The American University in Cairo
Department of Engineering
ENGR 312: Numerical Analysis and Computations
 
Term Project
Application of Cubic Splines Method in Solar Car Design


Contents:

  1 -  Abstract
  2 -  Introduction:

  3 -  Background   4 -  Procedures:   5 -  Results
  6 -  Case studies

Abstract:

The aim of this report is to introduce the steps and procedures of the ENGR 312 Analysis and computations project, entitled: Application of Cubic Splines. The project represents an introductory study for the MENG490 thesis students in their project to design a solar powered car. This report gives an introduction to solar cars and their history, in addition to the concept of solar-powered car sand their design. It also gives a background about some design concerns, such as: solar cells, aerodynamics and drag force.
By using a FORTRAN77 program to make cubic splines interpolation, some points were introduced to interpolate different points on the car surface. Finally, a final sketch for the car is done using MS Excel.
Such a design requires considering some important case studies; such as: drag calculations, aerodynamic body and other design cases to maximize the surface area in order to get the maximum power from the solar cells.

Key words: design of solar cars, cubic splines interpolation,
                   FORTRAN 77,aerodynamic


Cubic Splines:

Cubic splines interpolation is the most efficient method of interpolation. It includes entering a number of points. The following examples clarify the use of cubic splines.



FORTRAN 77 Code
c     ============================================
c     =    CUBIC SPLINES PROGRAM-PROJECT 312     =
c     =            SOLAR CAR PROJECT             =
c     ============================================
       dimension x(100), y(100), f(100), e(100), g(100), r(100),d2y(100)
       open (1, file="problem.txt", status='old')
c     ============================================
c     =              DATA ENTRY                  =
c     ============================================
       read (1,*) n
       do i=0,n-1
       read (1,*) x(i)
       read (1,*) y(i)
       end do
       read (1,*) xu
       close (1)
c     ============================================
c     =       PSEUDO CODE TRI-DIAGONAL           =
c     ============================================
       f(1)=2*(x(2)-x(0))
       g(1)=(x(2)-x(1))
       r(1)=(6/(x(2)-x(1)))*(y(2)-y(1))
       r(1)=r(1)+6/(x(1)-x(0))*(y(0)-y(1))
 
       do i=2,n-2
       e(i)=(x(i)-x(i-1))
       f(i)=2*(x(i+1)-x(i-1))
       g(i)=(x(i+1)-x(i))
       r(i)=6/(x(i+1)-x(i))*(y(i+1)-y(i))
       r(i)=r(i)+6/(x(i)-x(i-1))*(y(i-1)-y(i))
       end do
 
       e(n-2)=(x(n-2)-x(n-3))
 
       f(n-2)=2*(x(n-1)-x(n-3))
 
c     ============================================
c     =       PSEUDO CODE TDMA EVALUATION        =
c     ============================================
        Do M=2,n-2
       e(M)=e(M)/f(M-1)
       f(M)=f(M)-e(M)*g(M-1)
       r(M)=r(M)-e(M)*r(M-1)
       end do
       d2y(x(n-2))=r(n-2)/f(n-2)
       do k=n-3,1,-1
       d2y(x(k))=(r(k)-g(k)*d2y(x(k+1)))/f(k)
       end do
       d2y(x(0))=0
        d2y(x(n-1))=0
        do i=0,n-1
        open (2,file="output.txt", status='old')
        write (2,*) " The 2nd diffentials" , d2y(x(i))
        print *, d2y(x(i))
        end do
c     ============================================
c     =        PSEUDO CODE INTERPOLATION         =
c     ============================================
        flag=0
        i=1
        do i=1,n-1
        if((xu.GE.x(i-1)).and.(xu.LE.x(i)))then
        c1=d2y(x(i-1))/6/(x(i)-x(i-1))
        c2=d2y(x(i))/6/(x(i)-x(i-1))
        c3=y(i-1)/(x(i)-x(i-1))-d2y(x(i-1))*(x(i)-x(i-1))/6
        c4=y(i)/(x(i)-x(i-1))-d2y(x(i))*(x(i)-x(i-1))/6
        t1=c1*(x(i)-xu)**3
        t2=c2*(xu-x(i-1))**3
        t3=c3*(x(i)-xu)
        t4=c4*(xu-x(i-1))
 
        yu=t1+t2+t3+t4
 
        t1=-3*c1*(x(i)-xu)**2
        t2=3*c2*(xu-x(i-1))**2
        t3=-c3
        t4=c4
        dy=t1+t2+t3+t4
 
        t1=6*c1*(x(i)-xu)
        t2=6*c2*(xu-x(i-1))
        d2y=t1+t2
        flag=1
 
        end if
 
        if ((i.eq.n+1).or.(flag.eq.1)) exit
        end do
 
        if (flag.eq.0) then
        print *, "outside range"
        pause
        end if
c     ============================================
c     =                 DATA OUPUT               =
c     ============================================
       write (2,*) "The Interpolation =", yu
       print *, yu
        close (2)
       stop
       end

Points to consider when designing a solar car:

  1 -  The car should be designed in order to maximize the area exposed to sun light in order to achieve maximum power.
  2 -  The car shape should be so-called an aerodynamic shape in order to achieve minimum wind resistance, or the so-called drag force.
  3 -  The car should be as light as possible, because the power expected from the solar cells is not that much. In addition, most of this power will be utilized to overcome friction and drag.


Aerodynamics and Drag:

A body immersed in a flowing fluid is acted on by both pressure and viscous forces from the flow. The sum of the forces (pressure, viscous, or both) that acts normal to the free-stream direction is the lift, and the sum of that acts parallel to the free-stream  direction is defined as the drag.  These definitions are perhaps one of the famous conclusions of the famous Bernoulli’s equation, which is one of the fundamental laws governing the motion of fluids. It  relates an increase in flow velocity to a decrease in pressure and vice versa. Bernoulli's principle is used in aerodynamics to explain the lift of an airplane wing in flight. A wing is so designed that air flows more rapidly over its upper surface than its lower one, leading to a decrease in pressure on the top surface as compared to the bottom. The resulting pressure difference provides the lift that sustains the aircraft in flight. The velocity of a wind that strikes the bluff surface of a building is close to zero near its wall. According to Bernoulli's principle, this would lead to a rise in pressure relative to the pressure away from the building, resulting in wind forces that the structures must be designed to withstand.
Another important aspect of aerodynamics is the drag, or resistance, acting on solid bodies moving through air. The drag forces exerted by the air flowing over the airplane, for example, must be overcome by the thrust force developed by either the jet engine or the propellers. These drag forces can be significantly reduced by streamlining the body. For bodies that are not fully streamlined, the drag force increases approximately with the square of the speed as they move rapidly through the air. The power required, for example, to drive an automobile steadily at medium or high speeds is primarily absorbed in overcoming air resistance.
The following examples illustrate the importance of considering drag when designing a car

By comparing these three shapes, we notice that the shape of the airfoil is the one that shows minimum drag, because of its streamline shape. In addition, it shows less or almost no turbulence at the end.
  The difference between GM sunraycer (above) and the other car is that the stream line design of the GM gives it the minimum drag among all other cars. Such a design enables it to move at higher speeds and make good use of its solar power instead of wasting it in resisting drag.

Photo Voltaic Cells

Solar cells made from thin slices of crystalline silicon, gallium arsenide, or other semiconductor materials convert solar radiation directly into electricity. Cells with conversion efficiencies in excess of 30 percent are now available. By connecting large numbers of these cells into modules, the cost of photo-voltaic electricity has been reduced to 30 cents per kwh, about twice the rate that the largest U.S. cities were paying for electricity in 1989. Current use of solar cells is limited to remote, unattended low-power devices such as buoys and equipment aboard spacecraft.

Design Process

In order to design the exterior shape of the solar car, we had to consider the previously mentioned factors, which are:

  1 -  The design must maximize the amount of surface area exposed to sunlight to obtain maximum power.
  2 -  The design of the car must have an aerodynamic shape to minimize the amount  of drag to which the car is exposed.
  3 -  The car surface should have smooth gradual curves to have an aerodynamic body of low wind resistance.


Step #01: Deciding on a shape:

Solar cars have several unique shapes. The following figure shows the most famous and well-known shapes:

We finally decided to select the wing shape design. This is because we have found out that most of the universities tend to design this shape. In addition, it is perhaps the easiest one in manufacturing. In fact, our shape was not a simple copy for that one; we introduced some modifications.
Following is the design we set for ourselves:
Position Length (LTR) or Height from the ground
Entire frame  5.8 m
Wheels 25 cm diameter, 10 cm thickness
Driver cabin height 38 cm
Full height 1.48m
Tail length 3.4m

The tail is taken to be a straight line. The cross section of the car is taken as an ellipse of changing dimensions. Thus, the main concern for us became the parts in the front and the driver’s cabin. Using AutoCAD, we estimated some key points along the x-y plane, considering the left side of the coordinates as the y-axis, and the x-axis will be the direction along the ground.
 
After defining these dimensions, a rough sketch was drawn by AutoCAD. By offsetting and dimensioning, the following values were obtained.

X coordinate (cm) Y coordinate (cm)
        20                         89.7261
        40                         98.296
        70                         107.2946
        90                         111.0367
        110                       113.8179
        120                       115.0542
These numbers were processed into the computer program, asking the program to:
  a -  get the interpolations of every x point (with 1 cm increment from the first value).
  b -  get the angle of inclination of the tangent at the point

Then the output data is used to plot the front section surface.

The same will happen with the driver cabinet. However, since the cabinet will not be covered with photo-voltaic cells, there is no need to calculate the angle of inclination.

X coordinate cm      Y - coordinate cm
        120                      115.0542
        140                      136.0397
        160                       149.3282
        180                         150.6931
        200                         143.0437
        240                         130.2249
        260                         115.0293
 

Concerning the tail,  the angle of inclination is known since it is a simple straight line relation.

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