Kinematic Conceptual Design of In-Line Four-Cylinder Variable Compression Ratio Engine Mechanisms Considering Vertical Second Harmonic Acceleration

Home , Cylinder (engine), Engine configuration, Stroke (engine)

applied sciences

Article Kinematic Conceptual Design of In-Line Four-Cylinder Variable Compression Ratio Engine Mechanisms Considering Vertical Second Harmonic Acceleration

Seung Woo Kwak 1 , Jae Kyung Shim 2,* and Young Kwang Mo 1

1 Graduate School, Department of Mechanical Engineering, Korea University, Seoul 02841, Korea; [email protected] (S.W.K.); [email protected] (Y.K.M.) 2 School of Mechanical Engineering, Korea University, Seoul 02841, Korea * Correspondence: [email protected]; Tel.: +82-2-3290-3362

Received: 19 April 2020; Accepted: 27 May 2020; Published: 29 May 2020

Abstract: In the in-line four-cylinder engine, it is well known that the shaking force is due to the vertical second harmonic acceleration components of the pistons. This paper proposes a kinematic conceptual design method to determine the kinematic structure of a feasible in-line four-cylinder variable compression ratio (VCR) engine and its dimensions that would yield a lower vertical second harmonic acceleration at joints. Through type and dimensional synthesis, candidate VCR engine mechanisms are chosen and their dimensions satisfying design specifications are determined. Based on the analysis of the vertical second harmonic acceleration components at the joints, a feasible mechanism for an in-line four-cylinder VCR engine is selected. Then, the method finds the dimensions that yield a nearly minimized sum of the vertical second harmonic acceleration at each joint by adjusting the link lengths within specified tolerances. For validation, the result is compared with that of a constrained optimization using MATLAB. The proposed method would be useful at the conceptual design stage of multi-link multi-cylinder VCR and variable-stroke engine mechanisms where the second harmonic acceleration is an important design factor in the automotive industrial applications.

Keywords: kinematic conceptual design; mechanism design; variable compression ratio (VCR) engine mechanism; harmonic acceleration analysis; vertical second harmonic acceleration

1. Introduction The general performance and thermal efficiency of the internal combustion engine have been improved by various technologies such as turbochargers, fuel injection systems, and variable valve actuation systems [1]. In addition to the above, the variable compression ratio (VCR) engine technology has been considered as a method for improving fuel efficiency and reducing pollutants [2–5]. Various approaches have been suggested for the variation of the compression ratio, which include moving the crankshaft axis or the cylinder head, varying the combustion chamber volume or the piston deck height, and modifying the connecting rod geometry [6–10]. Numerous VCR engine mechanisms have been proposed [11–13], and their kinematic structures have been identified [14]. In general, as the VCR engine mechanism has more links than the conventional ﬁxed compression ratio engine, the design of a VCR engine mechanism could be a quite complicated problem: the kinematic structure and dimensions of the mechanism that ﬁts within the internal space of the engine must be determined, and then the dynamic characteristics and the balancing of shaking force and shaking moment need to be considered. The latter problem on the balancing of the conventional engine has been studied extensively [15–17]. The vertical second harmonic acceleration components of the

Appl. Sci. 2020, 10, 3765; doi:10.3390/app10113765 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 19

shaking force and shaking moment need to be considered. The latter problem on the balancing of the conventional engine has been studied extensively [15–17]. The vertical second harmonic acceleration Appl. Sci. 2020, 10, 3765 2 of 19 components of the pistons cause the shaking moments in the in-line three-cylinder, in-line five-cylinder, and V six-cylinder engines; the shaking forces in the in-line four-cylinder engine. In pistonsorder to cause balance the shaking the shaking moments forces, in the an in-lineextra three-cylinder,device, such as in-line a balancer, five-cylinder, which androtates V six-cylinder at double engines;engine speed, the shaking is used forces to balance in the the in-line shaking four-cylinder force [18,19]. engine. From In orderthis point to balance of view, the even shaking though forces, the ancomplete extra device, dynamic such analysis as a balancer, of a whichVCR engine rotates atunder double development engine speed, is isnot used possible to balance until the the shaking mass forceproperties [18,19 ].of From the parts this point are ofdetermined, view, even consider though theing completethe vertical dynamic second analysis harmonic of a VCRacceleration engine undercomponents development would isprovide not possible a good until starting the mass point properties at the conceptual of the parts design are determined, stage of consideringa new VCR theengine vertical mechanism. second harmonic acceleration components would provide a good starting point at the conceptualIn the designcase of stage the fixed of a new compre VCRssion engine ratio mechanism. engine, the crank can be shortened to decrease the magnitudeIn the caseof the of second the fixed harmonic compression acceleration ratio engine,of the piston, the crank but sh canortening be shortened the crank to length decrease causes the magnitudea change in of the the stroke. second For harmonic the VCR acceleration engine mechanis of the piston,m, since but more shortening links can the be crank used length to connect causes the a changepiston and in the the stroke. crank, Forthe second the VCR harmonic engine mechanism,acceleration sincecan be more reduced links by can adjusting be used the to connectlink lengths the pistonwithout and scarifying the crank, a desired the second stroke harmonic and top acceleration dead center can(TDC) be reducedposition. by adjusting the link lengths withoutIn scarifyingthis paper, a desireda kinematic stroke conceptual and top dead design center process (TDC) position.of in-line four-cylinder VCR engine mechanismsIn this paper,considering a kinematic the vertical conceptual second design harmonic process acceleration of in-line component four-cylinder at VCReach enginejoint is mechanismsproposed. The considering proposed method the vertical includes second the harmonictype and accelerationdimensional componentsynthesis of atcandidate each joint VCR is proposed.engine mechanisms, The proposed harmonic method includesacceleration the typeanalysis, and dimensional selection of synthesis a feasible of candidatemechanism, VCR and engine the mechanisms,minimization harmonic of the vertical acceleration second analysis, harmonic selection acceleration of a feasible sum in mechanism, the selected and mechanism. the minimization For the oftype the synthesis, vertical second the atlas harmonic of ki accelerationnematic chains sum inis theused selected to choose mechanism. appropriate For the candidates. type synthesis, The thedisplacement atlas of kinematic equations chains of candidate is used to choosemechanisms appropriate are derived candidates. and used The displacementto determine equationsthe initial ofdimensions candidate of mechanisms links that satisfy are derived prescribed and design used tospecifications. determine the Then, initial the dimensionsFourier series of expression links that satisfyfor the prescribeddisplacement design of each specifications. joint is determined Then, the numerically Fourier series and expressiondifferentiated for to the obtain displacement the vertical of eachharmonic joint isacceleration determined components. numerically Among and diff theerentiated candidates, to obtain a feasible the vertical mechanism harmonic is selected acceleration based components.on the vertical Among second theharmonic candidates, acceleration a feasible analysis mechanism when they is selected are used based in the on in-line the vertical four-cylinder second harmonicengine, and acceleration then by varying analysis the when link lengths they are of used the ininitially the in-line designed four-cylinder mechanism engine, within and prescribed then by varyingtolerances, the the link final lengths dimensions of the initially of the mechanism designed mechanism that yields withina nearly prescribed minimized tolerances, sum of the the second final dimensionsharmonic acceleration of the mechanism are determined. that yields a nearlyThe resu minimizedlt is compared sum of the with second that harmonic of a accelerationconstrained areoptimization determined. for Thethe validati result ison compared of the proposed with that method. of a constrained optimization for the validation of the proposed method. 2. Understanding of Harmonic Acceleration in the In-Line Four-Cylinder Engine 2. Understanding of Harmonic Acceleration in the In-Line Four-Cylinder Engine A slider-crank mechanism used in the conventional fixed compression ratio internal combustionA slider-crank engine mechanismis shown in usedFigure in 1. the The conventional displacement fixed of compressionthe piston pin, ratio s, can internal be derived combustion as engine is shown in Figure1. The displacement of the piston pin, s, can be derived as − θ 2 =+− θ lers cos  srsin 1 , (1)  rll e r cos θ2 s = rsin θ + 1 , (1)  −   r − l  where θ is the crank angle, r is the crank length, l is the connecting rod length, and e is the piston whereoffset.θ is the crank angle, r is the crank length, l is the connecting rod length, and e is the piston offset.

FigureFigure 1.1. Slider-crankSlider-crank mechanism.mechanism. Appl. Sci. 2020, 10, 3765 3 of 19

Equation (1) and its time derivatives can be directly used to determine the motion of the piston, however, in order to ﬁnd the eﬀects of changes in the design variables r and l on the acceleration thus on the inertia forces, an approximate form of Equation (1) is used [17–19]. When e = 0, for example, the binomial series expansion of the square root term in Equation (1) yields r r2 1r2 1r4 1 r6 1 cos2 θ = 1 cos2 θ cos4 θ cos6 θ . (2) − l − 2 l − 8 l − 16 l − · · ·

In most engines, since the crank-connecting rod ratio r/l is less than 1/3, Equation (2) can be estimated quite closely by the ﬁrst two terms. Hence, Equation (1) can now be approximated as ( ) l 1r s r sin θ + cos2 θ . (3) r − 2 l

Substituting the trigonometric identity cos2 θ = (1 + cos 2θ)/2 into Equation (3) gives the displacement of the piston pin as ( ) l 1r s r sin θ + (1 + cos 2θ) . (4) r − 4 l

Since the position of the piston pin is a periodic function of the crank angle, the Fourier series approximation of Equation (1) would yield the same as Equation (4). Assuming the angular velocity of the crank, ω is constant, the second time derivative of Equation (4) gives the acceleration of the piston pin as

.. r s rω2 sin θ + cos 2θ , (5) − l which consists of the harmonics of the crank angle θ = ωt where t is the time. Let meq be the equivalent concentrated mass of the piston and connecting rod at the piston pin, .. 2 then the inertia force at the piston pin, F = meqs, has its primary force component meqrω sin θ due to 2 2 − the ﬁrst harmonic acceleration and the secondary force component meqω r /l cos 2θ induced by the second harmonic acceleration. When the slider-crank mechanism is used in an in-line four-cylinder engine with crank throws at 90◦, 270◦, 270◦, and 90◦ phase angles, the harmonic acceleration components up to the fourth harmonic at the piston pins in cylinder 1 and cylinder 2 are calculated as shown in Figure2. The nth harmonic acceleration components expressed in terms of cos(nθ) and sin(nθ) in cylinders 1 and 4 appear as cos{n(θ + π)} and sin{n(θ + π)} in cylinders 2 and 3, each of which becomes cos(nθ) and sin(nθ) for − − odd n; cos(nθ) and sin(nθ) for even n. Hence, the inertia forces due to the odd harmonic acceleration components in cylinders 1 and 2, and in cylinders 3 and 4 would cancel out and sum to zero, however those induced by the even harmonic acceleration components do not cancel out but add up and result in unwanted vibration. Hence, the minimization of the second harmonic acceleration is a good guideline for the kinematic conceptual design of a new in-line four-cylinder VCR engine. Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 19 Appl. Sci. 2020, 10, 3765 4 of 19

) 60 60 ) 2 1st 2 1st Overall Overall 40 40 2nd 20 20 4th 3rd 4th 3rd 0 0 TDC BDC TDC BDC 2nd -20 -20

-40 -40 Piston acceleration (mm/rad acceleration Piston Piston acceleration (mm/rad acceleration Piston -60 -60 0 90 180 270 360 0 90 180 270 360 Crank angle (degree) Crank angle (degree) (a) (b)

FigureFigure 2. HarmonicHarmonic accelerationacceleration components components at theat pistonthe piston pin of pin slider-crank of slider-crank mechanism: mechanism: (a) Cylinder (a) Cylinder1 and (b) 1 Cylinder and (b) Cylinder 2. 2.

3.3. Kinematic Kinematic Conceptual Conceptual Design Design Process Process InIn this this section, section, the the general general kinematic kinematic conceptu conceptualal design design process process of of a a conventional conventional engine engine mechanismmechanism was was outlined, outlined, and and a a procedure procedure for for a a VCR VCR engine engine mechanism mechanism was was proposed. 3.1. Kinematic Conceptual Design Process of Conventional Engine Mechanisms 3.1. Kinematic Conceptual Design Process of Conventional Engine Mechanisms The kinematic design of mechanisms requires type synthesis, dimensional synthesis, and analysis. The kinematic design of mechanisms requires type synthesis, dimensional synthesis, and Regarding the kinematic design of the driving system for a conventional engine mechanism, since its analysis. Regarding the kinematic design of the driving system for a conventional engine kinematic structure is the slider-crank mechanism, the conceptual kinematic design is subject to the mechanism, since its kinematic structure is the slider-crank mechanism, the conceptual kinematic determination of the lengths of the crank and connecting rod, the piston offset, and the compression design is subject to the determination of the lengths of the crank and connecting rod, the piston height of the piston to achieve a target performance, and then is subject to kinematic and dynamic offset, and the compression height of the piston to achieve a target performance, and then is subject analysis. The guidelines for the main dimensions [1,19] and the balancing of the shaking force and to kinematic and dynamic analysis. The guidelines for the main dimensions [1,19] and the balancing shaking moment for engine mechanisms are well established [15–19]. of the shaking force and shaking moment for engine mechanisms are well established [15–19]. 3.2. Kinematic Conceptual Design Process of In-Line Four-Cylinder VCR Engine Mechanisms Considering 3.2.Vertical Kinematic Second Conceptual Harmonic AccelerationDesign Process of In-Line Four-Cylinder VCR Engine Mechanisms Considering Vertical Second Harmonic Acceleration In general, since a VCR engine mechanism requires more links and joints than the conventional engineIn general, to vary the since compression a VCR engine ratio, mechanism there can be requires many types more of links candidate and joints mechanisms. than the Hence, conventional the first enginestep of to the vary conceptual the compression design process ratio, ofthere a VCR can enginebe many is thetypes type of synthesis.candidate Formechanisms. this, graph Hence, theory the can firstbe applied step of tothe enumerate conceptual feasible design kinematic process of structures a VCR engine [20], or is appropriate the type synthesis. VCR engine For mechanismsthis, graph theorycan be can chosen be applied from the to atlasenumerate of kinematic feasible chains kinematic [14,20 structures,21]. Considering [20], or appropriate the number VCR of graphs engine to mechanismsbe enumerated can andbe chosen the complexity from the ofatlas the of dimensional kinematic chains synthesis [14,20,21]. of two-degree Considering of freedom the number variable of graphsmechanisms, to be enumerated it would be and enough the complexity to enumerate of the or dimensional choose basic synthesis one-degree of two-degree of freedom of candidate freedom variablemechanisms mechanisms, excluding it their would control be enough function to [22 en]:umerate the variable or choose action canbasic be achievedone-degree by of moving freedom one candidateof the ground mechanisms pivots. excluding their control function [22]: the variable action can be achieved by movingAfter one selecting of the ground candidate pivots. mechanisms, the next step is the dimensional synthesis to determine the jointAfter positions selecting andcandidate link lengths mechanisms, that satisfy the next desired step specifications is the dimensional such as synthesis the TDC to position determine and thestroke. joint Forpositions this, the and displacement link lengths equationsthat satisfy of desi thered selected specifications mechanisms such canas the be TDC used position to determine and stroke.the dimensions. For this, the displacement equations of the selected mechanisms can be used to determine the dimensions.The next step of the design process would be the kinematic and dynamic analysis of the initially designedThe next candidate step of mechanisms. the design process For the would dynamic be th analysis,e kinematic however, and dynamic the mass analysis properties of the of links initially and designedgas pressure candidate are required, mechanisms. which are For not the available dynamic at theanalysis, conceptual however, design the stage. mass Instead, properties the harmonic of links andacceleration gas pressure analysis are required, of candidates which is are the not proper available next step, at the since conceptual the distributed design stage. mass ofInstead, a link the can harmonicbe substituted acceleration by dynamically analysis of equivalent candidates point is the masses proper located next step, at neighboringsince the distributed joints [23 mass] and of the a linkshaking can be forces substitute in thed in-line by dynamically four-cylinder equivalent engine dependpoint masses upon thelocated second at neighboring harmonic acceleration. joints [23] andFor illustrationthe shaking purpose, forces onlyin the the in-line vertical four-cylin harmonicder acceleration engine depend of each jointupon will the be second considered harmonic in this acceleration.paper. The same For procedureillustration can purpose, be applied only to the the vert horizontalical harmonic acceleration. acceleration The vertical of each displacement joint will be of considered in this paper. The same procedure can be applied to the horizontal acceleration. The Appl. Sci. 2020, 10, 3765 5 of 19

the ith joint of a candidate VCR engine mechanism, fPiy (θ), which is a periodic function of period 2π of crank angle θ, can be represented by a Fourier series as

X∞ fPiy (θ) = a0 + (an cos nθ + bn sin nθ), (6) n=1 where R π a = 1 f ( )d 0 2π π Piy θ θ, − R π a = 1 f ( ) n d n = n π π Piy θ cos θ θ , 1, 2, , (7) − ··· R π b = 1 f ( ) n d n = n π π Piy θ sin θ θ , 1, 2, . − ··· Differentiating Equation (6) with respect to θ twice gives the Fourier series expression of the vertical acceleration (length/rad2) of the ith joint. The next step of the design process is the selection of feasible VCR engine mechanisms among candidates on the basis of the second harmonic acceleration analysis when they are used in the in-line four-cylinder engine. Then, for the selected feasible mechanism, evaluate the influence of the change in each link length on the sum of the vertical second harmonic acceleration at each joint and the stroke within prescribed tolerances, and adjust the link lengths to reduce the vertical second harmonic acceleration. With the adjusted link lengths, re-evaluate the influence of the link length changes and adjust the lengths of links, which are effective in reducing the vertical second harmonic acceleration sum and in satisfying the desired stroke.

4. Kinematic Conceptual Design of a Six-Link VCR Engine Mechanism In this section, the proposed process is applied to the kinematic conceptual design of a six-link VCR engine mechanism excluding the control function.

4.1. Type Synthesis Limiting the search of candidate VCR engine mechanisms to planar one-degree of freedom six-link mechanisms with revolute (R) joints and one prismatic (P) joint, there are two types of kinematic chains: Stephenson and Watt chains [20,22] whose unlabeled kinematic graphs are shown in Table1 in which the vertices represent the links and the edges represent the joints of the corresponding kinematic chain. If the control link to alter the compression ratio needs to be connected to the ground link (engine block) as in most machinery, the engine block must be a ternary link, which is connected to three other links: the piston by a P joint, the crank, and control link by R joints. By assigning appropriate links and joints to the vertices and edges of the unlabeled graphs, the labeled graphs and skeleton diagrams of the candidates, the Stephenson III and Watt II mechanisms, are obtained as shown in Table1. Notice that the vertex with a concentric circle in the labeled graph corresponds to the ground link of the mechanism. The compression ratio can be altered by moving the ground pivot, P3, along the line with arrows shown in the skeleton diagrams of Table1.

4.2. Dimensional Synthesis Using Displacement Equations For the dimensional syntheses of the Stephenson III and Watt II mechanisms, the displacement equations are derived. For this, the four-bar linkage in both mechanisms with links r0, r1, r2, and r3 shown in the skeleton diagram of Table1 is analyzed as follows [24]. The vector loop-closure equation of the four-bar linkage can be written as

r1 + r2 = r0 + r3, (8) or, equivalently as r = r + r r . (9) 2 0 3 − 1 Appl. Sci. 2020, 10, 3765 6 of 19

The x and y components of Equation (9) can be written respectively as

r cos θ = r cos θ + r cos θ r cos θ , (10) 2 2 0 0 3 3 − 1 1 r sin θ = r sin θ + r sin θ r sin θ , (11) 2 2 0 0 3 3 − 1 1 where ri is the length of ri and θi is the angle of ri measured from the positive x axis. Square Equations (10) and (11), and sum to obtain

2 2 2 2 r = r + r + r + 2r0r3(cos θ0 cos θ3 + sin θ0 sin θ3) 2 0 1 3 (12) 2r r (cos θ cos θ + sin θ sin θ ) 2r r (cos θ cos θ + sin θ sin θ ). − 0 1 0 1 0 1 − 1 3 1 3 1 3

Equation (12) can be written in terms of cosθ3 and sinθ3 as

A cos θ3 + B sin θ3 + C = 0, (13) where A = 2r r cos θ 2r r cos θ , 0 3 0 − 1 3 1 B = 2r r sin θ 2r r sin θ , (14) 0 3 0 − 1 3 1 C = r2 + r2 + r2 r2 2r r (cos θ cos θ + sin θ sin θ ). 0 1 3 − 2 − 0 1 0 1 0 1 Substitute the tangent half-angle identities into Equation (13) and solve the resulting equation to obtain   √ 2 2 2 1 B B C + A  θ3 = 2 tan− − ± − , π θ3 π. (15)  C A  − ≤ ≤ − Equation (15) states that there are two θ3 values for a given angle θ1, which correspond to two assembly modes. Dividing Equation (11) by Equation (10) gives the coupler angle θ2 as ! 1 r0 sin θ0 + r3 sin θ3 r1 sin θ1 θ = tan− − . (16) 2 r cos θ + r cos θ r cos θ 0 0 3 3 − 1 1

Now, the position of joint P2 can be written as " # " # " # P2x r0x r3 cos θ3 P2 = = + . (17) P2y r0y r3 sin θ3

For the candidate mechanisms in Table1 to be used in a VCR engine, there must be a crank connected to the ground link by an R joint in the four-bar linkage. In this research, it is assumed that the four bar is a crank and rocker. By Grashof criteria [25], the crank must be the shortest link and the following relation must hold for a crank-rocker four-bar linkage.

L + S < I2, (18) where L is the length of the longest link, S is the length of the shortest link, and I2 is the sum of the remaining two link lengths. The positions of the other joints in the each mechanism in Table1 can be determined as follows. Appl. Sci. 2020, 10, 3765 7 of 19 Appl. Sci. 2020Appl., 10 Sci., x FOR 2020 PEER, 10, x REVIEWFOR PEER REVIEW 6 of 19 6 of 19 Appl.Appl. Sci. Sci.Appl. 2020 2020Appl.Appl. Sci., ,10 10 20,Sci. Sci.,x 20x FOR FOR2020 ,2020 10 ,PEER ,xPEER ,10 10FOR, ,x xREVIEW FOR REVIEWFORPEER PEER PEER REVIEW REVIEW REVIEW 66 of of 19 19 6 of 6196 of of 19 19 Table 1.Table CandidateTable 1. Candidate 1. Candidatevariable variable compression variable compression compression ratio (VCR) ratio ratio (VCR)engine (VCR) enginemechanisms. engine mechanisms. mechanisms. TableTable 1. Table1. Candidate CandidateTableTable 1. Candidate 1.1. CandidatevariableCandidate variable variable compression compression variablevariable compression compressioncompression ratio ratio (VCR) (VCR) ratio ratioratio engine( VCRengine (VCR)(VCR)) enginemechanisms. mechanisms. engineengine mechanisms. mechanisms.mechanisms. StephensonStephensonStephenson Chain ChainChain Watt Watt Chain Watt Chain StephensonStephensonStephensonStephensonStephenson Chain Chain Chain ChainChain Watt Watt Chain WattChain Watt Watt Chain ChainChain

UnlabeledUnlabeled GraphUnlabeled Graph Graph UnlabeledUnlabeledUnlabeledUnlabeledUnlabeled Graph Graph Gr a GraphphGraph

StephensonStephensonStephenson III Mechanism III Mechanism Watt Watt II II Mechanism Watt II Mechanism StephensonStephensonStephensonStephensonStephenson III III Mechanism Mechanism III Mechanism IIIIII MechanismMechanism WattWatt II WattII Mechanism MechanismWattWatt II Mechanism IIII MechanismMechanism E P PS P E R r1 Labeled GraphLabeled Graph R R PS Labeled Graph Labeled Graph LabeledLabeled GraphLabeledLabeled Gr a GraphphGraph r3 r1 R R R R R R r6 R R R r6 TL r2 TL e e

P P6 6 E E PS PS

r6 r6 P5 Skeleton diagramSkeleton diagram P5 Skeleton diagramSkeleton diagram r5 SkeletonSkeleton diagramSkeleton diagramSkeleton diagram diagram TL r4 r5 P3 a r4 d P2 y P P1 TL 2 r3 r2 r2 r1 P0 x y r3 r0 P1 r0 P3 r1 P0 x

: ground link,: ground E: engine link, block,E: engine PS: piston,block, PS r1:: crank,piston, r 6r:1 :connecting crank, r6: connecting rod, TL: ternary rod, TL link,: ternary R: revolute link, R : revolute : ground link, E: engine block, PS: piston, r1: crank, r6:1 connecting6 rod, TL: ternary link, R: revolute : ground  link,:: groundground E: engine link,link,E Eblock,E:: engineengine PS : block, block,piston,PS PSPS r1:: piston,crank,piston,r 1 r rr6:1: : connecting crank,crank,r 6 rr6:: connectingconnecting rod, TL:TL ternary rod,rod, TLTL link,:: ternaryternary R:R revolute link,link, RR :: revoluterevolute joint, P: prismatic:: ground groundjoint, P link,: link, joint,prismatic E :P: engineiengine: position joint, block, block, P ofi: position joint P:S piston,: ipiston,. of joint1: r crank,: crank,i. 6 :r connecting: connecting rod, rod, :T ternaryL: ternary link, link,: revolute R: revolute joint, joint, PP: prismatic: prismatic joint, P P:i: positionposition of ofi joint jointi. i. joint, Pjoint,: prismaticjoint,joint, P: prismatic PP:: prismaticjoint,prismatic i Pjointi: position joint,joint,, Pi: position PP :iof: positionposition joint of i .joint ofof jointjoint i. ii.. 4.2. Dimensional4.2. Dimensional Synthesis UsSynthesising Displacement Using Displacement Equations Equations 4.2.4.2. Dimensional Dimensional4.2.4.2.1.4.2. 4.2.Dimensional Stephenson DimensionalDimensional Synthesis Synthesis Synthesis III Us SynthesisSynthesisUs Mechanisminging UsingDisplacement Displacement UsUs ingDisplacementing DisplacementDisplacement Equations Equations Equations EquationsEquations For the dimensionalFor the dimensional syntheses syntheses of the Stephens of the onStephens III andon Watt III andII mechanisms, Watt II mechanisms, the displacement the displacement ForFor the theThe dimensional dimensionalForFor position thethe dimensionaldimensional of syntheses syntheses joint P5 is syntheses synthesesof of the the Stephens Stephens ofof thethe on StephensonStephens III III and andon onWatt Watt IIIIII andIIand II mechanisms, mechanisms, WattWatt IIII mechanisms,mechanisms, the the displacement displacement thethe displacementdisplacement equationsequations areFor derived.the dimensional are Forderived. this, syntheses theFor four-barthis, ofthe t he linkagefour-bar Stephenson in linkage both III mechanisms inand both Watt mechanisms II withmechanis links withm r0s, , r the1links, r 2displacement, and r0, r r1,3 r2, and r3 equationsequations equationsare derived. areare For derived.derived. this, the ForFor four-bar this,this, thethe linkage four-barfour-bar in linkagelinkage both mechanisms inin bothboth mechanismsmechanisms with links withwith r0,0 r 1links,links1 r2,2 and rr00,, rrr11,3, 3 rr22,, andand rr33 shownequationsequations in shownthe are skeleton derived. arein the derived. diagramskeleton For this, For ofdiagram thethis, Ta blefour-bar" the 1 of isfour Ta#analyzed linkageble-bar" 1 linkageis analyzedasin followsboth #in mechanismsboth" as [24]. follows mechanisms [24]. with # linkswith lirnks, r , rr0,, rand1, r2 ,r and r3 shown in shownthe skeleton in the diagram skeleton ofdiagram TableP 1 5of xis Taanalyzedble 1r1 iscos analyzedasθ follows1 as [24].r 4followscos (θ2 [24].+ δ) shownshown inshown the in skeleton the in skeletonthe diagramskeleton diagram Pofdiagram Ta= bleof Table 1 of is Ta analyzed= 1ble is 1analyzed is analyzedas follows as+ follows as [24]. follows [24]. [24]. , (19) The vectorThe loop-closure vector loop-closure equation5 ofequation the four-bar of the linkage four-bar can linkage be written( can+ be as) written as TheThe vector vectorTheThe Thevector loop-closure loop-closure vectorvector loop loop-closureloop-closure-closure equation equation equation of equationequationofP 5th thy ee offour-bar four-bar the ofof r thfour1thsine elinkage four-barlinkagefour-bar-θbar1 linkage can can linkagelinkage rbe 4be sincan written written canθcanbe2 written bebe δas as written written as asas rr+=+ rr+=+, 12+=+rr 0312+=+ rr 03, (8) (8) where δ is the angle between links r rr12and+=+r , which rrrr 03rr+=+, can rr rr be obtained,, by the law of cosines as(8) (8) 2 rr12r41 rr12 r0312 2  r, 0  03r 03 3, (8) (8) (8) or, equivalentlyor, equivalently as as or, equivalentlyor, equivalently as as 2 2 2 ! or, equivalentlyor, equivalentlyor, equivalently as as as 1 r2 + r4 r5 rrrrδ ==+−cos− rrrr=+−. − . , (20) 2031=+−2031=+−2r2r4 (9) (9) rrrr2031=+−rrrrrrrr=+−. .. (9) (9) rrrr2031r22031 r2031 0  r. 3  r 1. (9) (9) (9) The x andThe y components x and y components of Equation of Equation(9) can be (9) written can be respectively written respectively as as and the position of P6 is TheThe x x andThe andThe Theyx y componentsand components xx and andy components yy componentscomponents of of Equation Equation of Equation ofof Equation(9) Equation(9) can can (9) be be can (9) (9)written written cancanbe written bebe respectively respectively writtenwritten respectively respectivelyrespectively as as  as asas θθθθ=+−" θθθθ# =+−e rrrr22003311cosrrrr22003311cosP cos6x  cos cos cos cos , cos , (10) (10) rrrrcosPθθθθθθθθ6 ==+−rrrr=+−cos cosθθθθθθθθ=+−==+− cos cosq cos cos , cos, , (21) rrrr2200331122003311cosrcosrrrr2200331122003311cos cos r cos cos cos  r cos2 cos cos  r cos , 2 cos , , (10)(10) (10)(10) 2P 26y 0 P 0y + 3r 3(e 1P x) 1 (10) 5 6 − − 5 rrrrsinθθθθ=+−rrrrsin sinθθθθ=+− sin sin sin sin , sin , (11) (11) e 22003311θθθθ=+−22003311θθθθ=+− (11) (11) and is the piston oﬀset.rrrrrrrr22003311sinsinθθθθrrrr=+−rrrr22003311sinsin sin sinθθθθ=+− sin sin sin sin sin sin sin , , sin sin , , (11) (11) (11) 22003311r2sin22003311 2 r 0 sin  0  r 3 sin  3  r 1 sin  1 , where ri iswhere the length ri is the of lengthri and θ ofi is r i theand angle θi is theof r anglei measured of ri measured from the positivefrom the x positive axis. x axis. where ri iiswherewhere the length rri i isis thethe of rlengthlengthi iand θ ofofi iis rr ithe i andand angle θθi i isis theofthe r i angle angleimeasured ofof rri i measured measuredfrom the positive fromfrom thethe x positive positiveaxis. xx axis.axis. whereSquarewhere r is the Equationsri Square islength the length Equationsof (10) r and ofand r θi and(11), (10) is the θandi isangle thesum(11), angle of toand r obtain measured ofsum ri measured to obtain from fromthe positive the positive x axis. x axis. SquareSquareSquare Equations EquationsSquareSquare Equations EquationsEquations (10) (10) and and (10) (11), (11),(10) (10)and and and and(11), sum (11),(11),sum and to and andtosum obtain obtain sumsum to obtain toto obtainobtain 22222222 rrrr2222=+++rrrr2222=+++2coscossinsin rr()2coscossinsinθθ rr()θθ + θθ + θθ 20132222=+++220132222=+++ 2 2 0302 ()θθ 030() 3θθ + θθ 0 3 + 3 θθ 0 3 rrrrrrrr2013=+++rrrrrrrr2013=+++2coscossinsin2coscossinsin rr 030rr()2coscossinsin2coscossinsinθθ rr rr 030() 3θθ + θθ 0 3 + 3 θθ 0 3 2013r22013 r 0  r 1  r 3 030 2 r 0 r 3 030 cos 0 3cos  3  0 sin 3  0 3 sin 0  3  3 (12) (12) −+−+2rr()() cos−+−+2θθrr cos()() cosθθ sin cos θθ sin sin θθ 2 sin rr cos 2 θθ rr cos cos θθ sin cos θθ sin sin θθ . sin(12)(12) . (12)(12) −+−+01()()−+−+θθ 001()() 1θθ 0 θθ 0 1 1 θθ 0 13 1 θθ 1 13 3 θθ 1 θθ 1 3 3 θθ 1 3 (12) −+−+22rr01rr()() cos cos−+−+22θθrr 0rr01 cos cos()() cos cos 1θθ 0 sin sin cos cos θθ 0 1sin sin sin sin 1 θθ 0 2 sin 2sin rr 13rr 1 cos cos 2 2 θθ rr1rr 13 cos cos cos cos 3 θθ 1 sin sincos cos θθ 1sin 3 sin sin sin 3 . θθ . 1 sin sin 3 . . 012r01 r 001cos 01 1 cos 0   0 sin 1  0113 1 sin 0  13  1 2 rr  1 13cos  13 3cos 1   1 sin 3  13 3 sin 1   . 3 Appl. Sci. 2020, 10, 3765 8 of 19

4.2.2. Watt II Mechanism

The positions of joints P5 and P6 are found respectively by " # " # " # P5x r0x r5 cos(θ3 + α) P5 = = + , (22) P5y r0y r5 sin(θ3 + α)

" #   P  e  P = 6x =  q  6  2 2 , (23) P6y  P y + r (e P x)  5 6 − − 5 where α is the angle between r3 and r5, which can be determined by the law of cosines as

2 2 2 ! 1 r3 + r5 r4 α = cos− − , (24) 2r3r5 and e is the piston oﬀset.

4.2.3. Initial Design of Candidate VCR Engine Mechanisms For the design speciﬁcations for the VCR engine mechanism given in Table2, the dimensions of the two candidates are determined using Equations (8)–(24) as shown in Figure3 and Table3.

Table 2. Design speciﬁcations for VCR engine mechanism.

Engine Type In-Line Four-Cylinder

Crank throw phase angle 90◦, 270◦, 270◦, 90◦ Bore 86 mm Stroke 86 0.025 mm ± TDC position 220 10 mm (Piston pin: 190 10 mm) ± ± Displacement 2 L Piston oﬀset 0 mm Radius of engine block internal space 82.5 mm (measured form crank center)

The initially designed mechanisms satisfy the desired stroke and piston pin position at TDC as shown in Table4, and there is no interference between the moving parts and the engine block.

Table 3. Dimensions of the initially designed VCR engine mechanisms.

Link Length and Pivot Position (Unit: mm) Link Stephenson III Mechanism Watt II Mechanism

r1 25.000 33.000 r2 57.000 43.000 r3 73.500 64.000 r4 44.000 25.000 r5 93.500 85.000 r6 140.000 100.000 r 78.000 62.300 0x − r y 53.000 35.600 0 −

Table 4. Analysis results of the initially designed mechanisms.

Stephenson III Watt II Design Speciﬁcations Stroke (mm) 85.993 86.008 86 0.025 ± Piston pin position at TDC (mm) 197.391 199.191 190 10 ± Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 19

Appl.Appl. Sci.Sci.2020 2020,,10 10,, 3765 x FOR PEER REVIEW 99 of of 19 19

(a) (b)

Figure 3. Initial design of VCR engine mechanisms: (a) Stephenson III and (b) Watt II.

Table 4. Analysis results of the initially designed mechanisms.

Stephenson III Watt II Design Specifications Stroke (mm)( a) 85.993 86.008 (b) 86 ± 0.025 Piston pin position at TDC (mm) 197.391 199.191 190 ± 10 FigureFigure 3.3. InitialInitial designdesign ofof VCRVCR engineengine mechanisms:mechanisms: ((aa)) StephensonStephenson IIIIII andand ((bb)) WattWatt II. II.

4.3. Harmonic Acceleration Analysis and Selection of the Final Mechanism Type 4.3. Harmonic AccelerationTable Analysis 4. Analysis and resultsSelection of theof the initially Final designedMechanism mechanisms. Type ForFor thethe harmonicharmonic accelerationacceleration analysisanalysis ofof thethe initiallyinitially designeddesigned mechanisms,mechanisms, thethe verticalvertical Stephenson III Watt II Design Specifications displacementdisplacement ofof eacheach jointjoint waswas expandedexpanded numericallynumerically toto determinedetermine thethe FourierFourier seriesseries expression,expression, Stroke (mm) 85.993 86.008 86 ± 0.025 andand thenthen didifferentiatedﬀerentiated twicetwice withwith respectrespect toto crank crank angle angleθ θ.1. ForFor thethe veriﬁcationverification ofof thethe numericalnumerical Piston pin position at TDC (mm) 197.391 1 199.191 190 ± 10 results,results, thethe verticalvertical harmonicharmonic acceleration acceleration components components of of the the piston piston pin pin ( P(P66)) ofof thethe initiallyinitially designeddesigned StephensonStephenson IIIIII mechanismmechanism inin aa singlesingle cylindercylinder engineengine werewere evaluatedevaluated asas shownshown inin FigureFigure4 ,4, and and the the 4.3. Harmonic Acceleration Analysis and Selection of the Final Mechanism Type resultresult agreedagreed well well with with the the previous previous research research [ 11[11].]. For the harmonic acceleration analysis of the initially designed mechanisms, the vertical displacement of each joint60 was expanded numerically to determine the Fourier series expression, Overall and then differentiated twice with respect to crank angle θ1. For the verification of the numerical 40 ) 6 results, the vertical harmonic2 acceleration components of the piston pin (P ) of the initially designed Stephenson III mechanism in a single cylinder engine were evaluated as shown in Figure 4, and the 20 result agreed well with the4th previous research 2nd[11]. 3rd 0 60 TDC BDC -20 Overall

Vertical harmonicVertical 40 ) 2 acceleration (mm/rad -40 20 4th 1st -60 2nd 3rd 0 0 90 180 270 360 Crank angle (degree) TDC BDC -20 FigureFigure 4.4. VerticalVertical harmonicharmonic accelerationacceleration componentscomponents atat pistonpiston pinpin (P(P66)) ofof thethe initiallyinitially designeddesigned Vertical harmonicVertical

StephensonStephenson IIIIII mechanismmechanismacceleration (mm/rad in in a a single single cylinder. cylinder. -40 1st -60 0 90 180 270 360 Crank angle (degree)

Figure 4. Vertical harmonic acceleration components at piston pin (P6) of the initially designed Stephenson III mechanism in a single cylinder. Appl. Sci. 2020, 10, x 3765 FOR PEER REVIEW 10 of 19

Figure 5 shows the vertical harmonic acceleration components at joints P2 and P5 in cylinder 1 Figure5 shows the vertical harmonic acceleration components at joints P and P in cylinder 1 and and cylinder 2 when the initially designed Stephenson III mechanism 2was used5 in an in-line cylinder 2 when the initially designed Stephenson III mechanism was used in an in-line four-cylinder four-cylinder engine with crank throws at 90°, 270°, 270°, and 90° phase angles. As in the case of the engine with crank throws at 90 , 270 , 270 , and 90 phase angles. As in the case of the slider-crank slider-crank mechanism in the ◦in-line◦ four-cylinder◦ ◦ engine in Section 2, the vertical inertia forces mechanism in the in-line four-cylinder engine in Section2, the vertical inertia forces due to the odd due to the odd harmonic acceleration components in cylinders 1 and 2, and cylinders 3 and 4 would harmonic acceleration components in cylinders 1 and 2, and cylinders 3 and 4 would cancel out, cancel out, and those induced by the even harmonic acceleration components added up. The same and those induced by the even harmonic acceleration components added up. The same canceling out canceling out of the vertical inertia forces due to the odd harmonic acceleration components of the vertical inertia forces due to the odd harmonic acceleration components occurred when the Watt occurred when the Watt II mechanism was used in the same in-line four-cylinder engine. II mechanism was used in the same in-line four-cylinder engine.

14 14 12 12 Overall 1st Overall 10 10 2nd 1st 2nd 8 8 ) )

2 6 2 6 4 4 2 2 0 0 -2 -2 -4 -4 Vertical Vertical harmonic harmonicVertical acceleration (mm/rad acceleration (mm/rad -6 -6 4th 4th -8 -8 3rd 3rd -10 -10 -12 -12 TDC BDC TDC BDC -14 -14 0 90 180 270 360 0 90 180 270 360 Crank angle (degree) Crank angle (degree) (a) (b)

60 60 Overall 50 50 Overall 40 40 1st

) 30 ) 30 2 2 2nd 1st 2nd 20 20

10 10

0 0

-10 -10 Vertical harmonic 4th harmonicVertical 4th acceleration (mm/rad -20 acceleration (mm/rad -20 3rd 3rd -30 -30

-40 -40 TDC BDC TDC BDC -50 -50 0 90 180 270 360 0 90 180 270 360 Crank angle (degree) Crank angle (degree) (c) (d)

Figure 5. Vertical harmonic accelerationacceleration components at joints P2 and P5 of the initially designed Stephenson III mechanism: (a)) JointJoint P22 in Cylinder 1; (b) Joint P22 inin Cylinder 2; (c) Joint P5 inin Cylinder Cylinder 1; and (d)) JointJoint P55 in Cylinder 2.

In order to determine the feasibility of the two candidate mechanisms when used in the in-line four-cylinder engine, the verticalvertical secondsecond harmonicharmonic accelerationacceleration atat eacheach jointjoint areare plottedplotted inin FigureFigure6 6.. As can be seen in Figure 6 6,, thethe verticalvertical secondsecond harmonicharmonic accelerationacceleration componentscomponents atat thethe jointsjoints inin the Watt IIII linkagelinkage werewere muchmuch higherhigher thanthan thosethose ofof thethe StephensonStephenson III mechanism and they were in phase, hence the vertical inertia forces might addadd up to a high value. In the case of the Stephenson III mechanism, the vertical second harmonic acceleration at joint PP22 and those at jointsjoints PP55 and PP66 were out of phase, and some vertical inertia forces wouldwould cancel out. For these reasons, between the two candidates, the Stephenson III mechanism was selectedselected asas aa feasiblefeasible mechanismmechanism forfor aa VCRVCR engine.engine. Appl.Appl. Sci. Sci. 20202020, 10, 10, x, 3765FOR PEER REVIEW 1111 of of 19 19

40 200

30 150 P P 6 5 P6 20 100 ) ) 2 2

10 50

P2 0 0 TDC BDC TDC BDC

-10 -50 acceleration (mm/rad acceleration (mm/rad Vertical second Vertical harmonic Vertical secondVertical harmonic -20 -100

P -30 2 -150

P5 -40 -200 0 90 180 270 360 0 90 180 270 360 Crank angle (degree) Crank angle (degree) (a) (b)

FigureFigure 6. 6. VerticalVertical second second harmonic harmonic acceleration acceleration at at each each joint joint in in candidate candidate mechanisms mechanisms in in the the in-line in-line four-cylinderfour-cylinder engine: engine: (a (a) )initially initially designed designed Stephenson Stephenson III III and and (b (b) )initially initially designed designed Watt Watt II. II.

4.4.4.4. Minimization Minimization of of Vertical Vertical Second Second Harmonic Harmonic Acceleration Acceleration SinceSince the the mass mass properties properties of of link linkss are are not not available available at at the the initial initial phase phase of of the the conceptual conceptual design design process,process, it it is is assumed assumed that that the the concentrated concentrated mass mass at at each each joint joint is is the the same same and and the the vertical vertical harmonic harmonic accelerationacceleration of of each each joint joint contributes contributes to to the the vertical vertical inertia inertia force force proportional proportional to to its its magnitude. magnitude. In In this this paper,paper, the followingfollowing steps steps are are proposed proposed to find to thefind optimum the optimum dimensions dimensions that yield that a nearly yield minimizeda nearly minimizedsum of the sum vertical of the second vertical harmonic second acceleration harmonic acceleration at each joint byat each varying joint the by initial varying link the lengths initial within link lengthsprescribed within tolerances prescribed on thetolerances stroke and on the link stroke lengths. and link lengths. S1S1 WithinWithin the the prescribed prescribed tolerances tolerances on on the the link link lengths lengths and and stroke, stroke, evaluate evaluate the the influence influence of of the the changechange in in thethe lengthlength of of each each link link on the on maximum the maximum sum of sum the vertical of the second vertical harmonic second accelerationharmonic accelerationcomponents components and the stroke. and the stroke. S2S2 FindFind links links that areare more more eff effectiveective in reducingin reduci theng maximumthe maximum sum ofsum the verticalof the secondvertical harmonic second harmonicacceleration acceleration components components when their when link lengths their arelink changed. lengths Then,are changed. adjust their Then, lengths adjust to reducetheir lengthsthe second to reduce harmonic the second acceleration harmonic sum. acceleration sum. S3S3 RepeatRepeat Step Step S1 S1 for for the the links links that that are are not not adjusted adjusted in in Step Step S2. S2. As As Step Step S2 S2 may may cause cause the the deviation deviation fromfrom the the desired desired stroke, stroke, find find links links that that are are effect effectiveive in in both both adjusting adjusting the the stroke stroke and and reduce reduce the the maximummaximum sum sum of of the the vertical vertical second second harmonic harmonic acceleration acceleration components. components. Then, Then, adjust adjust their their lengthslengths to to reduce reduce the the second second harmonic harmonic accele accelerationration sum sum and and to to satisfy satisfy the the desired desired stroke. stroke. The above steps are applied to the initial design of the Stephenson III mechanism as follows. The above steps are applied to the initial design of the Stephenson III mechanism as follows. Step 1 Step 1 is carried out and the result is plotted in Figure 7 by varying each link length by 0.1 mm is carried out and the result is plotted in Figure7 by varying each link length by 0.1 mm sequentially sequentially within the prescribed tolerances on the link lengths and stroke given in Table 5. within the prescribed tolerances on the link lengths and stroke given in Table5. As can be seen from Figure 7a,b, since the lengths of r4 and r5 of the ternary link are more As can be seen from Figure7a,b, since the lengths of r and r of the ternary link are more sensitive sensitive to the sum of the vertical second harmonic acceleration4 5 components, they are selected to to the sum of the vertical second harmonic acceleration components, they are selected to adjust their adjust their lengths in Step 2: the length of link r4 is taken to its minimum and r5 is adjusted to its lengths in Step 2: the length of link r is taken to its minimum and r is adjusted to its maximum within maximum within the prescribed tolerance4 on the link lengths given5 in Table 5. As Step S3, with the the prescribed tolerance on the link lengths given in Table5. As Step S3, with the adjusted lengths of r4 adjusted lengths of r4 and r5 in Step S2, the influences of the change in the length of each link except and r5 in Step S2, the influences of the change in the length of each link except r4 and r5 are plotted r4 and r5 are plotted in Figure 8. Figure 8c,d show that the stroke of the Stephenson III mechanism in Figure8. Figure8c,d show that the stroke of the Stephenson III mechanism with the adjusted link with the adjusted link lengths of r4 and r5 is 85.510 mm, which does not satisfy the design lengths of r and r is 85.510 mm, which does not satisfy the design specification 86 0.025 mm given specification4 86 ± 0.0255 mm given in Table 2. ± in Table2. From Figure 8c, it seems that the length of r1 could be increased to meet the desired stroke range. However, since lengthening r1 also increased the maximum sum of the vertical second harmonic accelerations as shown in Figure 8a, link r1 was excluded from the adjustment. Examining Figure 8a,c, r2 can be shortened to its minimum length to increase the stroke and to decrease the second harmonic acceleration sum, but another link needs to be adjusted to meet the desired stroke. Hence, with the minimum length of r2, the other three link lengths were decreased by 0.1 mm in sequence within the tolerances as shown in Table 6 and Figure 9. Appl. Sci. 2020, 10, 3765 12 of 19

From Figure8c, it seems that the length of r1 could be increased to meet the desired stroke range. However, since lengthening r1 also increased the maximum sum of the vertical second harmonic accelerations as shown in Figure8a, link r1 was excluded from the adjustment. Examining Figure8a,c, r2 can be shortened to its minimum length to increase the stroke and to decrease the second harmonic acceleration sum, but another link needs to be adjusted to meet the desired stroke. Hence, with the minimum length of r , the other three link lengths were decreased by 0.1 mm in sequence within the Appl. Sci. 2020, 10, x FOR PEER2 REVIEW 12 of 19 tolerances as shown in Table6 and Figure9.

37.0 45

36.8 43

36.6 41

36.4 39 r1 36.2 37 r2 r3 , Max value) Max , , Max value) Max , 2 2 36.0 r6 35 r4 (mm/rad (mm/rad 35.8 r0x 33 r5 r0y 35.6 31 Vertical second harmonic acceleration acceleration harmonic second Vertical Vertical second harmonic acceleration acceleration harmonic second Vertical

35.4 29

35.2 27 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Variation value (mm) Variation value (mm) (a) (b)

87.0 87.0

86.8 86.8

86.6 86.6

86.4 86.4 r1 86.2 86.2 r2 r3 86.0 86.0 r4 r6 Stroke (mm) Stroke (mm) Stroke 85.8 85.8 r5 r0x 85.6 85.6 r0y 85.4 85.4

85.2 85.2

85.0 85.0 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Variation value (mm) Variation value (mm) (c) (d)

FigureFigure 7.7. InﬂuenceInfluence onon thethe maximummaximum ofof verticalvertical secondsecond harmonicharmonic accelerationacceleration sumsum ofof lengthlength changechange

in:in: ((aa)) rr11,, rr33,, rr66, rr0x,, and and r0yy andand ( (b)) r2,, rr44,, and and rr55 ofof the ternary link; inﬂuenceinfluence on thethe strokestroke ofof lengthlength changechange in:in: ((cc)) rr11, r3,, rr66,, rr00x,x ,and and r0ry0 yandand (d ()d r)2r, 2r,4,r 4and, and r5 inr5 thein the in-line in-line four-cylinder four-cylinder engine. engine.

Table 5. Prescribed tolerances. Table 5. Prescribed tolerances. Link Lengths r 2 mm Link Lengths rii ±± 2 mm Stroke 8686 ± 11 mmmm ±

Table 6. Link length variation cases for Step S3.

Variation Link Length Case Number r2 r3 r4 r5 r6 r0y 1 56 73.5 42 95.5 140 −53 Decrease r3 with min. 1 : 56 : 42 95.5 140 −53 r2, r4, and max. r5 21 56 71.5 42 95.5 140 −53 1 56 73.5 42 95.5 138 −53 Decrease r6 with min. 2 : 56 73.5 42 95.5 : −53 r2, r4, and max. r5 21 56 73.5 42 95.5 140 −53 1 56 73.5 42 95.5 140 −53 Decrease r0y with 3 : 56 73.5 42 95.5 140 : min. r2, r4, and max. r5 21 56 73.5 42 95.5 140 −55 Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 19

The “:” in the table means the variation range as “-“. For example, the “1 : 21” and “73.5 : 71.5” for r3 Appl. Sci.in Case2020, 101 mean, 3765 that the variation Number “1, 2, 3, …, 21” correspond “73.5, 73.4, 73.3, …, 71.5” of r3.13 of 19

22 20.0

21 19.8

20 19.6 r3 19 19.4 r6 r1 , Max value) , Max value) 2 2 r0x 18 19.2 r2 r0y 17 19.0 (mm/rad (mm/rad

16 18.8 Vertical second harmonic acceleration second harmonic acceleration Vertical Vertical secondVertical harmonic acceleration 15 18.6 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Variation value (mm) Variation value (mm) (a) (b)

86.6 85.8

86.3 85.7

Desired stroke range 86.0 85.6 r3 r6 85.7 r1 85.5 r0x r2 Stroke (mm) Stroke (mm) r0y 85.4 85.4

85.1 85.3

84.8 85.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Variation value (mm) Variation value (mm) (c) (d)

FigureFigure 8.8. WithWith thethe minimumminimum lengthlength of ofr r44and and thethe maximummaximumr r55,, inﬂuenceinfluence onon thethe maximummaximum ofof verticalvertical secondsecond harmonicharmonic acceleration acceleration sum sum of of length length change change in: in:(a) r (1a )andr1 andr2 andr2 (band) r3, (rb6,) rr0x3,, andr6, rr00xy;, influence and r0y; inﬂuenceon the stroke on the of strokelength ofchange length in: change (c) r1 and in: r (2c )andr1 and (d) r32, andr6, r0x (,d and) r3, rr06y ,inr0 xthe, and in-liner0y infour-cylinder the in-line four-cylinderengine. engine.

Table 6. 86.225 Link length variation15.4 cases for Step S3. 86.200 Case 1 Case 1 15.3 86.175 Case 2 Case 2 86.150 Link Length Case 3 15.2 Case 3 86.125 Case Variation Number 86.100 15.1 r2 r3 r4 r5 r6 r0y 86.075 86.050 15.0 , Maxvalue)

86.025 12 56 73.5 42 95.5 140 53 14.9 86.000 Decrease r3 with min. Desired stroke range − Stroke (mm) Stroke 85.975 1 : 56 : 42 95.5 140 53 r2, r4, and max. r5 14.8 − 85.950 21(mm/rad 56 71.5 42 95.5 140 53 85.925 14.7 − 85.900 14.6 85.875 1 second harmonic acceleration Vertical 56 Satisfied73.5 design 42 95.5 138 53 85.850 Decrease r with min. − 2 6 :14.5 56 73.5 42 95.5 : 53 123456789101112131415161718192021r , r , and max. r 123456789101112131415161718192021− 2 4 Variation number5 21 56 73.5 42 Variation95.5 number140 53 − 1 56 73.5 42 95.5 140 53 Decrease r (awith) min. (b) − 3 0y : 56 73.5 42 95.5 140 : r , r , and max. r 2 4 5 21 56 73.5 42 95.5 140 55 Figure 9. Adjusted stroke and maximum value of vertical second harmonic acceleration sum− in each The “:” in the table means the variation range as “-“. For example, the “1 : 21” and “73.5 : 71.5” for r3 in Case 1 meancase: that(a) theadjusted variation stroke Number and“1, (b) 2, maximum 3, ... , 21” correspond vertical second “73.5, 73.4, harmonic 73.3, ... acceleration, 71.5” of r3. sum in the in-line four-cylinder engine. In Figure9a, the cases that satisfy the speciﬁed stroke range given in Table2 are marked with a In Figure 9a, the cases that satisfy the specified stroke range given in Table 2 are marked with a box. As shown in Figure9b, Case 3 of variation number 10 has the lowest maximum sum of vertical box. As shown in Figure 9b, Case 3 of variation number 10 has the lowest maximum sum of vertical second harmonic acceleration components. Hence, the ﬁnal dimensions were selected from this case. In Table7, the dimensions, stroke, piston pin position at TDC, and the maximum sums of the second Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 19

The “:” in the table means the variation range as “-“. For example, the “1 : 21” and “73.5 : 71.5” for r3 in Case 1 mean that the variation Number “1, 2, 3, …, 21” correspond “73.5, 73.4, 73.3, …, 71.5” of r3.

22 20.0

21 19.8

20 19.6 r3 19 19.4 r6 r1 , Max value) , Max value) 2 2 r0x 18 19.2 r2 r0y 17 19.0 (mm/rad (mm/rad

86.6 85.8

86.3 85.7

Desired stroke range 86.0 85.6 r3 r6 85.7 r1 85.5 r0x r2 Stroke (mm) Stroke (mm) r0y 85.4 85.4

85.1 85.3

84.8 85.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Variation value (mm) Variation value (mm) (c) (d) Appl. Sci. 2020, 10, 3765 14 of 19

Figure 8. With the minimum length of r4 and the maximum r5, influence on the maximum of vertical

second harmonic acceleration sum of length change in: (a) r1 and r2 and (b) r3, r6, r0x, and r0y; influence and fourth harmonic acceleration components in the in-line four-cylinder engine are shown and on the stroke of length change in: (c) r1 and r2 and (d) r3, r6, r0x, and r0y in the in-line four-cylinder comparedengine. with those of the initial design. The maximum of the vertical second and fourth harmonic acceleration sums were decreased by 58.333% and 34.882%, respectively.

86.225 15.4 86.200 Case 1 Case 1 15.3 86.175 Case 2 Case 2 86.150 Case 3 15.2 Case 3 86.125 86.100 15.1 86.075 86.050 15.0 , Maxvalue)

86.025 2 14.9 86.000 Desired stroke range Stroke (mm) Stroke 85.975 14.8

85.950 (mm/rad 85.925 14.7 85.900 14.6 85.875 second harmonic acceleration Vertical Satisfied design 85.850 14.5 123456789101112131415161718192021 123456789101112131415161718192021 Variation number Variation number

(a) (b)

Figure 9. Adjusted stroke and maximum value of vertical second harmonic acceleration sum in each Figure 9. Adjusted stroke and maximum value of vertical second harmonic acceleration sum in each case: (a) adjusted stroke and (b) maximum vertical second harmonic acceleration sum in the in-line case: (a) adjusted stroke and (b) maximum vertical second harmonic acceleration sum in the in-line four-cylinder engine. four-cylinder engine. Table 7. Comparison of the initial and ﬁnal mechanism. In Figure 9a, the cases that satisfy the specified stroke range given in Table 2 are marked with a Unit Initial Design Final Design Eﬀect (%) box. As shown in Figure 9b, Case 3 of variation number 10 has the lowest maximum sum of vertical r1 mm 25.000 25.000 r2 mm 57.000 56.000 r3 mm 73.500 73.500 r mm 44.000 42.000 Link 4 r5 mm 93.500 95.500 r6 mm 140.000 140.000 r0x mm 78.000 78.000 r y mm 53.000 53.900 0 − − Stroke mm 85.994 86.018 Piston pin position at TDC mm 197.391 185.124 Result Max. of 2nd harmonic acceleration sum mm/rad2 35.911 14.963 58.333 − Max. of 4th harmonic acceleration sum mm/rad2 10.714 6.977 34.882 −

The link geometry of the initially designed Stephenson III mechanism was compared with the final design with minimized vertical second harmonic acceleration in Figure 10. In Figure 11, the vertical harmonic acceleration components at the piston pin in the finally designed single-cylinder Stephenson III mechanism were compared with those of a conventional slider-crank mechanism whose crank length was 43 mm, offset was 0 mm, stroke was 86 mm, and piston pin position at TDC was 190 mm. The maximum vertical second harmonic acceleration at the piston pin in the final design of the Stephenson III mechanism was considerably lower than that of the slider-crank mechanism, each of which was calculated as 0.596 mm/rad2 and 12.859 mm/rad2. Even though the third and fourth acceleration components in the Stephenson III mechanism were calculated as 5.541 mm/rad2 and 1.157 mm/rad2, respectively, the vertical third harmonic acceleration would be cancelled out in the in-line four-cylinder engine and the vertical fourth harmonic acceleration was considerably lower than the second harmonic acceleration of the slider-crank mechanism. In Figure 12, the vertical second harmonic acceleration at each joint in the final design of the Stephenson III mechanism was compared with those of the slider-crank mechanism in the in-line four-cylinder engine configuration. In the case of the slider-crank mechanism, the peak value of the sum of the vertical second harmonic acceleration was 51.435 mm/rad2, while the peak value of the overall or sum of the vertical second harmonic acceleration components of the Stephenson III mechanism was 14.963 mm/rad2, which was about 29.091% of that for the slider-crank mechanism. Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 19 second harmonic acceleration components. Hence, the final dimensions were selected from this case. In Table 7, the dimensions, stroke, piston pin position at TDC, and the maximum sums of the second and fourth harmonic acceleration components in the in-line four-cylinder engine are shown and compared with those of the initial design. The maximum of the vertical second and fourth harmonic acceleration sums were decreased by 58.333% and 34.882%, respectively.

Table 7. Comparison of the initial and final mechanism.

Unit Initial Design Final Design Effect (%) r1 mm 25.000 25.000 r2 mm 57.000 56.000 r3 mm 73.500 73.500 r4 mm 44.000 42.000 Link r5 mm 93.500 95.500 r6 mm 140.000 140.000 r0x mm 78.000 78.000 r0y mm −53.000 −53.900 Stroke mm 85.994 86.018 Piston pin position at TDC mm 197.391 185.124 Max. of 2nd harmonic Result mm/rad2 35.911 14.963 −58.333 acceleration sum Max. of 4th harmonic mm/rad2 10.714 6.977 −34.882 acceleration sum

The link geometry of the initially designed Stephenson III mechanism was compared with the Appl. Sci. 2020, 10, 3765 15 of 19 final design with minimized vertical second harmonic acceleration in Figure 10.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 19

calculated as 5.541 mm/rad2 and 1.157 mm/rad2, respectively, the vertical third harmonic acceleration would be cancelled out in the in-line four-cylinder engine and the vertical fourth harmonic acceleration was considerably lower than the second harmonic acceleration of the slider-crank mechanism.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 19 60 60 Overall Overall 1st 40 40 ) )

2 2 2

2 calculated as 5.541 mm/rad and 1.157 mm/rad , respectively,2nd the vertical third harmonic 20 20 acceleration4th would be2nd (acancelled) 3rd out in the in-line four-cylinder4th engine (andb) the vertical fourth 3rd harmonic0 acceleration was considerably lower than0 the second harmonic acceleration of the FigureFigure 10. 10. StephensonStephenson III III VCR VCR engine engine mechanism: mechanism: ( (aa)) initial initial design designTDC and and ( (bb)) final ﬁnal design. design. BDC slider-crank mechanism.TDC BDC -20 -20 Vertical harmonic Vertical harmonic Vertical harmonic acceleration (mm/rad acceleration acceleration (mm/rad -40In60 Figure 11, the vertical harmonic acceleration-40 60components at the piston pin in the finally Overall designed single-cylinder1st Stephenson III mechanism were compared withOverall those of a conventional1st -6040 -6040 ) ) 2 slider-crank0 mechanism 90 whose 180 crank 270 length was 360 43 mm,2 090180270360 offset was 0 mm, stroke was 86 mm, and Crank angle (degree) 2ndCrank angle (degree) 20 20 piston pin4th position at TDC2nd was 1903rd mm. The maximum vertical 4thsecond harmonic acceleration at the 3rd 0 (a) 0 (b) piston pin in the final design of the Stephenson III mechanism wasTDC considerably lower BDC than that of TDC BDC 2 2 the slider-crank-20Figure 11. Vertical mechanism, harmonic each acceleration of which components was calculated at-20 piston as pin 0.596 in single mm/rad cylinder and engine: 12.859 (a )mm/rad final . Vertical harmonic Vertical harmonic Vertical harmonic acceleration (mm/rad Even acceleration (mm/rad designthough of the Stephensonthird and fourthIII mechanism acceleration and (b )components the slider-crank in mechanism.the Stephenson III mechanism were -40 -40 1st -60 -60 In 0Figure 12, 90 the vertical 180 second 270 harmonic 360 acceleration090180270360 at each joint in the final design of the Stephenson III mechanismCrank angle was(degree) compared with thos e of the slider-crankCrank anglemechanism (degree) in the in-line four-cylinder engine configuration.(a) In the case of the slider-crank mechanism,(b )the peak value of the sum of the vertical second harmonic acceleration was 51.435 mm/rad2, while the peak value of the FigureFigure 11.11. VerticalVertical harmonicharmonic accelerationacceleration components components at at piston piston pin pin in in single single cylinder cylinder engine: engine: ((aa)) ﬁnalfinal overall or sum of the vertical second harmonic acceleration components of the Stephenson III designdesign ofof thethe StephensonStephenson IIIIII mechanismmechanism andand ( b(b)) the the slider-crank slider-crank mechanism. mechanism. mechanism was 14.963 mm/rad2, which was about 29.091% of that for the slider-crank mechanism.

40In Figure 12, the vertical second harmonic acceleration60 at each joint in the final design of the Stephenson III mechanism was compared with those of the slider-crank mechanism in the in-line 30 P5 40 four-cylinder engine configuration.P6 Overall In the case of the slider-crank mechanism, the peak value of the 20 Piston pin ) ) 2 2 2 sum of the vertical second harmonic acceleration was 51.43520 mm/rad , while the peak value of the overall10 or sum of the vertical second harmonic acceleration components of the Stephenson III

0 2 0 mechanism wasTDC 14.963 mm/rad , which BDC was about 29.091% of that TDCfor the slider-crank BDCmechanism.

-10 40 -2060 acceleration acceleration (mm/rad acceleration acceleration (mm/rad Vertical second harmonic Vertical harmonic second Vertical -20 30 P -4040 P5 2 -30 P6 Overall 20 Piston pin ) ) 2 2 -40 -6020 10 0 90 180 270 360 090180270360 Crank angle (degree) Crank angle (degree) 0 0 TDC BDC TDC BDC (a) (b) -10 -20 acceleration acceleration (mm/rad acceleration acceleration (mm/rad Vertical second harmonic Vertical harmonic second Vertical Figure-20 12. Vertical second harmonic acceleration compon componentsents at each joint in four-cylinder engine: -40 ((a)) ﬁnalfinal designdesignP ofof thethe StephensonStephenson IIIIII mechanismmechanism andand ((bb)) thethe slider-crankslider-crank mechanism.mechanism. -30 2

-40 -60 5. Discussion0 90 180 270 360 090180270360 Crank angle (degree) Crank angle (degree) The conceptual design of a new VCR engine may depend on trial and error to find the feasible (a) (b) kinematic structure. The purpose of this research was to propose a simple method to determine a feasibleFigure mechanism 12. Vertical and second its dimensions harmonic acceleration that would compon yieldents a nearly at each minimized joint in four-cylinder sum of theengine: vertical second(a ) harmonicfinal design acceleration of the Stephenson at joints III mechanism without going and (b )through the slider-crank an optimization mechanism. procedure. For the validation of the proposed method, a constrained nonlinear optimization was carried out using ‘fmincon’5. Discussion solver of MATLAB optimization tool box to minimize the maximum sum of the second harmonicThe conceptualacceleration design at each of jointa new with VCR the engine design may specification depend on given trial and in Table error 2to and find the the Grashof feasible criteriakinematic for structure.the crank-rocker The purpose four bar of asthis the research constraints. was Theto propose computational a simple time method of the to optimization determine a wasfeasible 9.6 smechanism using a computer and its (Intel®dimensions i7 Quad that CPUwould 1.8 yield GHz, a 8nearly GB RAM) minimized with Windows sum of the10 (64vertical bit) operatingsecond harmonic system. Theacceleration results of at the joints optimization without going were comparedthrough an with optimization that of the procedure. proposed method For the validation of the proposed method, a constrained nonlinear optimization was carried out using ‘fmincon’ solver of MATLAB optimization tool box to minimize the maximum sum of the second harmonic acceleration at each joint with the design specification given in Table 2 and the Grashof criteria for the crank-rocker four bar as the constraints. The computational time of the optimization was 9.6 s using a computer (Intel® i7 Quad CPU 1.8 GHz, 8 GB RAM) with Windows 10 (64 bit) operating system. The results of the optimization were compared with that of the proposed method Appl. Sci. 2020, 10, 3765 16 of 19

5. Discussion The conceptual design of a new VCR engine may depend on trial and error to find the feasible kinematic structure. The purpose of this research was to propose a simple method to determine a Appl.feasible Sci. 2020 mechanism, 10, x FOR andPEER its REVIEW dimensions that would yield a nearly minimized sum of the vertical second16 of 19 harmonic acceleration at joints without going through an optimization procedure. For the validation inof Table the proposed 8 and Figure method, 13. aAs constrained shown in Table nonlinear 8, th optimizatione dimensions wasdetermined carried out by usingthe proposed ‘fmincon’ method solver andof MATLAB the optimization optimization were tool slightly box to minimize different, the because maximum the sum selected of the secondvariables harmonic in the acceleration proposed procedureat each joint were with the the link design lengths specification except givenr1 and in r0 Tablex, while2 and the the optimization Grashof criteria considered for the crank-rocker all the link lengthsfour bar as as variables. the constraints. The computational time of the optimization was 9.6 s using a computer (Intel®i7 Quad CPU 1.8 GHz, 8 GB RAM) with Windows 10 (64 bit) operating system. The results of the optimizationTable 8. Comparison were compared of the withresults that of the of theproposed proposed method method and the in Tableconstrained8 and Figureoptimization. 13. As shown in Table8, the dimensions determined by the proposedInitial methodResult of andProposed the optimization Result of were Effect slightly Unit different, because the selected variables in the proposedDesign procedureMethod were the linkOptimization lengths except (%)r1 and r0x, while the optimizationr1 consideredmm all the link25.000 lengths as variables.25.000 24.95798 - r2 mm 57.000 56.000 56.00000 - Table 8. Comparison of the results of the proposed method and the constrained optimization. r3 mm 73.500 73.500 73.44895 - r4 mm 44.000Initial 42.000Result of Result42.00000 of - Link Unit Effect (%) r5 mm 93.500Design Proposed95.500 Method Optimization95.49999 -

r6 r1 mm mm140.000 25.000 140.000 25.000 138.0000 24.95798 -- r mm 57.000 56.000 56.00000 - r0x 2 mm 78.000 78.000 80.00000 - r3 mm 73.500 73.500 73.44895 - r0y r mm mm−53.000 44.000 −53.900 42.000 − 42.0000055.00000 -- Link 4 Stroke r5 mm mm85.994 93.50086.018 95.500 95.4999985.975 -- Piston pin positionr6 mm mm 197.391 140.000 185.124 140.000 138.0000 184.424 - - r mm 78.000 78.000 80.00000 - Max. 2nd harmonic0x Result r0y mm/rad2 mm35.911 53.000 14.96353.900 55.00000 13.773 −7.940 - acceleration sum − − − Stroke mm 85.994 86.018 85.975 - Max. 4th harmonic Piston pin position2 mm 197.391 185.124 184.424 - Result mm/rad 10.714 6.977 7.114 +1.997 Max.acceleration 2nd harmonic sum acceleration sum mm/rad2 35.911 14.963 13.773 7.940 − Max. 4th harmonic acceleration sum mm/rad2 10.714 6.977 7.114 +1.997

40 40

30 30 P5 P5 P6 Overall P6 Overall 20 20 ) ) 2 2

10 10

0 0 TDC BDC TDC BDC

-10 -10 acceleration acceleration (mm/rad acceleration (mm/rad

Vertical harmonic second Vertical -20 harmonic second Vertical -20

P P -30 2 -30 2

-40 -40 0 90 180 270 360 0 90 180 270 360 Crank angle (degree) Crank angle (degree) (a) (b)

FigureFigure 13.13. VerticalVertical second second harmonic harmonic acceleration acceleration components components at joints at of joints the Stephenson of the Stephenson III mechanism III mechanismin the four-cylinder in the four-cylinder engine: (a) proposedengine: (a method) proposed and method (b) optimization and (b) optimization using commercial using commercial software. software. Even though the optimization method yielded optimum dimensions when a mechanism to be optimizedEven though was selected, the optimization it did not provide method information yielded optimum that could dimensions be used to selectwhen a a feasible mechanism mechanism to be optimizedamong many was candidates. selected, it The did proposed not provide method, informat on theion other that hand, could can be be used applied to select at the a kinematic feasible mechanismconceptual designamong stage many to candidates. determine a The feasible proposed mechanism method, and on its the dimensions other hand, simply can bybe plottingapplied theat thevertical kinematic second conceptual harmonic acceleration design stage component to determine at each a joint feasible and the mechanism influence ofand link its length dimensions changes. simplyThe influence by plotting of link the length vertical changes second on theharmonic second harmonicacceleration acceleration component can at be each utilized joint at and the finalthe influencelayout design of link stage length for a minutechanges. adjustment The influence of link of lengths link length and optimization changes on asthe well. second harmonic acceleration can be utilized at the final layout design stage for a minute adjustment of link lengths and optimization as well. For the completion of the kinematic design of a VCR engine, the control function needs to be considered. The proposed procedure can be initially applied to determine the dimensions of the mechanism for a specific compression ratio at which the vertical second harmonic acceleration needs to be taken into account more seriously. In order to vary the compression ratio, the ground pivot, P3, in Figure 10b needs to be moved to a new position. For a new position of P3, examine the Appl. Sci. 2020, 10, 3765 17 of 19

For the completion of the kinematic design of a VCR engine, the control function needs to be considered. The proposed procedure can be initially applied to determine the dimensions of the mechanism for a specific compression ratio at which the vertical second harmonic acceleration needs Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 19 to be taken into account more seriously. In order to vary the compression ratio, the ground pivot, P3, invertical Figure second 10b needs harmonic to be movedacceleration to a new components position. Forat the a newjoints position in the ofmechanismP3, examine with the the vertical same seconddimensions harmonic except acceleration for the position components of P3(r0x at, r the0y). If joints the acceleration in the mechanism is not withwithin the an same allowable dimensions range, exceptthe above for the procedure position ofcanP3( r0bex, r0reappliedy). If the accelerationto find dimensions is not within that an allowablecompromise range, between the above the procedurecompression can ratios. be reapplied to find dimensions that compromise between the compression ratios. TheThe four-barfour-bar linkagelinkage in the Stephenson Stephenson III III mechan mechanismism may may take take the the form form ofof a crank-rocker a crank-rocker or ordouble-crank double-crank type. type. In Inthis this paper, paper, the the Stephenson Stephenson III IIImechanism mechanism with with a crank-rocker a crank-rocker four four bar bar is isconsidered considered for for the the VCR VCR engine, engine, which which is is known known to to have lower levels ofof verticalvertical andand horizontalhorizontal excitationexcitation forces forces and and can can achieve achieve the samethe same level leve of boomingl of booming noise performancenoise performance as conventional as conventional engines withoutengines awithout balance shafta balance [26]. Theshaft Stephenson [26]. The III St VCRephenson engine III mechanism VCR engine with amechanism double-crank with four a bardouble-crank shown in Figure four bar 14 isshown proposed in Figure by Komatsubara 14 is proposed and by Kuribayashi Komatsubara [12]: and the Kuribayashi domain of motion [12]: the of thisdomain type of is smallmotion compared of this type to thatis small of the compared Stephenson to that III crank-rocker of the Stephenson version, III andcrank-rocker the second version, order ofand piston the second acceleration order at of the piston engine acceleration speed of 300 at rpmthe engine in the singlespeed cylinderof 300 rpm engine in the is lower single than cylinder that ofengine the slider-crank is lower than engine that of mechanism the slider-crank at compression engine mechanism ratios 8 andat compression 9.3, but higher ratios at 8 compression and 9.3, but ratiohigher 16.5. at Forcompression comparison, ratio the vertical16.5. For second comparison harmonic, the acceleration vertical componentssecond harmonic in the finalacceleration design ofcomponents the Stephenson in the III final mechanism design of with the a Stephenson crank-rocker III four mechanism bar shown with Figure a crank-rocker 10b were computed four bar atshown low and Figure high 10b compression were computed ratios at by low altering and high the co positionmpression of theratios ground by altering pivot, theP3. position As shown of the in 2 Tableground9, the pivot, vertical P3. secondAs shown harmonic in Table acceleration 9, the vertical components second inharmonic unit of m acceleration/sec at 300 rpmcomponents in a single in cylinderunit of m/sec engine2 at were 300 lower rpm in than a single those ofcylinder the slider-crank engine were engine lower mechanism than those at bothof the low slider-crank and high compressionengine mechanism ratios. at both low and high compression ratios.

FigureFigure 14.14. StephensonStephenson IIIIII VCRVCR engineengine mechanismsmechanisms withwith double-crankdouble-crank typetype four-barfour-bar linkage.linkage. Table 9. Vertical second harmonic acceleration components in the ﬁnally designed single-cylinder StephensonTable 9. Vertical III with second crank-rocker harmonic four acceleration bar at low andcomponents high compression in the finally ratios. designed single-cylinder Stephenson III with crank-rocker four bar at low and high compression ratios. Mechanism Unit Stephenson III (Crank-Rocker) Slider Crank Stephenson III Slider Mechanism Unit Compression ratio - 8(Crank-Rocker) 16.5 Crank - r mm 72.100 78.000 - Compression0x ratio - 8 16.5 - r0y mm 46.900 53.900 - r0x mm− 72.100 − 78.000 - Stroker0y mmmm 85.502−46.900 86.018−53.900 86.000- 2 2nd harmonic accelerationStroke at piston pin m/s mm 3.70185.502 0.58886.018 12.69186.000 Max. 2nd harmonic acceleration sum m/s2 5.542 3.692 12.691 2nd harmonic acceleration at piston pin m/s2 3.701 0.588 12.691 Max. 2nd harmonic acceleration sum m/s2 5.542 3.692 12.691

6. Conclusions This paper proposed a kinematic conceptual design process of VCR engine mechanisms considering the vertical second harmonic acceleration. From the graphs of six-link kinematic chains, Watt II and Stephenson III linkages were selected as candidate VCR engine mechanisms, and their initial dimensions satisfying the prescribed design specifications were determined. Fourier series analysis on the acceleration at each joint of the above two initially designed VCR engine mechanisms shows that the vertical second harmonic acceleration components of the Watt II linkage was much higher when they were used in the in-line four-cylinder engine, hence the Stephenson III mechanism was selected as the feasible VCR engine mechanism. Assuming that the concentrated mass at each joint in the Stephenson III mechanism was the same, the effects of link length changes on the maximum sum of vertical second harmonic acceleration components and the stroke were examined, and then links whose lengths would be altered to reduce the second harmonic acceleration and the other links to satisfy the stroke and to reduce acceleration were selected. By adjusting the link lengths from those of the initial design, the overall vertical second harmonic acceleration of the finally designed Stephenson III mechanism was reduced by 58.333% compared to that of the initial design and by 70.909% compared to the slider-crank mechanism. The proposed method can be applied to determine feasible mechanisms, their dimensions, and ideal mass distribution of the links in the automotive industrial applications, such as multi-link VCR and variable stroke engines of various multi-cylinder configurations, where the second harmonic acceleration needs to be considered at the kinematic conceptual design stage.

Author Contributions: Conceptualization, S.W.K. and J.K.S.; Methodology, Investigation, Software, Validation, Writing—Original Draft Preparation, S.W.K.; Formal Analysis, Y.K.M. and S.W.K.; Supervision, Writing—Review and Editing, J.K.S. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Heywood, J.B.; Welling, O.Z. Trends in Performance Characteristics of Modern Automobile SI and Diesel Engines. SAE Int. J. Engines 2009, 2, 1650–1662. 2. Shaik, A.; Moorthi, N.S.V.; Rudramoorthy, R. Variable compression ratio engine: A future power plant for automobiles—an overview. Proc. IMechE Part D 2007, 221, 1159–1168. [CrossRef] 3. Martyn, R. Beneﬁts and Challenges of Variable Compression Ratio (VCR). SAE Int. 2003.[CrossRef] 4. Gérard, D.; Besson, M.; Hardy, J.P.; Croguennec, S.; Thomine, M.; Aoyama, S.; Tomita, M. HCCI Combustion on a Diesel VCR Engine. SAE Int. 2008.[CrossRef] 5. Pesic, R.; Milojevic, S. Eﬃciency and ecological characteristics of a VCR diesel engine. Int. J. Automot. Technol. 2013, 14, 675–681. [CrossRef] 6. Stefan, P.; Kurt, I.Y.; Makus, S.; Knut, H. Variable compression in SI engines. SAE Int. 2001.[CrossRef] 7. Drangel, H.; Olofsson, E.; Raymond, R. The Variable Compression (SVC) and the Combustion Control (SCC)—Two Ways to Improve Fuel Economy and Still Comply with World-Wide Emission Requirements. SAE Int. 2002, 111, 1691–1706. 8. Ashish, J.C.; Vinayak, K.; Niranjan, S. State-of-the-art technology in variable compression ratio mechanism for spark ignition engine. Sadhan¯ a¯ 2018, 43, 211. 9. Ishikawa, S.; Kadota, M.; Yoshida, K.; Takahashi, K.; Kawajiri, S. Advanced Design of Variable Compression Ratio Engine with Dual Piston Mechanism. SAE Int. 2009.[CrossRef] 10. Jiadui, C.; Bo, W.; Dan, L.; Kai, Y. Study on the Dynamic Characteristics of a Hydraulic Continuous Variable Compression Ratio System. Appl. Sci. 2019, 9, 4484. 11. Moteki, K.; Aoyama, S.; Ushijima, K.; Hiyoshi, R.; Takemura, S.; Fujimoto, H.; Arai, T. A Study of a Variable Compression Ratio System with a Multi-Link Mechanism. SAE Int. 2003.[CrossRef] Appl. Sci. 2020, 10, 3765 19 of 19

12. Komatsubara, H.; Kuribayashi, S. Research and development of new variable compression ratio (VCR) engine mechanism (1st report, Basic characteristics and design of VCR engine mechanism). JSME 2018, 84, 1–13. [CrossRef] 13. Vianney, R.; Jacques, B.; Frederic, D. Study of a Gear-Based Variable Compression Ratio Engine. SAE Int. 2004. [CrossRef] 14. Hoeltgebaum, T.; Simoni, R.; Martins, D. Reconﬁgurability of engines: A kinematic approach to variable compression ratio engines. Mech. Mach. Theory 2016, 96, 308–322. 15. Arakelian, V.; Dahan, M.; Smith, M. A Historical Review of the Evolution of the Theory on Balancing of Mechanisms. In Proceedings of the International Symposium on History of Machines and Mechanisms Proceedings HMM 2000, University of Cassino, Italy, 11–13 May 2000; Marco, C., Ed.; Springer-Science + Business Media: Berlin, Germany, 2000; pp. 291–300. 16. Arakelian, V.H.; Smith, M.R. Shaking Force and Shaking Moment Balancing of Mechanisms: A Historical Review with New Examples. ASME J. Mech. Des. 2005, 127, 334–339. [CrossRef] 17. Heifetz, M.; Marsh, M. Engine Dynamics and Balancing. SAE Int. 1984.[CrossRef] 18. Norton, R.L. Design of Machinery, 3rd ed.; McGraw-Hill Higher Education: New York, NY, USA, 2009. 19. Robert, B.G. Automotive Handbook, 8th ed.; John Wiley & Sons: West Sussex, UK, 2010. 20. Tsai, L.W. Mechanism Design: Enumeration of Kinematic Structures According to Function; CRC Press: London, UK, 2000. 21. Mo, Y.K.; Shim, J.K.; Lim, D.J. Kinematic Structure Analysis of VariableCompression Ratio Engine Mechanisms. KSAE 2018, 26, 159–166. 22. Freudenstein, F.; Maki, E.R. Development of an optimum variable-stroke internal-combustion engine mechanism from the viewpoint of kinematic structure. ASME J. Mech. Trans. Autom. 1983, 105, 259–266. [CrossRef] 23. Arakelian, V.H.; Smith, M.R. Complete shaking force and shaking moment balancing of linkages. Mech. Mach. Theory 1999, 34, 1141–1153. [CrossRef] 24. Waldron, K.J.; Kinzel, G.L. Kinematics, Dynamics, and Design of Machinery; John Wiley & Sons: New York, NY, USA, 1999. 25. Erdman, A.G.; Sandor, G.N.; Kota, S. Mechanism Design: Analysis and Synthesis, 4th ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2001. 26. Sato, Y.; Masahiko, K.; Masayuki, H. A Study Concerning Booming Noise of a Multi-link Type Variable Compression Ratio Engine. SAE Int. 2009.[CrossRef]

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).