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Geometric Parameter Design of a Multiple-Link
Mechanism for Advantageous Compression Ratio andDisplacement Characteristics

2014-01-1627
Published 04/01/2014

Shugang Jiang and Michael H. Smith
A&D Technology Inc.
CITATION: Jiang, S. and Smith, M., "Geometric Parameter Design of a Multiple-Link Mechanism for Advantageous
Compression Ratio and Displacement Characteristics," SAE Technical Paper 2014-01-1627, 2014, doi:10.4271/2014-011627.
Copyright © 2014 SAE International

Abstract
Variable compression ratio and variable displacement
technologies are adopted in internal combustion engines
because these features provide further degrees of freedom to
optimize engine performance for various operating conditions.
This paper focuses on a multiple-link mechanism that realizes
variable compression ratio and displacement by varying thepiston motion, specifically the Top Dead Center (TDC) andBottom Dead Center (BDC) positions relative to the crankshaft.
It is determined that a major requirement for the design of thismechanism is when the control action changes monotonicallyover its whole range, the compression ratio and thedisplacement should change in opposite directions
monotonically. This paper presents an approach on how toachieve multiple-link mechanism geometric designs that fulfill
this requirement. First, a necessary and sufficient condition,
and a stronger sufficient condition are obtained on how the
TDC and BDC positions should change with respect to thecontrol action to fulfill the design requirement. Then Design ofExperiments (DoE) methodology is used for creating sets ofgeometric designs of the mechanism, for which kinematics arecalculated and checked against the conditions. A feasible
design that satisfies the conditions is selected and detailed
study on such characteristics as piston motion, stroke length,
displacement, combustion chamber volume, and compression
ratio etc. is performed. The design approach and obtained
results serve as a basis for further analysis and optimization of
the multiple-link mechanism.

Introduction
Variable compression ratio technology has long beenrecognized as a method to improve engine fuel economy [1, 2,
3, 4, 5, 6, 7]. At low power levels the engine operates at higher
compression ratio to capture the benefits of higher thermal
efficiency, while at high power levels the engine operates atlower 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. Variable stroke andconsequently variable displacement is also a desired feature[8, 9, 10, 11, 12, 13]. For fixed displacement engines, the load
variation is usually handled by throttling the intake air. For avariable displacement engine, when engine load demanddecreases, the displacement is decreased to at least partiallytake care of the need. With less throttling and decreased
stroke, the pumping loss and frictional loss are reduced, whichis advantageous for fuel economy. Variable displacement also
provides the potential for improved exhaust emissions.
There are various approaches to realize variable compression
ratio, such as moving the cylinder head, changing the
combustion chamber volume through a secondary piston,moving the crankshaft axis, and using multiple links forconnection between piston and crankshaft etc. There aredifferent types of mechanical realization of variabledisplacement as well. For the mechanisms that are capable ofrealizing both variable compression ratio and variabledisplacement, it makes sense to study both features in acomprehensive way to improve the engine performance. This
paper focuses on studying a multiple-link connecting rod
mechanism [5, 6, 7]. This mechanism was initially designed torealize variable compression ratio by varying the piston motion,
specifically the TDC and BDC positions of the engine. So far
the studies of the mechanism have focused on the variable
compression ratio characteristics. However, a good design ofthe geometry of this mechanism should also provide
advantageous displacement behavior of the engine. This paper
presents an approach to obtain geometric designs of themechanism that realize both advantageous compression ratio
and displacement characteristics.
The paper is organized as follows. First a description of themultiple-link mechanism is given. Equations describing thekinematics of the mechanism are derived. Then a necessaryDownloaded from SAE International by Shugang Jiang, Wednesday, March 19, 2014 07:57:45 PM

and sufficient condition, and a stronger sufficient condition are
obtained in the format of how the TDC and BDC positions
should change with respect to the control action in order thatthe mechanism will give the desired behavior. Next a Design ofExperiments (DoE) methodology is used for creating sets ofgeometric parameters of the mechanism, including crankradius, 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 conditions.

The equations governing the kinematics of the multiple-linkmechanism are derived as follows. First the motion of the four
bar linkage OABC need be obtained. From Figure 1, it isevident the position of crank pin A is:

(1)

(2)

A feasible design that satisfies the conditions is selected forfurther study. Such characteristics as detailed piston motion
profile, piston velocity and acceleration, displacement,
combustion chamber volume, compression ratio and crankangle range of the expansion stroke etc. are investigated forthe design.

Multiple-Link Mechanism
A schematic 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 C of a control shaft SC. When the
crankshaft OA 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, andsegment OC which can be considered a ground link, form afour-bar linkage. As a result, when the crankshaft rotates, thecontrol link acts as a rocker and rotates through a limited rangeof 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 the center of the eccentric journal C and thecontrol shaft axis S is called control shaft length in this study,
and denoted by ls. When the angle of the control shaft relative
to the engine block θs is changed, the position of the control
link will change, which subsequently tilts the lower linkclockwise or counterclockwise and thereby affects the motionof the piston and shifts the TDC and BDC positions. This leads
to change in both the combustion chamber volume and theswept volume. By manipulating the control shaft angle θs themechanism can realize different compression ratios anddisplacements for various engine 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 of the control
action θs. A major requirement is that the lengths of links of thefour bar linkage OABC need to satisfy the Grashof Condition,
which states that the sum of the lengths of the shortest andlongest 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

Denote the position of the control shaft axis S (XS, YS), then theposition of the control link point C is also easily obtained:

(3)

(4)

Note:

(5)

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Rewrite equation (5) and define a new vector

Lastly, the position of the piston pin E can be obtained. Denotethe crank offset δ, it is evident:

as:

(6)

Since
and
are already obtained by equations(1)(2)(3)(4) for each crank angle θ1 and control shaft angle θs,
is known. Denote the position of Z (XZ, YZ), then from thesecond part of equation (6) we have:

(16)

From
obtained as:, the position of piston pin E is

(17)

(7)

(8)

From equations (7) and (8), and also considering the feasible
ranges of the angles in Figure 1, we can obtain:(9)

Thus with the geometric dimensions of the multiple-linkmechanism known, and given the control shaft angle, we cancalculate the motion of each component for the whole crank
shaft revolution. In this study the piston motion is the focus.
With the piston motion profile calculated for the whole crankangle θ1 range from 0° to 360°, we can obtain the TDC and
BDC positions and their corresponding crank angles. We canalso calculate the velocity and acceleration of the piston, thestroke length, the displacement, the combustion chamber
volume, the compression ratio and other characteristics ofinterest.

Design Conditions
(10)

Note
be obtained as:, the position of control link point B can

(11)

(12)

Thus, the motion of the four bar linkage OABC is determined.
The position of connection point D can be calculated by making
use of the lower link dimensions. Note:

(13)

From

it is obtained:

While the multiple-link mechanism is initially designed torealize variable compression ratio, due to its realizing thefeature by changing the piston motion, it may lead to changes
in the stroke length, and consequently the displacement as
well. Since the piston motion is ultimately determined by thegeometry of the mechanism, this paper focuses on finding anapproach to obtain the geometric dimensions that giveadvantageous compression ratio and displacement behavior.
This is the basis for further design and optimization of themechanism.
Let us first investigate the desired behavior for compression
ratio and displacement. On one hand, when engine loaddecreases, obviously it is advantageous that the displacement
also decreases. When the demand in load change is handled
by varied displacement, less throttling is needed, which
reduces the pumping loss. A decreased stroke length also
reduces the frictional loss. On the other hand, compression
ratio should increase when engine load decreases, which helps
capture the benefits of higher thermal efficiency, andconsequently provides better fuel economy. Thus we see thatthe desired design of the multiple-link mechanism is when thecontrol action changes monotonically, the compression ratio
and the displacement should change monotonically in opposite
directions.

(14)

(15)

Denote compression ratio ε and displacement Vd - note they
are both functions of the control shaft angle θs - then for thewhole range of θs, we want to have either Scenario I:

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where d is the cylinder bore diameter, we have:

(18)
(27)

(19)

Also note the change of the combustion chamber volume iscaused by the change of the position of TDC, it is obtained:or Scenario II:

(28)
(20)

From (19) and (27) we know

(29)

(21)

Let us investigate Scenario I. Denote the combustion chamber
volume Vc, which is also a function of θs. Because

From (25) and (28) we know

(30)
(22)

Thus it is obtained:

it is obtained:

(31)

(23)

To satisfy (18), we needThis indicates that when the control shaft angle increases, both
TDC and BDC positions will move lower, and BDC will movelower at a bigger rate than that of TDC.
Substitute (27) and (28) into (24), and also note (31) and (22),
it is derived:(24)

From (19) and (24), it is obvious that we must have:

(32)

(25)

(19) and (25) indicate that to achieve a lower compression ratio
when θs increases, both Vd and Vc need to increase. Regarding
changes of TDC and BDC positions, denote the position of
TDC YTDC and that of BDC YBDC, which are functions of θs.
Since

(26)

With similar analysis, it is found that Scenario II gives the same
result as in (32). This is expected because Scenario I and II arein essence the same and they only differ by how we define thepositive direction of the control action θs. The condition shownin (32) indicates that in order for the multiple-link mechanism torealize 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
rate with respect to the control shaft angle for the BDC position
should be bigger than that of the TDC, but smaller than ε times
that of the TDC.

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The above process of deriving (32) indicates it is a necessary
condition for the desired mechanism behavior. However, we
can start from (32), reverse the process and end up with
Scenario I and II as well. Thus it is proved that (32) is anecessary and sufficient condition.
Using condition (32) to determine if a design will give thedesired mechanism behavior still involves finding out thecompression ratio over the whole range of the control shaft
angle, which needs the displacement and combustion chamber
volume information. To simplify, we can choose to use theminimum compression ratio ε0 the mechanism needs to realize,
and turn the condition into:(33)

This is a stronger condition for a design to fulfill therequirement. Since the minimum compression ratio a designneeds to realize is set in advance and known, condition (33)can be checked by using just the information of how the TDC
and BDC positions change with the control shaft angle, without
the need to calculate the displacement, the combustion
chamber volume, and the compression ratio over the whole
range of the control shaft angle. It is evident (33) is a sufficient
but not necessary condition.
One thing that deserves mentioning is that the conditions
derived above apply not only to the multiple-link mechanism
this paper studies. So long as a mechanism realizes thefeatures by changing the TDC and BDC positions of theengine, the conditions apply.
While the conditions provide insight for the design of thegeometric dimensions of the mechanism, it can be seen fromthe equations describing the kinematics of the multiple-linkmechanism that the individual and combined influence of thegeometric parameters on how the TDC and BDC positions willchange is not straightforward. An analytical solution of whatgeometric parameter set of the mechanism will satisfy theconditions could not be easily determined. This places achallenge on the design and optimization of the multiple-linkmechanism geometric dimensions. In this paper the approachof using DoE and numerical calculation is adopted.

Design of Experiments
DoE method is used for creating a series of multiple-linkmechanism geometric parameter combinations, which areinvestigated to obtain the mechanism designs that fulfill theconditions.
Referring to Figure 1, the multiple-link mechanism geometric
design parameters include the following:First, a specific range for each parameter is set up based on
experience, including reasonable physical constraints or limits.
Then, in the space spanned by all the parameters, DoE is used
to generate a series of parameter sets, and the properties ofthe mechanism for each parameter set are calculated. In thisstudy, an experiment plan is created which consists of
D-optimal points, V-optimal points and space-filling points. With
the design method selected, several hundreds parameter setsare created for investigation. For each of the createdparameter sets, the piston motion is calculated for the whole
range of the control shaft angle. Grashof Condition is checked
to make sure the geometry gives a truly operational multiplelink mechanism. After the TDC and BDC positions are
determined, the design conditions are checked to find out if thegeometry leads to a mechanism that gives both advantageouscompression ratio and displacement characteristics.
For this study, the range of the control parameter θs is set to befrom −45° to 45°, and the mechanism properties when θs is 0°
are chosen as the basis for calculation. The targetedcompression ratio range is from about 8 to about 14. The
combustion chamber volume is set in such a way that when θs
is 0°, the compression ratio realized is the designed value,
which is chosen to be 11 for this study.
For the DoE plan created above, it is found out that only a verysmall percentage of designs satisfy the conditions. A feasible
solution is selected for more detailed study. Piston motion
characteristics, including position, velocity and acceleration arecalculated. The displacement is calculated using the TDC and
BDC position information. The combustion chamber volume iscalculated based on how the TDC position changes from whenθs is 0°. The compression ratio is calculated subsequently.
Other characteristics such as maximum piston speed andacceleration rate, crank angle range of the expansion stroke
etc. are also studied.

Properties of the Selected Design
Table 1 lists the parameter values of a geometric design of themechanism that satisfies the conditions. Properties of thisdesign are presented below.
Table 1. Geometric Parameters of the Selected Design

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Figures 2(a), (b), and (c) show the piston position, velocity, andacceleration respectively during a crank revolution when thecontrol shaft angle changes from −45° to 45° in 5° increments.
The red lines correspond to −45° control shaft angle, the green
lines correspond to 0°, and the black lines correspond to 45°.
These figures are typical for mechanism geometric dimensions
that satisfy the design condition. It can be seen from Figure2(a) that when the control shaft angle θs increases, i.e., when
the control shaft rotates counterclockwise, both the TDC and
BDC move lower; the TDC and BDC move in the same
direction with the BDC moving at a higher rate than the TDC.
These observations are in consistence with (31). Also of notefrom Figure 2(a) is that when the control shaft angle changes,
the crank angle locations corresponding to the TDC and BDC
also change. This indicates that the crank angle ranges of theengine working cycle strokes might change as well. Figure 2(b)and 2(c) show that the piston velocity and acceleration changewith the control shaft angle in a more complex way; when θs
increases, the maximum piston speed and acceleration rate
(absolute values) both basically increase.

Figure 2. (cont.) Piston Position, Velocity and Acceleration

Figure 3 specifically shows the TDC and BDC positions withrespect to the control shaft angle. It is evident that when thecontrol shaft turns counterclockwise, both the TDC and BDC
positions are lowered, and the rate of change of the BDC
position is greater than that of the TDC position. This further
confirms the observations from Figure 2(a) and indicates thedesign conforms to (31).

Figure 3. TDC and BDC Positions

With the TDC and BDC positions obtained, it is trivial tocalculate the stroke length. Figure 4 shows that the stroke
length increases from 117.9 mm to 135.3 mm when the control
shaft rotates counterclockwise from −45° to 45°.

Figure 2.

To investigate how the conditions (32) and (33) apply to thisdesign, let us first find out how the displacement, thecombustion chamber volume, and the compression ratiochange with the control shaft angle, although checking
condition (33) does not need these information. Since the TDC
and BDC positions are already obtained for each control shaft
angle, the displacement can be easily calculated using (26),given the cylinder bore diameter, which is 89.0 mm for thisstudy. To calculate the combustion chamber volume for thewhole range of the control shaft angle, we need make use of
the targeted compression ratio when θs is 0°. At this control
shaft angle, the stroke length is 127.6 mm and thedisplacement is 0.794 liter. To realize the targeted compression

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ratio of 11, the combustion chamber volume is set to 0.0794
liter. Since the change in the combustion chamber volume iscaused by the change of the TDC position, when the control
shaft angle changes from 0°, based on how much change isincurred in the TDC position, the corresponding combustion
chamber volume can be calculated. After the combustion
chamber volume is obtained, the compression ratio is
subsequently obtained using (22).

Figure 6. Compression Ratio

Figure 4. Stroke Length

Figure 5 shows how the displacement and the combustion
chamber volume change with the control shaft angle. When thecontrol shaft rotates counterclockwise from −45° to 45°, thedisplacement increases from 0.733 liter to 0.842 liter, which isan increase of 14.8%; and the combustion chamber volume
increases from 0.058 liter to 0.121 liter, which is an increase of
109.5%. Figure 6 shows how the compression ratio changes
with the control shaft angle. In the same control shaft angle
range, the compression ratio decreases from 13.74 to 7.98.
The behavior of the displacement and the compression ratio
clearly indicates that this design belongs to Scenario I
described earlier and conforms to (18) and (19).

Based on how the TDC and BDC positions change with thecontrol shaft angle, the design criterion, i.e., the ratio of the rateof change of the BDC position to that of the TDC position, canbe easily calculated. Figure 7 shows that the design criterion
basically follows a decreasing trend when the control shaft
angle changes from −45° to 45°, and it is between 1 and thecompression ratio. This indicates that the design satisfies
condition (32). Note that the criterion is actually less than theminimum compression ratio, which is about 8, for the whole
range of the control shaft angle, which indicates that thestronger condition (33) is also satisfied in this case.

Figure 7. Design Criterion

Figure 5. Displacement and Combustion Chamber Volume

The above analysis shows that the design satisfies theconditions and gives compression ratio and displacement
behavior as desired. Some other characteristics indicated by
the data may also be of interest for further studies of themechanism performance. Maximum piston speed andacceleration rate (absolute values) are shown in Figure 8 and 9
respectively. When the control shaft rotates counterclockwise,
they both basically follow the increasing trend in the range
studied, with a small deviation for the acceleration rate near theend of the range.

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The study so far basically focuses on only the kinematics of themechanism. To further investigate the engine performance
when adopting a specific mechanism design, detailed study ofdynamics involving various subsystems, processes andcontrols, such as charging and discharging processes,
volumetric efficiency, fuel delivery, combustible mixture
formation, ignition timing, valve timing, combustion process
and efficiency, and power output etc., which are related to themechanism design and piston motion directly or indirectly,
need be performed. Related study in these areas is notcovered in this paper.

Summary/Conclusions
Figure 8. Maximum Piston Speed (Absolute Value)

This paper presents an approach on how to achieve multiplelink mechanism designs that realize advantageous behavior forboth compression ratio and displacement. Conditions are
obtained on how the TDC and BDC positions should changewith respect to the control action to fulfill the design
requirement. DoE methodology is used for creating sets ofgeometric designs of the mechanism, which are checkedagainst the conditions. Selected design that satisfies theconditions is further studied in detail. The design approach andobtained results serve as a basis and provide insights forfurther analysis and optimization of the multiple-linkmechanism.

References
Figure 9. Maximum Piston Acceleration (Absolute Value)

Also of interest is how the crank angle range of the expansion
stroke changes with respect to the control shaft angle. Figure
10 shows that when the control shaft rotates counterclockwise,
the crank angle range of the expansion stroke basically follows
a decreasing trend. When the mechanism is at the highest
compression ratio, the expansion stroke covers 192.4 crankangle degrees, while when the mechanism is at the lowest
compression ratio, the number reduces to 184.0 crank angle
degrees.

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