A REPORT ON
“DESIGN AND ANALYSIS OF DIFFERENTIAL GEARBOX”
A PROJECT REPORT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF TECHNOLOGY (MECHANICAL ENGINEERING ENGINEERING))
BY PRATIK PATEL (06m43)
UNDER THE GUIDANCE OF Mr. S. S. PATEL
AT U.V.PATELCOLLEGE OF ENGINEERING, GANPATUNIVERSITY, GANPAT VIDYANAGAR, KHERVA 2009
CERTIFICATE
This project work is the bonafide work done by PRATIK PATEL, Roll No. 06 ME 43, student of VIII semester of Mechanical Engineering Department of U. V. Patel College of Engineering, under the guidance of Mr. S.S.PATEL SIR towards the partial fulfillment of the requirements for the Degree of Bachelor of Technology (Mechanical) of Ganpat University, GanpatVidyanagar.
Guide : S.S.PATEL
Internal Examiner: Internal Examiner: ____________________ ____________________
External Examiner: External Examiner: ____________________ ____________________
Head of Head of Department: Department: S. M. PATEL
CERTIFICATE
This project work is the bonafide work done by PRATIK PATEL, Roll No. 06 ME 43, student of VIII semester of Mechanical Engineering Department of U. V. Patel College of Engineering, under the guidance of Mr. S.S.PATEL SIR towards the partial fulfillment of the requirements for the Degree of Bachelor of Technology (Mechanical) of Ganpat University, GanpatVidyanagar.
Guide : S.S.PATEL
Internal Examiner: Internal Examiner: ____________________ ____________________
External Examiner: External Examiner: ____________________ ____________________
Head of Head of Department: Department: S. M. PATEL
Acknowledgement This Project shall be incomplete if I fail to convey my heart-felt gratitude to those people whom I received considerable support and encouragement during this project creation. Lots of People have helped, provided technical, commercial commercial and also behavioral acumen at all levels of the project and it`s my luck for get the kind of support of them. I have no words to thanks to our special project faculty Mr. S. S. Patel sirfor giving us his innumerous knowledge among our group of for project partners. They are becoming as Way to take us from dark to bright future and specially they play their important role for Motivation and to endeavor for succeed in project creation.
Regards,
PRATIK PATEL
Abstract My Project “DESIGN AND ANALYSIS OF DIFFERENTIAL GEARBOX” mainly focuses on the mechanical design and analysis of gearbox as transmit the power. I had developed this work as my semester project with a view to get familiar with the technologies as well as application of theories into practical work done by industries. My project contains the design and material selection of the gearbox for different type of vehicles also. For better efficiency, improvement of power transmit transmit rate is important important phenomenon. phenomenon.
Index Sr No.
1 2 2.1 2.2 3 3.1 3.2 3.3 4 4.1 4.2 5
Contents Acknowledgement Abstract Index List of figures List of tables Introduction Defination Function of a System Use of Gearbox Transmission System Manual Transmission Automatic Transmission Semi-Automatic Transmission Gearbox Specification About Gear Ratio Explanation of the term About Gearbox
Page No i. ii. iii. iv. v. 1 3 4 4 5 6 7 8 9 10 10 11
5.1 5.2 5.3 6 6.1 6.2 6.3 6.4 7 7.1 7.2 7.3 8 8.1 8.2 9 10
Gearbox glossary Gearbox Material Types of Gearbox Simple Differential Gearbox Introduction of Simple Differential Gearbox Material Used Types of Differential Gearbox Use of Differential Gearbox Design of Simple Differential Gearbox Design of gears Design of Bearings Design of Shafts Analysis of Gearbox Analysis of Gear Shaft Analysis of pinion Shaft Bibliography References
12 13 13 15 16 16 17 17 18 19 24 27 32 33 35 37 39
List of figure Fig No 3.1.1 3.2.1 3.3.1 6.1.1 7.2.1 7.3.1 7.3.2 7.3.3
Name Of Figure Manual Transmission Automatic Transmission Semi-Automatic Transmission Simple Differential Gearbox Deep Groove Ball Bearing Basic Design of Shaft Combine Failure of Shaft Keyway of Shaft
Page No 6 7 8 16 24 27 28 28
List of Table Table No 7.1.1 7.2.1 7.3.1
List Of Table Pinion and Gear Bearing selection Shaft Select
Page No 23 26 31
I NTRODUCTION
r e t p a 1. Introduction h c Gearboxes are used in almost every industry right from power to marine, and also include agriculture, textile, automobiles, aerospace, shipping etc. There are different types of gearboxes available for varying uses. These gearboxes are constructed from a variety of materials depending on their end use and the kind of industry they are being used in. The product has numerous industrial applications for providing high torque and smooth speed reductions. These gearboxes are also manufactured keeping certain specifications in mind, which will also vary depending on the application.
DEFINATION
2 Defination
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9 A gearbox, also known as a gear case or gearhead, is a gear or a hydraulic system
responsible for transmitting mechanical power from a prime mover (an engine or electric motor) into some form of useful output. It is referred to the metal casing in which a number of gears are sealed.
9 A gearbox is also a set of gears for transmitting power from one rotating shaft to another.
They are used in a wide range of industrial, automotive and home machinery application. 9 Gearheads are available in different sizes, capacities and speed ratios. Their main
function is to convert the input provided by an electric motor into an output of lower RPM and higher torque.
2.1Functions of a Gearbox 9 A gearbox is precisely bored to control gear and shaft alignment. 9 It is used as a housing/container for gear oil. 9 It is a metal casing for protecting gears and lubricant from water, dust and other
contaminants.
2.2Use of Gearbox ¾ A variety of gearboxes find applications in a number of industries depending on the end
use. Some of the industries using gearboxes include : ¾ Agricultural ¾ Industrial ¾ Construction ¾ Mining ¾ Petrochemicals ¾ Food processing
T RANSMISSION S YSTEM
3 Transmission System 9 There are three types of transmission system. ¾ Manual Transmission System ¾ Automatic Transmission System ¾ Semi-automatic Transmission System
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3.1
Manual Transmission A manual transmission is often referred to as a stick shift or standard transmission, essentially used in the automobile industry. This type of transmission features gear ratios that are selected by engaging pairs of gears inside the transmission.
Manual Transmission
Types of Manual Transmission Synchronized Systems: This type of transmission system or gearbox does not
require synchronization with the driver while changing gears. Unsynchronized Systems: This type of transmission system or gearbox allows
free spinning of gears with their relative speeds synchronized by the driver for avoiding clashing and grinding of gears. Types of Gearbox Available
There are different types of gearbox available and with different mountings. Some of them are: ¾ Floor Mounted Shifter ¾ Column Mounted Shifter ¾ Sequential Manual Shifter
Advantages of a Manual Transmission System
Cheaper than the automatic transmission system. Better fuel economy than other transmission systems. Requires low maintenance. Does not require active cooling.
3.2Automatic Transmission
9 Automatic transmission is a type of gearbox, especially used in the automobile industry
for changing gear ratios automatically. It does not require manual shifting of gears. The gearbox has a set of selected gear range.
9 The system is hydraulically operated and makes use of a torque converter and a set of
planetary gears.
Automatic Transmission
9 Parts of an Automatic Transmission System
An automatic transmission consists of the following parts : ¾ Torque Converter : It is a device connecting engine and transmission. The
instrument takes place of a mechanical clutch, allowing the engine to remain running. A torque converter that provides a variable amount of torque multiplication at low engine speeds. ¾ Planetary Gearbox Set : The bands and clutches of this gear set are actuated
with the help of hydraulic servos controlled by the valve body, thereby providing two or more gear ratios. ¾ Valve Body : The system receives pressurized fluid from a main pump operated
torque converter. The pressure coming from this pump is regulated and used to run a network of spring-loaded valves, check balls and servo pistons. The valves make use of pump pressure and the pressure from a centrifugal controller on the output side. 9 Use of Automatic Transmission ¾ Automobiles ¾ Forklift trucks
¾ Lawn mowers
3.3 Semi-Automatic Transmission 9 Semi-automatic transmission is a system that makes use of electronic sensors, processors
and actuators for gear shifting. The system removes the need of a clutch pedal required for gear changing. This system is widely used in the automobile industry.
Semi-Automatic Transmission 9 Types of Semi-Automatic Transmission ¾ Direct Shift Gearbox ¾ Dual Clutch Gearbox
The system allows for only forward and backward shift into higher and lower gears. It does not make use of the traditional H-pattern, normally used in automobiles. The system is also equipped with sensors that sense the direction of the shift. The input combines with the sensor placed in the gearbox and senses the current speed and selected gear. The unit also determines the torque required for smooth functioning. The system also reduces fuel consumption significantly.
GEARBOX SPECIFICATION
4 Gearbox Specifications
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9 There are a number of performance specifications which must be considered while
choosing a gearbox for different industrial application.
9 Some of the important specifications are : ¾ Gear ratio: The ratio may be specified as x: 1, where x is an integer.
¾ Output torque ¾ Maximum input power ¾ Maximum input speed ¾ Gearing arrangement ¾ Reducer output ¾ Shaft Alignment
4.1 About Gear Ratio 9 The gear ratios can be defined as the relationship between the number of teeth on two
different gears meshed together or the circumference of two pulleys connected with a drive belt. 9 Generally the number of teeth on a gear is also proportional to the circumference of the
gear wheel, so the bigger the wheel, the more teeth it has. Therefore the gear ratio can also be explained as the relationship between the circumferences of both wheels.
4.2 Explanation of the Term 9 The concept of gear ratio can be well explained with the help of an example as follows : 9 Suppose a smaller gear has 12 teeth, while the larger gear has 24 teeth. Therefore the
gear ratios between the smaller and the larger teeth are 12/24 or 1:2. 9 The first number in the ratio is generally the gear that power is applied to. The ratio also
means that for one revolution of the smaller gear, the larger gear has made 1/2 or 0.50 revolutions. This further implies that the larger gear turns slowly as compared to the smaller one.
ABOUT GEARBOX
5.1 Gearbox Glossary
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9 ANSI: American National Standards Institute. 9 Addendum: It can be defined as the radial or perpendicular distance between the pitch
circle and the top of the teeth. 9 Alignment: Accurate alignment of shafts on which gears are mounted is an important
factor in gear life and performance. Shafts are set parallel in spur and helical gearboxes,
and perpendicular in most bevel and worm gearboxes. D is-alignment of shafts could lead to premature gear failure or other performance issues including noise. 9 BrinellHardness Number (BHN): It is a measure of the hardness of a material such as
steel. 9 Backlash: It is the extent by which the width of a tooth space exceeds the thickness of
the engaging tooth on the pitch circles. 9 Backlash Variation: It can be defined as the difference between maximum and
minimum backlash occurring in a complete revolution of the larger of a pair of mating gears. 9 Bevel Gears: These are conical shaped gears designed to operate on intersecting axes. In
most gearboxes the shafts will intersect at a 90o angle. Straight bevel gears and spiral bevel gears are the two common types. 9 Bore: It is the diameter of the hole in a sprocket, gear, bushing, etc. 9 Bottom Diameter: Also known as a root diameter, it is the diameter of a circle measured
across the bottoms of opposite tooth spaces. 9 Bull Gear: A bullgear is the larger gear in two or more gear set, the smaller one, known
as a pinion. 9 Burning: Cutting a steel plate with the help of a torch. 9 Bushing: A mechanical device/tool used for mounting a sprocket or gear on a shaft. 9 CD: Center Distance. 9 CP: Circular Pitch. 9 Distance (CD): It is the shortest distance between non-intersecting axes of engaged
gears. 9 Circular Pitch (CP): It is the distance along the pitch line between corresponding
profiles of adjacent teeth. 9 Circular Thickness: It is the thickness of the tooth on the pitch circle. 9 Coupling Sprockets, Chains, and Covers: A product line used for connecting two non-
continuous shaft ends. 9 DP:Diametral Pitch. 9 Dedendum (DED): It is the radial or perpendicular distance between the pitch circle and
the bottom of the gullet. 9 Diametral Pitch (DP): It is the ratio of the number of teeth to the number of inches in
the pitch diameter.
5.2 Gearbox Materials 9 A range of gearboxes are constructed from a variety of materials depending on the
industry or the product in which they are being used for. Finest quality materials are used to manufacture gearboxes for ensuring reliability, ease of maintenance and long life. The specialty gearboxes materials undergo vibration and endurance test to ensure that the end product is of premium quality. ¾ Aluminum Gearbox ¾ Cast Iron Gearbox ¾ Bronze Iron Gearbox ¾ Stainless steel Gearbox
5.3 Types of Gearbox 9 A variety of gearboxes are manufactured from different superior quality materials and
with different performance specifications depending on their industrial application. These gearboxes are available in a range of capacities, sizes and speed ratios, but the main function is to convert the input of a prime mover into an output with high torque and low RPM. A variety of gearbox find application in a large number of industries including agriculture, aerospace, mining, paper and pulp industry.
9 Some of the popular types of gear boxes in use are as follows
¾ Bevel Gearbox ¾ Helical Gearbox ¾ Planetary Gearbox ¾ Sequential Gearbox ¾ Spiral Bevel Gearbox
¾ Worm Reduction Gearbox ¾ Cycloidal Gearbox ¾ Offset Gearbox ¾ Right Angle Bevel Gearbox ¾ Shaft Mounted Gearbox ¾ Worm Gearbox ¾ Crane Duty Gearboxes
SIMPLE DIFFERENTIAL GEARBOX
6. Simple differential Gear box
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Simple differential Gearbox
6.1 Introduction of Simple differential Gear box 9 Bevel gearboxes are special speed reducers with their shafts lying perpendicular to each
other and therefore used mainly in right-angle applications. The gearbox is a kind of right angle gear and is suitable for a right angle solution with a low ratio. These gearboxes save more energy as compared to worm gears and are available in varying gearratios.
6.2 Materials Used 9 These gearboxes are constructed from a variety of materials. Some of the popularly used
materials are : ¾ Cast Iron ¾ Aluminum Alloy ¾ Steel 9 The ratio of a bevel gearbox can be determined by dividing the number of teeth in the
larger gear by the number of teeth in the smaller one. These gearboxes generate varying level of torque and can also be customized to suit individual requirements.
6.3 Types of Bevel Gearboxes
9 Straight Bevel Gearboxes : The gearbox has straight and tapered teeth, and are also the
easiest to manufacture. These gearboxes are mainly used for low speed applications. 9 Spiral Bevel Gearboxes : The gearbox has curved and oblique teeth. These gearboxes
are mainly used for high performance and highspeed applications.
6.4 Use of Bevel Gearboxes 9 Bevel gearboxes are used in diverse fields including : ¾ Printing Presses ¾ Newspaper Conveyors ¾ Material Handling ¾ Paper Converting ¾ Conveyors ¾ Food Processing ¾ Rubber Processing ¾ Wrapping Machines
DESIGN OF SIMPLE DIFFERENTIAL GEARBOX
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7.1Design of Simple differential gear box
Given Data Input Shaft Power = 10KW Speed Of Input Shaft N p = 1500rpm Speed Of Input Shaft N g = 500rpm
Gear Material
For Pinion, DirecteHardeing Steel C40 Tensile Strength σ p= 620 N / mm 2 Hardness = 175 BHN Bending Strength σ bp =
= 206.67
N / mm
For Gear, Cast iron Grade 25 Tensile strength σg = 245 N / mm
2
Hardness = 245 BHN Bending Strength σ bg = Teeth on pinion Z p = 14
Gear Ratio I =
= =3
Calculate Teeth On Gear
i=
3= Zg = 42
=81.67 N / mm2
2
From the No of Teeth Select Pressure Angle
Pressure angle = 20 ̊
Select Module m = 5
Now Calculated the diameter of Gear & Pinion
Dg = m × zg =210 mm So, r g = 105 mm
Dg = m × z p = 70 mm So, r p = 35 mm
Center Distance A0 = = 110.7 mm
Width b = 0.25 × A0 = 27.67 mm
tan r = =
r = 71.56 ̊
tan γ =
γ = 18.43 ̊
Mean Radius for Pinion r mp =
-
= 30.62 mm
Mean Radius for Gear r mg =
-
= 100.62 mm
Torque transmission on Pinion
Mtp = = 63661.97 Nmm
Torque on Gear
Mtg =Mtp × = 190985.93 N.mm
Z p = = 214.75
Zg = = 132.85
Materials are different and hence the product of σ b Y should be compared For full depth involute profile Y = 0.484 –
Y p = 0.484 –
= 0.29
Yg = 0.484 –
= 0.462 Now, σ bg × Yg = 37.73 N / mm σ bp × Y p = 59.93 N / mm
2
Here, σ bg × Yg<σ bp × Y p
Gear is Weak
Sw =
= 3775.67 N Now, Calculate tengetial force Pt, Mtp = Pt = 1818.91 N
Velocity, V =
= 5.49 m / s Cv =
= 6.7050
2
Now calculate effective force,
Peff =
here , K s = Multiplying factor = Cs × K m Peff = = 3870 N
Check For safety, S b ≥ Peff ×Fos 8474.79 ≥ 3870 N
Hence the design is safe.
Pitch Cone Diameter Number of Teeth Module Pitch Angle Cone Distance Face Width Addenda Dedenda Clearance
Pinion D p = 70mm Z p = 14 m = 5 ϒ
= 18.43® A0 = 110.7mm b = 27.67 mm ha = m = 5mm hf = 1.2m = 6mm c =0.2m = 1mm Table-1
Gear Dg = 210mm Zg = 42 m = 5 ┌ = 71.56®
A0 = 110.7mm b = 27.67 mm ha = m = 5mm hf = 1.2m = 6mm c =0.2m = 1mm
7.2 Design of Bearings 9 Here I had used a two angular contact deep grew ball Bearings, one on each side are
utilized at the bevel shaft.
Deep groove ball bearing
7.2.1 Selection Procedure of Angular Contact Ball Bearing
The Torque produced at the Worm shaft is as found out earlier P= 10 ×
=
T = 63662 N.mm Tangential load Pt =
=
= 1818.91 N
From Pt Axial load, Pa = Pt ×
= 6318.33 N Radial load, Pr
=
Pt ×
Sinφ n Cosφn Sinλ + µCosλ
= 1732.18 N Taking moment about the bearing, For, Vertical Plane R v = 100 × 6318.34 N = 631.83 KN
For Horizontal Plane R n = 100 × 1732.18 N = 173.218 KN
Resultant,
R
= 655 KN Here, P = Fr = R = 655 KN
Lifeof thebearingL 10 :
L10
=
60 × N × Lh10 106
= = 900 million rev. Basic dynamic load rating C:
Let,Factorof SafetybeN f
=
2.5
1/3
C = P × (L 10) ×Nf = 655 × (900)
1/3
× 2.5
=15.8KN
From the above value of Dynamic load calculated and the minimum acceptable diameter (25mm).We can select the angular contact bearing 6304 from below table.
Bearing No 6300 6301 6302 6303 6304 6305 6306 6307 6308 6309 6310 6311 6312 6313 6314 6315 6316
Principal Dimensions Bore ‘d’ Outside Width mm Diameter 10 35 11 12 37 12 15 42 13 17 47 14 20 52 15 25 62 17 30 72 19 35 80 21 40 90 23 45 100 25 50 110 27 55 120 29 60 130 31 65 140 33 70 150 35 75 160 37 80 170 39
Basic Capacity Static Dynamic ‘C0’ kN ‘C’ kN 3.40 8.06 4.15 9.75 5.40 11.40 6.55 13.50 7.80 15.90 11.60 22.50 16.00 28.10 19.00 33.20 24.00 41.00 31.50 52.70 38.00 61.80 45.00 71.50 52.00 81.90 60.00 92.30 68.00 104.00 76.50 112.00 86.50 124.00 Table - 2
Permissible rpm Grease Oil Lubrication Lubrication 20,000 26,000 19,000 24,000 17,000 20,000 16,000 19,000 13,000 16,000 11,000 14,000 9,000 11,000 8,500 10,000 7,500 9,000 6,700 8,000 6,300 7,500 5,600 6,700 5,000 6,000 4,800 5,600 4,500 5,300 4,300 5,000 3,800 4,500
7.3 Design for Shaft
Design of Shaft
General Considerations of Shaft Design 1. To minimize both deflections and stresses, the shaft length should be kept as short as possible and overhangs minimized.
2. A cantilever beam will have a larger deflection than a simply supported (straddle mounted) one for the same length, load, and cross section, so straddle mounting should be used unless a cantilever shaft is dictated by design constraints. (Figure 9-2 shows a situation in which an overhung section is required for serviceability.) 3. A hollow shaft has a better stiffness/mass ratio (specific stiffness) and higher natural frequencies than a comparably stiff or strong solid shaft, but will be more expensive and larger in diameter. 4. Try to locate stress-raisers away from regions of large bending moment if possible and minimize their effects with generous radii and relief. 5. General low carbon steel is just as good as higher strength steels (since deflection is typical the design limiting issue). 6. Deflections at gears carried on the shaft should not exceed about 0.005 inches and the relative slope between the gears axes should be less than about 0.03 degrees. 7. If plain (sleeve) bearings are to be used, the shaft deflection across the bearing length should be less than the oil-film thickness in the bearing. 8. If non-self-aligning rolling element bearings are used, the shaft’s slope at the bearings should be kept to less than about 0.04 degrees.
9. If axial thrust loads are present, they should be taken to ground through a single thrust bearing per load direction. Do not split axial loads between thrust bearings as thermal expansion of the shaft can overload the bearings. 10. The first natural frequency of the shaft should be at least three times the highest forcing frequency expected in service, and preferably much more. (A factor of ten times or more is preferred, but this is often difficult to achieve).
Combine Failure of shaft
Keys & Keyways
Keyway of shaft
Grade En 8 Steel Material Type of Steel : Constructional Steel 080M40 9 Taking moment about the bevel pinion
M1 = M1 = 245.25 Nm 9 Taking moment about the bevel Gear
M2=
M2 = 392.4 Nm 9 Considering maximum bending moment i.e. M 2
Now, the average shear stress and bending stress acting on the shaft are =
= =
= Now, For En 8 material Syp= 620 MPa Se = 300 MPa
9 For rotating shaft Sodenberg’s equation as per Fatigue criteria,
Let us take
= 1.5,
d = 214 mm. for shaft 2, 9 Here the M2 of shaft 1 is same for M 1 of shaft 2,
M1 = 392.4 Nm 9 As per shaft 1 calculate M 2
M2 = 145.6 Nm 9 Considering maximum bending moment i.e. M 1
Now, the average shear stress and bending stress acting on the shaft are =
=
=
= Now, For En 8 material Syp= 620 MPa Se = 300 MPa
9 For rotating shaft Sodenberg’s equation as per Fatigue criteria,
Let us take
= 1.5,
d = 75 mm.
Table for selecting length and key way Diameter (d) Tolerance Length(L) 16 j6 28 18 19 20 j6 36 22 24 25 j6 42
Key Way 3×3 4×4 4×4 4×4 4×4 5×5 5×5
t 1.8 2.5 2.5 2.5 2.5 3 3
28 30 32 35 37 40 42 45 48 50 55 56 60 63 65 70 71 75 80 85 90 95 100 110 120 125 130 140 150 160 170 180 190 200 220
k6
58
k6
82
m6
105
m6
130
m6
165
m6
200
m6
240
m6
280
5×5 5×5 6×6 6×6 6×6 10 × 8 10 × 8 12 × 8 12 × 8 12 × 8 14 × 9 14 × 9 16 × 10 16 × 10 16 × 10 18 × 11 18 × 11 18 × 11 20 × 12 20 × 12 22 × 14 22 × 14 25 × 14 25 × 14 28 × 16 28 × 16 28 × 16 32 × 18 32 × 18 36 × 20 36 × 20 40 × 22 40 × 22 40 × 22 45 × 25
3 3 3.5 3.5 3.5 5 5 5 5 5 5.5 5.5 6 6 6 7 7 7 7.5 7.5 9 9 9 9 10 10 10 11 11 12 12 13 13 13 15
From the table length of shaft 1 is L 1 = 105 mm and length of shaft 2 is L 2 = 280 mm. Dimensions and basic capacity of single row deep groove ball bearings
A NALYSIS OF GEARBOX
8 Analysis of Gearbox 8.1 Analysis of Gearshaft Choose beraing D as the origin At point m the components of forces can be written by the vector P g Pg = Pgx i + Pgy j + Pgz k Pg= -Prg i - Pag j + Ptg k
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Negative sign indicate that the force is in negative direction of the axis Pg= -239.24 i – 719.92 j + 2079.1 k
Similarly assuming bearing reaction at C as F cx, Fcy,Fczwe get Fc = Fcx i + Fcy j +Fczk
Also reaction at bearing D will be F dx&Fdz (Fdy=0)
Fd = Fdx i + Fdzk
Besides the forces there is a torque acting on the shaft about y-axis.Assuming this torque anticlockwise
T = Mtg j Convention used Anticlockwise Clockwise
+ ve
- ve
Once the forces & torques are define their position have to be decided R m = 100.62 i+ (-60-30.62) j+ 0 k
Where R m gives the position of point m about origin R m = 100.62 i – 90.62 j Similarly,
R c = 0 i+ (-60-90) j+ 0 k R c = -150 j
For any given system to be in equilibrium,their moments about any point should be zero Taking moment about D we get:
R m ×Pg + Rc × Fc + T= 0
R×P= = i(R yPz – R zPy) - j(R xPz – R zPx) + k(R xPy – R yPx)
R m × Pg = = - 187064.39 i- 209438.28 j- 94118.27 k
R c × Fc = = - 150
I - 0 j+ 150
k
Substituting we get , (-187064.39 – 150 F cz)i + (-209438.28 + M tg)j + (-94118.27 + 160 F cx)k = 0 Equating we get Fcz =
= -1247.09 N Mtj = -209438.28 N.mm
Fcx = = 627.455 N
Also summation of all the forces must be zero
Fc +Fd + Pg= 0 627.455 i + F cy j – 1241.09 k + f ox i + 0 j + F dz k – 2392.24 I – 719.92 j + 2079.1 k = 0 (Fox +388.215) I + (F cy-719.92) j + (F dz +832.06) k = 0 Fox = -388.215 N Fcy = 719.92 N Fdz= -832.01 N
The reaction at C & D a Fc = 627.455 i + 719.92 j -1247.09 k Fd = -388.215 I – 832.01 N Analysis of pinion shaft P p = Pap i + Pfp j – Ptp k PP = 239.24 i +719.92 j -2079.1 k Fa = Fax i + Fay j + Faz k F b = F by j + F bz k T = Mtp i (assume anticlockwise direction when seen from bearing B) Position of these four R a = -90 i R m = -(90 + (170 – r mg )) i + (-30.62) j r mg = 100.62 R m = -160.602 i – 30.62 j Taking moment about B, we get R a ×Fa + R m × P p + T= 0
R a × Fa = = - j(-90 Faz) + k (-90 F ay) = - 90 Faz j - 90 F ay k
R m × P p = = 62942.12 i – 333668.38 j – 108295.06 k Substituting ( Mtp + 62942.12) i + (90 F az - 333668.38) j – (90 F ay - 108295.06) k = 0 Mtp = 62942.12 N.mm (Clockwise when seen from bering B or anticlock when seen from A)
Fcz =
= 37070.42 N
Fcx = = -1203.27 N Summation of all the forces must be zero
Fa +F b + P p= 0 Fax i – 1203.27 j + 3707.42 k + 0i+ F b j + F bz k + 239.24 i + 719.92 j - 2079.1 k = 0 (Fax +239.24) i + (F by-483.35) j + (F bz +1628.32) k = 0 Fax = -239.24 N F by = 483.35 N F bz= -1628.32 N The reaction at A & B are Fa = 239.24 i + 1203.27 j - 3707.42 k F b = 483.35 j - 1628.32 k (indicates the corrected directions).
BIBLIOGRAPHY
9 9 (2005) About gearbox[online]. Available: http://www.gearsandgearbox.edu/ 9 (2009) introduction [Online]. Available: http://en.wikipedia.org/ 9 (2002)bearing design[online]. .Available: http://www.bearingdesign.org 9 (2009) shaftdesign on Wikipedia. [Online]. Available: http://en.wikipedia.org/ 9 (2007) gearsrating [Online]. Available: http://everongearbox.com/
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R EFERENCES
9 Machine design – 2 by R.B Patil 9 Transmission System Design by FarazdakHaidari 9 Design of Machine Elements by V.B Bhandari 9 Design Data Book of Engineers
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