A Study of a Variable Compression Ratio and Displacement
Mechanism Using Design of Experiments Methodology
Shugang Jiang, Michael H. Smith, Masanobu Takekoshi

Abstract
Due to the ever increasing requirements for engine performance, variable
compression ratio and displacement are drawing great interest, since these features
provide further degrees of freedom to optimize engine performance for various
operating conditions. Different types of mechanisms are used to realize variable
compression ratio and displacement. These mechanisms usually involve relatively
complicated mechanical design compared with conventional engines. While theflexibility in these mechanisms introduces additional engine design and control
possibilities, it also increases the challenges for the development and optimization,
since how the geometry of the mechanisms affects the engine performance is notstraightforward, and building prototype engines of various mechanical design
parameters takes a long time with high cost.
This paper presents a study of a multiple-link mechanism that realizes variable
compression ratio and displacement by varying the piston motion, specifically the
Top Dead Center (TDC) and Bottom Dead Center (BDC) positions relative to the
crankshaft. The study focuses on obtaining desirable behavior for both compression
ratio and displacement. It is determined a major requirement for the design of themechanism is that when the control action changes monotonically over its whole
range, the compression ratio and displacement should change in opposite directions
monotonically. This paper gives an approach on how to achieve multiple-linkmechanism designs that fulfill this requirement. First, a condition is obtained on how
the TDC and BDC positions should change with respect to the control action to fulfill
the design requirement. Then Design of Experiments (DoE) methodology is used for
creating sets of geometric designs of the mechanism. Kinematics of the mechanism
are calculated for each design and checked against the condition. Based on theoutcome, a feasible design is selected and detailed study on such characteristics aspiston motion, stroke length, combustion chamber volume, displacement, crankangle range of the expansion stroke, and compression ratio is performed. Directions
for future work in simulation and testing of the mechanism are given.

1.

Introduction

Ever increasing requirements for engine performance and stricter exhaust emissions
and fuel economy standards have been driving engine technology advancement.
Variable compression ratio technology has long been recognized as a method toimprove engine fuel economy [1]-[3]. As the compression ratio increases, the engine
thermal efficiency increases. However, too high compression ratio will lead to engine
knock. With variable compression ratio, at low power levels the engine operates at ahigher compression ratio to capture the benefits of high fuel efficiency; at high power

levels the engine operates at a lower compression ratio to prevent knock. By
continuously changing the compression ratio, an engine enables optimum
combustion efficiency at all engine speed and load conditions, realizing better engine
performance, lower fuel consumption, and lower exhaust emissions.
There are various approaches to realize variable compression ratio, such as moving
the cylinder head, moving the crankshaft axis, and using multiple links for connection
between piston and crankshaft. While the main purpose of these mechanisms is torealize variable compression ratio, they might also lead to changes in many otheraspects of the engine, such as piston motion, stroke length, charging anddischarging efficiency, and combustion processes. These characteristics need to beconsidered when designing the variable compression ratio mechanism anddetermining its detailed dimension values. Optimization of related engine control
parameters need be performed accordingly as well.
This paper focuses on studying a multiple-link connecting rod mechanism [4]-[6].
This mechanism realizes variable compression ratio by varying the piston motion,
specifically the TDC and BDC positions of the engine. So far major studies of themultiple-link mechanism have focused on the variable compression ratio
characteristics. However, a good design of the geometry of the mechanism should
realize not only the desired compression ratio, but also advantageous displacement
behavior of the engine. Since higher compression ratio is used for low and medium
power levels, where decreased displacement is advantageous, the desired design ofthe multiple-link mechanism is that when the control action changes monotonically,
the compression ratio and the displacement change monotonically in opposite
directions.
This paper presents an approach to obtain geometric designs of the mechanism that
fulfill the requirement mentioned above. First a condition is obtained in the format of
how the TDC and BDC positions should change with respect to the control action inorder that the mechanism will give the desired behavior. Equations describing thekinematics of the components in the mechanism are derived. The piston position canbe calculated during the whole revolution of the crankshaft for specific mechanism
dimensions and control action. To obtain geometric designs of the mechanism thatsatisfy the condition, Design of Experiments (DoE) methodology is used for creatingsets of geometric parameters, including crank radius, crank offset, upper link length,
lower link dimensions, control link length, control link position, and control shaft
length. Piston motion corresponding to each parameter set is calculated and the
behavior is checked against the condition. A feasible design that satisfies thecondition is selected for further study. Such characteristics as detailed piston motion
profile, piston velocity and acceleration, crank angle range of the expansion stroke,
combustion chamber volume, displacement, and compression ratio are investigated
for the design. These results serve as a basis for further analysis and optimization of
the mechanism geometric design.
The paper is organized as follows: the multiple-link mechanism is described and its
equations of motion are derived in Section 2. Section 3 presents the condition on
how the TDC and BDC positions should change with respect to the control action to
fulfill the design requirement for achieving desired compression ratio anddisplacement behavior. In Section 4, a Design of Experiments (DoE) plan isestablished for the mechanism dimensions, and the kinematics of the mechanism
are calculated for each design and checked against the condition. Further detailed
study of a design that satisfies the condition is presented in Section 5. Section 6
provides directions for future studies of the mechanism using simulation softwareand testing. The last section summarizes the paper.

2.

Multiple-link Mechanism

Configuration of the multiple-link mechanism is shown in Figure 1. Origin (O) is the
crankshaft axis. The crank pin (A) and piston pin (E) are connected by means of an
upper link (DE) and a lower link (ABD). The lower link is also connected, via a control
link (BC), to the eccentric journal of a control shaft (SC). When the crankshaft
rotates, the lower link revolves around the crankshaft axis and pivots around the
crank pin. Meanwhile, motion of the lower link is constrained by the control link. Notethe crankshaft (OA), the lower link (AB), the control link (BC), and segment (OC)which can be considered a ground link, form a four-bar linkage. As a result, when the
crankshaft rotates, the control link acts as a rocker and rotates through a limited
range of angles around the eccentric journal of the control shaft. The connection
point (D) of the lower and upper links traces roughly an oval shape and the piston pin(E) undergoes reciprocal motion.
The control shaft (SC) is supported by the engine block. The distance between thecenter of the eccentric journal and the control shaft axis is denoted as ls , and will becalled control shaft length in this study. When the angle of the control shaft relative tothe engine block ( θ s ) is changed, the position of the control link will change, which
subsequently tilts the lower link clockwise or counterclockwise and thereby affects
the motion of the piston and shifts the TDC and BDC positions. This leads to change
in both the combustion chamber volume and the swept volume, and consequently
changes the compression ratio. By manipulating the control shaft angle θ s theengine can realize different compression ratios for various operating conditions.
Note that in order for the multiple-link mechanism to operate as desired, certain
constraints need to be satisfied for its geometric dimensions over the whole range ofthe control action θ s . A major requirement is that the lengths of links of the four bar
linkage (OABC) need to satisfy the Grashof Condition, which states that the sum of
the length of the shortest and longest links need be less than that of the remaining
links. This constraint needs to be verified during design of the mechanism
dimensions.

Figure 1 Multiple-link Mechanism
The equations governing the motion of the multiple-link mechanism are derived asfollows. In Figure 1, denote the position of the control shaft axis (S) as ( X S , YS ). The
position of control link point (C) is then:(1)
X C = X S + ls cos θ s
YC = YS + ls sin θ s(2)Note for the four bar linkage (OABC), we have:
OA + AB = OC + CB

(3)

Rewrite equation (3) and define a new vector Z as:
Z = ( X Z , YZ ) = OA − OC = CB − AB(4)

Since OA and OC are known at each crank angle θ1 and control shaft angle θ s , Z
can be calculated and we have(5)
X Z = l1 cosθ1 − X c
YZ = l1 sin θ1 − Yc

From equation (4) we also have(6)

X Z = l3 cos θ3 − l2 cos θ 2(7)

YZ = l3 sin θ3 − l2 sin θ 2(8)

By eliminating θ 2 from equations (7) and (8), we obtain(9)l 2 − l 2 + X Z2 + YZ2
X Z cos θ3 + YZ sin θ3 = 3 2
2l3
Define the right hand side of equation (9) as m , we have the solution as
(10)


Y 
m

θ3 = tan −1  Z  ± cos −1 
 X 2 +Y2  XZ ZZ 
The two solutions of θ3 correspond to the two modes of assembly for the four bar
linkage (OABC), which are symmetric about line OC. The solution for Figure 1 is:
(11)


Y 
m

θ3 = tan −1  Z  + cos −1 
 X2 +Y2  XZ ZZ 

Subsequently, θ 2 can be calculated from equation (8) as:(12) l sin θ3 − YZ 
θ 2 = π − sin −1  3l2

Since OB = OC + CB , the position of B can be calculated as:(13)
X B = X C + l3 cos θ3
YB = YC + l3 sin θ3(14)

Thus, the motion of the four bar linkage (OABC) is determined. The motion of point(D) can be calculated by making use of the lower link dimensions. Note:
2
2
2
(15)
−1  l2 + l4 − l5 
θ 4 = θ 2 − cos  2l2l4 
the Position of (D) can be obtained from OD = OA + AD as:
X D = l1 cos θ1 + l4 cos θ 4

YD = l1 sin θ1 + l4 sin θ 4

(16)
(17)

Lastly, the position of the piston pin (E) can be obtained. Denote the crank offset asδ , then(18)δ − XD θ6 = cos −1  l6 
and subsequently from OE = OD + DE , the piston pin position is obtained as:
YE = YD + l6 sin θ 6(19)Thus, with the geometric dimensions of the multiple-link mechanism known, we cancalculate the motion of each component for the whole crank shaft revolution, given
the control shaft angle. In this study the piston motion is the focus. With the piston
motion profile calculated for the crank angle ( θ1 ) range from 0° to 360°, we canobtain the TDC and BDC positions and their corresponding crank angles. We canalso calculate the velocity and acceleration of the piston, the stroke length, the crankangle range of the expansion stroke, the displacement, the combustion chamber
volume, and the compression ratio.

3.

Multiple-link Mechanism Design Criterion

The main purpose of the multiple-link mechanism is to realize variable compression
ratio. However, because the mechanism realizes this feature by changing the piston
motion, it may lead to changes in the stroke length, and consequently thedisplacement, as well. When compression ratio and displacement change, various
characteristics of the engine, such as the charging and discharging processes,
volumetric efficiency, combustion process and efficiency, and power output all canchange. When designing a multiple-link mechanism, all these influences need to beconsidered. Since compression ratio and stroke length (and displacement) areultimately determined by the geometry of the mechanism, this paper focuses onfinding an approach to obtain the geometric dimensions that give advantageouscompression ratio and displacement behavior. This is the basis for further design andoptimization of the mechanism.
As discussed before, when engine load decreases, compression ratio should
increase to capture the benefits of higher fuel efficiency; while for a decreased
engine load demand, a decreased displacement is advantageous, since fordecreased displacement and stroke length, the pumping loss and frictional loss are
reduced. With variable displacement, less throttling of the intake air will be needed totake care of the load variation. This is a desired property for the engine.
From the above discussion, we can see that a desired design of the multiple-linkmechanism is that when the control action changes monotonically, the compression
ratio and the displacement should change monotonically in opposite directions.
Denote compression ratio as ε and displacement as Vd ─ note they are both
functions of the control shaft angle θ s ─ then for the whole range of θ s , we want tohave either Scenario I:(20)
dε (θ s )
<0
dθ s
(21)
dVd (θ s )>0dθ sor Scenario II:(22)dε (θ s )
>0
dθ s
(23)
dVd (θ s )<0dθ s
Let us investigate Scenario I. Denote the combustion chamber volume as Vc , whichis also a function of θs . Because(24)
V (θ )ε (θ s ) = 1 + d s
Vc (θ s )we have

dε
=
dθ s

Vc

dVd
dV
− Vd c
dθ s
dθ s
2
Vc

(25)

To satisfy (20), we need
(26)
dVd
dV
< Vd cdθ s
dθ s
From (21) and (26), it is obvious that we must have(27)
dVc (θ s )>0dθ sThat is, to achieve lower compression ratio when θ s increases, both Vd and Vc needto increase. Regarding changes of TDC and BDC positions, denote the position of
TDC as YTDC and that of BDC as YBDC , which are functions of θ s . Since(28)
1
Vd = πd 2 (YTDC − YBDC )4
where d is the cylinder bore diameter, we have(29)
dVd 1 2  dYTDC dYBDC = πd −dθ s 4dθ s  dθ sAlso note the change of the combustion chamber volume is caused by the change ofthe position of TDC, we have
(30)
dVc
1
dY
= − πd 2 TDC
dθ s
4
dθ s
From (21) and (29) we know
(31)
dYTDC dYBDC
>
dθ s
dθ s
From (27) and (30) we know
(32)
dYTDC
<0
dθ s
Thus we have
(33)
dYBDC dYTDC
<
<0
dθ s
dθ s
This indicates that when the control shaft angle increases, both TDC and BDC
positions will move lower, and BDC will move lower at a bigger rate than that of TDC.
Substitute (29) and (30) into (26), and also note (33), it is derived
(34)
dY
dYTDC
1 < BDC
<ε
dθ s
dθ s
With similar analysis, it is found that Scenario II gives the same result as in (34). Thisis expected because Scenario I and II are in essence the same and they only differ
by how we define the positive direction of the control action θ s . The condition shownin (34) indicates that in order the multiple-link mechanism can realize the desired
compression ratio and displacement behavior, when the control shaft angle changes,
both TDC and BDC positions need to move in the same direction; the change ratewith respect to the control shaft angle for the BDC position should be bigger than thatof the TDC, but smaller than ε times that of the TDC.
Vc

To simplify the calculation, instead of finding the compression ratio corresponding toeach control shaft angle, for the condition in (34) we can choose to use the minimum
compression ratio ε 0 the mechanism needs to realize, and thus get:
(35)
dY
dYTDC
1 < BDC
< ε0
dθ s
dθ s
This is a stronger condition for a design to fulfill the requirement. In the following
section we will use this condition as a criterion for selection of the mechanism’s
geometric dimension.

4.

Design of Experiments

The previous section presents the condition for a mechanism design to givedesirable compression ratio and displacement characteristics. However, it can beseen from Section 2 that the individual and combined influence of the geometricparameters and the control parameter on the motion of the piston, specifically TDC
and BDC positions, is not straightforward. How they affect the compression ratio anddisplacement could be complicated and may not be easily determined. This places achallenge on the design and optimization of the multiple-link mechanism dimensions.
In this section, a DoE method is used for creating a series of multiple-link mechanism
parameter combinations, which are investigated to obtain the mechanism designs
that fulfill the design requirement.
Refering to Figure 1, the multiple-link mechanism geometric design parameters
include the following:
l1 , l2 , l3 , l4 , l5 , l6 , δ , X s , Ys , ls
First, a specific range for each parameter is set up based on experience, including
reasonable physical costraints or limits. Table 1 shows the geometric parameter
ranges adopted for this study. Then, in the space spanned by all the parameters,
DoE is used to generate a series of parameter combinations, and the properties ofthe mechanism for each combination are calculated. For this study, we set the rangeof the control parameter θ s to be from -45° to 45°, and choose the mechanism
properties when θ s is 0° as the basis for calculation. The targeted compression ratio
range is from about 8 to about 14. The combustion chamber volume is set in such a
way that when θ s is 0°, the compression ratio realized is the designed value, which is
chosen to be 11 for this study. For each parameter set, the piston motion is
calculated for the whole range of θ s . At each control shaft angle, TDC and BDC
positions are determied, and the combustion chamber volume is calculated based on
how the TDC position changes from when θ s is 0°. The displacement is calculatedusing the TDC and BDC position information. The compression ratio and othercharacteristics are calculated subsequently.
Table 1 Geometric Parameters and Their Ranges
Parameterl1

Minimum(mm)
42

Maximum
(mm)
52

l2
l3

115
120

135
140

l4
l5

58
135

72
155

l6

150

170

δ
Xs
Ys
ls

-4
-126

8
-110

-136

-120

20

40

EasyDoE ToolSuite® from IAV Automotive Engineering, Inc. is used to establish atest plan, which is actually a design plan for the geometric parameters in this study.
Using this utility program, ranges and steps are set for each parameter, and a model
structure is selected. Creation of the test plan is also dependent on the design
method selected. D-optimality minimizes the generalized variance of the parameter
estimates. V-optimality minimizes the variance of the predicted response. In thisstudy, the test plan is created such that 60% are D-optimal points, 30% are V-optimal
points and 10% are space-filling points. With the design method selected, 158
parameter sets are created for investigation. For each of the created parameter set,
it is checked whether the geometry gives a truly operational multiple-link mechanism
for the whole range of the control action. i.e., the design needs to satisfy the Grashof
Condition. Figure 2 shows the DoE test plan results and the distribution of l1 , l2 , l3 inthree dimensions.

Figure 2 DoE Test Plan Results

5.

Test Plan Results Analysis

Behavior of the multiple-link mechanism is calculated for each parameter set of thetest plan over the whole range of the control action. Criterion obtained in Section 3 ischecked against each parameter set. For the DoE plan created in Section 4, only asmall percentage of designs satisfy the condition. A feasible solution is selected fromthose designs for further study. Table 2 lists the parameter values of this solution.
Table 2 Selected Mechanism Parameters
Parameter
l1
l2
l3

Value (mm)
42
115
140

l4
l5

58
155

l6

150

δ
Xs
Ys
ls

-4
-110
-120
30

Properties of the selected mechanism design are presented as follows. Figure 3 (a),(b), and (c) show the piston position, velocity and acceleration during a crankrevolution respectively, while the control shaft angle changes from -45° to 45° in 10°
increments. The red lines correspond to -45° control shaft angle, while the black lines
correspond to 45°. These Figures are typical for mechanism geometric dimensions
that satisfy the design condition. It can be seen that when the control shaft angle
changes, the TDC and BDC move in the same direction but at different rates. Whenthe control shaft rotates counterclockwise, i.e., when θ s increases, both TDC and
BDC move lower; the maximum velocity and acceleration rates increase. Also
deserves note is that the crank angle locations corresponding to the TDC and BDC
also change.

(a)

(b)

(c)
Figure 3 Piston Position, Velocity and Acceleration

Figure 4 shows the TDC and BDC positions with respect to the control shaft angle. It
indicates clearly that when the control shaft turns counterclockwise, both TDC and
BDC positions are lowered, and the rate of change of the BDC position is greaterthan that of the TDC position. Figure 5 shows that the design criterion, i.e., the ratioof the rate of change of the BDC position to that of the TDC position, is between thecompression ratio and 1, which indicates that the design satisfies the design
condition.

Figure 4 TDC and BDC Positions

Figure 5 Design Criterion
Figure 6 and 7 show how the compression ratio and stroke length change withrespect to the control shaft angle, respectively. It is clear that this case belongs to
Scenario I described in Section 3. When the control shaft rotates counterclockwise

from -45° to 45°, the compression ratio decreases from 13.74 to 7.98, and the stroke
length increases from 117.9 mm to 135.3 mm. Correspondingly, when the bore
diameter of the cylinder is 89 mm, the displacement increases from 0.7335 liter to0.8420 liter, and the combustion chamber volume increases from 0.0576 liter to0.1207 liter. Figure 8 shows how the displacement and combustion chamber volume
change with the control shaft angle.

Figure 6 Compression Ratio

Figure 7 Stroke LengthFigure 8 Displacement and Combustion Chamber Volume
Figure 9 shows how the crank angle range of the expansion stroke changes withrespect to the control shaft angle. When the control shaft rotates counterclockwise,
the curve basically follows a decreasing trend. The Figure shows that when themechanism is at the highest compression ratio, the expansion stroke covers 192.4
crank angle degree, while when the mechanism is at the lowest compression ratio,
the number reduces to 184.0 crank angle degree.

Figure 9 Crank Angle Range of the Expansion Stroke
Figure 10 and Figure 11 show the maximum piston speed and acceleration (absolutevalues) respectively. When the control shaft rotates counterclockwise, they bothbasically follow the increasing trend in the range, with a small deviation for theacceleration near the end of the range.

Figure 10 Maximum Piston Speed (Absolute Value)

Figure 11 Maximum Piston Acceleration (Absolute Value)

6.

Future Work

The approach presented in this paper gives multiple-link mechanism designs that
fulfill the desired compression ratio and displacement behavior. As shown in Section
5, major kinematics characteristics of the mechanism can be calculated. To further
investigate the engine performance when adopting specific mechanism design,
various simulation and testing studies need to be performed. Detailed simulation of
the engine performance involves various subsystems, processes and controls, suchas charging and discharging processes, fuel delivery, combustible mixture formation,
ignition timing, valve timing, combustion properties etc., which are related to themechanism design and piston kinematics directly or indirectly. Commerciallyavailable engine simulation software, such as GT-Power, might be used for thispurpose, provided it could take into account the specific piston motion and thesubsequent influences on various aspects of the engine performance. Related study
in this area will be performed and presented in the future.

Besides performing necessary simulations, it is desirable to build prototype enginesof various multiple-link gemmetric design parameters and carry out tests. However,this approach would take a long time with high cost. To deal with this challenge, A&D
Co. Ltd. developed a highly flexible testing engine, whose piston is electrohydraulically controlled and can realize arbitrary motion profiles within the physicallimits of the system. Figure 12 shows the configuration of the testing engine system.
The system consists of two ADX controllers, which are high performance real time
hardware platforms that realize specific control functions. One ADX serves as auniversal engine controller that implements such functions as ignition, injection etc.
The other ADX mainly implements hydraulic control of the piston and intake andexhaust valves. Based on the targeted engine testing condition, the hydraulic control
ADX will create necessary encoder signals that correspond to the rotational behaviorof the engine emulated. These signals are sent to the ignition/injection control ADX
for engine operation control. The hydraulic control ADX uses the piston position
sensor measurement as feedback, and control the piston motion to desired
trajectories by manipulating the hydraulic actuator that is attached to it. During thetesting engine operation, the hydraulic system will actuate or load the piston in orderthat it will follow the target curve of motion. Although greatly different from themechanical design of a real engine, by realizing the same piston motion, the testing
engine in a certain sense duplicates the operation and performance of the realengine. The testing results will provide further insights for improving the mechanism
design. By changing the target piston motion curve, the system can be used forstudying the characteristics of multiple-link mechanism engine of various geometric
designs. Preliminary testing shows that the system gives excellent performance inrealizing desired piston motion trajectories. Detailed work in this area will be
implemented and presented in the future.

Figure 12 Highly Flexible Testing Engine System

7.

Conclusion

This paper presents an approach on how to achieve multiple-link mechanism
designs that realize desirable behavior for both compression ratio and displacement.
A condition is obtained on how the TDC and BDC positions should change with

respect to the control action to fulfill the design requirement. DoE methodology isused for creating sets of geometric designs of the mechanism, which are checkedagainst the condition. Selected design that satisfies the condition is further studied indetails. The design approach and obtained results serve as a basis and provide
insights for further analysis and optimization of the multiple-link mechanism.

References[1]

Charles Mendler, Roland Gravel, “Variable Compression Ratio Engine”, SAE
2002-01-1940[2]

Martyn Roberts, “Benefits and Challenges of Variable Compression Ratio(VCR)”, SAE 2003-01-0398[3]

Amjad Shaik, N Shenbaga Vinayaga Moorthi and R Rudramoorthy, “Variable
Compression Engine: a Future Power Plant for Automobiles – an Overview”,
Proc. IMechE Vol. 221 Part D: J. Automobile Engineering[4]

Katsuya Moteki, Shunichi Aoyama, Kenshi Ushijima, et al., “A Study of a
Variable Compression Ratio System with a Multi-Link Mechanism”, SAE 200301-0921[5]

Naoki Takahashi, Shunichi Aoyama, Katsuya Moteki, et al., “A Study
Concerning the Noise and Vibration Characteristics of an Engine with MultipleLink Variable Compression Ratio Mechanism”, SAE 2005-01-1134[6]

Ryosuke Hiyoshi, Shunichi Aoyama, Shinichi Takemura, et al., “A Study of aMultiple-link Variable Compression Ratio System for Improving Engine
Performance”, SAE 2006-01-0616

Acknowledgements
The authors would like to thank IAV Automotive Engineering, Inc. for providing the
EasyDoE ToolSuite® for the study.