sustainability
Article Vibration Suppression of a Single-Cylinder Engine by Means of Multi-objective Evolutionary Optimisation
Suwin Sleesongsom 1,* and Sujin Bureerat 2
1 Department of Aeronautical Engineering and Commercial Pilot, International Academy of Aviation Industry, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand 2 Sustainable and Infrastructure Development Center, Department of Mechanical Engineering, Faculty of Engineering, KhonKaen University, KhonKaen City 40002, Thailand; [email protected] * Correspondence: [email protected]; Tel.: +66-02-329-800
Received: 31 May 2018; Accepted: 15 June 2018; Published: 18 June 2018
Abstract: This paper presents a new design strategy for the passive vibration suppression of a single-cylindrical engine (SCE) through multi-objective evolutionary optimisation. The vibration causes machine damages and human pain, which are unsustainable problemsthat need to be alleviated. Mathematical forced vibration analyses of a single-cylinder engine, including dynamic pressure force due to ignition combustion, are presented. A multi-objective design problem is set to find the shape and size variables of the crank and connecting rod of the engine. The objective functions consist of the minimisation of the crank and connecting rod mass, and the minimisation of vibration response while the SCE is subject to inertial force and pressure force. Moreover, design constraints include crank and rod safety. The design problem is tackled by using an adaptation of a hybrid of multi-objective population-based incremental learning and differential evolution (RPBIL-DE). The optimum results found that the proposed design strategy is a powerful tool for the vibration suppression of SCE.
Keywords: vibration suppression; single-cylinder engine; multi-objective evolutionary algorithms; dynamic analysis; crank–slider
1. Introduction A single-cylinder engine (SCE) is one of the most widely used engines, especially in motorcycles, which are the most popular two-wheel automotive in this world. It is also included in a variety of applications particularly for agricultural proposes such as the driving pump, walking tractor, lawnmower, etc. In contrast with the applications, the vibration of this engine is the main problem at present. Two causes of vibration are from moving links in a crank–slider mechanism and ignition pressure due to combustion process. These can be a cause of machine damages, human discomfort, and user-accumulated fatigue and pain. SCE vibration can be alleviated in two ways i.e., balancing and isolation. Balancing a SCE can be classified as active and passive balancing [1]. Active balancing is a method for reducing shaking force and moment by introducing dummy pistons and geared revolving counter weights, etc. Passive balancing, on the other hand, is a method used to reduce shaking force and moment by the addition or removal of mass from various portions of the moving links. Research work toward this area has been continually made. Lowen et al. [2] summarised the techniques for the force and moment balancing of linkages. Zhang and Chen [3] have applied vibration suppression of a four-bar linkage by using the weighted sum method, which is a means to convert multi-objective optimisation to become a problem with one design objective. The counterweights’ mass parameters were set as design variables in this passive balancing. Snyman et al. [4] have applied
Sustainability 2018, 10, 2067; doi:10.3390/su10062067 www.mdpi.com/journal/sustainability Sustainability 2018, 10, 2067 2 of 19 an unconstrained optimisation problem to minimise the transmission of engine vibration due to inertial forces to the supporting structure where the case study is a mounted four-cylinder V-engine rotating at idling speed by an active balancing method. The individual balancing masses and associated phase angles of counter rotating balancing masses were chosen as design variables. Chiou et al. [5] proposed an optimum design in which disk counterweights were added to reduce shaking force and moment of the drag-link drive of mechanical presses. Sleesongsom [6] proposed applying multi-objective optimisation to reduce the engine mount translation and rotation displacements of SCE where the normalised normal constraint method [7], in combination with sequential quadratic programming, is an optimiser. The use of finite element analysis and optimisation codes for connecting rod [8], crankshaft [9], and piston design [10] has been conducted. In addition, the finite element technique has been used to optimise the crankshaft parameters of a single-cylinder motorcycle engine to reduce vibration without considering the gas pressure force inside the combustion chamber [11]. The second vibration suppression technique for the engine is vibration isolation. The challenge for designers and engineers is how to properly select vibration isolators in order to minimise the force transmission to the engine base [11–13] and the powertrain mounting system [14,15]. Further work focuses on optimisation of engine mounting systems and blocks can be found in References [16–23], while the literature of using meta-heuristic algorithms (MHs) or evolutionary algorithms (EAs) for engine mounting and engine part design can be seen further in Reference [24]. Both methods has been studied and used in industry, but the new design technique still lacks development. Recent works of automotive technology have focused on designing the motor of an electric vehicle (EV) to increase its efficiency and reduce vibration [25]. This kind of automotive uses an electric motor as a power or hybrid with the traditional engine. This research focuses on optimising the flux-weakening performance and reducing the vibration of an Interior permanent magnet (IPM) motor for EVs using the evolutionary algorithm (EA), which focuses on the source of vibration similar to our present research. Furthermore, this kind of designing problem is multi-objective optimisation, but the authors compromise it to be a single objective. So, in the present research, we focus on using a multi-objective evolutionary algorithm (MOEA) to alleviate the vibration of a single-cylinder engine. This research proposes a new design strategy for the vibration suppression of a single-cylinder engine using a multi-objective evolutionary algorithm (MOEA). In this design, design variables including the shape and sizing parameters of the engine are proposed to suppress the inertia force and pressure force, which are the main vibration causes of this kind of engine. The MOEA optimiser is the hybrid of multi-objective population-based incremental learning and differential evolution (RPBIL-DE). The new design technique can reduce the vibrations that cause machine damages, human discomfort, and user-accumulated fatigue and pain, which can lead to sustainable development.
2. Single-Cylinder Engine Model Herein, vibration analysis of a single-cylinder engine system is simplified for ease in the computation of an optimisation process. The kinematic and dynamic force analyses of a crank–slider with external ignition forces are carried out, while the obtained reactions will be used as external forces for the engine box and mounting system.
2.1. Kinematic and Kinetic Analyses
Figure1 shows a crank–slider with the crank radius R and connecting rod L. The parameters θ2 and θ3 are the angular positions of links 2 and 3, respectively, while x is the position vector of the . .. piston. Given that θ2, θ2, and θ2 are known input variables, we can have the relation:
R sin(θ2) = L sin(θ3) (1) Sustainability 2018, 10, x FOR PEER REVIEW 3 of 19
Sustainability 2018, 10, 2067 3 of 19 Rsin(2 ) Lsin(3 ) (1)
Figure 1.Single-cylinder engine model. Figure 1. Single-cylinder engine model. Having determined the first and second-order derivatives of Equation (1) with respect to time t Having determined the first and second-order derivatives of Equation (1) with respect to time t and rearranged the derivative equations, the parameters. 3 and..3 can be obtained as: and rearranged the derivative equations, the parameters θ3 and θ3 can be obtained as: 3 arcsin(Rsin(2 )/ L) θ3= arcsin (R sin(θ2)/L) .3 [R.2 cos(2 )]/[Lcos(3 )] (2) θ = [Rθ cos(θ )]/[L cos(θ )] 3 2 2 2 3 2 (2) .. . .. . 3 {L23 sin(3 ) R[2 cos(2 ) 22 sin(2 )]}/[Lcos(3 )] θ3 = Lθ3 sin(θ3) + R[θ2 cos(θ2) − θ2 sin(θ2)] /[L cos(θ3)] The position of the piston can be written as: The position of the piston can be written as: x Rcos2 Lcos(3) (3)
The velocity and acceleration of thex = pistonR cos θcan2 + thenL cos be(θ3 determined) by differentiating Equation(3) (3): The velocity and acceleration of the piston can then be determined by differentiating Equation (3): x R sin L sin( ) . 2 2 . 3 3 . (4) xx= −RR[θ22sincosθ2 −Lθ3 sin(θ3]) L[2 cos( ) sin( )] .. . 2 2 .. 2 2 . 23 3 .. 3 3 (4) x = −R[θ cos θ2 + θ2 sin θ2] − L[θ cos(θ3) + θ3 sin(θ3)] 2 3 For the kinematic analysis of a crank–slider, if we have the input values ofθ2, , and , the . ..2 2 angularFor position, the kinematic velocity analysis, and of acceleration a crank–slider, of link if we 3, haveas well the as input the position, values of velocityθ2, θ2, and, andθ2 ,acceleration the angular position,of piston velocity,4, can be and computed acceleration using of equations link 3, as (1) well–(4). as the position, velocity, and acceleration of piston 4, canFor be computed dynamic force using analysis Equations in (1)–(4). this paper, the crank–slider system is thought of as being kinematicallyFor dynamic driven force by input analysis angular in this velocity paper, and the accelerat crank–sliderion at systemthe input is link thought 2. The of kinematic as being kinematicallyanalysis can be driven accomplished by input angularas previously velocity detailed. and acceleration A free-body at the diagram input of link a 2.crank The–slider kinematic at a analysisparticular can motion be accomplished phase is shown as previouslyin Figure 2. detailed. The piston A free-body is subject diagram to external of aforces crank–slider due to gas at apressure particular P, while motion the phase moment is shown M2 is applied in Figure at2 .link The 2, piston so as to is subjectmeet its to prescribed external forcesmotion. due The to force gas pressureanalysis canP, while be computed the moment usingM the2 is following applied at system link 2, of so equations as to meet: its prescribed motion. The force analysis can be computed using the following system of equations: [A]{F} = {RHS} (5) Where: [A]{F} = {RHS} (5) Sustainability 2018, 10, x FOR PEER REVIEW 4 of 19
Sustainability 2018, 10, 2067 4 of 19 I 22 I 22 022 021 021 r r r r 0 0 1 O2 / G2 ,y O2 / G2 ,x B / G2 ,y B / G2 ,x 12 Where: 022 I 22 I 21 021 021 A I2×2 0 r I2×2 r r 02×r2 0 02×10 02×1 h 12 i h B / G3 ,y B / G3 ,x i C / G3 ,y C / G3 ,x −r r −r r 0 0 1 O2/G2,y O2/G2,x B/G2,y B/G2,x 1×2 0 0 0 I 0 0 22 −I 22 I22 0 21 0 2×2 2×2 2×1 1 2×1 2×1 A = h i h i 01×2 rB/G ,y rB/G ,x −rC/G ,y rC/G ,x 0 0 3 3 3 3 " # 0 02×2 02×2 −I2×2 02×1 F = {F12,xF12,yF32,xF32,yF43,xF43,yF14,yM2}T 1 m a 2 G2 T F = {F12,xF12,yF32,xF32,yF43,xF43,yF14,yM2} IG 2 2 m2aG RHS m3aG 2 I α3 I G2 2 = G3 3 and: RHS m3aG3 . m aI αF 4 CG3 3 P and: . m4aC − FP Fij is the constrained force acting at body i by body j, mi is the mass of body i, IGi is the moment Fij is the constrained force acting at body i by body j, mi is the mass of body i, IGi is the moment of of inertia with respect to the axis at the centroid of body i, ri/j is the relative position vector of point i inertia with respect to the axis at the centroid of body i, ri/j is the relative position vector of point i with respect to point j, and aC and aG2 are the acceleration vector of link 4 (piston) and the centre of with respect to point j, and aC and aG2 are the acceleration vector of link 4 (piston) and the centre of gravity i in the x–y coordinates, respectively. The gas pressure P (kPa) in one cycle for some engine gravity i in the x–y coordinates, respectively. The gas pressure P (kPa) in one cycle for some engine has has been proposed by Asadi et al. [8] as follows: been proposed by Asadi et al. [8] as follows: 101.3 0 2 101.3 0 ≤ θ ≤ π 1.21 2 − 7.537.53x x1.21 π≤2 θ 2≤ 2π 2 = ≤ ≤ ( ) P P 29502950 22π θ22 1313/6/ 6π (kPa)kPa (6(6)) −1.21 29.8x 1.21 9/4π ≤ θ2 ≤ 3π 29.8x 9 / 4 2 3 101.3 3π ≤ θ2 ≤ 4π 101.3 3 2 4 Where x in above equation is in Equation (3). WhereThe x in external above forceequationFP due is in to E gasquation pressure (3). can be computed by: The external force FP due to gas pressure can be computed by: FP = (P − Patm)Ap (7) FP = (P− Patm)Ap (7) 2 WhereWhere Patmatm isis atmosphere atmosphere pressure pressure (kPa) (kPa) and and AApp isis the the piston piston area area (m (m2).).
Figure 2. FreeFree-body-body diagram of a crank crank–slider.–slider.
2.2. Engine Vibration System Sustainability 2018, 10, 2067 5 of 19
2.2. Engine Vibration System A mounted engine system will be modeled as a simple spring-mass system with the rigid mass Sustainability 2018, 10, x FOR PEER REVIEW 5 of 19 having six degrees of freedom. Linear spring behavior is assumed as shown in Figure3, while force and displacementA mounted relation engine can system be written will be modeled as: as a simple spring-mass system with the rigid mass having six degrees of freedom. Linear spring behavior is assumed as shown in Figure 3, while force and displacement relation can be written as: F = k(r − r0) = kδr (8) Sustainability 2018, 10, x FOR PEER REVIEW 5 of 19 F k(r r0 ) kr (8) Where k is springA mounted stiffness, enginer0 systemis the will position be modeled of the as a unstretched simple spring- spring,mass systemr is with the the position rigid mass vector of the Where k is spring stiffness, r0 is the position of the unstretched spring, r is the position vector of the spring underhavingspring the undersix force degrees theF force, andof freedom. F,δ andr is δ a rLinear is spring a spring spring translational translational behavior is vector. assumed vector. as shown in Figure 3, while force and displacement relation can be written as:
F k(r r0 ) kr (8)
Where k is spring stiffness, r0 is the position of the unstretched spring, r is the position vector of the spring under the force F, and δr is a spring translational vector.
Figure 3. Spring displacement vector in three-dimensional spaces. Figure 3. Spring displacement vector in three-dimensional spaces. A rigid body attached with a number of linear springs is given in Figure 4. From the figure, the position vector of the i-th spring can be expressed with respect to the centroid position as: A rigid body attached with a number of linear springs is given in Figure4. From the figure, r r r (9) the position vector of the i-thFigure spring 3. Spring can displacement be expressedi vectorc in withci three- respectdimensional to spaces. the centroid position as:
Where ri is the position vector of spring I, rc is the position vector of the mass centre, and rci is the potionA rigidvector body of spring attached i with with respect a number to the ofr centroid.i linear= rc +springs rci is given in Figure 4. From the figure, the (9) positionWhen vector the bodyof the is i- inth motion,spring can the be derivation expressed of with the vectorsrespect into Equationthe centroid (9) canposition be written as: as: Where ri is the position vector of spring I, rc ris ther positionr vector of the mass centre, and rci is the r i rc cir (10(9)) potion vector of spring i with respect to the centroid.i c ci Where ri is the position vector of spring I, rc is the position vector of the mass centre, and rci is the Whenpotion the bodyvector isof inspring motion, i with respect the derivation to the centroid. of the vectors in Equation (9) can be written as: When the body is in motion, the derivation of the vectors in Equation (9) can be written as: δri = δrc + δrci (10) ri rc rci (10)
Figure 4. Vector position of spring position relative tothe center of mass.
Figure 4. Vector position of spring position relative tothe center of mass. Figure 4. Vector position of spring position relative tothe center of mass. Sustainability 2018, 10, 2067 6 of 19
As the centroid and the i-th point are at the same body, we can have:
δri = δrc + δθ × rci (11)
Where δθ is the vector of rotation displacements of the body. The translation and rotation vectors can be defined as: ux θx δrc = uy , δθ = θy (12) uz θz
Where ui is the translation in i-th direction and θi is the angular displacement in the i-th axis. The rigid body has six degrees of freedom, as shown in Figure5. By substituting Equation (12) into Equation (11), we have:
ux u u + θ r − θ r 1 0 0 0 r −r y x y ci,z z ci,y ci,z ci,y uz δri = uy + θzrci,x − θxrci,z = 0 1 0 −rci,z 0 rci,x = Tid (13) θx uz + θxrci,y − θyrci,x 0 0 1 rci,y −rci,x 0 θy θz
Where Ti is called a transformation matrix for the i-th spring and d is the displacement vector of the body. As a result, elastic potential energy of the i-th spring is:
1 1 1 U = k δrTδr = dT(k TTT )d = dTK d (14) i 2 i i 2 i i i 2 i If the spring-mass system has n linear springs, the total elastic potential energy can be computed as:
n ! 1 T 1 T U = d ∑ Ki d = d Kd (15) 2 i=1 2
Where K is the stiffness matrix of the system. The kinetic energy or the work due to inertial forces can be computed as: 1 . T . 1 . T . 1 . T . T = mδr δr + δθ Iδθ = d Md (16) 2 c c 2 2 Where: m 0 0 0 0 0 0 m 0 0 0 0 " # m 0 0 0 m 0 0 0 M = = (17) 0 I 0 0 0 Ixx −Ixy −Ixz 0 0 0 −Iyx Iyy −Iyz 0 0 0 −Izx −Izy Izz m is body mass, and I is the matrix of moments of inertia. Adding the work done by external forces to the system, a vibration model of a three-dimensional (3D) spring-mass system can be expressed as:
.. Md + Kd = F (18)
Damping can be added to the model using a proportional damping matrix or a Reylize damping i.e., C = αM + βK (19) Sustainability 2018, 10, x FOR PEER REVIEW 7 of 19
Damping can be added to the model using a proportional damping matrix or a Reylize damping i.e.: Sustainability 2018, 10, 2067 C = αM + βK 7(19 of) 19 Where α and β are the proportional damping constants to be specified. The dynamic model then Wherebecomesα: and β are the proportional damping constants to be specified. The dynamic model then becomes: .. . Md + Cdd+ Kd = FF((tt)) (20(20))
In this work, numerical solutions of the system of differential equations in Equation (20) can be carried out by using Newmark Newmark’s’s integration technique [26].].
Figure 5. DegreeDegree of freedom of rigid body in three three-dimensional-dimensional spaces.
3. Hybrid RPBIL RPBIL-DE-DE for Multi Multi-Objective-Objective Optimi Optimisationsation The multi multi-objective-objective design design problems problems of trusses of trusses [27,28 [27] ,and28] mechanism and mechanismss [29,30] [ 29have,30] been have solved been withsolved the with hybridi thes hybridisationation of real-code of real-code population population-based-based incremental incremental learning and learning differential and differential evolution (RPBILevolution-DE (RPBIL-DE).). This optimizer This is optimizer found to isbe found one of to the be high one- ofperformance the high-performance multi-objective multi-objective optimisers, andoptimisers, is therefore and is thereforeselected to selected solve toour solve problem our problem in this in study.this study. The The algorithm algorithm is isextended extended from from References [[3131,32,32]]by by integrating integrating into into it the it differentialthe differential evolution evolution (DE) operators (DE) operators in the main in procedurethe main procedureof real-code of real population-based-code population incremental-based incremental learning learning (RPBIL), (RPBIL) leading, leading to a hybrid to a hybrid algorithm algorithm [27]. [This27]. This technique technique is developed is developed to avoidto avoid a prematurea premature convergence convergence searching searching of of RPBIL RPBIL duedue toto the probability of of matrix matrix updating updating rel relyingying on on the the current current best best solution solution.. The The mutation mutation and and crossover crossover of DEof DE are are incorporated incorporated into into a RPBIL a RPBIL procedure. procedure. This This hybridi hybridisationsation has has been been proved proved that that it it can can increase increase the p populationopulation diversity for multi multi-objective-objective optimi optimisation.sation. Additionally, Additionally, t thehe non non-dominated-dominated solution solutionss can be chose chosenn using a clustering technique that is detailed in Reference [33].]. The RPBIL-DERPBIL-DE and DE operator partsare shown in Algorithm 1, where F is a scaling factor, pc is a crossover probability, and operator partsare shown in Algorithm 1, where F is a scaling factor, pc is a crossover probability, and CR CRis the is the probability probability of selecting of selecting an elementan element of an of offspringan offspringc in c binomial in binomial crossover. crossover.
Algorithm 1.Multi-objective RPBIL-DE [27].
Input: NG(number of generation), NP (population size), nI(number of subinterval), NT(number of trays),
objective function name (fun), Pareto archive size (NA) Output: xbest, fbest
Initialisation: Pij= 1/nI for each tray, where Pij is a probability matrix Main steps : Generate a real-code population X from the probability trays and find f = fun(X) : Find a Pareto archive A Sustainability 2018, 10, 2067 8 of 19
Algorithm 1. Multi-objective RPBIL-DE [27].
Input: NG(number of generation), NP (population size), nI(number of subinterval), NT(number of trays), objective function name (fun), Pareto archive size (NA) Output: xbest, fbest Initialisation: Pij= 1/nI for each tray, where Pij is a probability matrix Main steps : Generate a real-code population X from the probability trays and find f = fun(X) : Find a Pareto archive A 1: For i = 1 to NG 2: Separate the non-dominated solutions into NT groups using a clustering technique, and find the centroid rG of each group 3: Update each tray Pij based on rG 4: Generate a real-code population X from the probability trays 5: For j = 1 to NP recombine X and A using DE operators 5.1: Select p from A randomly 5.2: Select q and r from X randomly, q 6= r 5.3: Calculate c = p + F(q −r) (DE/best/1/bin) 5.4: Set ci into its bound constraints. 5.5: If rand For more details of RPBIL-DE, see Reference [27]. 4. Design Problems A simplified forced vibration model is used in this study instead of the more complicated model as presented in Reference [34]. Figure6 displays the vibration model of a mounted single-cylinder engine where the mass matrix (including mass and moments of inertia) and the mass centre of the engine system are set to be constant. The engine box is attached to the ground by using four liner springs as shown. The origin of the reference rectangular coordinates is located at the engine box mass centre. For a computational procedure, forces and moments due to the moving links of a crank–slider are computed separately. Then, the dynamic force vector is obtained as: Fx F F y 21,x F F + F F(t) = z = 21,y 41,y (21) M 0 x My RO/G × F21 + RC/G × F41 Mz x Where RC/G = RO/G + 0 . 0 Sustainability 2018, 10, 2067 9 of 19 The parameters according to the kinematic, force, and vibration analyses are given in Table1. TheSustainability external 2018 force, 10, x FOR due PEER to pressure REVIEW inside the cylinder followed Equation (6). The fidelity of9 of the 19 optimisation result in the next section is affected by the pressure force and inertia force, which will be stud studiedied in the next section. International System System of of Units Units (SI) (SI) are are used used unless unless otherwise otherwise specified. specified. Figure 7 7 displays displays the the top top and and front front views views of of the the crank. crank. The The parameters parameters used used to to define define the the crank crank dimensions and shapeshape areare ttCC,, llPP, RC11,, RRCC2,2 ,RR2,2 ,rCr,C and, and ψψ. .If If the the values values of of those those parameters parameters are are known, known, the mass centre and moment of inertia of the crank can be calculated. Figure Figure 88 showsshows thethe connectingconnecting rod where nine design parameters areare usedused toto definedefine thethe shapeshape andand dimensionsdimensions ofof thethe rodrod asas ll11,, ll22, b1,, b2,, RR11,, RR22, ,t,t ,rrp1p1, and, and rp2rp2. It. Itshould should be be noted noted that that the the crank crank and and rod rod are are created created for for design design demonstration demonstration in this paper.paper. ForFor practicalpractical applications,applications, their their shapes shapes may may be be defined defined differently. differently. From From Figures figures7 and 7 and8, l8,P l=P l=1 ,l1 and, andR 2R=2 =R RCC22,, soso 1414 parametersparameters areare assigned assigned as as elements elements of of a a design design vector vector as asx x= = { R{RC1C1,,R RCC22, rC,, T R2,, ψψ,, llP,, ttCC, ,RR1,1 ,rp1rp1, r,p2r,p2 l2,, lt2,, bt1,, band1, andb2}Tb. 2} . Figure 6. A s single-cylinderingle-cylinder engine and its engine box.box. Table 1. System parameters. Table 1. System parameters. Parameters Symbols Quantities ParametersTotal engine mass Symbolsm 14.528 Quantities kg Total enginePiston massmass mm4 0.214.528 kg kg 2 PistonMoment mass of inertia Ixx, Iyy, Izzm, I4xy, Ixz, Iyz 0.0768, 0.0640, 0.0812,0.2 kg0, 0, 0 kg-m 2 MomentCentre of of inertia gravity Ixx,Iyy,IzzR,IG xy,Ixz,Iyz 0.0768, 0.0640,[0,0,0]T 0.0812, m 0, 0, 0 kg-m T CentreCrank of shaft gravity centre RRO/GG [−0.760, −0.0232,[0,0,0] 0.0100]m T m T CrankMount shaft centrestiffness RO/Gk [−0.760,4 × 10−60.0232, N/m 0.0100] m 6 MountCrank stiffness length Rk 0.14 × m10 N/m ConnectingCrank length rod length LR 0.3 m0.1 m Connecting rod length L 0.3 m Material density ρ 7850 kg/m3 Material density ρ 7850 kg/m3 Piston diameter d 100 mm Piston diameter d 100 mm Sustainability 2018, 10, 2067 10 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 10 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 10 of 19 Figure 7. Model of crank and design parameters. Figure 7. Model of crank and design parameters. Figure 7. Model of crank and design parameters. Figure 8. Model of the connecting rod and design parameters. Figure 8. Model of the connecting rod and design parameters. The multi-objectiveFigure design 8. Model problem of the for connecting this work rod is posed and design to find parameters. a design variable vector x such that:The multi-objective design problem for this work is posed to find a design variable vector x such that: The multi-objective design problem for this work is posed to find a design variable vector x Min: f = {f1(x),f2(x)}T such that: (22) Min: f = {f1(x),f2(x)}TT (22) Subject to: Min: f = {f 1(x),f 2(x)} (22) Subject to: Subject to: σCrank ≤ σall σσCrankCrank ≤≤ σσallall σRod ≤ σall σRod ≤ σall σRod ≤ σall λRod ≥ 1 λRod ≥ 1 λRod ≥ 1 rrp1p1 ≥≥ rp2r p2+ 0.005+ 0.005 m m rp1 ≥ rp2 + 0.005 m l2 ≥ b1 + 0.005 m l2 ≥ b1 + 0.005 m Sustainability 2018, 10, 2067 11 of 19 l2 ≥ b1 + 0.005 m b1 ≥ b2 + 0.02 m rp1 ≥ rp2 + 0.005 m l1 ≥ l2 R1 ≥ rp1 + 0.002 m rC1 ≥ R2 + 0.002 m RC1 ≥ rc1 + 0.002 m R1 ≥ R2 + 0.002 m xl ≤ x ≤ xu Where σCrank is the maximum stress on the crank, σall is an allowable stress, and σRod is the maximum stress on the connecting rod. The bound constraints are set as xl = {0.03, 0.05, 0.015, 0.01, π/6, 0.03, 0.01, 0.03, 0.02, 0.01, 0.02, 0.002, 0.02, and 0.01}T, and xu = {0.045, 0.09, 0.04, 0.03, π, 0.05, 0.03, 0.05, 0.03, T 0.03, 0.04, 0.005, 0.04, and 0.03} . The buckling factor for the rod λRod is defined as the ratio of critical load to applied load. The first three design constraints are set for structural safety, while the other constraints are assigned for manufacturing tolerances and practicality. The objective functions used in T this study are set as f = {urms + θrms, mass} . The root mean squares (RMS) of the vibration translations (urms) and rotations (θrms) over the period t∈ [0, tmax] can be computed as: r 1 Z u = (u2 +u2 + u2)dt (23) rms T x y z and: r 1 Z θ = (θ2+θ2 + θ2)dt (24) rms T x y z In the function evaluation process, with the given input design vector x being decoded, the shape and sizing parameters are repaired to meet constraints 4–12, and the inertial properties of the crank and rod can then be computed (the rest of constraints will be handled by using the non-dominated sorting scheme [35]. Then kinematic and dynamic force analyses are carried out as detailed in Section2. A simple finite element model using a three-dimensional (3D) beam element is applied to determine the maximum stresses on the crank and rod. A buckling factor is also calculated in the cases of the rod. Also, the obtained dynamic forces are used as external excitation for the vibration model of the engine. Having obtained a dynamic response, the objective functions can then be computed. Three multi-objective optimisation problems with the same design objectives and constraints but different engine rotational speeds are posed as: OPT1: min {urms + θrms, mass}, constant crank angular speed 1000 rpm OPT2: min {urms + θrms, mass}, constant crank angular speed 1500 rpm OPT3: min {urms + θrms, mass}, constant crank angular speed 2000 rpm The RPBIL-DE is used to tackle each design problem, with 10 runs starting with the same initial population. The population size is set to be 100, while the total number of iterations is 150. The crank and connecting rod are made of alloy steel AISI 4140H with a Young’s modulus of 3 211.65 GPa, σyt = 417.1 MPa, and density of 7850 kg/m . For each finite element analysis, the maximum compressive force over the period of time [0, tmax] will be used for buckling calculation. Sustainability 2018, 10, 2067 12 of 19 SustainabilitySustainability 20182018,, 1010,, xx FORFOR PEERPEER REVIEWREVIEW 1212 ofof 1919 5.5. Pressure Pressure Force Force and and Inertia Inertia Force Force ValidationValidation TheThe gas gas pressurepressure pressure forceforce force andand and inertiainertia inertia forceforce force exertexert exert onon on thethe the engineengine engine boxbox box similarsimilar similar toto anan to externalexternal an external force.force. force. TheThe Thefidelityfidelity fidelity ofof thethe of thebothboth both forcesforces forces isis veryvery is very importantimportant important inin inthethe the vibrationvibration vibration analysisanalysis analysis ofof of thethe the singlesingle single-cylinder--cylindercylinder engine, engine, whichwhich wewe willwill dodo byby consideringconsiderconsideringing thethe forcesforces versusversus thethe crankcrank angle.angle. TheThe gasgas pressurepressupressurere force,force, inertiainertia force,forceforce,, andand totaltotal forceforce inin oneone cyclecycle is is codedcoded byby using using MATLAB MATLAB commercialcommercial softwaresoftware overover thethe interval interval [0,[0, ttmaxmax]],], , asas as showshow shownnn inin in FF Figuresigureiguress 99––1111 atat 1000 rpm.rpm. The maximum gasgas pressurepressure forceforceforce exertedexertedexerted on onon the thethe piston pistonpiston headhead occurredoccurred atat thethe maximum maximum torque,torque, butbut the the maximum maximum tensile tensile force force occurred occurred duringduring thethe maximum maximum revolutionrevolution speedspeed [[8][8]8]... Figure Figure 9 99 shows showshowss that thatthat the thethe maximum maximummaximum gas gasgas pressurepressurepressure force forceforce is isis 22,374 22,37422,374 N, N,N, which whichwhich occurs occursoccurs inin thethe combustioncombustion process.process. The The i inertialinertialnertial force force duedue toto theththee slider–cranksliderslider––crankcrank mechanismmechanism in in thethe xx directiondirection isis showshow inin Figure Figure 1010;; meanwhile, meanwhile, thethe maximum maximum inertiainertia inin positive positive direction direction isis 1141N, 1141N, while while the the negativenegative inertiainertia forceforce isis 22862286 N.N. FigureFigure 111111 showsshows thethe totaltotal forceforce duedue toto thethe gasgas pressurepressure forceforce andand inertiainertia forceforce thatthathatt givegive thethe maximummaximum gas gas pressure pressure force force a asass 20,364 20,364 N, N, while while the the maximum tensiletensile forceforce isis 28672867 N.N. AllAAllll ofofof thethethe diagrams diagramdiagramss indicate indicateindicatesimilar similarsimilar trends trendtrendss to toto the thethe work workwork by byby Reference ReferenceReference [8 [8]],[8] while,, whilewhile the thethe magnitude magnitudemagnitude of of allof allall of theofof thethe forces forceforce aress areare different, differendifferen ast,t, aasas result aa resultresult of theooff thethe differences differencedifference inss the iinn thethe system systemsystem parameters. parameters.parameters. 44 xx 1010 2.52.5 22 1.51.5 11 0.50.5 Gas Pressure Force(N) Gas Pressure Force(N) 00 -0.5-0.5 00 22 44 66 88 1010 1212 1414 CrankCrank Angle(rad)Angle(rad) FigureFigure 9. 9. GasGas pressure pressure forceforce versusversus crankcrank angleangle atat 10001000 rpm.rpm. 15001500 10001000 500500 00 -500-500 Inertia Inertia Force(N) Inertia Inertia Force(N) -1000-1000 -1500-1500 -2000-2000 -2500-2500 00 22 44 66 88 1010 1212 1414 CrankCrank Angle(rad)Angle(rad) FigureFigure 10.10. InertiaInertia force force due due to to crank crank slider slider versusversus crankcrank angleangle atat 10001000 rpm.rpm. Sustainability 2018, 10, 2067 13 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 13 of 19 4 x 10 2.5 Sustainability 2018, 10, x FOR PEER REVIEW 13 of 19 2 4 x 10 2.5 1.5 2 1 1.5 Total Force(N) 0.5 1 Total Force(N) 0 0.5 -0.5 0 2 4 6 8 10 12 14 0 Crank Angle(rad) Figure 11. Total force-0.5 due to gas pressure force and inertia force versus crank angle at 1000 rpm. Figure 11. Total force due0 to gas2 pressure4 force6 and inertia8 force10 versus12 crank14 angle at 1000 rpm. Crank Angle(rad) 6. Design Results 6. Design ResultsFigure 11. Total force due to gas pressure force and inertia force versus crank angle at 1000 rpm. We implemented the RPBIL-DE for solving the design problems, ran 10 OPT1-3 runs, and chose Wethe best implemented front based theon the RPBIL-DE hypervolume forsolving indicator the of each design design problems, problem. ran According 10 OPT1-3 to its runs, definition, and chose 6. Design Results the bestthe frontlarger basedthe hypervolume on the hypervolume, the better indicatorthe Pareto of front. each Figure designs 12 problem.–14 show According the best front to its at definition,each the largerengineWe the speed. implemented hypervolume, The results the the RPBILfrom better -minimiDE the for Paretossolvinging vibrations front. the design Figures (RMS) problems 12 and–14 the, show ran mass 10 the OPT1 of best the-3 frontruns single, and at cylinder each chose engine speed.enginethe The best (kg) results front at based the from engine on minimising the speed hypervolume of vibrations1000 indicatorrpm are (RMS) in of the each and range design thes of mass problem. [0.06402 of the According0.06402] single and cylinder to its[2.688 definition, engine4.998] (kg) at thekg,the engine respectively.larger speed the hypervolume ofThe 1000 vibration rpm, the are and better in engine the the ranges mass Pareto a oft the [0.06402front. engine Figure 0.06402] speeds 12 –of14and 1500 show [2.688 rpm the are 4.998]best in thefront kg, range respectively.at eachs of engine speed. The results from minimising vibrations (RMS) and the mass of the single cylinder The vibration[0.06252 0.06252] and engine and [2.688 mass 4.986] at the kg engine. At the speed engine of speed 1500 of rpm 2000 are rpm in, the results ranges are of in [0.06252 the ranges 0.06252] ofengine [0.06012 (kg) 0. at06 the012] engine and [2.688 speed 4.957] of 1000 kg for rpm vibration are in the magnitude ranges of and [0.06402 engine 0.06402] mass, respectively. and [2.688 4.998]Some and [2.688 4.986] kg. At the engine speed of 2000 rpm, the results are in the ranges of [0.06012 0.06012] selectedkg, respectively. design solutionsThe vibration of each and design engine problemmass at the in Fengineigures speed12–14 of and 1500 the rpm corresponding are in the range cranks of– and [2.688 4.957] kg for vibration magnitude and engine mass, respectively. Some selected design sliders[0.06252 of 0.06252] each front and are [2.688 illustrated 4.986] inkg F. igureAt thes 15engine–17. speed of 2000 rpm, the results are in the ranges solutionsof [0.06012Dynamic of each 0.06 designanalyses012] and problem of [2.688 the crank 4.957] in Figures–sliders kg for vibrationin12 F–igure14 ands magnitude 15 the–17 are corresponding carried and engine out, and mass crank–sliders the, respectively. results are of shown eachSome front are illustratedinselected Figures design 18 in– Figures23. solutionsFigure 15s –18, 17 of .20 each, and design 22 display problem the components in Figures 12 of– 14the andtranslational the corresponding displacements crank of– Dynamicthesliders six engines,of each analyses front while ofare F theigure illustrated crank–sliderss 19, 21 in, andFigure 23 in sdisplay Figures15–17. the 15 components–17 are carried of the out, rotational and the displacements results are shown in Figuresof theDynamic 18six– engines.23. Figuresanalyses From 18of our ,the 20 design crankand 22– slidersresult displays in, when F theigure focus componentss 15–ing17 areon carriedvibration of the out translational amplitude, and the results, it displacementsis found are shown that of the sixourin engines,F igurestechnique 18 while– 23.can Figure control Figuress 18, the 19 20 vibration, ,21 and and 22 23 displayamplitude display the to thecomponents oscillate components in of a smallthe of translational thestrip rotational throughout displacements displacements the Pareto of of the sixfront,the engines. six while engines, the From whilechanging our Figure design of masss 19, results, 21is ,in and accordance when 23 display focusing with the componentsthe on shape vibration design of the amplitude, parameters rotational it displacementsof is the found moving that our of the six engines. From our design results, when focusing on vibration amplitude, it is found that techniqueparts of can a single control cylinder the vibration. amplitude to oscillate in a small strip throughout the Pareto front, our technique can control the vibration amplitude to oscillate in a small strip throughout the Pareto while the changing of mass is in accordance with the shape design parameters of the moving parts of front, while the changing of mass is in accordancePareto front with of OPT1 the shape design parameters of the moving 5 a singlepart cylinder.s of a single cylinder. 6 4.5 Pareto5 front of OPT1 5 6 4 4.5 54 3.5 , Engine mass , 2 f 4 3 4 3 2 3.5 , Engine mass , 2 1 f 3 2.5 0.064 0.064 0.064 0.064 0.064 0.064 0.064 3 2 f , Vibration amplitude 1 1 2.5 Figure 12. The best Pareto front of OPT1. 0.064 0.064 0.064 0.064 0.064 0.064 0.064 f , Vibration amplitude 1 Figure 12. The best Pareto front of OPT1. Figure 12. The best Pareto front of OPT1. Sustainability 2018, 10, 2067 14 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 14 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 14 of 19 Pareto front of OPT2 Pareto front of OPT2 5 6 5 6 Sustainability 2018, 10, x FOR PEER4.5 REVIEW5 14 of 19 4.5 5 Pareto front of OPT2 5 4 6 4 4 4.5 5 4 3.5 , Engine mass , 2 3.5 f , Engine mass , 3 2 f 3 4 2 2 3 4 3 3.5 1 , Engine mass , 2 1 f 3 2.5 2.5 0.0625 0.0625 0.0625 0.06252 0.0625 0.0625 0.0625 0.0625 0.0625 3 0.0625 0.0625 0.0625f , Vibration0.0625 0.0625amplitude0.0625 0.0625 0.0625 0.0625 f1, Vibration amplitude 1 1 Figure 13. The best Pareto front of OPT2. 2.5 Figure 13. The best Pareto front of OPT2. Figure0.0625 13.0.0625 The0.0625 best0.0625 Pareto0.0625 front0.0625 of0.0625 OPT2.0.0625 0.0625 f , Vibration amplitude Pareto1 front of OPT3 5 Pareto front of OPT3 6 Figure 13. The best Pareto front of OPT2. 5 6 5 Pareto front of OPT3 4.5 5 5 4.5 6 5 4 4.5 4 4 4 4 3.5 , Engine mass , 4 2 3.5 3 f , Engine mass , 2 3 f 2 3.5 , Engine mass , 2 2 3 f 3 3 2 1 3 1 2.5 1 2.50.0601 0.0601 0.0601 0.0601 0.0601 0.0601 0.0601 0.0601 0.0601 0.0601 0.0601 0.0601 f0.0601, Vibration0.0601 amplitude0.0601 0.0601 0.0601 0.0601 2.5 f1, Vibration amplitude 0.0601 0.0601 0.0601 10.0601 0.0601 0.0601 0.0601 0.0601 0.0601 f , Vibration amplitude Figure 14. The1 best Pareto front of OPT3. Figure 14. The best Pareto front of OPT3. FigureFigure 14. 14.The The bestbest Pareto front front of of OPT3. OPT3. 1 2 3 1 2 3 1 2 3 4 5 6 4 4 5 6 6 Figure 15. Some selected design solutions of OPT1. Figure 15. Some selected design solutions of OPT1. Figure 15. Some selected design solutions of OPT1. Sustainability 2018, 10, 2067 15 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 15 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 15 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 15 of 19 1 2 3 1 2 3 1 2 3 4 5 6 4 5 6 4 5 6 Figure 16. Some selected design solutions of OPT2. FigureFigure 16. 16.Some Some selectedselected design solutions of of OPT2. OPT2. Figure 16. Some selected design solutions of OPT2. 1 2 3 1 2 3 1 2 3 4 5 6 4 5 6 4 5 6 Figure 17. Some selected design solutions of OPT3. Figure 17. Some selected design solutions of OPT3. FigureFigure 17.17. SomeSome selected design design solutions solutions of of OPT3. OPT3. -4 x 10 -4 2 x 10 -4 1 2x 10 1 2 , m 0 2 x 1 , m 2 u 0 x 3 u , m 0 23 x -2 u 4 -20 0.02 0.04 0.06 0.08 0.1 0.12 34 -2 0 0.02 0.04 time,0.06 sec 0.08 0.1 0.12 5 0 0.02 0.04 0.06 0.08 0.1 0.12 45 -5 time, sec 6 x 10 -5 time, sec 5 5 x 10 6 -5 5x 10 6 5 , m 0 y , m u 0 y u , m 0 y -5 u -50 0.02 0.04 0.06 0.08 0.1 0.12 -5 0 0.02 0.04 time,0.06 sec 0.08 0.1 0.12 0 -5 0.02 0.04 time,0.06 sec 0.08 0.1 0.12 x 10 -5 time, sec 1 x 10 -5 1x 10 1 , m 0 z , m u 0 z u , m 0 z -1 u -10 0.02 0.04 0.06 0.08 0.1 0.12 -1 0 0.02 0.04 time,0.06 sec 0.08 0.1 0.12 0 0.02 0.04 time,0.06 sec 0.08 0.1 0.12 time, sec Figure 18. The components of the translational displacements of the six engines in Figure 15. Figure 18. The components of the translational displacements of the six engines in Figure 15. FigureFigure 18. 18.The The components components ofof thethe translationaltranslational displacements displacements of of the the six six engines engines in inFigure Figure 15. 15 . Sustainability 2018, 10, 2067 16 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 16 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 16 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 16 of 19 -4 x 10-4 2x 10-4 2x 10 1 2 1 0 12 0 2 , degree x 23 , degree 0 x 3 , degree x -2 34 -20 0.02 0.04 0.06 0.08 0.1 0.124 -20 0.02 0.04 0.06 0.08 0.1 0.1245 0 0.02 0.04 time,0.06 sec 0.08 0.1 0.12 5 time, sec 56 0.01 time, sec 6 0.01 6 0.01 0 0 , degree y , degree 0 y , degree y -0.01 -0.01 0 0.02 0.04 0.06 0.08 0.1 0.12 -0.01 0 0.02 0.04 0.06 0.08 0.1 0.12 time, sec 0 0.02 0.04 time,0.06 sec 0.08 0.1 0.12 0.05 time, sec 0.05 0.05 0 0 , degree z , degree 0 z , degree z -0.05 -0.05 0 0.02 0.04 0.06 0.08 0.1 0.12 -0.05 0 0.02 0.04 0.06 0.08 0.1 0.12 time, sec 0 0.02 0.04 time,0.06 sec 0.08 0.1 0.12 time, sec Figure 19. The components of the rotational displacements of the six engines in Figure 15. FigureFigure 19. 19.The The components components ofof thethe rotationalrotational displacements of of the the six six engines engines in inFigure Figure 15. 15 . Figure 19. The components of the rotational displacements of the six engines in Figure 15. -4 x 10-4 2x 10-4 2x 10 1 2 1 , m 0 12 x , m 0 2 u x , m u 0 23 x u -2 3 -20 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.0834 -20 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.084 0 0.01 0.02 0.03 time,0.04 sec 0.05 0.06 0.07 0.0845 -5 time, sec 5 x 10-5 time, sec 56 5x 10-5 6 5x 10 6 5 , m 0 y , m 0 u y , m u 0 y u -5 -50 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -50 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0 0.01 0.02 0.03 time,0.04 sec 0.05 0.06 0.07 0.08 -5 time, sec x 10-5 time, sec 1 x 10-5 1 x 10 1 , m 0 z , m 0 u z , m u 0 z u -1 -1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 time, sec 0 0.01 0.02 0.03 time,0.04 sec 0.05 0.06 0.07 0.08 time, sec Figure 20. The components of the translational displacements of the six engines in Figure 16. Figure 20. The components of the translational displacements of the six engines in Figure 16. FigureFigure 20. 20.The The components components ofof thethe translational displacements displacements of of the the six six engines engines in inFigure Figure 16. 16 . -4 x 10-4 5 x 10-4 5 x 10 1 5 1 0 12 0 2 , degree x , degree 0 23 x 3 , degree -5 x -5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 34 -5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 4 0 0.01 0.02 0.03 time,0.04 sec 0.05 0.06 0.07 0.08 45 -3 time, sec 5 x 10-3 time, sec 56 5 x 10-3 6 5 x 10 6 5 0 0 , degree y , degree 0 y , degree -5 y -5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -5 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 time, sec 0 0.01 0.02 0.03 time,0.04 sec 0.05 0.06 0.07 0.08 time, sec 0.05 0.05 0.05 0 0 , degree z , degree 0 z , degree -0.05 z -0.05 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -0.05 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 time, sec 0 0.01 0.02 0.03 time,0.04 sec 0.05 0.06 0.07 0.08 time, sec Figure 21. The components of the rotational displacements of the six engines in Figure 16. Figure 21. The components of the rotational displacements of the six engines in Figure 16. FigureFigure 21. 21The. The components components ofof thethe rotational displacements displacements of of the the six six engines engines in inFigure Figure 16. 16 . Sustainability 2018, 10, 2067 17 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 17 of 19 Sustainability 2018, 10, x FOR PEER REVIEW 17 of 19 -4 x 10 -4 2 x 10 2 1 1 , m 0 2 x , m 0 2 u x u 3 -2 3 -20 0.01 0.02 0.03 0.04 0.05 0.06 4 0 0.01 0.02 0.03 0.04 0.05 0.06 4 time, sec 5 -5 time, sec 5 x 10 -5 6 5 x 10 6 5 , m 0 y , m 0 u y u -5 -50 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 time, sec -5 time, sec x 10 -5 1 x 10 1 , m 0 z , m 0 u z u -1 -10 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 time, sec time, sec Figure 22. The components of the translational displacements of the six engines in Figure 17. Figure 22. The components of the translational displacements of the six engines in Figure 1717.. -4 x 10 -4 5 x 10 5 1 1 0 2 0 2 , degree x , degree 3 x -5 3 -50 0.01 0.02 0.03 0.04 0.05 0.06 4 0 0.01 0.02 0.03 0.04 0.05 0.06 4 time, sec 5 -3 time, sec 5 x 10 -3 6 5 x 10 6 5 0 0 , degree y , degree y -5 -50 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 time, sec time, sec 0.05 0.05 0 0 , degree z , degree z -0.05 -0.050 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 time, sec time, sec Figure 23. The components of the rotational displacements of the six engines in Figure 17. Figure 23. The components of the rotational displacements of the six engines in Figure 1717.. 7.7. ConclusionsConclusions 7. Conclusions TheThe vvibrationibration suppressionsuppression ofof aa singlesingle--cylindercylinder engineengine byby meansmeans ofof multimulti--objectiveobjective evolutionaryevolutionary The vibration suppression of a single-cylinder engine by means of multi-objective evolutionary optimisationoptimisation is is proposed. proposed. Simple Simple kinematic kinematic and and dynamic dynamic force force analyses analyses of of a a single single cylinder cylinder are are optimisation is proposed. Simple kinematic and dynamic force analyses of a single cylinder are presented.presented. The The multi multi--objectiveobjective design design problems problems are are posed posed to to minimise minimise thethe massmass of of the the engine engine presented. The multi-objective design problems are posed to minimise the mass of the engine mechanismmechanism andand vibrationvibration amplitudesamplitudes ofof thethe engineengine system.system. WeWe implementedimplemented RPBILRPBIL--DEDE toto solvesolve thethe mechanism and vibration amplitudes of the engine system. We implemented RPBIL-DE to solve designdesign problems.problems. TheThe obtainedobtained designdesign solutionssolutions areare illustratedillustrated andand analysed.analysed. TheThe computationcomputation resultsresults the design problems. The obtained design solutions are illustrated and analysed. The computation revealreveal thatthat thethe proposedproposed designdesign processprocess foforr formingforming thethe movingmoving partsparts ofof aa singlesingle--cylindercylinder engineengine isis results reveal that the proposed design process for forming the moving parts of a single-cylinder practical.practical. OurOur techniquetechnique cancan controlcontrol thethe vibrationvibration amplitudeamplitude toto oscillateoscillate inin aa smallsmall stripstrip throughoutthroughout engine is practical. Our technique can control the vibration amplitude to oscillate in a small strip thethe ParetoPareto frontfront byby optimioptimissationation ofof thethe movingmoving partpartss ofof aa singlesingle--cylindercylinder engineengine.. WithWith thethe useuse ooff throughout the Pareto front by optimisation of the moving parts of a single-cylinder engine. With the RPBILRPBIL--DE,DE, multiple multiple solutions solutions for for decision decision--makingmaking can can be be obtained obtained within within one one optimisation optimisation run. run. use of RPBIL-DE, multiple solutions for decision-making can be obtained within one optimisation run. FutureFuture workwork willwill bebe thethe useuse ofof threethree--dimensionaldimensional finitefinite elementelement analysisanalysis forfor calculatingcalculating thethe designdesign Future work will be the use of three-dimensional finite element analysis for calculating the design constraints.constraints. ThisThis couldcould bebe aa timetime--consumingconsuming process,process, whichwhich mmayay requirerequire aa surrogatesurrogate--assistedassisted MOEA.MOEA. constraints. This could be a time-consuming process, which may require a surrogate-assisted MOEA. Also,Also, differentdifferent shapesshapes ofof thethe crankcrank andand connectingconnecting rodrod shouldshould bebe studied.studied. Also, different shapes of the crank and connecting rod should be studied. Author Contributions: Suwin Sleesongsom and Sujin Bureerat conceived and designed the numerical Author Contributions: Suwin Sleesongsom and Sujin Bureerat conceived and designed the numerical experiments; Suwin Sleesongsom performed the numerical experiments; Suwin Sleesongsom and Sujin Bureerat experiments; Suwin Sleesongsom performed the numerical experiments; Suwin Sleesongsom and Sujin Bureerat analyzed the data; Suwin Sleesongsom wrote the paper. analyzed the data; Suwin Sleesongsom wrote the paper. Sustainability 2018, 10, 2067 18 of 19 Author Contributions: Suwin Sleesongsom and Sujin Bureerat conceived and designed the numerical experiments; Suwin Sleesongsom performed the numerical experiments; Suwin Sleesongsom and Sujin Bureerat analyzed the data; Suwin Sleesongsom wrote the paper. Acknowledgments: The authors are grateful for the financial support provided by the Thailand Research Fund, and King Mongkut’s Institute of Technology Ladkrabang. Conflicts of Interest: The authors declare no conflict of interest. References 1. Paul, B. Kinematics and Dynamics of Planar Machinery; Pretice-Hall: New Jersey, NJ, USA, 1979. 2. Lowen, G.G.; Tapper, F.R.; Berkof, R.S. Balancing of linkages-an update. Mech. Mach. Theory 1983, 18, 213–220. [CrossRef] 3. Zhang, S.M.; Chen, J.H. The Optimum balance of shaking force and shaking moment of linkage. Mech. Mach. Theory 1994, 30, 589–597. 4. Snyman, J.A.; Heyns, P.S.; Vermeulen, P.J. Vibration isolation of a mounted engine through optimization. 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