Simulating Vehicle Dynamics for designing an Electric Vehicle from an Conventional Vehicle and Calculating Power and Energy Requirements for Different dynamics conditions
AIM :-
To convert an conventional vehicle to an electric vehicle and simulate its dynamic system to design an appropriate EV using MATLAB and SIMULINK
OBJECTIVE :-
- Analyse an conventional vehicle to convert it into appropriate electric vehicle and calculate its Traction.
- Analyse the speed of electric vehicle by reading Tyre Specification in ideal conditions.
- Calculate different Forces acting on the vehicle and Simulate it using drive cycle to analyse Power and Energy Requirenments to design an electric vehicle using data post-processing.
- And Calculate Average energy consumption of the EV.
- Analysing Gradeability factor using Different Studies/Cases.
- Simulating Electric vehicle dynamics using different values of Gradeability.
- Analyse the Power curves for different Tyre nominal Pressure using Given Conditions.
EV DESIGN :-
Vehicle design is a very complex domain since it involves a lot of attributes, like: interior space, dynamic performance, active and passive safety, connectivity, etc.. The same principles apply to either electric vehicles or conventional vehicles (with internal combustion engines) design.
In a series of articles we are going to focus on the VEHICLE DYNAMICS. The same calculation methods can be applied whether we design an electric vehicle from scratch or if we do a conversion from a conventional vehicle.
EV market analysis :-
The entry point in the electric vehicle design are the performance parameters. In our case we are going to focus on:
- top speed
- 0-100 kph acceleration time
- range
In order to come up with suitable parameters, we can study the current market of electric vehicles and also some statistical data. In the table below you can find a list of some electric vehicles and their main parameters.
| Vehicle | Range [km] |
Battery Type |
Battery capacity [kWh] |
Battery Voltage [V] |
Motor Max Power [kW] |
Motor Max Torque [Nm] |
Motor Type |
Top Speed [kph] |
Time 0-100 kph [s] |
| BMW i3 |
257 | Li-ion | 42.2 | 360 | 127 | 250 | PM Sync. |
150 | 7.3 |
| Mitsubishi I-miev |
160 | Li-ion | 16 | 330 | 49 | 196 | PM Sync. |
130 | 12.1 |
| Kia Soul EV |
249 | Li-ion | 30 | 375 | 81.4 | 285 | PM Sync. |
145 | 11 |
| Nissan Leaf |
273 | Li-ion | 40 | 350 | 110 | 320 | AC Sync. |
146 | 7.9 |
| Renault Zoe |
303 | Li-ion | 41 | 400 | 80 | 225 | AC Sync. |
137 | 11.4 |
| Tesla Model S 75D |
375 | Li-ion | 75 | 350 | 305 | 525 | AC Sync. |
228 | 4.2 |
*Sync. (Synchronous)
Source: bmw.co.uk, mitsubishi-cars.co.uk, kia.co.uk, media.nissan.eu, renault.co.uk, teslamotors.com
Electric vehicle conversion:-
In our case we are going to focus on the the conversion of a sports car with internal combustion engine into an fully electric vehicle. The model of choice will be 16MY Jaguar F-type, with the main parameters defined in the table below:
| Engine | 3-litre V6 DOHC V6, aluminium-alloy cylinder block and heads |
|
| Maximum torque [Nm] | 450 | |
| Engine speed @ maximum torque [rpm] | 3500 | |
| Maximum power [HP] | 340 | |
| Engine speed @ maximum power [rpm] | 6500 | |
| Transmission type | automatic, ZF8HP, RWD | |
| Gear ratio | 1st | 4.71 |
| 2nd | 3.14 | |
| 3rd | 2.11 | |
| 4th | 1.67 | |
| 5th | 1.29 | |
| 6th | 1.00 | |
| 7th | 0.84 | |
| 8th | 0.67 | |
| Final drive (i0) | 3.31 | |
| Tire symbol | 295/30ZR-20 | |
| Vehicle mass (kerb) [kg] | 1741 | |
| Aerodynamic drag, Cd [-] | 0.36 | |
| Frontal area, A [m2] | 2.42 | |
| Top speed [kph] | 260 | |
| Acceleration time 0-100 kph [s] | 5.3 | |
From data available on-line, we can also extract the static engine torque values at full load, function of engine speed:
| Engine speed points (full load) [rpm] | 1000 | 2020 | 2990 | 3500 | 5000 | 6500 |
| Engine static torque points (full load) [Nm] | 306 | 385 | 439 | 450 | 450 | 367 |
The engine torque and power at full load is plotted in the image below:
Jaguar F-type 3-litre V6 DOHC V6 torque and power at full load
The high level requirements of the electric vehicle is to have the same (or better) dynamic performance compared with the internal combustion engine version.
Based on the analysis of the current EV market and the performance parameters of the base vehicle (internal combustion engine), in the table below, we are going to summarise the high level requirements of our electric vehicle conversion.
| Battery | Nominal voltage [V] | 400 |
| Chemistry | Lithium-ion | |
| Powertrain | Electric motor type | PM Synchronous |
| Vehicle | Range [km] | 250 |
| Top speed [kph] | 260 | |
| Time 0-100 kph [s] | 5.3 | |
| Kerb weight [kg] | 1741 |
In order to achieve the required performance, the electric vehicle will be all-wheel drive (AWD), with an drive unit on each axle. The drive unit will contain the power electronics (inverter), electric motor, transmission (fixed gear) with decoupling element (clutch / dog clutch) and differential.
VEHICLE WHEEL:
- Conventional Steel Wheels
- Modern Alloy Wheels
- Forged versus Cast Wheels
- Chrome Coated Wheels
- Bottom Line.
What should be the maximum speed of the motor used in this electric Vehicle (16MY Jaguar F-type) capable to run at 190 kmph, if the fixed gear ratio is 7 and tire size is 245/45 R18 ? How to read a tire?
A tire's dimensions can be determined by reading the markings on the sidewall. An example tire size is 245/45R18, which tells us the width, sidewall height, internal construction and matching wheel diameter.
Section Width
The first three digits indicate the section width, or cross section, of the tire in millimeters.
The section width is defined as the widest point of the inside sidewall to the widest point of the outside sidewall when mounted on an appropriate wheel . For this ZE950 tire, the section width is 245 millimeters or 9.6 inches.
Sidewall Aspect Ratio
The two numbers following the section width identify the sidewall aspect ratio, and is commonly referred to as the tire's profile or series. The aspect ratio is presented as a percentage of the section width. In 245/45R18, the 45 denotes the sidewall height is 45 percent of its section width of 245 millimeters, making it 110 millimeters tall or 4.34 inches.
Internal Construction
The letter following the aspect ratio indicates the type of internal construction.
- R: Radial - most common construction for modern tires.
- D: Diagonal - usually found in light truck tires and spare tires.
- B: Belted - an outdated construction method that is no longer used.
The last numbers in the sequence indicate the wheel diameter (in inches) that matches the tire. This means that the 245/45R18 tire will mount properly onto an 18-inch wheel.
GIVEN-
speed v = 90kmph,
fixed gear ratio G = 7,
tire size is 245/45 R18
FORUMLA-
To find the maximum speed of electric motor angular velocity
omega = G*v/r
Where, G = 7 ,v = 190 kmph , r = 245/45 R18
To convert the units v = 190x5/18
v = 52.7 m/s
tire diameter = Aspect ratio+2xwidth
= 25.4*18+2x245
d = 947.2mm
tire radius = r = d/2
r = 947.2/2
r = 473.6mm
r = 0.4736m
Sub the values in formula,
ω = G*v/r
ω = 7x52.7/0.4736
= 778.92rad/s
By converting the unit
ω = 778.92x60/2π">π
Angular velocity = 7441.98 rpm
Traction force calculation:-
One key requirement of our battery electric vehicle conversion is to reach 100 kph from standstill in 5.3 seconds. From this requirement, knowing the vehicle weight, we can calculate what is the total traction force and torque. Also, we can check if the wheel (tire) friction can sustain the required traction force.
The input data in our calculation is:
- vehicle total weight (vehicle kerb weight x mass factor + driver weight): 1908 kg
- vehicle initial speed: 0 kph
- vehicle final speed: 100 kph
- vehicle initial time: 0 s
- vehicle final time: 5.3 s
- tire radius: 0.4736m
- tire friction coefficient: 1.0
- gravitational acceleration: 9.81 m/s2
According to Newton’s second law of motion:
From above we can write the equation of the required traction force as:
where:
Ft [N] – total traction force
mv [kg] – total vehicle mass
vf [m/s] – final speed
vi [m/s]– initial speed
tf [s] – final time
ti [s] – initial time
Replacing the input data into equation , gives the total traction force required to achieve 0-100 kph in 5.3 s:
The total required traction torque can be calculated as:
where:
Tt [N] – total traction torque
rw [m] – wheel radius
Replacing the input data into equation , gives the total traction torque required to achieve 0-100 kph in 5.3 s:
In terms of friction, we can calculate the available friction force as:
where:
Ff [N] – friction force
W [N] – vehicle weight
g [m/s2] – gravitational acceleration
μf [-] – friction coefficient (wheel-road)
Replacing the input data into equation , gives the available friction force:
Since the available friction force is bigger than the total traction force, assuming that there is no slip between the wheels and road, the total traction force can be applied at the wheels in order to achieve the 0-100 kph acceleration time.
Drive Cycles:-
A drive cycle is one of the methods used by a vehicle's powertrain control module (PCM) to determine whether an emissions system repair was performed successfully. It involves a special test drive that duplicates the scenario of a person starting the car and making a short freeway trip, as if driving to work.
WLTC drive cycle-
The average energy consumption of the vehicle Eavg [Wh/km] will be calculated on a homologation cycle. For our example we are going to use the WLTC drive cycle. The test procedure WLTP (Worldwide harmonized Light vehicles Test Procedure) contains several driving cycles:
- Class 1 – low power vehicles with PWr <= 22
- Class 2 – vehicles with 22 < PWr <= 34
- Class 3 – high-power vehicles with PWr > 34
Where PWr [kW/Tonne] is the power-to-weight ratio, defined as the ratio between the rated engine power and kerb weight.
Forces Acting on the Electric Vehicle and its Power and Energy Requirments :-
The Total traction Force T [N] is the sum of the inertial force, road slope force, road load (friction) force and aerodynamic drag force.
where:
Fi [N] – inertial force
Fs [N] – road slope force
Fr [N] – road load force
Fa [N] – aerodynamic drag force
The inertial force is given by the equation:
where:
m [kg] – total vehicle mass
a [m/s2] – vehicle acceleration
The vehicle acceleration can be calculated as:
where:
dv [m/s] – speed difference
dt [s] – time difference
The road slope force is given by the equation:
where:
g [m/s2] – gravitational acceleration
alpha [rad] – road slope angle
The road load (friction) force is given by the equation:
where:
c_r [-] – road rolling resistance coefficient
The aerodynamic drag force is given by the equation:
where:
ρ [kg/m3] – air density at 20 °C
c_d [-] – air drag coefficient
A [m2] – vehicle frontal area
v [m/s] – vehicle speed
The total power Ptot [W] is calculated as the product between the total road forces and the vehicle speed:
By integrating the total power over time (for the whole duration of the cycle), we get the total energy consumption Etot [J]:
m = 1908.05; % [kg]
g = 9.81; % [m/s^2]
c_d = 0.36; % [-]
A = 2.42; % [m^2]
rho = 1.202; % [kg/m^3]
alpha = 0; % [rad]
c_r = 0.011; % [-]Vehicle Simulation over WLTC Drive Cycle :-
TheSimulink block diagram model is run for 1800 s, which is to total duration of the WLTC drive cycle. A Clock block is used to generate a time step of 1 s. This time step is set because the input data (WLTC speed profile) is sampled at 1 s.
Since the sample time is 1 s, which means Δt = 1, the vehicle acceleration will be calculated as the difference between the current speed value and the previous speed value. In the WLTC driving cycle, the road slope is considered 0 rad, therefore will not have any influence on the energy consumption.
Depending on the sign of the total power, we can distinguish between the acceleration and braking (deceleration) phases of the vehicle. The integration of the power, for a Δt = 1, gives the energy. The acceleration and braking energies are calculated separately and then summed up to give the total energy.
By dividing the last calculated value of the total energy to the total length of the WLTC drive cycle (23.266 km), we get the average energy consumption of the vehicle, 179.3 Wh/km.
Data post-processing:-
The same model can be used for any other vehicle, the only change needed being the input parameters update. Also, the simulation can be run for different drive cycles, like FTP or NEDC or for custom cycles which can include also a road gradient.
Time [sec]
In above plot,
blue = Acceleration energy
Red = Total Energy
Orange = Braking Energy
The average energy consumption over WLTC drive cycle is 179.3 Wh/km. This value will be used to calculate the total energy required for the high voltage battery.
CONDITION-1:-
GRADIABILITY:-
The grade (also called slope, incline, gradient, mainfall, pitch or rise) of a physical feature, landform or constructed line refers to the tangent of the angle of that surface to the horizontal. It is a special case of the slope, where zero indicates horizontality. A larger number indicates higher or steeper degree of "tilt". Often slope is calculated as a ratio of "rise" to "run", or as a fraction ("rise over run") in which run is the horizontal distance (not the distance along the slope) and rise is the vertical distance.
The grades or slopes of existing physical features such as canyons and hillsides, stream and river banks and beds are often described. Grades are typically specified for new linear constructions (such as roads, landscape grading, roof pitches, railroads, aqueducts, and pedestrian or bicycle circulation routes). The grade may refer to the longitudinal slope or the perpendicular cross slope.
Δh = rise
l = slope length
α = angle of inclination
Equation of Gradiability:-
Grades are related using the following equations with symbols from the figure at top.
Tangent as a ratio-
This ratio can also be expressed as a percentage by multiplying by 100.
Angle from a tangent gradient-
If the tangent is expressed as a percentage, the angle can be determined as:
If the angle is expressed as a ratio (1 in n) then:
STUDY-1:
1. To find the ratio of hill climbing power required by fully loaded tata ultra truck to the half loaded one.
GIVEN DATA:
FORMULA USED:
HILL CLIMBING FORCE:
Fhc = m x g x sinθ">θ
where, m = Total mass
g = Gravity
POWER:
Power is a rate at which work is done, or energy is used. It is equal to the amount of work done divided by the time it takes to do the work. The unit of power is the Watt (W), which is equal to a Joule per second (J/s).
Total power = Total force x speed
SOLUTION:
Gross vehicle weight = Kerb weight + Pay load
Gross vehicle weight = 11990kg
gravity = 9.8m/s^2
Gradeability = 25.6%
Gradeability = 15 degree
θ">θ = 15 degree
Force = m x g x sin θ">θ
F = 11990x9.8xsin(15)
F = 11990 x 9.8 x 0.6503
F = 117502 x 0.6503
F = 76411.5506N
Total power = Total force x speed
speed = 30 kmph
By converting the unit speed = 30x5/18 = 8.33 mps
power = 76411.5506 x 8.33
power = 636762.922 W
The half loaded truck,
Pay load = 8255
= 8255/2
= 4127.5 kg
kerb weight = 3735 kg
Gross vehicle weight = Kerb weight + Pay load
Gross vehicle weight = 3735+4127.5
= 7862.5 kg
Force = m x g x sin θ">θ
F = 7862.5x9.8xsin(15)
F = 7862.5 x 9.8 x 0.6503
F = 77052.5 x 0.6503
F = 50107.241 N
Total power = Total force x speed
speed = 30 kmph
By converting the unit speed = 30x5/18 = 8.33 mps
power = 50107.241 x 8.33
power = 417393.318 W
The ratio of hill climbing power required by fully loaded tata ultra truck to the half loaded one,
Power Ratio = Power of fully loaded truck/Power of half loaded truck
Power Ratio = 636762.922/417393.31
The ratio of hill climbing power = 1.53
STUDY-2:
A flyover is planned to be constructed on a national highway. Keeping in mind about the gradeability for trucks, how much can be the minimum length of the flyover, considering the minimum height at the summit should be 5.5 meters? State your assumptions.
GIVEN DATA,
Minimum height h = 5.5m
θ">θ = 15 degree
tanθ">θ x 100 = gradeability
Percentage of gradeability = h/d x 100
TO FIND: The minimum length of the flyover
SOLUTION
sinθ">θ = h/L
sin(θ">θ) = 5.5/L
0.258 = 5.5/L
L = 5.5/0.258
L = 21.32
L = 21.3
Minimum length of flyover = 21.3m
By pythagoras theorem,
b =?
a^2+b^2 = c^2
(5.5)^2+b^2 = (21.3)^2
b^2 = (21.3)^2-(5.5)^2
b = 15.8m
Minimum Run of flyover = 15.8m
Total Run of flyover = 31.6m
SIMULATING ELLECTRIC VEHICLE DYNAMICS WITH DIFFERENT GRADEABILITY:-
Now, we will input the inclination angle alph in the previous used Simulink Model of an Electric Vehicle using WLTC drive Cycle
Inclination angle inputed are 0, 0.08724, 0.174533
Simulation Results:-
Traction Force vs Time:-
Traction Power vs Time:-
Total Energy vs Time:-
From, The above Results there is the positive Shift in the Plots of Traction force, Traction Power and Total Energy Requirnments which indicates that there is increase in the Power Requirnments and Energy Consumption of the Electric Vehicle.
CONDITION-2:-
Assuming Total Nominal Pressure of the Tire Decreases by 15% which increases the coefficient of rolling resistance by two times then what is the effect in the electric vehicle Tractive Force and Power requirnments??
clear all
close all
clc
% Input Parameters
m = 1908.05; % [kg]
g = 9.81; % [m/s^2]
c_d = 0.36; % [-]
A = 2.42; % [m^2]
rho = 1.202; % [kg/m^3]
alpha = 0; % [rad]
c_r1 = 0.011; % [-]
c_r2 = 0.022;
% Input parameters of WLTC Drive Cycle
% Velocity(km/h)
v = [0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 1.7 5.4 9.9 13.1 16.9 21.7 26.0 27.5 28.1 28.3 28.8 29.1 30.8 31.9 34.1 36.6 39.1 41.3 42.5 43.3 43.9 44.4 44.5 44.2 42.7 39.9 37.0 34.6 32.3 29.0 25.1 22.2 20.9 20.4 19.5 18.4 17.8 17.8 17.4 15.7 13.1 12.1 12.0 12.0 12.0 12.3 12.6 14.7 15.3 15.9 16.2 17.1 17.8 18.1 18.4 20.3 23.2 26.5 29.8 32.6 34.4 35.5 36.4 37.4 38.5 39.3 39.5 39.0 38.5 37.3 37.0 36.7 35.9 35.3 34.6 34.2 31.9 27.3 22.0 17.0 14.2 12.0 9.1 5.8 3.6 2.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.2 1.9 6.1 11.7 16.4 18.9 19.9 20.8 22.8 25.4 27.7 29.2 29.8 29.4 27.2 22.6 17.3 13.3 12.0 12.6 14.1 17.2 20.1 23.4 25.5 27.6 29.5 31.1 32.1 33.2 35.2 37.2 38.0 37.4 35.1 31.0 27.1 25.3 25.1 25.9 27.8 29.2 29.6 29.5 29.2 28.3 26.1 23.6 21.0 18.9 17.1 15.7 14.5 13.7 12.9 12.5 12.2 12.0 12.0 12.0 12.0 12.5 13.0 14.0 15.0 16.5 19.0 21.2 23.8 26.9 29.6 32.0 35.2 37.5 39.2 40.5 41.6 43.1 45.0 47.1 49.0 50.6 51.8 52.7 53.1 53.5 53.8 54.2 54.8 55.3 55.8 56.2 56.5 56.5 56.2 54.9 52.9 51.0 49.8 49.2 48.4 46.9 44.3 41.5 39.5 37.0 34.6 32.3 29.0 25.1 22.2 20.9 20.4 19.5 18.4 17.8 17.8 17.4 15.7 14.5 15.4 17.9 20.6 23.2 25.7 28.7 32.5 36.1 39.0 40.8 42.9 44.4 45.9 46.0 45.6 45.3 43.7 40.8 38.0 34.4 30.9 25.5 21.4 20.2 22.9 26.6 30.2 34.1 37.4 40.7 44.0 47.3 49.2 49.8 49.2 48.1 47.3 46.8 46.7 46.8 47.1 47.3 47.3 47.1 46.6 45.8 44.8 43.3 41.8 40.8 40.3 40.1 39.7 39.2 38.5 37.4 36.0 34.4 33.0 31.7 30.0 28.0 26.1 25.6 24.9 24.9 24.3 23.9 23.9 23.6 23.3 20.5 17.5 16.9 16.7 15.9 15.6 15.0 14.5 14.3 14.5 15.4 17.8 21.1 24.1 25.0 25.3 25.5 26.4 26.6 27.1 27.7 28.1 28.2 28.1 28.0 27.9 27.9 28.1 28.2 28.0 26.9 25.0 23.2 21.9 21.1 20.7 20.7 20.8 21.2 22.1 23.5 24.3 24.5 23.8 21.3 17.7 14.4 11.9 10.2 8.9 8.0 7.2 6.1 4.9 3.7 2.3 0.9 0.0 0.0 0.0 0.0 0.0 0.0 0.5 2.1 4.8 8.3 12.3 16.6 20.9 24.2 25.6 25.6 24.9 23.3 21.6 20.2 18.7 17.0 15.3 14.2 13.9 14.0 14.2 14.5 14.9 15.9 17.4 18.7 19.1 18.8 17.6 16.6 16.2 16.4 17.2 19.1 22.6 27.4 31.6 33.4 33.5 32.8 31.9 31.3 31.1 30.6 29.2 26.7 23.0 18.2 12.9 7.7 3.8 1.3 0.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.5 2.5 6.6 11.8 16.8 20.5 21.9 21.9 21.3 20.3 19.2 17.8 15.5 11.9 7.6 4.0 2.0 1.0 0.0 0.0 0.0 0.2 1.2 3.2 5.2 8.2 13.0 18.8 23.1 24.5 24.5 24.3 23.6 22.3 20.1 18.5 17.2 16.3 15.4 14.7 14.3 13.7 13.3 13.1 13.1 13.3 13.8 14.5 16.5 17.0 17.0 17.0 15.4 10.1 4.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 2.1 4.8 9.1 14.2 19.8 25.5 30.5 34.8 38.8 42.9 46.4 48.3 48.7 48.5 48.4 48.2 47.8 47.0 45.9 44.9 44.4 44.3 44.5 45.1 45.7 46.0 46.0 46.0 46.1 46.7 47.7 48.9 50.3 51.6 52.6 53.0 53.0 52.9 52.7 52.6 53.1 54.3 55.2 55.5 55.9 56.3 56.7 56.9 56.8 56.0 54.2 52.1 50.1 47.2 43.2 39.2 36.5 34.3 31.0 26.0 20.7 15.4 13.1 12.0 12.5 14.0 19.0 23.2 28.0 32.0 34.0 36.0 38.0 40.0 40.3 40.5 39.0 35.7 31.8 27.1 22.8 21.1 18.9 18.9 21.3 23.9 25.9 28.4 30.3 30.9 31.1 31.8 32.7 33.2 32.4 28.3 25.8 23.1 21.8 21.2 21.0 21.0 20.9 19.9 17.9 15.1 12.8 12.0 13.2 17.1 21.1 21.8 21.2 18.5 13.9 12.0 12.0 13.0 16.0 18.5 20.6 22.5 24.0 26.6 29.9 34.8 37.8 40.2 41.6 41.9 42.0 42.2 42.4 42.7 43.1 43.7 44.0 44.1 45.3 46.4 47.2 47.3 47.4 47.4 47.5 47.9 48.6 49.4 49.8 49.8 49.7 49.3 48.5 47.6 46.3 43.7 39.3 34.1 29.0 23.7 18.4 14.3 12.0 12.8 16.0 19.1 22.4 25.6 30.1 35.3 39.9 44.5 47.5 50.9 54.1 56.3 58.1 59.8 61.1 62.1 62.8 63.3 63.6 64.0 64.7 65.2 65.3 65.3 65.4 65.7 66.0 65.6 63.5 59.7 54.6 49.3 44.9 42.3 41.4 41.3 42.1 44.7 48.4 51.4 52.7 53.0 52.5 51.3 49.7 47.4 43.7 39.7 35.5 31.1 26.3 21.9 18.0 17.0 18.0 21.4 24.8 27.9 30.8 33.0 35.1 37.1 38.9 41.4 44.0 46.3 47.7 48.2 48.7 49.3 49.8 50.2 50.9 51.8 52.5 53.3 54.5 55.7 56.5 56.8 57.0 57.2 57.7 58.7 60.1 61.1 61.7 62.3 62.9 63.3 63.4 63.5 64.5 65.8 66.8 67.4 68.8 71.1 72.3 72.8 73.4 74.6 76.0 76.6 76.5 76.2 75.8 75.4 74.8 73.9 72.7 71.3 70.4 70.0 70.0 69.0 68.0 68.0 68.0 68.1 68.4 68.6 68.7 68.5 68.1 67.3 66.2 64.8 63.6 62.6 62.1 61.9 61.9 61.8 61.5 60.9 59.7 54.6 49.3 44.9 42.3 41.4 41.3 42.1 44.7 48.4 51.4 52.7 54.0 57.0 58.1 59.2 59.0 59.1 59.5 60.5 62.3 63.9 65.1 64.1 62.7 62.0 61.3 60.9 60.5 60.2 59.8 59.4 58.6 57.5 56.6 56.0 55.5 55.0 54.4 54.1 54.0 53.9 53.9 54.0 54.2 55.0 55.8 56.2 56.1 55.1 52.7 48.4 43.1 37.8 32.5 27.2 25.1 26.0 29.3 34.6 40.4 45.3 49.0 51.1 52.1 52.2 52.1 51.7 50.9 49.2 45.9 40.6 35.3 30.0 24.7 19.3 16.0 13.2 10.7 8.8 7.2 5.5 3.2 1.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.8 3.6 8.6 14.6 20.0 24.4 28.2 31.7 35.0 37.6 39.7 41.5 43.6 46.0 48.4 50.5 51.9 52.6 52.8 52.9 53.1 53.3 53.1 52.3 50.7 48.8 46.5 43.8 40.3 36.0 30.7 25.4 21.0 16.7 13.4 12.0 12.1 12.8 15.6 19.9 23.4 24.6 25.2 26.4 28.8 31.8 35.3 39.5 44.5 49.3 53.3 56.4 58.9 61.2 62.6 63.0 62.5 60.9 59.3 58.6 58.6 58.7 58.8 58.8 58.8 59.1 60.1 61.7 63.0 63.7 63.9 63.5 62.3 60.3 58.9 58.4 58.8 60.2 62.3 63.9 64.5 64.4 63.5 62.0 61.2 61.3 62.6 65.3 68.0 69.4 69.7 69.3 68.1 66.9 66.2 65.7 64.9 63.2 60.3 55.8 50.5 45.2 40.1 36.2 32.9 29.8 26.6 23.0 19.4 16.3 14.6 14.2 14.3 14.6 15.1 16.4 19.1 22.5 24.4 24.8 22.7 17.4 13.8 12.0 12.0 12.0 13.9 18.8 25.1 29.8 33.8 38.2 43.4 48.9 53.8 57.8 61.5 65.0 68.4 71.6 73.0 74.3 76.2 77.9 79.5 81.0 82.3 83.5 84.6 85.5 86.3 87.1 88.1 89.1 90.1 91.0 91.7 92.3 92.8 93.1 93.1 93.1 93.1 93.1 93.1 93.1 93.1 93.1 93.1 93.2 93.2 93.3 93.7 94.2 95.0 95.8 96.4 96.8 97.0 97.1 97.2 97.3 97.4 97.4 97.4 97.4 97.3 97.3 97.3 97.3 97.2 97.1 97.0 96.9 96.7 96.4 96.1 95.7 95.5 95.3 95.2 95.0 94.9 94.7 94.5 94.4 94.4 94.3 94.3 94.1 93.9 93.4 92.8 92.0 91.3 90.6 90.0 89.3 88.7 88.1 87.4 86.7 86.0 85.3 84.7 84.1 83.5 82.9 82.3 81.7 81.1 80.5 79.9 79.4 79.0 78.7 78.7 78.8 79.1 79.4 79.6 79.8 79.8 79.6 79.3 78.9 78.5 78.2 77.9 77.7 77.7 77.8 77.9 78.1 78.3 78.3 78.4 78.4 78.4 78.2 78.0 77.7 77.3 76.9 76.6 76.2 75.7 75.2 74.7 74.4 74.3 74.4 74.6 74.9 75.1 75.3 75.5 75.8 75.9 76.0 76.0 76.0 75.9 75.9 75.8 75.7 75.5 75.2 75.0 74.7 74.1 73.7 73.3 73.5 74.0 74.9 76.1 77.7 79.2 80.3 80.8 81.0 81.0 81.0 81.0 81.0 80.9 80.6 80.3 80.0 79.9 79.8 79.8 79.8 79.9 80.0 80.4 80.8 81.2 81.5 81.6 81.6 81.4 80.7 79.6 78.2 76.8 75.3 73.8 72.1 70.2 68.2 66.1 63.8 61.6 60.2 59.8 60.4 61.8 62.6 62.7 61.9 60.0 58.4 57.8 57.8 57.8 57.3 56.2 54.3 50.8 45.5 40.2 34.9 29.6 27.3 29.3 32.9 35.6 36.7 37.6 39.4 42.5 46.5 50.2 52.8 54.3 54.9 54.9 54.7 54.1 53.2 52.1 50.7 49.1 47.4 45.2 41.8 36.5 31.2 27.6 26.9 27.3 27.5 27.4 27.1 26.7 26.8 28.2 31.1 34.8 38.4 40.9 41.7 40.9 38.3 35.3 34.3 34.6 36.3 39.5 41.8 42.5 41.9 40.1 36.6 31.3 26.0 20.6 19.1 19.7 21.1 22.0 22.1 21.4 19.6 18.3 18.0 18.3 18.5 17.9 15.0 9.9 4.6 1.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.2 4.4 6.3 7.9 9.2 10.4 11.5 12.9 14.7 17.0 19.8 23.1 26.7 30.5 34.1 37.5 40.6 43.3 45.7 47.7 49.3 50.5 51.3 52.1 52.7 53.4 54.0 54.5 55.0 55.6 56.3 57.2 58.5 60.2 62.3 64.7 67.1 69.2 70.7 71.9 72.7 73.4 73.8 74.1 74.0 73.6 72.5 70.8 68.6 66.2 64.0 62.2 60.9 60.2 60.0 60.4 61.4 63.2 65.6 68.4 71.6 74.9 78.4 81.8 84.9 87.4 89.0 90.0 90.6 91.0 91.5 92.0 92.7 93.4 94.2 94.9 95.7 96.6 97.7 98.9 100.4 102.0 103.6 105.2 106.8 108.5 110.2 111.9 113.7 115.3 116.8 118.2 119.5 120.7 121.8 122.6 123.2 123.6 123.7 123.6 123.3 123.0 122.5 122.1 121.5 120.8 120.0 119.1 118.1 117.1 116.2 115.5 114.9 114.5 114.1 113.9 113.7 113.3 112.9 112.2 111.4 110.5 109.5 108.5 107.7 107.1 106.6 106.4 106.2 106.2 106.2 106.4 106.5 106.8 107.2 107.8 108.5 109.4 110.5 111.7 113.0 114.1 115.1 115.9 116.5 116.7 116.6 116.2 115.2 113.8 112.0 110.1 108.3 107.0 106.1 105.8 105.7 105.7 105.6 105.3 104.9 104.4 104.0 103.8 103.9 104.4 105.1 106.1 107.2 108.5 109.9 111.3 112.7 113.9 115.0 116.0 116.8 117.6 118.4 119.2 120.0 120.8 121.6 122.3 123.1 123.8 124.4 125.0 125.4 125.8 126.1 126.4 126.6 126.7 126.8 126.9 126.9 126.9 126.8 126.6 126.3 126.0 125.7 125.6 125.6 125.8 126.2 126.6 127.0 127.4 127.6 127.8 127.9 128.0 128.1 128.2 128.3 128.4 128.5 128.6 128.6 128.5 128.3 128.1 127.9 127.6 127.4 127.2 127.0 126.9 126.8 126.7 126.8 126.9 127.1 127.4 127.7 128.1 128.5 129.0 129.5 130.1 130.6 131.0 131.2 131.3 131.2 130.7 129.8 128.4 126.5 124.1 121.6 119.0 116.5 114.1 111.8 109.5 107.1 104.8 102.5 100.4 98.6 97.2 95.9 94.8 93.8 92.8 91.8 91.0 90.2 89.6 89.1 88.6 88.1 87.6 87.1 86.6 86.1 85.5 85.0 84.4 83.8 83.2 82.6 82.0 81.3 80.4 79.1 77.4 75.1 72.3 69.1 65.9 62.7 59.7 57.0 54.6 52.2 49.7 46.8 43.5 39.9 36.4 33.2 30.5 27.3 26.3 24.4 22.5 20.5 18.2 15.5 12.3 8.7 5.2 0.0 0.0 0.0 0.0 0.0 0.0];
% Acceleration(km/h/sec)
a = [0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.85 2.60 4.10 3.85 3.50 4.30 4.55 2.90 1.05 0.40 0.35 0.40 1.00 1.40 1.65 2.35 2.50 2.35 1.70 1.00 0.70 0.55 0.30 -0.10 -0.90 -2.15 -2.85 -2.65 -2.35 -2.80 -3.60 -3.40 -2.10 -0.90 -0.70 -1.00 -0.85 -0.30 -0.20 -1.05 -2.15 -1.80 -0.55 -0.05 0.00 0.15 0.30 1.20 1.35 0.60 0.45 0.60 0.80 0.50 0.30 1.10 2.40 3.10 3.30 3.05 2.30 1.45 1.00 0.95 1.05 0.95 0.50 -0.15 -0.50 -0.85 -0.75 -0.30 -0.55 -0.70 -0.65 -0.55 -1.35 -3.45 -4.95 -5.15 -3.90 -2.50 -2.55 -3.10 -2.75 -1.80 -1.80 -1.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.10 0.95 2.95 4.90 5.15 3.60 1.75 0.95 1.45 2.30 2.45 1.90 1.05 0.10 -1.30 -3.40 -4.95 -4.65 -2.65 -0.35 1.05 2.30 3.00 3.10 2.70 2.10 2.00 1.75 1.30 1.05 1.55 2.00 1.40 0.10 -1.45 -3.20 -4.00 -2.85 -1.00 0.30 1.35 1.65 0.90 0.15 -0.20 -0.60 -1.55 -2.35 -2.55 -2.35 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0.00 0.00 0.00 0.00 1.10 2.20 2.05 1.75 1.45 1.25 1.15 1.25 1.60 2.05 2.55 3.05 3.45 3.70 3.70 3.50 3.25 2.90 2.55 2.20 1.80 1.40 1.00 0.80 0.70 0.65 0.65 0.55 0.50 0.55 0.65 0.80 1.10 1.50 1.90 2.25 2.40 2.25 1.80 1.35 1.00 0.75 0.55 0.35 0.10 -0.25 -0.75 -1.40 -1.95 -2.30 -2.30 -2.00 -1.55 -1.00 -0.45 0.10 0.70 1.40 2.10 2.60 3.00 3.25 3.40 3.45 3.25 2.80 2.05 1.30 0.80 0.50 0.45 0.50 0.60 0.70 0.75 0.75 0.75 0.85 1.00 1.15 1.35 1.55 1.60 1.60 1.60 1.65 1.70 1.70 1.75 1.70 1.55 1.45 1.35 1.25 1.15 0.95 0.70 0.50 0.25 0.00 -0.20 -0.30 -0.40 -0.45 -0.50 -0.65 -0.75 -0.85 -0.95 -1.00 -0.95 -0.80 -0.65 -0.50 -0.40 -0.30 -0.20 -0.30 -0.40 -0.55 -0.75 -0.85 -0.95 -1.00 -0.90 -0.70 -0.55 -0.35 -0.20 -0.10 0.00 0.10 0.15 0.20 0.35 0.50 0.65 0.80 1.00 1.15 1.25 1.20 1.05 0.90 0.70 0.40 0.05 -0.25 -0.70 -1.20 -1.60 -1.85 -1.85 -1.55 -1.10 -0.60 -0.20 -0.05 -0.05 -0.20 -0.35 -0.45 -0.45 -0.30 -0.05 0.30 0.60 0.85 1.05 1.20 1.35 1.40 1.40 1.30 1.15 1.05 0.90 0.80 0.80 0.80 0.80 0.80 0.80 0.75 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% Rolling Resistance
F_r1 = m*g*c_r1*cos(alpha);
F_r2 = m*g*c_r2*cos(alpha);
% Slope Resistance
F_s1 = m*g*sin(alpha);
F_s2 = m*g*sin(alpha);
% Aerodynamics Resistance
F_d1 = (1/2)*rho*c_d*A.*v.*v;
F_d2 = (1/2)*rho*c_d*A.*v.*v;
% Acceleration force
F_a1 = m.*a;
F_a2 = m.*a;
% Traction force required
T1 = F_r1+F_s1+F_d1+F_a1;
T2 = F_r2+F_s2+F_d2+F_a2;
% Total Power(kW)
P1 = (T1.*v)/1000;
P2 = (T2.*v)/1000;
% Creating plots
plot(v, P1,'b')
hold on
plot(v, P2,'r')
xlabel('Speed(Km/hr)')
ylabel('Power(W)')
legend('Normal Pressure of the Tyre','Assuming 15% Decreased Pressure of the Tyre')
title('Comparison of the Power Curves For Different Tyre Pressure Conditions')
grid on
OUTPUT:-
Magnified Plot of Power vs Speed-
RESULT :-
From, the above Plot it can be analysed that there is Positive shift in the curve which indicates that Power Requirenment increases with the decrease in the Tyre Nominal Pressure.