the rear camber, and −3.9◦ front and −4.07◦ rear on the caster angle). Furthermore,Based on the as results the bump of the steer spring variation and damper was reduced assembly to zero movement it was easy presented to deduce in thatFigures the 28only and toe 31 angle, it is clearthat thatthe front the race wheels car had had a was linear the all-around input given behavior by the and driver had from good thehandling steering capabilities. wheel, and As the this rear is a toe race angle car, we was did eq notual test to orthe look static at anytoe rideangle metrics; (0.5°). norAs the did bumpwe aim camber to tune and any bump of them. caster In anyvariations race car, were the handlingreduced metricsto zero as are well, more the important camber thanand the ride metrics, as the ride is influenced by several components that the given suspension Appl. Sci. 2021, 11, 4167 23 of 25

Appl.Appl. Sci. Sci. 2021 2021, ,11 11, ,x x FOR FOR PEER PEER REVIEW REVIEW 2323 of of 25 25

system does not have, such as suspension bushes and anti-roll bars, and any attempt to castertunecaster them anglesangles could duringduring be made corneringcornering only by wouldwould changing bebe equalequal the dampingtoto thethe staticstatic or springonesones (1.5°(1.5° characteristics, frontfront andand 1°1° and onon thethe rearpresentedrear camber, camber, paper and and did− −3.9°3.9° not front front aim and and to tune − −4.07°4.07° for rear rear them. on on the the caster caster angle). angle).

FigureFigure 31. 31. Front FrontFront spring springspring and and damper damper movement—skid movement—skid pad. pad.

FigureFigure 32. 32. Lateral LateralLateral force—skid force—skidforce—skid pad. pad.

6. ConclusionsBasedBased onon thethe resultsresults ofof thethe springspring andand damperdamper assemblyassembly movementmovement presentedpresented inin FiguresFiguresWhile 2828 andand for passenger31,31, itit isis clearclear cars thatthat there thethe are racerace a multitudecarcar hadhad aa linear oflinear suspension all-aroundall-around system behaviorbehavior solutions, andand hadhad in goodthegood case handlinghandling of single-seater capabilities.capabilities. race AsAs cars, thisthis due isis aa to racerace the car,car, specific wewe diddid regulations, notnot testtest oror the looklook range atat anyany of ridesolutionsride met-met- rics;isrics; limited, nornor diddid most wewe of aimaim the toto existing tunetune anyany solutions ofof them.them. being InIn anyany based racerace on car,car, the thethe classical handlinghandling four-bar metricsmetrics suspension areare moremore importantmechanism,important thanthan which thethe ride hasride themetrics,metrics, major asas disadvantage thethe rideride isis influencedinfluenced of the contradiction byby severalseveral componentscomponents between the thatthat wheel thethe giventrackgiven and suspensionsuspension the camber systemsystem angle doesdoes variations. notnot have,have, suchsuch asas suspensionsuspension bushesbushes andand anti-rollanti-roll bars,bars, andand anyForany attempt reasonsattempt to ofto tune complexity,tune themthem couldcould maintenance, bebe mademade only andonly cost, byby changingchanging it is preferable thethe dampingdamping to use oror a purelyspringspring characteristics,mechanicalcharacteristics, suspension and and the the presented presented system, although paper paper did did the not not active aim aim suspensiontoto tune tune for for them. them. would ensure a superior behavior. The proposed suspension system offered the advantages of the purely mechanical 6.suspension;6. Conclusions Conclusions while functional, it was close to the behavior of the active suspension, and could be implemented on both the front and rear axles of the single seater (no matter which WhileWhile for for passenger passenger cars cars there there are are a a multitud multitudee of of suspension suspension system system solutions, solutions, in in the the of them is motor axle). casecase ofof single-seatersingle-seater racerace cars,cars, duedue toto thethe spspecificecific regulations,regulations, thethe rangerange ofof solutionssolutions isis The design, simulation, and optimization process in a virtual prototyping environment limited,limited, mostmost ofof thethe existingexisting solutionssolutions beingbeing basedbased onon thethe classicalclassical four-barfour-bar suspensionsuspension is aimed at evaluating and improving the kinematic behavior/movement of the suspension mechanism,mechanism, whichwhich hashas thethe majormajor disadvantagedisadvantage ofof thethe contradictioncontradiction betweenbetween thethe wheelwheel system, which directly affect vehicle ride and handling performance. The optimization tracktrack and and the the camber camber angle angle variations. variations. study was based on a DOE screening—interactions—D-optimal investigation strategy, and ForFor reasonsreasons ofof complexity,complexity, maintenance,maintenance, andand cost,cost, itit isis preferablepreferable toto useuse aa purelypurely mechanicalit resulted in suspension a significant system, improvement although of the measuredactive suspension target variations would ensure by only a requiringsuperior minormechanical changes suspension to the initial system, setup although of the system the active (front suspension or rear). would ensure a superior behavior. The proposed suspension system offered the advantages of the purely me- behavior.Based The on theproposed suspension suspension performance system shown offered in the Figures advantages 28 and 31of andthe thepurely chassis me- chanical suspension; while functional, it was close to the behavior of the active suspen- lateralchanical forces suspension; presented while in Figures functional, 29 and it32 was, both close the suspensionto the behavior optimization of the active viability suspen- and sion, and could be implemented on both the front and rear axles of the single seater (no suspensionsion, and could system be implemented performance couldon both be the acknowledged front and rear for axles various of the road single conditions seater (no on matter which of them is motor axle). whichmatter the which experimental of them is data motor recordings axle). were performed, such as straight line, corners, and TheThe design,design, simulation,simulation, andand optimizationoptimization processprocess inin aa virtualvirtual prototypingprototyping environ-environ- mentment isis aimedaimed atat evaluatingevaluating andand improvingimproving thethe kinematickinematic behavior/movementbehavior/movement ofof thethe Appl. Sci. 2021, 11, 4167 24 of 25

bumps (vibrators). A linear suspension behavior was obtained, without sudden changes (which could induce a non-linear behavior of the single seater). The single-seater race car that was equipped with the suspension system depicted in this work managed to pass all the Formula Student tests, and thus was able to participate in any Formula Student competition, thus proving the integrity of the tuning strategy, and the high performance of the proposed suspension system.

Author Contributions: Conceptualization, Vlad T, ot, u; methodology, Cătălin Alexandru; software,

Vlad T, ot, u; validation, Vlad T, ot, u and Cătălin Alexandru; writing—original draft preparation, Vlad

T, ot, u; writing—review and editing, Vlad T, ot, u and Cătălin Alexandru; supervision, Cătălin Alexandru. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Conflicts of Interest: The authors declare no conflict of interest.

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