DIFFERENTIAL SYSTEM A Term Project Report

MAY 30, 2014 MOHAMMAD ALI JINNAH UNIVERSITY

Islamabad Campus

Course: Machine Design

Instructor: Salman Warsi

Differential System

1 | P a g e

GROUP MEMBERS

Umair Zamir Mughal

(ME-113-007)

Zeeshan Nawaz

(ME-113-001)

Zaheer ul haque

(ME-113-008)

Rana Bilal Khalid

(ME-113-015)

Zeeshan Raheed

(ME-113-017)

Differential System

2 | P a g e

Abstract Differential systems are a vital part involved in the smooth running of an automobile. This

project aims at analyzing the said system in an efficient manner. The design and analysis was

conducted on Microsoft Excel and creo parametric 2.0 with promising results. The differential

system given below is designed for Suzuki Wagon R 1.0 in an approximate fashion.

Differential System

3 | P a g e

Table of Contents Abstract ........................................................................................................................................... 2

Introduction .................................................................................................................................... 4

Concept ........................................................................................................................................... 5

Geometrical Description ............................................................................................................. 5

Working ....................................................................................................................................... 5

Case 1 (Straight Path) .............................................................................................................. 5

Case 2 (Curved Path) ............................................................................................................... 5

Methodology ................................................................................................................................... 6

Calculations ..................................................................................................................................... 7

Symbols Used .............................................................................................................................. 7

Formulae Used: ........................................................................................................................... 1

Results ............................................................................................................................................. 1

Geometry Calculations (Metric Units) ........................................................................................ 1

Stresses Calculations (English Units) ........................................................................................... 3

Calculations for velocity distribution .......................................................................................... 4

Discussion........................................................................................................................................ 6

Differential System

4 | P a g e

DIFFERENTIAL SYSTEM

Introduction Differential systems are the key components in every automobile. The purpose is to provide

variable angular velocity at the wheels by extracting torque form the power shaft and delivering

it at variable ratios to axles without which there is chance of tipping over.

Differentials are of various types among which the common ones are Open Differential, Limited

Slip Differential (LSD) and Locking Differentials. We designed the open differential because of it

being the simplest one. Although more mature designs are available but due to the fact that

only the demonstration of a differential system was intended, open differential was chosen.

The objectives of this project were:

State and understand the purpose of differential.

Design of straight bevel gears.

Identify and description of various gears used in the differential

Different speeds on rear wheels at same torque.

The differential was designed for Suzuki Wagon R 1.0. Specifications available at the companies’

at website were used to design and calculate gear dimensions and the analysis. Throughout the

analysis AGMA/ANSI1 standards were followed. All the calculations regarding the geometry and

stress analysis were performed on Microsoft Excel 2013. The design was also reproduced on

creo parametric 2.0.

1 ANSI/AGMA 2005-D03 (Revision of ANSI/AGMA 2005--C96)

Differential System

5 | P a g e

Concept

Geometrical Description Torque is supplied from the engine, via the transmission, to a drive shaft which runs to the final

drive unit that contains the differential. A straight bevel pinion gear takes its drive from the end

of the shaft, this meshes with the large bevel ring gear, known as the crown wheel. The crown

wheel and pinion may mesh in perpendicular orientation. The crown wheel gear is attached to

the differential which contains the 'sun' and 'planet' wheels or gears, which are a cluster of four

opposed bevel gears in perpendicular plane, so each bevel gear meshes with two neighbors,

and rotates counter to the third, that it faces and does not mesh with. The two sun wheel gears

are aligned on the same axis as the crown wheel gear, and drive the axle half shafts connected

to the vehicle's driven wheels. The other two planet gears are aligned on a perpendicular axis

which changes orientation with the ring gear's rotation.

Working The system works on relative friction at the two tires. The two possible scenarios are explained

below:

Case 1 (Straight Path)

In case a vehicle is following a straight path, there is no relative friction between the two tires

and as a result there is no differential movement of the planetary system of gears other than

the minute movements necessary to compensate for slight differences in wheel diameter in the

off road. The result is almost equal velocities at both axle shafts.

Case 2 (Curved Path)

In this case when the car takes a turn at either sides, there is relatively more resistance towards

that particular side. Also, the turning side tire has a smaller curve to follow in comparison with

the opposite side. As a result of this friction, the angular velocity of the turning side decrease

and a net moment is produced, which in larger magnitude can tilt the car.

This scenario engages the planetary gears and the smaller gears start rotating at their

respective axis providing a pivot to the axle shafts gears. The result is a distributed angular

velocity and eventually more stability.

Differential System

6 | P a g e

Methodology Although in actuality the system uses zerol or hypoid gears, here in our case we used straight

bevel gears to keep the analysis as simple as possible.

The design started with the selection of a suitable vehicle which in our case was Suzuki Wagon

R 1.0 as it is the latest car launched in Pakistan and it being an efficient off road automobile. We

used its information that was readily available from the catalog. For the design purposes details

at maximum like angular frequency and torque available at that very frequency was used.

Starting with the description of nomenclature we searched for the relations and graphs useful

to identify and calculate the useful aspects from AGMA standards for the design of bevel gears2.

Since a sum of six gears was involved in the calculations therefore in order to save time we

implemented the spread sheet in the form of a calculator that could be used by anyone in

future as well. To check the authenticity of the spread sheet analysis the values were matched

with the handwritten analysis and gave pretty fruitful results.

Results obtained from the spread sheet aided to perform further analysis of bending and

surface stress calculations. Making the good use of the skills another calculator was thus made

with the aid of spreadsheets.

From certain results obtained then were used to make the 3D model of the differential

assembly on creo-parametric.

2 ANSI/AGMA 2005-D03 (Revision of ANSI/AGMA 2005--C96)

Differential System

7 | P a g e

Calculations

Symbols Used

Shaft angle = ∑

Pressure angle=

Pinion teeth= 𝑁𝑝

Gear teeth= 𝑁𝑔

Power=P

Angular Velocity= ω

Gear Pitch dia=D

Pinion pitch cone angle = α𝑝

Gear Pitch cone angle = α𝑔

Cone Distance= C𝐷𝑖𝑠𝑡

Face width=F

Module = m

Addendum pinion= a𝑝

Addendum gear= a𝑔

Dedendum pinion= b𝑝

Dedendum gear= b𝑔

Addendum angle pinion= aθ𝑝

Addendum angle gear= aθ𝑔

Dedendum angle pinion= 𝑏𝜃𝑝

Dedendum angle gear= 𝑏𝜃𝑔

Outer cone angle pinion = 𝑂𝐶 𝛼𝑝

Outer cone angle gear = 𝑂𝐶 𝛼𝑔

Outer diameter pinion = 𝑂𝐷𝑝

Outer diameter gear = 𝑂𝐷𝑔

Root cone angle pinion= RC α𝑝

Root cone angle gear= RC α𝑔

Velocity at contact point between

gears = v

Angular velocity of crown

gears= ω𝑖𝑛

Angular velocity of left gear = ω1

Angular velocity of right gear = ω2

Pitch radius of left gear= r1

Pitch radius right gear= r2

Power in from transmission= P𝑖𝑛

Power out to left half shaft= P𝑜𝑢𝑡 1

Power out to right half shaft= P𝑜𝑢𝑡 2

Torque transmitted to left

wheel= T1

Torque transmitted to right

wheel= T2

Number of teeth of left gear= N1

Number of teeth of right gear= N2

Differential System

1 | P a g e

Formulae Used:

α𝑝 = tan−1× [sin ∑

𝑁𝑔

𝑁𝑝+cos ∑

]

α𝑔 = ∑ − α𝑝

C𝐷𝑖𝑠𝑡 =𝐷

2 sin α𝑔

𝐹 =𝐿

3 ∴ 𝐿 =

𝐷

2 sin α𝑔

𝑚 =1

𝑃𝑑

a𝑝 = 2𝑚 − 𝑎𝑔

a𝑔 = 0.54𝑚 + [4.60𝑚

𝑁𝑔 cos α𝑝

𝑁𝑝 cos α𝑔

]

b𝑝 = 2.188 𝑚 − a𝑝

b𝑔 = 2.188 𝑚 − a𝑔

b𝜃𝑝= a𝜃𝑝

= tan−1 [ b𝑝

C𝐷𝑖𝑠𝑡]

b𝜃𝑔= a𝜃𝑔

= tan−1 [ b𝑔

C𝐷𝑖𝑠𝑡]

OC α𝑝= α𝑝 + a𝜃𝑝

OC α𝑔= α𝑔 − a𝜃𝑔

RC α𝑝= α𝑝 − b𝜃𝑝

RC α𝑔= α𝑝 − b𝜃𝑔

OD𝑝 = 𝑑 + (2 α𝑝 cos α𝑝)

OD𝑔 = 𝐷 + (2 α𝑔 cos α𝑔)

𝑇 =𝑃×6600

𝜔

K𝑣 = (𝐴

√𝐴+200 v𝑡)

𝑏 = 2T𝑝 P𝑑 K𝑎 K𝑚 K𝑠

d𝑝 𝐹𝐽 K𝑣 K𝑥

𝑐 =

C𝑝 C𝑏 √ 2T𝑑

𝐹𝐼 𝑑2 ( T𝑝

T𝑑 )

2 C𝑎 C𝑚

C𝑣 C𝑠 C 𝑓 C𝑥

Differential System

1 | P a g e

Results

Geometry Calculations (Metric Units)

Diameteral Pitch 0.352941176

Pitch Diameter (Gear) 204

Gear Teeth 72

Pitch cone angle (pinion) 14.03624347

Pitch cone angle (gear) 75.96375653

Cone Distance 102.5216767

Module 71.96666667

Adendum (Gear) 2.833333333

Adendum (Pinion) 2.833333333

Dedundum (Pinion) 3.541666667

Dedundum (Gear) 3.541666667

Differential System

2 | P a g e

Face Width 30.756503

Dedundum angle (Pinion) 1.102976949

Dedundum angle (Gear) 1.102976949

Adendum angle(Pinion) 1.102976949

Adendum angle (gear) 1.102976949

Outer Cone Angle (Pinion) 15.13922042

Outer Cone Angle (gear) 74.86077958

Root Cone Angle (Pinion) 12.93326652

Root Cone Angle (gear) 74.86077958

Outside Diameter (Pinion) 56.49050372

Outside Diameter (Gear) 204

Differential System

3 | P a g e

Stresses Calculations (English Units)

Pitch dia gear 8.037

Pitch dia pinion 2.007

Geometric factor pinion 0.253

Geometric factor gear 0.221

Face width 1.38

Torque (pinion) 681.4259297

Dynamic factor 0.422492412

Application factor 2

Load distr. Factor 1.6

Size Factor 1

Crowning factor 1.5

bending stress pinion 5154

bending stress gear 5898

surface stress 648115.7751

Differential System

4 | P a g e

Factor of Safety pinion(bending) 6.743401785

Factor of Safety gear (bending) 7.097999322

Factor of Safety (surface) 6.364434031

Calculations for velocity distribution

No. of teeth of axial gears = 18

Radius of axial gears = 51 mm

Transmission frequency = 6200 rpm

Transmission frequency = 50 kW

CASE 1 straight road

Input frequency 1550 rpm

Frequency for left axle 1550 rpm

Frequency for left axle 1550 rpm

Power for left axle 23.5 kW

Power for right axle 23.5 kW

Torque for left axle 144.6993548 Nm

Torque for left axle 144.6993548 Nm

Differential System

5 | P a g e

CASE 2 Turning right 75:25

Input frequency 1550 rpm

Frequency for left axle 1162.5 rpm

Power for left axle 387.5 rpm

Power for left axle 17.62500002 kW

Power for right axle 5.875000007 kW

Torque for left axle 144.699355 Nm

Torque for left axle 144.699355 Nm

CASE 1 Turning left 25:75

Input frequency 1550 rpm

Frequency for left axle 387.5 rpm

Power for left axle 1162.5 rpm

Power for left axle 5.875 kW

Power for right axle 17.625 kW

Torque for left axle 144.699355 Nm

Torque for left axle 144.699355 Nm

Differential System

6 | P a g e

Discussion The data analysis shows a factor of safety which seems too good to be true. The reason behind

it being it so is that many of our initial assumptions and estimates weren’t correct as no clear

data was available. Everything was estimated on the basis of experience and via research from

the internet.

The calculator that we devised proved to be an ingenious way out of the tedious process of

calculating and recalculating. The data shown in the report is only of one gear set i.e. pinion and

ring gear. For the rest the same has to be repeated with minor changes such as gear diameters

and gear ratios.