GENETIC ALGORITHM BASED OPTIMUM DESIGN OF AN AUTOMOTIVE COMPOSITE

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GENETIC ALGORITHM BASED OPTIMUM DESIGN OF AN AUTOMOTIVE COMPOSITE Powered By Docstoc
					 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
          INTERNATIONAL JOURNAL OF MECHANICAL
 6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 1, January- April (2012), © IAEME
           ENGINEERING AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 3, Issue 1, January- April (2012), pp. 110-119
                                                                             IJMET
© IAEME: www.iaeme.com/ijmet.html
Journal Impact Factor (2011) : 1.2083 (Calculated by GISI)                ©IAEME
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 AN OVERVIEW OF GENETIC ALGORITHM BASED OPTIMUM
 DESIGN OF AN AUTOMOTIVE COMPOSITE (E-glass / epoxy and
             HM-carbon / epoxy) DRIVE SHAFT
                                  Hargude N.V1, Ghatage k.D2
 1
     Department of Mechanical Engineering,PVPIT Budhgon 416416, Sangli, Maharashtra, India,
     2
       Department of Mechanical Engineering, RIT Sakhrale 415414, Sangli, Maharashtra, India,
                   E-mail: nvhargude@gmail.com; ghatagekishor89@gmail.com

 ABSTRACT

         Substituting composite structures for conventional metallic structures has many
 advantages because of higher specific stiffness and higher specific strength of composite
 materials. Laminated composites, with their advantage of higher specific stiffness, gained
 substantiality in the field of torque carrying structures through many applications. Composite
 drive shafts offer the potential of lighter and longer life drive train with higher critical speed.
 Present work is an overview of attempt of optimization of design parameters of a composite
 drive shaft, which replaces a conventional steel shaft in an automobile power-train, using
 Genetic Algorithm (GA). The parameters such as ply thickness, number of plies and stacking
 sequence are optimized for E-glass / epoxy and HM-carbon / epoxy shafts using GA with the
 objective of weight minimization of the composite shaft which is subjected to constraints such
 as torque transmission, torsional buckling load and fundamental natural frequency. The
 weight reduction can be achieved considerably.

 Keywords: Genetic algorithm, Stacking sequence, Composite drive shaft.

 1. INTRODUCTION
          Advanced composite materials seem ideally suited for long, power drive shaft
 applications. Their elastic properties can be tailored to increase the torque and the rotational
 speed at which they operate. The advanced composite materials such as Boron, 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). An automotive drive
 shaft transmits power from the engine to the differential gear of a rear wheel drive vehicle as
 shown in Fig.1. The torque capability of the drive shaft for passenger cars should be larger
 than 3500 Nm and the fundamental bending natural frequency should be higher than 9200
 rpm to avoid whirling vibration. Since the fundamental bending natural frequency of a one-
 piece drive shafts made of steel or aluminium is normally lower than 5700 rpm when the
 length of the drive shaft is around 1.5 m, the steel drive shaft is usually manufactured in two
 pieces to increase the fundamental bending natural frequency because the bending natural
 frequency of a shaft is inversely proportional to the square of beam length and proportional to
 the square root of specific modulus.



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6340(Print), ISSN 0976 – 6359(Online) Volume 3, Issue 1, January- April (2012), © IAEME
          The two-piece steel drive shaft consists of three universal joints, a centre supporting
bearing and a bracket, which increases the total weight of an automotive vehicle and
decreases fuel efficiency. Also, metallic structure has less specific modulus, specific strength
and its corrosion resistance is less as compared with composite materials. Advantages of
composite drive shafts includes: significant weight reduction, reduced bearing & journal
wear, symmetric composite assures dynamic balance & increased operating speeds,
electrically conductive or non-conductive, custom end-fitting configurations, corrosion
resistant, reduced noise, vibration & harshness (NVH), long fatigue life. Almost all
automobiles (at least those which correspond to design with rear wheel drive and front engine
installation) have transmission shafts shown in Fig. 1. The weight reduction of the drive shaft
can have a certain role in the general weight reduction of the vehicle and is a highly desirable
goal, if it can be achieved without increase in cost and maintaining quality and reliability. An
efficient design of composite drive shaft could be achieved by selecting the proper variables,
which can be identified for safe structure against failure and to meet the performance
requirements. As the length and outer radius of drive shafts in automotive applications are
limited due to spacing, the design variables include the inside radius, layers thickness, number
of layers, fiber orientation angle and layers stacking sequence. In optimal design of the drive
shaft these variables are constrained by the lateral natural frequency, torsional vibration,
torsional strength and torsional buckling. It is possible to reduce the weight of the drive shaft
considerably by optimizing the design parameters by satisfying the all design constraints.
          In the previous study by the authors [12], Genetic algorithm (GA) is 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, no work has been
reported (to the best of the knowledge of the authors) on composite drive shafts using the GA
approach. The first composite drive shaft was developed by the Spicer: U-Joint Division of
Dana Corporation for the Ford econoline van models in 1985. The General Motors pickup
trucks which adopted the Spicer product enjoyed a demand three times that of projected sales
in its first year (1988).
          In the present work an attempt has been overviewed to evaluate the suitability of
composite material such as E-glass / epoxy and HM-carbon / epoxy for the purpose of
automotive transmission applications. A one-piece composite drive shaft for rear wheel drive
automobile is designed optimally by using GA for E-glass / epoxy and HM-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.




Fig. 1. The conventional two-piece steel drive shaft for a rear wheel driving vehicle.

2. SPECIFICATION OF THE PROBLEM
        The torque transmission capability of the drive shaft for passenger cars, small trucks,
and vans should be larger than 3500 Nm (Tmax) and fundamental natural bending frequency of
the drive shaft should be higher than 6500 rpm (Nmax) to avoid whirling vibration. The drive
shaft outer diameter do should not exceed 100 mm due to space limitations. Here outer
diameter of the shaft is taken as 90 mm. The drive shaft of transmission system is to be
designed optimally to the specified design requirements.




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3. DESIGN OF COMPOSITE DRIVE SHAFT
Assumptions:
The following assumptions are made in calculations [1]:
• 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 centre coincides with
    the geometric centre;
• 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.

3.1. Selection of cross section and materials:
         The drive shaft can be solid circular or hollow circular. Here hollow circular cross-
section was chosen because the hollow circular shafts are stronger in per kg weight than solid
circular and the stress distribution in case of solid shaft is zero at the centre and maximum at
the outer surface while in hollow shaft stress variation is smaller. In solid shafts the material
close to the centre are not fully utilized.
Table 2. Mechanical properties of E-glass / epoxy and HM carbon / epoxy

                             E-glass /          HM carbon /
      Property
                             epoxy              epoxy
      E11 (GPa)              50.0               190.0
      E22 (GPa)              12.0               7.7
      G12 (GPa)              5.6                4.2
      ν12                    0.3                0.3
       T     C
      σ 1 = σ 1 (MPa)        800.0              870.0
       T     C
      σ 2 = σ 2 (MPa)        40.0               54.0
      τ12 (MPa)              72.0               30.0
                 3
      ρ (kg/m )              2000.0             1600.0
      Vf                     0.6                0.6
         The E-glass / epoxy, high strength carbon / epoxy and high modulus carbon / epoxy
materials are selected for composite drive shaft. Table 2 shows the properties of the E-glass /
epoxy and high modulus carbon / epoxy materials used for composite drive shafts.
         E11 , E22 , G12 , σT1 , σC1 , σT2 and σC2 represent lamina properties in longitudinal and
transverse directions (Fig. 2) respectively. ν12 , τ12 , ρ and Vf are the Poisons ratio, shear stress
and fiber volume fractions.
         The designer must take into account the factor of safety when designing a structure.
Since, composites are highly orthotropic and their fractures were not fully studied the factor
of safety is taken as 2.

3.2. Torque Transmission of the Composite drive Shaft:
3.2.1.Stress-Strain Relationship for Unidirectional Lamina:
         Since the lamina is thin and no out-of-plane loads are applied, it is considered as the
plane stress
problem and 3-D problem can be reduced into 2-D problem. For unidirectional 2-D lamina,
the
stress-strain relationship in terms of physical material direction is given by,


                                      ,        (1)



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where σ, τ, γ and ε represent stresses and strains in material directions. The matrix Q is
referred as the reduced stiffness matrix for the layer and its terms are given by:

                 ,                                ;


                 ,                        .           (2)

For an angle-ply lamina, where fibbers are oriented at an angle with the positive X-axis
(Longitudinal axis of shaft), the stress strain relationship is given by,


                                      ;               (3)


where σ and ε represent normal stresses and strains in X, Y and XY directions respectively
and bar over   matrix denotes transformed reduced stiffness. Its terms are individually given
by:

                                      ;
                                          ;
                                                          ;
                                      ;
                                                          ;
                                                      ;


with C = cosθ and S = sinθ.




Fig. 2. Shows relation between material coordinate system and X – Y coordinate system

3.2.2 Force and moment resultants:
         For a symmetric laminate, the B matrix vanishes and the in plane and bending
stiffness are uncoupled.


                                          ;           (4)




                                              ;       (5)



where Nx , Ny , Nxy and Mx , My , Mxy in (4), (5) referred as forces and moments per unit width.



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                                  ;           (6a)                               ;        (6b)
                                          .   (6c)

where Aij , Bij and Dij are extensional, coupling and bending stiffnesses having i, j = 1, 2...6
respectively, hk is the distance between the neutral fiber to the top of the Kth layer.
Strains in the reference surface is given by:


                                      ,       (7)



Where


                                              (8)

The in-plane elastic constants for a balanced symmetric shaft, with total thickness t are;

                    ;                               ;

where Ex and Ey are the Young’s modulus of the shaft in axial and hoop direction

            ;             ;

where Gxy and νxy are the rigidity modulus in xy plane and Poisson’s ratio of the composite
shaft.
When a shaft is subjected to torque T, the resultant forces Nx , Ny , Nxy in the laminate by
considering the effect of centrifugal forces are:

        ;                     ;                 .

where ρ is the density, t is the thickness, r mean radius and ω is the angular velocity of the
composite shaft.
The stresses in Kth ply are given by;


                                              (9)




                                              (10)

After evaluating 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.

3.3. Torsional Buckling Capacity:




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         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 for the torsional
buckling load Tcr of a thin walled orthotropic tube, which is expressed below:




where Ex and Ey are the Young’s modulus of the composite shaft in axial and hoop direction, r
and t are the mean radius and thickness of the composite shaft.
This equation has been generated from the equation of isotropic cylindrical shell and has been
used for the design of drive shafts. From the equation, the torsional buckling capability of
composite shaft is strongly dependent on the thickness of composite shaft and the average
modulus in the hoop direction.

3.4. Lateral Vibration:
        Natural frequency fnt based on the Timoshenko beam theory is given by:

                      ;                    (11)

                            ,              (12)


where fnt and p are the natural and first natural frequency. Ks is the shear coefficient of the
natural frequency (< 1), fs is a shape factor (equals to 2) for hollow circular cross-sections.

Critical speed:
              .

4. DESIGN OPTIMIZATION OF COMPOSITE DRIVE SHAFT

         First, fibers are selected to provide the best stiffness and strength beside cost
consideration. It is the best selection, indeed, to use carbon fibers in all layers but due to their
high prices a hybrid of layers of carbon-epoxy and E-glass-epoxy could be utilized. Since the
fiber orientation angle that offers the maximum bending stiffness which leads to the
maximum bending natural frequency is to place the fibers longitudinally at zero angle from
the shaft axis, on the other hand, the angle of ±45º orientation realizes the maximum shear
strength and 90º is the best for buckling strength [4]. The main design goal is to achieve the
minimum weight while adjusting the variables to meet a sufficient margin of safety, which is
translated in a critical speed (natural frequency) higher than the operating speed, a critical
torque higher than the ultimate transmitted torque and a nominal stress (the maximum at fiber
direction) less than the allowable stress after applying any of the failure criteria like the
maximum stress criteria [4].
         Due to the physical geometry (larger radius) of the drive shafts used in the mentioned
applications including automotive applications, the shear strength which specify the load
carrying capacity, is of minor design importance since the failure mode is dominated by
buckling, therefore the main design factors are the bending natural frequency and the
torsional buckling strength, which are functions of the longitudinal and hoop bending
stiffness, respectively [4]. The variable of the laminate thickness has a big effect on the
buckling strength and slight effect on bending natural frequency. A discrete variable
optimization algorithm could be employed for optimization of ply thickness and orientation.
Rangaswamy et al. [2] used Genetic Algorithm and Rastogi [9] used GENESIS/I-DEAS
optimizers for the optimization of variables in the design of drive shaft in automotive
applications. Darlow and Creonte [10] employed the general-purpose package OPT, version
3.2 in optimizing the lay-up of a graphite-epoxy composite drive shaft for helicopter tail rotor.


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        Most of the design optimization methods assume that the design variables are
continuous. In structural optimization, almost all design variables are discrete. A simple
Genetic Algorithm (GA) is 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 easily
handled by GA.

4.1. Comparison between GA and other methods:
     GA differs from traditional optimization algorithm in many ways. A few are listed here
[8]:

1. GA does not require a problem specific knowledge to carry out a search. GA uses only the
values of the objective function. For instance, calculus based search algorithms use derivative
information to carry out a search;
2. 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 GAs process a number of
designs.

4.2. 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:
         ,
Or
                       ,                  (13)

4.3. Design Variables:
The design variables of the problem are
     • Number of plies [n];
     • Stacking Sequence [θ ];
                            k
    • Thickness of the ply [tk].
The limiting values of the design variables are;
     1] n ≥ 0
     2] -90       90
     3] 0.1       0.5
where k = 1, 2,…, n and n = 1, 2, 3,…, 32.
The number of plies required depends on the design constraints, allowable material
properties, thickness of plies and stacking sequence. Based on the investigations it was found
that up to 32 numbers of plies are sufficient.

4.4. Design Constraints:
1. Torque transmission capacity of the shaft:
2. Bucking torque capacity of the shaft:

3. Lateral fundamental natural frequency:


The constraint equations may be written as:
                 , If
                    = 0 Otherwise;
                 , If
                    = 0 Otherwise;
                  , If


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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
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                   = 0 Otherwise
 =             .

Using the method of Rajeev and Krishnamoorthy [7], the constrained optimization can be
converted to unconstrained optimization by modifying the objective function as:
               . For all practical purposes K1 is penalty constant and is assumed to be 10.

5. COMPUTER PROGRAM
         An attempt has been made by Rangaswamy et al [1] to develop a powerful and
efficient computer program using C language to perform the optimization process, and to
obtain the best possible design. The flow-chart describing the step-by-step procedure of
optimizing the composite drive shaft using GA is shown in Fig. 3.




                           Fig.3. Flow chart of GA based optimal design

6. SPECIMEN FABRICATION



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        Two methods are proposed for manufacturing the composite drive shaft either
cocurring carbon fibre epoxy composite layer on the inner surface of an aluminium tube or
wrapping on the outer surface [3]. In most of the manufacturing methods following stacking
sequence have been adopted [14].

1. [450]4 All layers are of glass/epoxy.
2. [450]4 All layers are of carbon/epoxy.
3. [900]4 All layers are of glass/epoxy.
4. [900]4 All layers are of carbon/epoxy.
5. [(450)2 glass/(900)2 carbon].
6. [(45o)2 carbon/(90o)2 glass].

Woven roving Fabric fibers used in both [0/90] and [±45] lay-up. The thicknesses of the
composites were measured to be:
Carbon/epoxy layer thickness=0.35mm and glass/epoxy layer thickness =0.37 mm.

7. RESULTS AND DISCUSSION

7.1. Effect of fiber orientation angle on natural frequency
         The structure consists of four layers stacked as [+45oglass/-45oglass/0ocarbon/90oglass]. The
fibers must be oriented at zero degrees to increase the natural frequency by increasing the
modulus of elasticity in the longitudinal direction of the shaft. This explains why the carbon
fibers, with their high modulus were oriented at the zero angle. The drive shaft loses 44.5% of
its natural frequency when the carbon fibers are oriented in the hoop direction at 90o instead
of 0o. The cost factor plays a role in selecting only one layer of carbon/epoxy [13].

7.2. Effect of fiber orientation angle on buckling torque
         The best fiber orientation angle for maximum buckling strength is 90o. At this angle,
the fibers are oriented in the hoop direction, thereby increasing the hoop modulus (Eh). It can
be observed that, by changing the angles of the 3rd or the 4th layer, the critical buckling
torque of the drive shaft is not substantially affected by the fiber orientation angles [13]. This
is attributed to the fact that the modulus, Ex, has its maximum value at the zero degree fiber
orientation angle, and the modulus, Eh, has its maximum value at a 90o angle.

7.3. Effect of layers stacking sequence on buckling torque
         This normal bending stiffness is correspondent to the component, D22, of the bending
stiffness matrix [D]. Therefore, the value of D22 specifies the buckling strength. A.R. Abu
Talib et al [13] concluded that the best case scenario stacking sequence is [45/-45/0/90], and
the worst case scenario is [0/90/-45/45]. The best stacking offers buckling torque of 2303.1
Nm and the worst stacking offers a torque of 1242 Nm, with a loss in buckling resistance
capability equal to 46.07%.

7.4. Summary of GA results
         The objective of weight reduction has been achieved by implementation of genetic
algorithms by authors. The use of E-glass / epoxy composite in drive line has resulted in
48.36% of weight saving than that of steel drive shaft of same dimensions. Resulted lighter
shaft of thickness 6.8mm has a torque transmission capability of 3525.4 Nm. The High
modulus carbon / epoxy composite drive shaft of thickness 2.04mm resulted in 86.90% of
weight saving has a torque transmission capability of 3656.7Nm which is much higher than
that of conventional steel drive shaft[1].

8. CONCLUSIONS
1. A procedure to optimum design of composite drive shaft made up of E-glass / epoxy and
high modulus carbon / epoxy multilayered composites have been discussed.



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2. GA is suggested as an effective optimization tool for optimal design of composite drive
shaft for better stacking sequence, better torque transmission capacity and bending vibration
characteristics.
3. 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 conventional steel
shaft.
4. Results obtained are encouraging and GA can be suggested effective and efficient tool for
other complex and realistic designs often encountered in engineering applications.

9. REFERENCES

1. Thimmegowda Rangaswamy, and Sabapathy Vijayarangan “Optimal Sizing and Stacking
Sequence of Composite Drive Shafts” journal of Material science, Vol.11, No.2, 2005.
2. R.R Ajith, T. Rangaswamy, S. Vijayarangan and G. Chandramohan “Genetic Algorithm
Based Optimal Design Of Composite Shaft” International journal of Material Science and
Engineering, December2004.
3. Dai Gil Lee and Hak Sung Kim “Design and manufacture of an automotive hybrid
aluminum/composite drive shaft” journal of composite structure, Vol 63, 2004 pp.87-99.
4. M. A. Badie, A. Mahdi, and A. R. Abutalib “Automotive composite drive shafts:
Investigation of the design variable effects” International Journal of Engineering and
Technology, Vol. 3, No.2, 2006, pp. 227-237.
5. Durk Hyun Cho, Dai Gil Li, Jin Ho Choi “Manufacture of one-piece automotive drive
shafts with aluminum and composite materials” journals of Composite structure, Vol. 38, No.
l-4, 1997 pp. 309-319.
6. M.A.K. Chowdhuri , R.A. Hossain, Design Analysis of an Automotive Composite Drive
Shaft, International Journal of Engineering and Technology Vol.2(2), 2010, 45-48.
7. Rajeev., S., Krishnamoorthy, C. S. Discrete Optimization of Structure Using Genetic
Algorithms J. Structural Engg. ASCE 118 1992: pp. 1233 – 1250.
8. Goldberg, D. E. Genetic Algorithms in Search, Optimization and Machine Learning,
Reading MA, Addison-Wesley, 1989.
9.Rastogi, N. (2004), Design of composite driveshafts for automotive applications, SAE,
Technical Paper Series, 2004-01-0485.
10. Darlow, M. S. and Creonte, J. (1995), Optimal design of composite helicopter power
transmission shafts with axially varying fibre lay-up, Journal of the American Helicopter
Society 40 (2): 50-56.
11. Rao, S. S. Mechanical Vibrations. Addision-Wesely Publishing Company, NY: pp. 537 –
541.
12. Vijayarangan, S., et. al. Design Optimization of Leaf Springs Using Genetic Algorithms
Inst. Engrs. India Mech. Engng. Div. 79 1999: pp. 135 – 139.
13. A.R. Abu Talib et al “Developing a hybrid, carbon/glass fiber-reinforced, epoxy
composite automotive drive shaft” journal of Materials and Design 31 (2010) 514–521




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