Effect of Variation in Velocity of Cyclist and Coefficient of Drag on the Drag Force

Objective:

To plot graph showing the effect of variation in Velocity of Cyclist and Coefficient of Drag on the Drag Force

Problem Statement:

Given the factors involved in the computation of the Drag Force, which are:

1. Velocity of the Cyclist (v)
2. Drag coefficient(c_d)
3. Frontal area(A)
4. Density of air(rho)

We need to calculate Drag Force on the cyclist while riding the cycle and verify the effect of variation of:

1. Velocity of the Cyclist(v)
2. Drag coefficient(c_d)

Solution Procedure:

On a flat road, aerodynamic drag is by far the greatest barrier to a cyclist's speed, accounting for 70 to 90 percent of the resistance felt when pedaling.

Drag depends on the density of the air, the square of the velocity, the air's viscosity and compressibility, the size and shape of the body, and the body's inclination to the flow. In general, the dependence on body shape, inclination, air viscosity, and compressibility are very complex.

The drag equation states that drag (D) is equal to the drag coefficient (Cd) times the density (rho) times half of the velocity (V) squared times the reference or frontal area (A).

Mathematically, it can be defined as follows:

Drag Force= 0.5*(Frontal Area) *(Density of air in kg/m³) *(Drag Coefficient) *((Velocity)²)

D = Cd * A * 0.5 * (rho) * V²

The reference area A is typically defined as the area of the orthographic projection of the object on a plane perpendicular to the direction of motion. For non-hollow objects with simple shapes, such as a sphere, this is the same as a cross-sectional area.

Coefficient of Drag (Cd) deals with complex dependencies in the form of a single variable. The coefficient of drag changes with different shapes:

Sno.	Shapes	Coefficient of Drag
1	Sphere	0.47
2	Half-sphere	0.42
3	Flat plate	1.28
4	Bullet	0.295
5	AirFoil	0.045
6	Long Cylinder	0.82
7	Short Cylinder	1.15
8	Cube	1.05
9	Angled cube	0.80

# TO CALCULATE DRAG FORCE AGAINST A BICYCLE- Challenge for Week 1
# Submitted by M K Pragya
#_______________________________________________________________________________________________________________________
#Formula Used: D = Cd * A * 0.5 * (rho) * V^2


#Part A: Write a program to plot Velocity vs Drag Force


#Assuming the values of frontal area(A), Density of air(rho) and taking coefficient of drag to be constant for Part A and variable for Part B

import math
import matplotlib.pyplot as plt

#Inputs

c_d=0.8				                           # random value for drag coefficient
A=0.1  				 						   #taking a constant value for frontal area(m^2)
rho=1.2										   #taking standard value for density of air (kg/m^3)
velocities=[1,2,3,4,5,6,7,8,9,10,11,12,13,14]  #velocities of the bicycle(m/s)
drag_forces=[]                                 #empty array to store the calculated value of drag force

#Calculations

for v in velocities:
	drag_forces.append(0.5*c_d*rho*A*v*v)

#Plotting

plt.plot(velocities,drag_forces, color='blue', linewidth = 3, marker='o', markerfacecolor='red', markersize=6,label="Area=0.1 m/s^2, coefficient of drag=0.8, density of air=1.2 kg/m^3")
plt.xlabel('Velocity',fontname="Times New Roman",fontsize=16)
plt.ylabel('Drag Force',fontname="Times New Roman",fontsize=16)
plt.legend()
plt.title('Graph for Velocity vs Drag Force',fontname="Times New Roman",fontsize=20,fontweight='bold')
plt.show()

#--------------------------------------------------------------------------------------------------------------------------------------------------------------

#Part B: Write a program to plot Drag Co-efficient vs Drag force

#Inputs

#taking a constant value for the frontal area(m^2) and standard value for the density of air (kg/m^3) with value same as mentioned above

v=6                                                              #taking a random value for velocity(m/s)
c_d2=[0.47,0.42,1.28,0.295,0.045,0.82,1.15,1.05,0.80]            #Drag Coefficients for different shapes of the given Frontal Area
drag_forces2=[]													 #empty array to store the calculated value of drag force

#Calculations:

for c in c_d2:
	drag_forces2.append(0.5*c*rho*A*v*v)

#Plotting:

plt.plot(c_d2,drag_forces2, color='red', linewidth = 3,marker='o', markerfacecolor='yellow', markersize=6, label="Area=0.1 m/s^2, velocity=6m/s, density of air=1.2 kg/m^3")
plt.xlabel('Coefficient of Drag',fontname="Times New Roman",fontsize=16)
plt.ylabel('Drag Force', fontname="Times New Roman",fontsize=16)
plt.legend()
plt.title('Graph for Coefficient of Drag vs Drag Force', fontname="Times New Roman",fontsize=20,fontweight='bold')
plt.show()

Result:

The velocity of the cyclist is directly proportional to the drag force and the drag force follows a parabolic curve with an increase in velocity.

The coefficient of drag is directly proportional to the drag force and the drag force increases linearly with an increase in the drag coefficient.

Discussion:

Velocity is the parameter which the cyclist can control according to the circumstance. Reducing speed is not a feasible option for the usability of cycles. The other parameters can be looked after for ensuring the best performance for the cyclist in a given condition. While improvements in physical structure have improved aerodynamic performance, the cyclist is the largest obstacle to improvement as the human body is not very streamlined. Body positioning is important. Reducing the frontal area helps riders increase their speed and efficiency over time. The shape thus formed due to the positioning of the cyclist decides the coefficient of drag. The lower drag coefficient is appreciated for desired results.