Graphical Analysis

Graphical analysis was carried out to compare and illustrate the change of turning radius with change in other parameters such as slip angle, wheel base and track width.Entire analysis has been carried out using MATLAB software.In the analysis carried out, the following graphs have been plotted and analysed;For Two Wheel Steering mechanism:-1) Turning Radius vs Wheel base2) Turning Radius vs Track Width3) Turning Radius vs Inner Slip AngleFor Four Wheel Steering mechanism:-1) Turning Radius vs Wheel base2) Turning Radius vs Track Width3) Turning Radius vs Inner Slip AngleDuring the analysis, the following assumptions were made:-1) The centre of mass of vehicle lies at the midpoint of the wheel base along the axis of vehicle2) There is no oversteer/understeer of vehicle during turn3) There are only four variables that affect steering/turning of vehicle. These are turning radius, wheel base, track width and slip angles4) While relationship between two variables is being plotted, other variables are kept constant5) Track width and wheel base are taken as the same values of Maruti 8006) Inner slip angle is assumed as 25 degrees.7) During Four wheel steering, the inner and outer slip angles on both rear and front wheels are equal in magnitude and opposite in phase

The following notations have been used in the MATLAB programs:-R - Turning Radius (in meters)b Track width (in meters)L Wheel base (in meters)

Graph 1:-Wheel base vs Turning Radius in Four Wheel SteeringAssumptions:-1) Inner Slip Angle is constant ( = 25 degrees)2) Track Width is constant ( = 1.2 meters)MATLAB Program:-L=linspace(1.5,2.5,10);R=((cot(25*pi/180))/2.*L)+0.6plot(L,R)xlabel('Wheel base (in meters)')ylabel('Turning Radius (in meters)')title('Wheel base vs Turning Radius when slip angle is constant')

Analysis:- From graph, we observe that there is a directly proportional relationship between Turning Radius and Wheel base. We have assumed dimensions of standard vehicle i.e Maruti 800 in which Wheel base is 2.175 meters. The value of Turning Radius corresponding to this Wheel base is 2.932 meters Increase in wheel base is desired due to aesthetic and ergonomic reasons. However as demonstrated by graph, excessive wheel base will cause large turning radius which will lead to difficulty in steering and effective cornering

Graph 2:- Wheel base vs Turning Radius in Two Wheel SteeringAssumptions:-1) Inner Slip Angle is constant ( = 25 degrees)2) Track Width is constant ( = 1.2 meters)MATLAB Program:-L=linspace(1.5,2.5,10);R=sqrt((L/2).^2+5.2643^2);plot(L,R)xlabel('Wheel base (in meters)')ylabel('Turning Radius (in meters)')title('Wheel base vs Turning Radius keeping Steering Angle constant')

Analysis:- From graph, we observe that there is a directly proportional relationship between Turning Radius and Wheel base. We have assumed dimensions of standard vehicle i.e Maruti 800 in which Wheel base is 2.175 meters. The value of Turning Radius corresponding to this Wheel base is 5.375 meters Increase in wheel base is desired due to aesthetic and ergonomic reasons. However as demonstrated by graph, excessive wheel base will cause large turning radius which will lead to difficulty in steering and effective cornering Comparing the graphs 1 & 2 we see that there is a substantial increase in the Turning Radius using Four wheel steering and Two wheel steering, from 2.932 meters to 5.375 meters This represents a 45.45% increase in the Turning Radius while keeping all other factors constant

Graph 3Inner Slip Angle vs Turning Radius in Four Wheel SteeringAssumptions:-1) Wheel base is constant ( = 2.175 meters)2) Track width is constant ( = 1.2 meters)MATLAB program:-x=linspace(15*pi/180,35*pi/180,10);R=0.5*(2.175*(cot(x))+1.2);plot(x*180/pi,R)xlabel('Inner Slip Angle (in degrees)')ylabel('Turning Radius (in meters)')title('Inner Slip Angle vs Turning Radius keeping Wheel base constant')

Analysis:- From graph, we observe that there is an inverse relationship with a non-linear slope between Turning Radius and Inner Slip Angle We have assumed Inner Slip Angle as 25 degrees The value of Turning Radius corresponding to this Inner Slip Angle is 2.937 meters Due to reasons of stability of vehicle during turn, wear of tire etc, the Inner Slip Angle cannot be reduced indiscriminately in order to reduce Turning Radius despite inverse relationship between the two Ideally the Inner Slip Angle should not exceed 35 degrees

Graph 4Inner Slip Angle vs Turning Radius in Two Wheel SteeringAssumptions:-1) Wheel base is constant ( = 2.175 meters)2) Track width is constant ( = 1.2 meters)MATLAB program:-x=linspace(0,50*pi/180,10);y=(((1.2/2+2.175*cot(x)).^2)+(2.175/2)^2).^0.5;plot(x*180/pi,y)xlabel('Inner Slip angle (in degrees)')ylabel('Turning Radius (in meters)')title('Slip Angle vs Turning Radius keeping Wheel base constant')

Analysis:- From graph, we observe that there is an inverse relationship with a non-linear slope between Turning Radius and Inner Slip Angle We have assumed Inner Slip Angle as 25 degrees The value of Turning Radius corresponding to this Inner Slip Angle is 5.435 meters Due to reasons of stability of vehicle during turn, wear of tire etc, the Inner Slip Angle cannot be reduced indiscriminately in order to reduce Turning Radius despite inverse relationship between the two Ideally the Inner Slip Angle should not exceed 35 degrees Analysing and comparing graphs 3 & 4, we observe that there is a marked increase in the Turning Radius using Four wheel and Two wheel steering from 2.937 meters to 5.435 meters This corresponds to an increase of 45.96% in the Turning Radius while keeping all other factors constant

Graph 5Track width vs Turning Radius in Four Wheel SteeringAssumptions:-1) Wheel base is constant ( = 2.175 meters)2) Inner Slip Angle is constant ( = 25 degrees)MATLAB Program:-b=linspace(0.8,2,10);R=(cot(25*pi/180))*2.175/2+b/2;plot(b,R)xlabel('Track Width (in meters)')ylabel('Turning Radius (in meters)')title('Track Width vs Turning Radius keeping Slip angle constant')

Analysis:- From graph, we observe that there is a directly proportional relationship between Turning Radius and Track Width We have assumed dimensions of standard vehicle i.e Maruti 800 in which Track Width is 1.2 meters. The value of Turning Radius corresponding to this Track Width is 2.932 meters Increase in track width is desired due to as it provides greater comfort and rigidity to vehicle. However as demonstrated by graph, excessive track width will cause large turning radius which will lead to difficulty in steering and effective cornering and parking of vehicle

Graph 6Track width vs Turning Radius in Two Wheel SteeringAssumptions:-3) Wheel base is constant ( = 2.175 meters)4) Inner Slip Angle is constant ( = 25 degrees)MATLAB Program:-b=linspace(0.8,2,10);R=(((b/2+2.175*cot(25*pi/180)).^2)+(2.175/2)^2).^0.5;plot(b,R)xlabel('Track Width (in meters)')ylabel('Turning Radius (in meters)')title('Track Width vs Turning Radius keeping Slip angle constant')

Analysis:- From graph, we observe that there is a directly proportional relationship between Turning Radius and Track Width We have assumed dimensions of standard vehicle i.e Maruti 800 in which Track Width is 1.2 meters. The value of Turning Radius corresponding to this Track Width is 5.375 meters Increase in track width is desired due to as it provides greater comfort and rigidity to vehicle. However as demonstrated by graph, excessive track width will cause large turning radius which will lead to difficulty in steering and effective cornering and parking of vehicle On comparison of graphs 5 & 6, we see that there is a significant increase in Turning Radius in Four wheel steering and Two wheel steering, from 2.932 meters to 5.375 meters This implies an increase to the order of 45.45% in Turning Radius keeping other factors constant

Conclusions of Graphical Analysis:-1) There was a huge decrease in the Turning Radius of vehicle, taking all other factors and variables into account, while converting vehicles steering mechanism from a Two Wheel Steer to a Four Wheel Steer 2) By graphical analysis, we have determined this decrease in Turning Radius to be 45.45%3) Turning Radius had been reduced from 5.375 meters in Two Wheel Steer to 2.932 meters in For Wheel Steer4) We have also observed the various relationships between Turning Radius and other factors, such as;a) Wheel base (direct relationship)b) Inner Slip Angle (inverse relationship)c) Track width (direct relationship)