cam shape ion by genetic algorithm

Cam shape optimisation by genetic algorithm

J. Lampinen*

Laboratory of Information Processing, Department of Information Technology, Lappeenranta University of Technology,

P.O. Box 20, FIN-53851 Lappeenranta, Finland

Accepted 23 December 2002

Abstract

This article overviews a genetic algorithm based computer-aided approach for preliminary design and shape optimisation of cam profiles

for cam operated mechanisms. The primary objective of the work was to create a complete systematic approach for preliminary cam shape

design including cam shape design automation and true cam shape optimisation with respect to the simulated computer models of cam

mechanisms. Typically, shape optimisation of a cam cross-section is a multiobjective optimisation problem of two-dimensional geometric

shape in a heavily constrained environment. In order to illustrate the genetic algorithm based cam shape optimisation approach, a cam shape

design example is described, in which a cam shape designed by genetic algorithm is compared with its more conventionally designed

counterpart.

q 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Genetic algorithms; Shape optimisation; Cam mechanism

1. Introduction

Shape optimisation based on genetic algorithm (GA) [1],

or based on evolutionary algorithms (EA) in general, is a

relatively young and potential field of research. However,

the author knows currently over 100 articles, where

GA-based geometrical boundary shape optimisation has

been under investigations. The large majority of these

articles have been published since 1995, and only a few

articles have been published before 1990. The interest

towards researching evolutionary shape optimisation tech-

niques appears to be just started to grow, rather than reached

a stable and mature state.

Currently the most popular application area of EA-based

shape optimisation seems to be the shape optimisation in

connection with computational fluid dynamics (CFD),

especially aerodynamic shape optimisation in the field of

aircraft design, for example [2–10]. Also many shape

optimisation problems raised from the field of electrical

engineering have been under recent investigations, for

example [4,11–14].

While the intensity of evolutionary shape optimisation

research is currently the highest in the fields of aerospace

engineering and electrical engineering, difficult shape

optimisation problems are common in many other areas as

well. In the future, one of the most potential areas

for EA-based shape optimisation applications is undoubtedly

mechanical engineering, since designing machine com-

ponents typically includes shape determination and optim-

isation for functional surfaces of the components. In the field

of mechanical engineering, in addition to cam shape

optimisation discussed in this article, evolutionary

shape optimisation approach have been applied also for

shape optimisation of: a strain gauge load cell [15], a

cantilever beam [16], a torque arm [16], a spherical pressure

vessel [16] and a conical pivot bearing journal [17].

Just mentioning a few examples.

This article focuses on overview of the results of a

research project in which we have developed a computer-

aided design and optimisation method for shape design

of an internal-combustion engine valvecam [18–23] and

a cam used to operate a diesel fuel injection equipment

[23,24].

0010-4485/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0010-4485(03)00004-6

Computer-Aided Design 35 (2003) 727–737

www.elsevier.com/locate/cad

* Address: Laboratory of Information Processing, Department of

Information Technology, Lappeenranta University, University of

Technology, P.O. Box 20, FIN-53851, Lappeenranta, Finland. Tel.: þ358-

5-621-2813, fax: þ358-5-621-2899.

E-mail address: [email protected] (J. Lampinen).

http://www.elsevier.com/locate/cad

1.1. Cam mechanism

In the field of mechanical engineering, the cam

mechanism along with the crank mechanism is the most

common type of mechanism for converting a rotational

movement to a controlled reciprocating movement.

An example of a cam mechanism is shown in Fig. 1.

The disc type cam shown in Fig. 1 is an eccentric and

asymmetric lobe of its shape. The cam is mounted to a

rotating camshaft that is driven by the operating device.

In the case of an internal-combustion engine, the operating

device is the engine’s main power output shaft via the

engine’s timing gear. The cam follower converts the

rotational movement of cam to a reciprocating movement

and transmits it to the operated device. The movements of

the cam follower are dependent on the boundary shape of the

cam profile—the cam controls the reciprocating movements

by its shape. In the case of an internal-combustion engine,

the operated device is the valve mechanism (Fig. 2).

The target of the cam shape optimisation is to optimise

the function (movements) of the operated device without

violating geometric and physical constraints of cam

designing. Cam shape optimisation is typically a heavily

constrained multiobjective optimisation problem in which

multiple, design task dependent, material strength and

durability, cam manufacturing technology, geometric,

kinetic and dynamic constraints should be satisfied

simultaneously. For example, the minimum local concave

radius of the cam shape may not be smaller than the radius

of the grinding stone used for manufacturing the cam.

The minimum local convex radius is constrained by the

contact pressure (Hertz’s pressure) between the cam and

cam follower. Typical cam design objectives are maximum

rise and return rates for the cam follower movements,

minimum instantaneous contact force between the cam and

follower simultaneously with minimum dynamic force

fluctuation. Typically also one or more task dependent

measures of operated device performance are among the

objectives. Thus, the objectives are the target functional

properties of the cam mechanism and the device operated by

it, and the constraints are imposed by the restrictive design

conditions.

1.2. Valve mechanism

The function of the valvetrain assembly (Fig. 2) is to

open and close the inlet and exhaust gas valves in the

internal-combustion engine. The function of the cam is to

control valvetrain movements by its cam action and thus by

its shape. The return force for the valvetrain components is

applied by the valve spring. The valve spring is also

required for maintaining the contact between the cam and its

follower. Since the maximum acceleration of the valvetrain

components can be as high as 2000 ms22, or even higher,

the components of the valvetrain are under extreme

dynamic forces. Due to the dynamic forces and the

flexibility of the components, the entire valvetrain can be

mechanically considered as an oscillating system (Fig. 2),

in which the conversion from stored to free energy causes

forced vibration.

For computing the dynamic forces affecting the valve-

train during the cam operating cycle, a valvetrain simulator

program, based on the mathematical model of the

valvetrain, is used. Fig. 3 illustrates the dynamic force

fluctuation in valve mechanism due to flexure of its

Fig. 1. Cam mechanism is used to convert the rotational movement of a

camshaft to a controlled reciprocating movement of the cam follower.

The cam controls the follower movements by its shape. The cam follower

transmits the movement to the operated device. Thus, optimal cam shape is

required for optimal operation of a cam-operated system.

Fig. 2. Cam operated valve mechanism of an internal-combustion engine

(valvetrain) and its equivalent mass/spring system. 1-mass system is shown

here for sake of clarity only. Typically we have used 4–10 mass systems for

a computer simulation model of the valvetrain.

J. Lampinen / Computer-Aided Design 35 (2003) 727–737728

components under dynamic forces by comparing computer

simulation results with the results calculated assuming that

the components of the mechanism are rigid bodies without

flexure.

A proper control of dynamic forces affecting the cam

mechanism is one of the main objectives of cam design.

Because the cam controls the valvetrain by its shape, it must

be designed so that unnecessary dynamic force fluctuation

will be avoided.

2. Cam shape parameterisation

The cam shape optimisation task is converted to a

parameter value optimisation task by using B-spline curves

[19,23,25,26] for shape representation. The cam boundary

shape is represented by a B-spline curve as shown in Fig. 4.

Shape is defined by 40 floating-point values assigned to

corresponding control points of the B-spline curve. Thus the

co-ordinates of the B-spline curve control points are serving

here as cam shape design variables. To optimise the shape,

the values assigned to control points must be optimised.

Then, in principle at least, any non-linear parameter

optimisation method can be used to solve this optimisation

task and thus to optimise the shape of the cam cross-section.

However, in practice a global optimisation algorithm, like

GA, appears to be essential for obtaining solutions of

acceptable quality [23].

As illustrated in Fig. 4, actually the shape of the cam

follower displacement (lift) curve is the direct subject to

shape optimisation instead of direct optimisation the cam

boundary shape itself. Since there is an unambiguous

geometric relationship between the cam boundary shape and

the corresponding cam displacement curve, the cam

boundary shape can be determined on the basis of the

displacement curve and the geometric dimensions of the cam

mechanism. Thus, the cam boundary shape is optimised

indirectly by optimising the cam follower displacement

curve. The purpose of this arrangement is to simplify the

required computations and reduce the overall computational

cost for cam shape optimisation.

Generally, we have found that the B-spline method is

beneficial for cam shape parameterisation because its

computational implementation is efficient and free of

problems with numerical stability. Also it is advantageous

that the degree of the curve and the number of control points

can be selected independently. So, it is possible to select the

degree of the curve just high enough to satisfy curve

smoothness and continuity requirements with still providing

a high number of control points and a high degree of

freedom for curve shape modifications. For a more detailed

discussion and description of the shape parameterisation

method, see Refs. [19,23].

3. Cam shape optimisation by GA

When individual continuous shapes are represented by

fixed and limited number of floating-point values the

individuals can be represented also as a population of GA.

Each individual of the population describes one complete

cam shape with 40 floating-point values. So, a chromosome,

composed of 40 floating-point valued genes, represents

each individual cam shape. Each gene represents one

floating-point value assigned to the control point of the

B-spline curve.

The shape optimisation itself is now a straightforward

process (Fig. 5). Individual shapes, represented by a vector

of 40 floating-point valued shape control points, will be

evaluated with the fitness-function. The crossover and

mutation operations of GA are responsible for generating

new alternative shapes (in their parametric form).

For this work a floating-point encoded GA with

arithmetic crossover and mutation operations was applied.

The applied arithmetic crossover operation generates ith

component, xi, of the trial solution vector, X, on basis of

Fig. 3. The typical force fluctuation in a valve mechanism during the cam

operating cycle. The dynamic force affecting valvecam follower of a large

diesel engine during the cam operating cycle at engine running speed

500 r/min is shown. Computer simulation results using simulation models

with and without rigid body assumption are compared.

Fig. 4. Cam follower displacement curve parameterisation using 6th degree

B-splines. The cam shape is unambiguously defined by 40 floating-point

values assigned to the control points of curve. The cam follower

displacement curve defines indirectly the cross-sectional shape of the

corresponding cam profile since there is a direct geometric relationship

between the cam shape and its displacement curve.

J. Lampinen / Computer-Aided Design 35 (2003) 727–737 729

the parent vectors A and B, as follows:

xi ¼ ri·ai þ ð1 2 riÞbi; i ¼ 1;…; 40; ð1Þ

where ri is an independent, uniformly distributed random

number from the range (0,1). The parent vectors, A and B, are

randomly chosen individuals from the current population.

After performing the crossover operation, the generated

offspring vector, X, will be subject to following mutation

operation before evaluating the generated trial solution

xi ¼ r1;i·xi þ ð1 2 r1;iÞðr2;i ·ðu 2 lÞ þ lÞ;

i ¼ 1;…; 40;

ð2Þ

where r1;i and r2;i are independent, uniformly distributed

random values generated from the range (0,1), while u and

l are the predefined upper and lower boundaries for the

positions of the B-spline control points. In case

for normalised cam displacement curve (Fig. 4) settings l ¼

20:1 and u ¼ 1:1 are typically appropriate.

With the described operations, crossover probability

Pc ¼ 0:25 and mutation probability Pm ¼ 0:015 were

applied. The population size of 40 individuals was used

here. Since each individual candidate cam shape

was represented in the population by a vector of 40

floating-point values, the population can be viewed as a

40 £ 40 matrix. For the deciding which ones of the current

population members and the generated trial solutions will

survive to the next generation’s population, an elitist rank

based replacement rule was applied.

The targets of the cam shape optimisation, the optimal

characteristics of cam follower movements (displacement,

velocity, acceleration, jerk, timing of movements, etc.)

depend on the requirements of the particular device

operated by the cam mechanism. In practice there are also

multiple restrictive design conditions involved, such as

strict geometric limitations. Cam shape design for a

high-speed cam mechanism is typically a multiobjective

optimisation problem in a heavily constrained environ-

ment. Typically the objective function is also highly

multimodal and non-linear. Auxiliary information, like the

derivatives of the objective function, is not available. The

fitness-function is available only in the form of a

computer program, not in analytical form. These are

some of the main reasons why we use GA for cam shape

optimisation.

Concerning the other algorithms that we originally

considered: both enumeration based approaches and pure

random search were out of question due to enormous

computational effort they would require for providing high

resolution solutions needed here. Unfortunately, without

exception, all considered deterministic optimisation

approaches rely on one or more assumptions concerning

the properties of the objective function, that cannot be

assumed to be satisfied in our case, e.g. linearity, continuity,

unimodality, separability, availability of auxiliary infor-

mation, etc. Since in general our approach requires taking

the objective function as a black box, and only

the availability of the objective function value can

be guaranteed, no further assumptions were within possi-

bilities. Thus, the only realistic alternative was applying a

stochastic global optimisation approach. Since GAs have

already widely demonstrated capabilities for effective,

efficient and robust global optimisation in cases for many

black-box type computer models, including many shape

optimisation models, we considered GAs as the most

attractive alternative for our purposes.

Fig. 6 represents the convergence history of an example

cam design process. Typically evaluation of 10.000–40.000

trial cam shapes, case dependently, was required for each

cam shape optimisation process. As shown in Fig. 6, a

considerable speed-up of the optimisation process was

achieved by starting with the initial population containing

a set of prototype cam designs, instead of starting with a

population of randomly generated shapes. A set of

different cam shapes, having simple sinusoidal motion

characteristics, was used for initialisation in the case

illustrated in Fig. 6. While these simple cam shapes are

practically always rather far from the optimum shape, they

are already cam shapes rather than saw-disks or gear-

wheels—as in case for random initialisation. Thus they

provide a much better set of starting points for further

optimisation, as Fig. 6 is suggesting.

Thus, when the GA is applied to design a cam shape, a

design engineer starts the designing process with describing

the design targets and design constraints. The population of

the GA is initialised with a selection of 40 different but

already reasonably well performing prototype cam shapes.

Fig. 5. Flow chart of a cam shape optimisation process based on genetic

algorithm.


Then the GA process starts its attempts for finding a cam

shape that is as close to the design targets as possible and

meets all constraints.

In the case of shape optimisation of an internal-

combustion engine valvecam, the base of the system is

a mathematical model of cam-operated mechanism, a

simulator program. The fitness-function of the GA is

based on evaluation of the simulation results. The GA

is used to produce automatically alternative cam shapes for

the valve mechanism simulator program, to run the

simulator, and finally to evaluate the cam shapes on the

basis of the simulator output data. All objective and

constraint functions involved can be evaluated either on

basis of the simulation results for the trial cam shape or

directly on basis of the trial cam shape itself. Finally, one

single fitness-value for the individual cam shape is

computed as a weighted linear combination of all objective

functions. A penalty function method is applied for handling

multiple (also weighted) constraint functions.

Thus, the objective function to be minimised by the GA

was of the following form

fcostðXÞ ¼Xn

i¼1

wi·fiðXÞ þXnþm

i¼nþ1

wi·giðXÞ; ð3Þ

where wi denotes weights assigned to each objective and

constraint function involved. Each of n objective functions,

fiðXÞ; and m constraint functions, giðXÞ; involved is heavily

dependent on the particular cam design case at hand and no

general set of objectives or constraints can be given.

In general, it is assumed here that the design engineer

provides the particular set of objective and constraint

Fig. 6. Convergence history for a valvecam shape optimisation process with a comparison of population initialisation methods. By using a non-random

initialisation with a selection of initial cam designs, improved convergence rate and better optimisation results were seen.


functions ad hoc. In order to provide examples of

functions, that the design engineer may wish to define,

some objective and constraint functions can be mentioned.

The following ones, among others, have been used in the

case for the shape design example given later on in this

article:

The objective function,

f1ðXÞ ¼ Fmax; ð4Þ

is applied to minimise the value of the maximum force peak,

Fmax; affecting in the pushrod during the simulated cam

operating cycle.

The objective function,

f2ðXÞ ¼ð2p

0ðFRðaÞ2 FSðaÞÞ

2da; ð5Þ

is applied to minimise the amount of dynamic force

fluctuation during the cam operating cycle. Function

FSðaÞ express the simulated force affecting in the cam

follower as a function of cam rotation angle, a. FRðaÞ is

the corresponding function computed under rigid body

assumptions. Examples of both functions are illustrated in

Fig. 3.

The constraint function,

g3ðXÞ ¼ maxð0; ðrall 2 rminÞÞ; ð6Þ

is applied to ensure, that the minimum local cam radius of

curvature (convex radius), rmin; exceeds the allowed

minimum value, rall:

The constraint function,

g4ðXÞ ¼ maxð0; ðvC 2 vCmaxÞÞ; ð7Þ

is applied to ensure, that the valve closing velocity, vC; falls

below the allowed maximum velocity, vCmax.

In addition to the above examples, several other

objective/constraint functions were involved. While it is

not appropriate, within the scope of this article, to

provide here all the functions involved, the further

details of the problem specific objective and constraint

functions, that we have applied, can be found from

Ref. [23].

By using GA in this way it is not only possible to

design and to optimise cam shape, but GA also

automates the design process [23]. In fact, the design

automation was one of the primary motivations for

developing the GA-based approach for cam shape

optimisation, since the conventional cam shape determi-

nation processes for high-speed cam mechanisms are

typically both laborious and time consuming. For a more

detailed discussion, see Refs. [18,21,23].

4. Distributed computation

A well-known drawback of GA is its non-efficient

use of processing capacity. In this case, because a

floating-point encoded GA is used, the operations of the

GA (selection, crossover, mutation, etc.) are not compu-

tationally expensive when compared with binary encoded

GA. But the values of the fitness-function are relatively

costly to compute, because the computationally expensive

simulator of the cam mechanism is used as a part of the

fitness-function for evaluating the candidate cam shapes.

The fitness evaluation represents more than 97% of the

total usage of CPU-time.

In practice, for achieving a solution of acceptable

level, it is sometimes necessary to repeat the evaluation

of the target function over 40,000 times. If an efficient

PC (AMD K6 400 MHz or equivalent workstation) is

used, it means in practice that the optimisation process

takes typically 50–150 h depending on the used cam

mechanism simulation model. Without parallelization the

process takes still about 13–40 h if an efficient main-

frame computer is used (1 processor in use) for

computation. The need for speeding up the computation

is obvious.

Because of that, we have distributed the computation

of fitness-values in a local area network (LAN) of

PC-workstations. The distribution model used is coarse-

grained and modified from a so-called standard distri-

bution model, in which only the fitness evaluation is

distributed to the slave processors. The method is based

on a population maintained by a master process (Fig. 7).

The master process sends the individuals to be evaluated

Fig. 7. Our model for coarse-grained distribution in LAN. Model is based

on of standard type distribution model, but uses shared interface files as a

buffer of fitness evaluation tasks. The main process and the slave processes

are only loosely coupled via these files and not synchronised to each other.


by the slave processes to a shared disk file. A slave

process read the fitness evaluation task from this file,

evaluates the fitness-value, and then return the result back

to the main process via another shared disk file. Thus,

evaluation of fitness-values is distributed via a local area

network to slave processes, which are working asynchro-

nously with respect to the main process. The implemen-

tation allows that the number of slave processes can be

freely selected and also freely altered during the

optimisation process. The shared disk files are serving

as a logical interface between the master and slave

processes. A detailed description of our distribution model

can be found in Refs. [21,23].

5. A cam shape design example

In this section, the cam designed and optimised with

GA is compared with the cam designed with a more

conventional method (trial and error method). The

particular cam studied in this example is an inlet

valvecam for a large diesel engine designed for marine

diesel and electric power generation applications. Both

cam designs are compared using the same simulation

model of the valve mechanism. Both cams have the

same timing of valve movements and the same

maximum displacement. In order to illustrate dynamic

properties of the cam optimised with GA, we have

designed the GA-optimised cam, that is used here as an

example, to produce a displacement curve, which is

similar to the displacement curve of the compared

conventionally designed cam. When control of an

internal-combustion engine’s gas exchange is considered,

there is no significant difference between the compared

cams. Also both the cams satisfy all the design

constraints involved. So, we may concentrate on

comparing the properties of the cams from the

perspective of design objectives.

5.1. Comparing cam kinetics

In Figs. 8–11 the comparisons of the basic kinetic

properties of compared cams are shown. There is a direct

geometric relationship between the cam boundary shape

and the cam displacement curve. So, instead of cam shape

we may concentrate here on the cam follower displace-

ment curve. As shown in Fig. 8, there is only a slight

difference in cam follower displacement curves of the

compared cams. Actually in order to find any significant

difference, we must observe the derivatives of the cam

displacement curves.

The derivatives of the cam follower displacement

curves are important because of their physical meaning.

When the cam follower displacement curve represents the

position of the cam follower as a function of the angular

Fig. 9. Comparison of the velocity curves between a conventionally designed

cam shape and a cam shape designed and optimised by a genetic algorithm.

Fig. 10. Comparison of the acceleration curves between a conventionally

designed cam shape and a cam shape designed and optimised by a genetic

algorithm.

Fig. 11. Comparison of the pulse (jerk) curves between a conventionally


algorithm.

Fig. 8. Comparison of the displacement curves between a conventionally


algorithm.


position of the camshaft, the first derivative of the cam

follower displacement curve represents the follower

velocity (Fig. 9) and the second derivative represents

the follower acceleration (Fig. 10). The 3rd derivative of

the displacement curve (Fig. 11) represents the follower

pulse (jerk). In practice, this leads into requirement, that

at least 6th degree B-spline curve must be used for the

displacement curve and cam shape representation in order

to maintain the required smoothness also for the

acceleration and jerk curves.

So, in designing of cam shapes for high-speed cam

mechanisms, not only the cam shape is important, but

also its derivatives must be considered in order to

maintain good dynamic behaviour of the cam mechan-

ism. Because of this, the cam shape optimisation for a

high-speed cam mechanism is always a difficult task. In

case of an internal-combustion engine valvecam, any

high values of impulse curve must be avoided because

their existence means high force shocks in cam

mechanism during the operation. However, correctly

timed impulses can be applied effectively for control-

ling the dynamic force fluctuation in the cam

mechanism. Coarsely simplifying, the oscillation started

by an impulse can be cancelled later on by another

impulse, which is timed to opposite phase with respect

to the oscillation started by the first impulse. Appar-

ently, the jerk curve of the cam designed by the GA

(Fig. 11) reveals GA’s tendency to take advantage of

correctly timed impulses more effectively than a human

designer does.

5.2. Comparing cam dynamics

The most difficult problem in valvecam shape optim-

isation is to control the dynamic forces that affect

the valvetrain force transmitting components. In case of

high-speed cam mechanisms, minimising the force

fluctuation is also one of the most important design

targets. It is important for avoiding unnecessary vibration,

impact loads, friction, wear, noise and false motions of

cam mechanism. Unfortunately the control of force

oscillation in a system with 4–10 oscillating (non-linear)

spring/mass subsystems is a difficult optimisation task

when the optimisation must be done by altering the cam

shape to generate a more favourable form of system

excitation force.

Figs. 12 – 15 represent the results of valvetrain

simulations of compared cams in four different engine

speeds. In order to ensure a good dynamic behaviour

through the operating speed range, four engine running

speeds shown in Figs. 12–15 were simulated during the

fitness evaluation of every individual cam. It would be

possible to simulate more than four running speeds, but

in this case shown in Figs. 12–15 the operating speed

range of the engine is quite narrow, and four simulated

speeds is adequate.

Fig. 13. The comparison of the dynamic behaviour between a convention-

ally designed cam shape and a cam shape designed and optimised by a

genetic algorithm.



genetic algorithm.



genetic algorithm.



genetic algorithm.


As shown in Figs. 12–15, the dynamic force fluctuations

are noticeably lower with a cam optimised by GA

(consider Eq. (5)). Also the peak force value is noticeably

lower (consider Eq. (4)). As shown in Figs. 8–11, there is

only a slight difference in the shapes of the compared cams.

Despite the fact that the motions of the compared cams are

almost identical (Figs. 8–11), the cam designed and

optimised by GA generates less force fluctuation

(Figs. 12–15). The reduced dynamical force oscillation is

achieved without sacrificing high valve opening and closing

rates, which are important for the thermodynamic efficiency

of the engine.

We have also found, that the valve mechanism is often

less sensitive to changes of valve clearance when cams

optimised by GA are used. In certain cases the valve

mechanism is also less sensitive to changes of running

speed.

Both cam design methods compared, the evolutionary

approach and the conventional trial-and-error method,

were both based on using exactly the same simulator

program and they were used here for solving exactly the

same real life cam design problem. Thus, it is justified to

compare their capabilities for optimisation of cam

mechanism simulation models. In this particular case

the quality of the compared conventionally designed cam

was remarkably good. Despite that the GA-based method

found a clearly better solution (with respect the objectives

defined by Eqs. (4) and (5)). However, because a

simulated cam mechanism was used, it is neither relevant

nor possible to conclude that the cam designed by

evolutionary approach would perform better with respect

to the corresponding physical reality, too. Anyway, it can

be concluded that it is likely to perform better if the

quality of the used simulation model is at least

reasonably good.

Another point to consider is that the cam shape

design by manually re-designing the trial cams and

simulating them all over again takes a week or two for

an experienced design engineer. Correspondingly, the

design by the GA-based system takes hours rather than

days. The labour of an engineer is required only for

starting the optimisation process and for handling the

optimisation result. The optimisation process itself is

automatic.

6. Conclusions

This article overviewed a novel genetic algorithm

based approach for preliminary cam shape design and

optimisation based on predictive computer simulations of

cam mechanisms. The method described can be applied to

optimise cam shapes in principle for any type of cam

mechanism. The approach does not limit defining, ad hoc,

any objective or constraint function needed for the

particular cam design task at hand. Furthermore,

the approach is not limited to the usage of any particular

computer simulator program or limited to designing any

particular cam operated system only.

The primary objective of the work was to create a

complete systematic approach for preliminary cam shape

design including cam shape design automation and true

cam shape optimisation with respect to the simulated

computer models of cam mechanisms. The objective is

important since in mechanical engineering, a wide

variety of cam mechanisms are used to convert the

rotational movement of the camshaft to a controlled

reciprocating movement of the cam follower, which

transmits the movement to an operated device. The cam

controls the cam follower movements by its shape. To

optimise the movements of the cam follower in order to

optimise the functioning of the operated device, the

cross-sectional shape of the cam must be optimised to

produce the optimal kinetic and dynamic characteristics

of the movements.

The shape optimisation method described and dis-

cussed here is straightforward. The boundary shape of

the cam profile is represented by a B-spline curve,

which makes it possible to express the cam shape in

parametric form. By using this technique, the cam

shape can be defined unambiguously with a limited

number of floating-point valued parameters. The idea is

to convert a shape optimisation task to a parameter

value optimisation task. A floating-point encoded

genetic algorithm is used to solve this non-linear

global optimisation problem. The individuals of the

population are the alternative cam shape designs in a

parametric form.

The fitness-function of the genetic algorithm is based on

evaluation of a simulated cam mechanism model

that contains also a model of the operated device.

The alternative trial cam shape designs are first simulated

on simulated cam mechanism. Then the fitness-value for

each individual cam is calculated on the basis of the

simulation results by cam design case dependent objective

and constraint functions.

By using GA it is not only possible to design and to

optimise boundary shapes of objects, GA may also be

applied in order to implement automatic shape design

process. GA can be viewed as an optimisation tool and

also as a tool for implementing automatic computer-

aided design systems. By an example on applying the

method for solving a real world cam shape optimisation

task, it is demonstrated that the method is capable of

finding automatically better or at least comparable

solutions with respect to a more conventional manual

trial-and-error design approach. Despite the fact that the

cam used for comparisons is a well designed represen-

tative of the trial-and-error based conventional design

method, we managed to find a significantly better

solution when compared on the basis of valve mechan-

ism simulations.


Of course, the conclusion is fully valid only within the

context of simulated cam mechanisms. The quality of

the optimisation result with respect to the physical reality

depends on the quality of the simulator program.

However, the target of preliminary cam design is finding

an optimum cam shape with respect to the given

simulated computer model of cam mechanism. The results

discussed in this article suggest that a systematic

evolutionary optimisation approach is capable of finding

a better solution with respect to the simulation model and

given set of objectives and constraints, than an intuitive

trial-and-error method.

Generally, the results are suggesting that a computer and

a systematic optimisation approach is capable of using

the existing cam mechanism simulator programs more

effectively than a human and intuitive trial-and-error

experimentation does. In addition, the evolutionary

approach is automatic and does not require an experienced

design engineer for generating and evaluating each single

trial cam design.

A general conclusion of this article is that the described

novel method can be used effectively for automatic cam

shape determination and systematic seeking for optimum

cam shapes by using existing predictive computer simu-

lation models of cam mechanisms.

Acknowledgements

The author would like to acknowledge the contribution

of Professor Jarmo T. Alander, from University of Vaasa

(Finland), to this work. A substantial part of the work

described in this article was funded by the Finnish

Technology Development Centre TEKES, Wartsila NSD

Corporation and the Foundation of Emil Aaltonen, which is

acknowledged with gratefulness.

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Jouni Lampinen is a Professor of Information

Processing at Lappeenranta University of

Technology, Lappeenranta, Finland. Pre-

viously, from 1995 to 1999, he have been as

a researcher of industrial applications of

evolutionary algorithms at the genetic algor-

ithms research group of the University of

Vaasa, Vaasa, Finland. He have graduated

1990 as Automotive Engineer from the

Tampere Institute of Technology, Tempere,

Finland. Later on, 1998, he received the degree

of M.Sc. (Economics) and the degree of D.Sc.

(Economics) 2000, both from the University of Vaasa, having computer

science as the major subject. His Ph.D. thesis, “Cam Shape Optimization by

Genetic Algorithm”, investigates a genetic algorithm based approach for

cam shape determination and optimisation. His main research interests are

engineering optimisation, evolutionary computation, soft-computing and

computer-aided design.


cam shape ion by genetic algorithm

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