|
Academic Open Internet Journal |
Volume 15, 2005 |
Design optimization
of composite drive shafts using genetic algorithm
T.Rangaswamy1 and S.Vijayarangan2
1Research
scholar & Corresponding author
2Professor
& Co-author
Department
of Mechanical Engineering,
Telephone:
91 - 422 – 2572177, 2572477; Fax: 91- 422 – 2573833
E-mail: trangaswamy@yahoo.com
Abstract
The
orientations of fiber direction in layers and number of layers and the
thickness of the layers as well as material of composites play a major role in
determining the strength and stiffness. Steel drive shaft of an Automobile is
made in two sections connected by a support structure, bearings and U-joints
and hence over all weight of assembly as well as vibration problems will be
more. Also, they have less specific modulus, specific strength and low damping
capacity. In his context, the replacement of steel by composite material along
with optimum design will be a good contribution in the process of weight
reduction of drive shafts. A formulation and solution technique using genetic
algorithm (GA) for design optimization of composite drive shaft which is
subjected to the constraints such as torque transmission, torsional buckling
capacities and fundamental lateral natural frequency is presented here. On
applying the GA, the optimal ply thickness, number of plies required and
stacking sequence have been obtained, which contributes towards achieving the
minimum weight with adequate strength and stiffness. The weight savings of the composite shafts
were 48.36 % and 86.90 % of the steel shaft respectively.
Keywords: Stacking sequence; genetic
algorithm; optimization; composite drive shaft; weight reduction
1 INTRODUCTION
The advanced composite materials such as Graphite,
Carbon, Kevlar and Glass with suitable resins are widely used because of their
high specific strength (strength/density) and high specific modulus
(modulus/density) [1]. Weeton et al. [2] described the application
possibilities of composites in the field of automotive industry as elliptic
springs, drive shafts, leaf springs etc., Beard more et.al [3] explained the
potential for composites in structural automotive applications from a
structural point of view. Andrew Pollard [4] proposed the polymer Matrix
composites in driveline applications. A GA proposed by Goldberg [5] based on
natural genetics has been used in this work. In the previous study by the
authors [6], a GA was applied for the design optimization of steel leaf
springs. Although design optimization of steel springs and composite leaf
springs has been the subject for quite few investigators [7].
In the present work an attempt is made to
evaluate the suitability of composite material such as E-Glass/Epoxy and HS-Carbon/Epoxy
for the purpose of automotive transmission applications. A one-piece composite
drive shaft for rear wheel drive automobile was designed optimally by using GA
for E-Glass/ Epoxy and HS-Carbon/Epoxy composites with the objective of
minimization of weight of the shaft which is subjected to the constraints such
as torque transmission, torsional buckling strength capabilities and natural
bending frequency
2 SPECIFICATION
OF THE PROBLEM
The torque transmission capability of the drive shaft
for passenger cars, small trucks, and vans should be larger than 3,500Nm(Tmax)
and fundamental natural bending frequency of the shaft should be higher than
6,500 rpm(Ncrt) to avoid whirling vibration. The outer diameter (do)
should not exceed 100 mm due to space limitations and here do is
taken as 90 mm. The drive shaft of transmission system was designed optimally
to the specified design requirements [9].
3
DESIGN OF COMPOSITE DRIVE SHAFT
3.1 Assumptions
The shaft rotates at a constant speed about its
longitudinal axis. The shaft has a uniform, circular cross section. The shaft
is perfectly balanced, i.e., at every cross section, the mass center coincides
with the geometric center. All damping and nonlinear effects are excluded. The
stress-strain relationship for composite material is linear & elastic;
hence, Hook’s law is applicable for composite materials. Since lamina is thin
and no out-of-plane loads are applied, it is considered as under the plane
stress
3.2
Selection
of Cross-Section and Materials
The E-Glass/Epoxy and HS Carbon/Epoxy materials are
selected for composite drive shaft. Since, composites are highly orthotropic
and their fractures were not fully studied. The factor of safety was taken as
2 and the fiber volume fraction as 0.6.
|
Fig.1. Conventional two-piece drive
shaft arrangement for rear wheel vehicle driving system [9] |
Table
1 Mechanical properties for each
lamina of the laminate[9]
|
3.3 Torque transmission capacity
of the composite drive shaft The lamina is
thin and if no out-of-plane loads are applied, it is considered as the plane
stress problem. Hence, it is possible to reduce the 3-D problem into 2-D problem.
For unidirectional 2-D lamina, the stress-strain relation ship is given
by
|
(1) The matrix Q is
referred as the reduced stiffness
matrix for the layer and its terms are |
|||
|
|
|
|
|
When a shaft is subjected to torque T, the resultant
forces in the laminate by considering the effect of centrifugal forces are
 |
 |
 ; |
where
The stresses in K th ply are given by the following matrix.
Knowing the stresses in each ply, the failure of the laminate is determined
using the First Ply Failure criteria. That is, the laminate is assumed to
fail when the first ply fails. Here maximum stress theory is used to find
the torque transmitting capacity[9 ].
3.4 Torsional Buckling Capacity
Since long thin
hollow shafts are vulnerable to torsional buckling, the possibility of the
torsional buckling of the composite shaft was checked by the expression[10]
for the torsional buckling load Tcr of a thin walled orthotropic
tube and which is expressed below.
(3)
This equation (3) has been generated from the equation
of isotropic cylindrical shell and has been used for the design of drive shafts.
From this equation, the torsional buckling capability of a composite shaft
is strongly dependent on the thickness of composite shaft and the average
modulus in the hoop direction.
3.5 Lateral Vibration
Natural frequency
based on the Timoshenko beam theory is given by,
(4)
The critical speed of the shaft is
(5)
4 FORMULATION OF THE OPTIMIZATION PROBLEM USING GENETIC ALGORITHM
Formulation of an optimal design problem involves identification
of the design variables, objective function and design constraints. The design
constraints are not mandatory for all types of optimization problems. Most
of the methods used for design optimization assume that the design variables
are continuous. In structural optimization, almost all design variables are
discrete. GA used to obtain the optimal number of layers, thickness of ply
and fiber orientation of each layer. All the design variables are discrete
in nature and are easily handled by GA. With reference to the middle plane,
symmetrical fiber orientations are adopted
4.1 Comparison between GA and other methods
GA differs from
traditional optimization algorithm in many ways. A few are listed here [5].
- GA does not
require a problem specific knowledge to carryout a search. GA uses only
the values of the objective function. For instance, calculus based search
algorithms use derivative information to carryout a search.
- GA uses a
population of points at a time in contrast to the single point approach
by the traditional optimization methods. That means at the same time GA
process a number of designs.
4.2 Comparison between biological GA terms
Chromosome - a small rod like body found in the living cells, which
is responsible for the transmission of generic information denotes coded design
vector in GA. Gene - which is a
part of the chromosome carrying the hereditary information denotes each bit
in the coded design vector in GA. Population
- denotes a number of coded design variables in a cell. where as Generation
denotes the population of design vectors, which are obtained after one computation
in other wards the process of termination of the loop was carried out by fixing
the maximum number generations. These max.number generations is fixed after
trail runs.
4.3 Objective Function
The objective
for the optimum design of the composite drive shaft is the minimization of
weight, so the objective function of the problem is given as
: Weight of the shaft,
;
(6)
4.4 Design Variables
The design variables of the problem are number of plies,
stacking Sequence and thickness of the ply. The limiting values of the design
variables are given as follows
|
1]. N > 0 n = 1,2,3…32 |
2].
k =1, 2,……
n |
3].
|
The number of plies required depends on the design constraints,
allowable material properties, and thickness of plies and stacking sequence.
Based on the investigations it was found that up to 32 numbers of plies are
sufficient.
| 4.5 Design Constraints |
| 1].Torque
transmission capacity of the shaft :
|
2].Tortional
Buckling capacity of the shaft:
|
3].
Fundamental natural frequency of
the shaft :
|
The constraint
equations may be written as:
| 1].
If T < Tmax = 0 Otherwise |
2].
If Tcr < Tmax = 0 Otherwise |
3].
If Ncrt
< Nmax =
0 Otherwise |
(7)
4.6 Penalized Objective Function
GA is generally used to solve unconstrained optimization
problems with bounds on design variables. For constrained problems, one must
transform the constrained problem into unconstrained one by using suitable
penalty function. In this study, penalty function suggested by Rajeev and
Krishnamurthy [11] is used.
That is, Φ
=m (1+k1C) (8)
m is the objective function; k1 is parameter that
has to be judiciously selected depending on the influence of a violated function
C. k1 = 10 is found to be suitable for the problem considered in
this paper; C is the constraint violation function and is computed as afore
mentioned.
4.7 Input GA parameters
| . |
Total string length = String
length for number of plies {5}+16*String length for fiber orientation {8}
+ String length for thickness of ply {6} =139.
Table 2
| GA Parameters |
|
| Number of Parameters |
:n/2+2, if n is even |
| :(n+1)/2+2,if n is odd |
|
| Total string length |
:139 |
| Population size |
:50 |
| Maximum generations |
:150 |
| Cross-over probability |
:1 |
| Mutation probability |
:0.003 |
| String length for number of plies |
:5 |
| String length for fiber orientation |
:8 |
| String length for ply thickness |
:6 |
4.8 Computer program
A tailor made computer program using C language has been
developed to perform the optimization process, and to obtain the best possible
design. Fig.2 shown is GA flow chart.
5 RESULTS AND DISCUSSION
Variation of objective function
value and number of layers of HS Carbon/Epoxy and E-Glass/Epoxy shafts
with number of generations are shown in figs. 3-6. For the first 106
generations of E-glass/epoxy shaft and 46 generations of HS-Carbon/epoxy shaft,
the weight is found to be fluctuating. The fluctuation is reduced to a minimum
from generation nos. 90-106 in E-Glass/Epoxy shaft and 31- 46 in HS Carbon/Epoxy
shaft, but later they get converged. This is because the population is filled
with the best individuals, and further operations results in no change in
fitness value.
Fig. 2 Flow chart of GA based optimal design |
Fig.3 Variation of Weight of E-Glass/Epoxy Drive shaft with number of generations
|
Fig.4 Variation of No. of Layers of E- Glass/Epoxy Drive Shaft with number of generations |
|
 |
 |
Table 3 Optimal design values of
composite shafts with steel shaft
| |
do (mm) |
L (mm) |
tk (mm) |
n (plies) |
t (mm) |
Optimum Stacking sequence |
T (Nm) |
Tcr (Nm) |
Ncrt (rpm) |
wt. (kg) |
(%) save |
| Steel[9] |
90 |
1250 |
3.32 |
1 |
3.32 |
------ |
3501 |
43857 |
9323 |
8.6 |
--- |
| E-Glass/ Epoxy |
90 |
1250 |
0.4 |
17 |
6.8 |
|
3525 |
29856 |
6514 |
4.4 |
48.36 |
| HS carbon /Epoxy |
90 |
1250 |
0.12 |
17 |
2.04 |
|
3879 |
3772 |
7495 |
1.12 |
86.90 |
Table 4 Weight comparison of the
shaft including end pieces
| |
Weight of the shaft (tube only) Kg |
Weight of end Pieces,kg |
Total weight of the drive shaft,Kg |
| Steel |
8.6 |
1.118 |
9.718 |
|
E-Glass/epoxy |
4.4 |
1.118 |
5.518 |
| HS carbon/Epoxy |
1.12 |
1.118 |
2.238 |
6 CONCLUDING REMARKS
1. A procedure to design a composite drive shaft is
suggested.
2. Drive shaft made up of E-Glass/ Epoxy and HS Carbon/Epoxy multilayered composites have been
designed.
3. The designed drive shafts are optimized using GA
for better stacking sequence, better torque transmission capacity and bending vibration characteristics.
4. The usage of composite materials and optimization
techniques has resulted in considerable amount
of weight saving in the range of 48 to 86% when compared to steel shaft.
5. These results are encouraging and suggest that GA
can be used effectively and efficiently in other complex and realistic designs
often encountered in engineering applications
REFERENCES
[1] Jones, R.M., Mechanics of Composite Materials, 2e, McGraw-Hill Book Company,
(1990).
[2] John W. Weeton,
et. al, Engineers guide
to composite materials, A.S. for Metals,
(1986).
[3] P.Beardmore, and Johnson C.F, "The potential
for composites in structural automotive applications", Journal of Composites Science and Technology, 26, (1986), pp.251-281.
[4] Andrew pollard,
“PMCs in driveline applns.”,
(1989), GKN
[5] Goldberg DE.
“Genetic algorithms in search, opt. and m/c learning”,
[6] S.Vijayarangan, et.al “Design optimization of leaf
springs using genetic algorithms”, Inst Engrs.
[7] S.Vijayarangan
and I. Rajendran “Optimal design of a composite leaf spring using genetic
Algorithm” Computers and structures, 79(2001), pp.1121-1129
[8] Mallick, P.K. and Newman. “Composite. Materials
Technology”, Hanser publishers, 1990, pp.206- 210.
[9] T.Rangaswamy and S, Vijayarangan, “Stacking sequence
optimization of composite drive shafts using genetic algorithms”, International congress on computational mechanics
and simulation (ICCMS–04), IIT Kanpur,
India, Dec. 2004, pp.663-670.
[10] Zinberg, H.Symonds, M. “The Development of an Advanced Composite Tail
Rotor Driveshaft”, American Helicopter Society 26th Annual Forum, June 1970.
[11] Rajeev, S and Krishnamoorthy, C.S, “Discrete optimization of structure using genetic algorithms”, J Struct. Engg. ASCE, 118(1992), pp.1233-1250
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