Dhiral P. Sawlani

Abstract

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The primary requirement of front suspension in bicycles is to absorb shocks, provide better handling, riding comfort and safety to the rider. A newly designed front wheel with provisions of shock absorption directly through the bicycle rim can be an inclusion to the effectiveness of the design of the conventional bicycle rim with front shock absorbers. The newly designed bicycle wheel is based on the conventional bicycle rim with an arrangement of Leaf Springs Strips instead of conventional spokes. A Pentagonal Hub is used that supports the Axle of the bicycle and Leaf Springs Strips. The Leaf Spring Strips act as supporting member as well as a damping member. The wheel is modeled using NX DESIGN. The analysis of the newly designed wheel is carried out using NX NASTRAN. For the analysis purpose, different loading conditions on the bicycle wheels are considered. The results of Analysis are obtained in the form of Stress generated in the bicycle rim and leaf springs strips and are compared with the Safe Design Stress Parameters to ensure the Safety of Design.

Keywords Bicycle Rim · Bicycle Wheel · Shock · Absorb · Leaf Spring Strips · Pentagonal Hub

Abbreviations

m Mass of Rider

Mb Mass of Bicycle

M Total Combined Mass

W Total Combined Weight

g Gravitational Acceleration

Wf Total Weight on Front Bicycle Wheel

Nf Normal Reaction Force on Front Wheel

Nr Normal Reaction Force on Rear Wheel

Sut Ultimate Tensile Strength

Syt Yield Tensile Strength

? Safe Design Stress

D. P. Sawlani (&), Member

Department Of Mechanical Engineering,

Government Engineering College Surat,

Surat 395009, Gujarat, India

E-mail:- [email protected]

F Pedal Force

Ø Angle of Inclination

Fs Factor Of Safety

Crr Co-efficient of Rolling Resistance

Cd Drag Co-efficient

? Density of Air

A Projected Area of bicycle and rider.

P Pedal Power

Pw Power on Wheels

T Pedaling Torque

Tw Pedaling Torque

D Diameter of Bicycle Rim

R Radis of Bicycle Rim

N Rotation per minute (RPM)

K.E. Kinetic Energy of Wheel

T.E. Total Energy of Wheel

I Moment of Inertia of Wheel

E Total Energy

v Linear Velocity

u Initial linear velocity

a Linear Acceleration

t Time

s Distance

? Angular Velocity

?0 Initial Angular Velocity

? Angular Acceleration

? Angular Distance

Tb Braking Torque

Fi Impact Force

Si Slow Down Distance

Introduction

Design

The design of front wheel is simple and sturdy in construction. In the current design of the front wheel consist of Leaf Spring Strips, a Steel Pentagonal Hub, and Conventional Bicycle Rim. The wheel design is based on the principle of self-adjustment and damping capacity of the leaf springs. One end of leaf spring strip is connected to the pentagonal hub through bolting and the other end of the strip is connected to the rim periphery through riveting.

The new design of wheel consists of Axle, Pentagonal Hub, Spring Steel Strips, Conventional Bicycle Rim, and Tubeless Rubber Tire as shown in Figure 2. The connection of the spring steel strips with the bicycle rim and pentagonal hub is as shown in Figure 1. The spring steel strips work as a supporting as well as shock absorbing members. The pentagonal hub has threaded holes to accommodate bolts for connection of strips. One end of the spring steel strip is connected to the pentagonal hub and fixed by using bolts. The other end of the spring steel strip

is connected to the bicycle rim through nuts and bolts.

Thus, there are two fixed point of connection for the spring steel strips and they act like a damping member when loads and shocks are experienced during motion of the bicycle. The design is simple, sturdy and has an additional feature of shock absorption. This wheel can be beneficial for several applications such as Wheelchairs, normal low-cost bicycles, mountain bicycles. Design of this wheel leads to the new type of suspensions for a bicycle that would provide shock absorption directly through the wheel itself.

Working

The working of the wheel is simple. The spring steel strips work as damping members for the shocks produced. When shocks and loads are acting on the wheel, the shape of the strips changes and absorb the shocks coming to the rider by displacement of spring steel strips according to the intensity of load. The shock is transferred from rider’s handle to front forks; front forks to the axle; axle to the hub; hub to the strips; strips to the rim; rim to the tire and vice versa if shocks come from ground to tire.

Here, the hub is attached to axle and axle is connected to the body of the cycle through front forks. While experiencing shocks the axle center displaces in the opposite direction in which the shock is experienced. The spring steel strips get some elastic deformation with the application of load and shocks in motion.

This elastic deformation of spring steel strips produces damping effect that results in absorption of shock directly through the wheel. The deformation of the wheel is due to the mass of rider as well as torque generated on the wheels due to pedaling as shown in Figure 2.

Simulation

The simulation of the bicycle wheel is carried out in NX Design. The FEA is basically divided into three parts PRE, SOLVER and POST. The Finite Element Analysis is an effective computational method to obtain results for the wheel.

The PRE process part includes the creation of the model, preparation of geometry of mesh, allocation of material to different parts. The meshing of the wheel is conducted by 3D Tetrahedral Mesh. The mesh size is optimized automatically by the software. The Meshing of the wheel is shown Figure 3.

Materials allocated to different components are as follows;

Axle – Steel

Pentagonal Hub – Steel

Strips – Spring Steel (AISI Steel 1008 HR)

Tire – Rubber (poly-urethane-soft)

The components are connected with each other through mesh mating – glue coincident type. Then a file is generated which is sent to the solver (NX NASTRAN). This file is known as the Solution.

The NX NASTRAN Solver uses Finite Element Method (FEM). Element Iterative Solver is used. Structural type of analysis is conducted for the solution of the wheel. Linear Statics – Global Constraints solution type is employed for the solution of the load conditions on the wheel.

A different set of loads are contemplated for the solution of the wheel. The Solution is given to the solver to solve and results are obtained.

POST process includes displaying of results in to form of graphs and deformation of the model in the form of graphics.

Analysis

Analysis of newly designed wheel is carried out by considering the following effects;

· Effect of Driving Force and Torque,

· Effect of Braking Torque,

· Effect of Weight of Rider on the Wheel

(Static Loads)

· Effect of Driving Torque and Total Combined Weight (Dynamic Driving Loads).

· Effect of Braking Torque and Total Combined Weight (Dynamic Braking Loads).

Different Mass of the rider are assumed to be m1 = 70 kg, m2 = 85 kg, m3 = 100 kg and mass of the bicycle is assumed to be 15 kg for the analysis of the wheel. The Total Combined Mass is taken into consideration for analysis i.e. M1 = 85 kg, M2 = 100 kg,M3 = 115 kg.

As the bicycle is in motion, the spring steel strips are the members that constantly deflect due to shocks and loads. So, failure may occur at spring steel strips. Thus, assuming Factor of Safety for spring steel as 2 (Fs = 2) 1

Spring Steel

Sut = 331 N/mm2 2 Syt = 285 N/mm2 2

The Stresses generated on the wheel should be less than the Safe Design Stress. For Stationary Condition (i.e. Rider on the bicycle and no motion of bicycle), the designing is done considering Syt. Therefore, Safe Design Stress for the stationary condition is 142.5 N/mm2.

For Dynamic Condition (i.e. Rider on the bicycle and bicycle in motion), the designing is done considering Sut. Therefore, Safe Design Stress for the dynamic condition is 165 N/mm2.

For analysis, the Elemental Nodal Stress generated should be less than the Safe Design Stress.

Effect of Driving Torque

Assuming that the bicycle is moving with constant velocity on a road, then the pedal force on the rear wheel should be more than the resisting force opposing the bicycle motion. These resisting forces opposing the bicycle motion are Gravity, Rolling Resistance, Aerodynamic Drag and internal bicycle friction. Here Total Combined Mass (M) is taken into consideration for Calculations.

The Driving/Pedaling force produced by the rider through pedaling is obtained by the given equation (1) consider the resisting forces on bicycle and rider.

F = Mg* sinØ + Crr * Mg + 0.5 Cd ?Av2 3 (1)

This above equation is taken into consideration when Angle of Inclination (Ø) is to be considered. So when no inclination is considered the equation (2) is as follows:

F = Crr * Mg + 0.5 Cd ?Av2 3 (2)

This above equation gives pedaling force required to get the cycle in motion. The torque produced at the wheels can be determined from the given equation (3):

T = F * R (3)

The torque generated on the rear wheel of the bicycle is equal to the torque obtained on the front wheel of the bicycle. These values of pedaling torque that produce stress in the wheel as shown in the Figure 4.

Elemental Nodal Stress are taken into consideration to know whether the stresses generated in the elements due to pedaling torque are under the Safe Design Stress Limit. The values of Elemental Nodal Stress Due to Pedaling Torque is shown in Table 1.

Effect of Braking Torque

The braking torque is the torque produced at the wheels while braking. The effect of the braking torque is higher than the effect of the pedaling torque. So it is necessary to consider Effect of Braking Torque for safe design of the wheel. Here Total Combined Mass (M) is taken into consideration for Calculations.

The Braking Torque (Tb) can be determined by the following equation (3):

E = Tb * ? (3)

Here, it is assumed that the bicycle is in motion on the road at a constant velocity of 5.56 m/s (20 km/h). Assuming 8 seconds as the time taken by the bicycle to stop. For calculation of braking torque, total energy of the wheel is calculated by the following expression:

E = Kinetic Energy (K.E.) + Energy of rotating body

= 0.5mv2 + 0.5I ?2

Moment of Inertia (I) = Mb * R

Angular velocity of wheel (?) = V/R = 19.04 rad/s

Now, consider the kinematic equations of motion of linear form and converting them to polar form to determine the Angular Acceleration of wheel (?).

Linear Form

Polar Form

v = u + at

? = ?0 + ?t

s = ut + 0.5 at2

? = ?0t + 0.5?t2

v2 = u2 + 2as

?2 = ?02 + 2??

Angular Acceleration (?) = ? / t = 2.38 rad/s2

The angular distance (?) traveled by the wheel during braking is given by solving the polar form of kinematic equation (4) given below:

? = ?0t + 0.5?t2 ? (4)

? = 76.16 radians

The Braking Torque on the wheel can be determined by the following equation (5):

Tb = E / ? (5)

The braking torque generated due to braking of the bicycle and their values of braking torque are shown in Figure 4.

Elemental Nodal Stress is taken into consideration to know whether the stresses generated in the elements due to braking torque are under the Safe Design Stress Limit. The values of Elemental Nodal Stress Due to Braking Torque is shown in Table 2.

The values of stress generated in the wheel due to Pedaling torque and Braking Torque.

Mass (M)

kg

Pedaling Torque (T) Nm

Nodal Stress due to Driving Torque

N/mm2

Nodal Stress due to Braking Torque

N/mm2

M1

4.46

3.128

23.97

M2

4.68

3.283

26.10

M3

4.90

3.437

28.24

Nodal stress generated in the wheel due to pedaling as well as braking should be less than the Safe Design Stress i.e. 165 N/mm2. Thus, the design is safe when driving torque acts on the wheel.

Effect of Total Combined Weight

Assuming that cycle is in the stationary condition and the loads acting on the wheels of the bicycle are due to Rider’s weight as well as Bicycle’s weight. Also assuming the weight of the bicycle as 15 kg and the masses of the rider as stated above as m1, m2 and m3.

The masses of the rider as well as mass of the bicycle are summed up and then the combined mass (M) is converted into weight through given equation (6) below:

W = M * g (6)

Table 3 shows the masses of rider, mass of bicycle and Total combined mass. The Total combined mass is converted to Total combined weight (W).

Mass of Rider

(m)

kg

Mass of Bicycle (mb) kg

Total Combined Mass

(M)

kg

Total Combined Weight

(W)

N

1.

70

15

85

833.85

2.

85

15

100

981

3.

100

15

115

1128.15

The Normal Reaction Force that acts on the wheels of the bicycle is due to the Total Combined Weight as shown in the equation (7) below. The Normal Reaction Force produced acts in vertically upward direction as shown in Figure 5.

W=N (7)

From the Center of Gravity, the Total Combined Weight acting on the wheels gets distributed. On the rear wheel the Total Combined Weight Acting is 2/3 times of Total Combined Weight (W) and on the front wheel is 1/3 times of Total Combined Weight (W) 5 as shown in Figure 5.

As the loads due to Total Combined Weight (W) on the front wheel is 1/3 times, the weight acting on the front wheel is as shown in Table 4.

Nodal Stresses generated in the front wheel due to the weight of the rider as well as the weight of the bicycle should be less than the Safe Design Stress for Static Loads i.e. 142.5 N/mm2.

Total Combined Weight (W)

N

Total Weight on the Front Wheel (Wf)

N

Nodal Stress in front wheel

N/mm2

833.95

277.95

52.01

981

327

64.05

1128.15

376.05

70.37

The Maximum Stress generated in the front wheel for maximum weight on the front wheel is 70.37 N/mm2 as shown in Table 4 which is less than the Safe Design Stress i.e. 142.5 N/mm2 . Thus the design is safe.

Effect of Driving Torque and Total Combined Weight

For the dynamic analysis of the wheel, the Effect of Driving Torque and Total Combined Weight is considered.

Considering the above theory of Effect of Driving Torque, the Driving/Pedaling force produced by the rider through pedaling is obtained by the equation (2) consider the resisting forces on bicycle and rider.

Power produced by the rider of different weight by pedaling can be given by the equation (8) given below:

P = F * V 3 (8)

The power obtained on the wheel is always less than the power obtained by pedaling because of the drivetrain losses which are assumed to be 3% of the Pedal Power. Total power available on the wheel can be given by the equation (9). Table 5 shows the power available by pedaling and power available at wheels.

Pw = (1-0.03) * P (9)

Pedal Power (P)

Watts

Power on wheels (Pw)

Watts

85.06

82.51

89.18

86.51

93.24

90.44

The equation (10) gives the torque produced by the pedal power and its values for different weight of riders is as shown in Table 6. The RPM of the wheel for a constant velocity of 5.56 m/s can be given by equation (11):

Pw = 2?NT / 60

Tw = Pw * 60 / 2?N (10)

V = ?DN / 60

N = V * 60 / ?D (11)

Power on wheels (Pw)

Watts

Torque on the wheels (Tw)

Nm

82.51

4.52

86.51

4.74

90.44

4.95

Considering the above theory of Effect of Total Combined Weight, the mass of the rider as well as the mass of the bicycle are summed up and then the combined mass (M) is converted into combined weight (W) through equation (6). The combined weight acting on the front wheel is 1/3 times of the Total Combined Weight. 5

For the Dynamic Analysis of the Wheel, both the Driving Torque on the wheel as well as Combined Weight are considered to be acting simultaneously. The Nodal Stress generated due to Driving Torque and Combined Weight on Front Wheel loads should be less than the Safe Design Stress for Dynamic Loads i.e. 165 N/mm2. Nodal Stresses generated in the wheel is shown in Table 7.

Combined Weight on Front Wheel (Wf)

N

DrivingTorque on the wheel (Tw)

Nm

Nodal Stresses due to both loads

N/mm2

277.95

4.52

55.17

327

4.74

67.36

376.05

4.95

73.83

The Nodal Stresses generated in the Front wheel of the bicycle are less than the Safe Design Stress for dynamic Loads. Thus, the design is safe.

Effect of Braking Torque and Total Combined Weight

For the Dynamic Analysis of the Wheel, the Effect of Braking Torque and Total Combined Weight are considered.

Considering the above theory of Effect of Braking Torque, the effect of the braking torque is higher than the effect of the pedaling torque. The values of the Braking Torque can be determined by equation (5) and the values of braking torque are shown in Figure 4.

Considering the above theory of Effect of Total Combined Weight, the mass of the rider as well as the mass of the bicycle are summed up and then the combined mass (M) is converted into combined weight (W) through equation (6). The combined weight acting on the front wheel (Wf) is 1/3 times of the Total Combined Weight (W). 5

For the Dynamic Analysis of the Wheel, both the Braking Torque on the wheel as well as Combined Weight are considered to be acting simultaneously. The Nodal Stress generated due to Braking Torque and Combined Weight on Front Wheel should be less than the Safe Design Stress for Dynamic Loads i.e. 165 N/mm2. Nodal Stresses generated in the wheel is shown in Table 8.

Combined Weight on Front Wheel (Wf)

N

Braking Torque on the wheel (Tw)

Nm

Nodal Stresses due to both loads

N/mm2

277.95

20.29

66.19

327

23.34

80.36

376.05

26.38

88.81

The Nodal Stresses generated in the Front wheel of the bicycle are less than the Safe Design Stress for Dynamic Loads. Thus, the design is safe.