Chapter 10 Shafts.pdf

11/20/2023 1 20 Nov 23 CHAPTER 10 SHAFT 1 THEORY OF MACHINE AND MACHINE DESIGN DEPARTMENT OF EDUCATION AND TRAINNING HOCHIMINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION TS Phan Công Bình binhpc@hcmu[.]

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CHAPTER 10: SHAFT

1

THEORY OF MACHINE AND MACHINE DESIGN

DEPARTMENT OF EDUCATION AND TRAINNING

HOCHIMINH CITY UNIVERSITY OF TECHNOLOGY

AND EDUCATION

TS Phan Công Bình binhpc@hcmute.edu.vn binhpc.tpm@gmail.com

Outcome

1 Getting the overview of shaft

2 Understanding the mechanical properties of shaft

3 Calculating and design shaft

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Theorical contents

I Overview

II Basic criteria of calculation

III Fundamentals of calculation and design

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TS Phan Công Bình binhpc@hcmute.edu.vn binhpc.tpm@gmail.com 20-Nov-23

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Overview Coupling component

- Shaft ( Trục )

- Bearing ( Ổ trục )

- Couplings ( Khớp nối )

Most of mechanical machinery and equipment contain shaft

Rotating component

Pulley, gear, sprocket,…

assemble with shaft

Almost shafts are used to support and transmit torque

1 Application

Overview

a Supporting component ( bending )

+ Shaft with a neutral wheel

+ Shaft assemble with neutral gear

b Transmitting torque ( twist )

+ Driving shaft

+ Transmitting shaft

Overview

a Based on load

+ Axles ( Trục tâm ) are subjected only to bending

+ Shaft ( Trục truyền ) are intended not only to support revolving parts but also to

transmit torque

2 Classification

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Overview

a Based on load

+ Parallel drive ( trục truyền chung ) are applied to transmit torque to multi-component

at the same time

Parallel drive

Overview

b Centerline ( đường tâm )

+ Straight shaft ( Trục thẳng )+ Crank shaft ( Trục khuỷu )+ Flexible wire shaft ( Trục mềm )

Crank shaft

Straight shaft

Flexible wire shaft

(Trục mềm)

Overview

c Structure ( Kết Cấu )

+ Plain shaft: constant diameter ( Trục trơn )

+ Step shaft: Changing diameter ( Trục bậc )

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Overview

d Cross-section ( Tiết diện )

+ Solid shaft (Trục đặc)

+ Hollow shaft ( Trục rỗng)

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3 Shaft material

Technical requirement:

 Reducing the stress concentration (giảm tập trung ứng suất)

 Manufacturing and Maintenance Technology (có tính công nghệ)

* Common carbon steel:

CT3, CT5, C30, C40, C45, C50

Common material :

* Common alloyed steel:

40CrNi, 40CrNi2MoA, 30CrMnTi, 30CrMnSiA

20Cr, 12CrNi3A, 18CrMnTi

C45 has 0.45% carbon the most common steel in the world

 Carbon steel (thép carbon)

 Alloyed steel (thép hợp kim)

Overview

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4 Structure

Requirements of shaft structure

a Journal (ngõng trục): Using to assemble bearings

b The clamped parts (thân trục): Used to mount

the rotating component

c Transitions location: Between 2 different diameter on

shaft

d Another surface: Key sitting ( rãnh then ), fillet

( góc lượn ),…

+ Strength: Satisfying the technical requirements

+ Technology: Ability of processing disassemble, assemble and

maintenance

+ Reduce stress concentration : Fillet suitable to structure

t p d

Overview

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5 Pattern failures and Design method

a Break (gãy)

Overload or fatigue

Strength

validation

Overview

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b Deformation (biến dạng)

rigidity

Overview

Not enough to rigidity (Không đủ độ cứng)

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c Vibration (dao động)

Vibration

Rotating Component is off centerline (lệch tâm)

Overview

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I Overview

II Basic criteria of calculation

III Fundamentals of calculation

and design

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Basic criteria calculation of shaft

1 Calculating strength

Sequence of calculation and design

a Preliminarily calculation (Tính sơ bộ)

b Calculating strength (Tính sức bền tĩnh)

c Examining (Tính kiểm nghiệm)

- No shaft structure

- According to torque (twist only)

- Calculating diameter preliminarily

- Selecting bearing preliminarily, sketch out ( phát thảo ) shaft structure

- Determining the acting force (torque and bending) and Reactive force

- Calculating exactly shaft diameters at critical cross sections and selecting

parameter based on standard)

- Design shaft structure

- Examining overload (kiểm nghiệm quá tải)

- Examining strength and fatigue (kiểm nghiệm độ bền mỏi)

- (chỉ tính momen xoắn)

- (Tính đường kính sơ bộ)

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a Preliminary calculation (Tính sơ bộ)

′

where, T is a torque acting on shaft, N.mm

Selecting material and allowable torque [ ′], MPa

=> The preliminary shaft d, mm

Selecting preliminary bearing and sketch out shaft structure

Vị trí xác định đường kính:

+ Đầu trục đối với trục vào và ra + Thân trục lắp bánh dẫn với trục trung gian

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b Calculating strength

- Determining all forces acting on shaft

- Drawing the diagram of bending or torque moment

- Obtaining exactly diameter of shaft by using the IV strength theory

- Completing the shaft structure

- Calculating the support reactions ( Phản lực ) at pins ( Gối đỡ )

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Based on IV strength theory ( Thuyết bền 4 )

0.1

0.2

The value Mtđdetermining by formula

đ =

đ

đ 0.1

- Diameter of journal must be standardized in (TLTK)

b Calculating strength

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Examble:

Design shaft I with the parameter in picture Material C35 has allowable twist

torque [ ′]=20 Mpa and =20 Mpa

Belt force acting on shaft Fđ= 791.8N and

Force acting on gearing Fr1= 1178.9N,Ft1= 3239N

Ft1=3239N

O

y

z x

T1=181385Nmm

n1=298v/p

P1=5.66kW

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f=90 l=145

+ Calculating diameter preliminary

≥ = 35.6 mm;

Selecting diameter based on standard d = 36 mm,

- Selecting diameter of shaft based on standard

+ Sketch out shaft structure

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Selecting djaccording standard

+ Determining the equivalent moment at cross-section j Mtđ

+ Calculating shaft diameter at cross-section

0.1

+ Completing structure

Ability of processing disassemble, assemble and maintenance

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c Examining

• Examining over load

where,

=

+ ≥ [ ]

• Examining strength and fatigue

, bending and torque at danger cross − section

, ℎ allowable stress when overload and elastic ( chảy )

safety factor 1.5~2.5 , safety factor of bending and torque

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2 Calculating rigidity.

a Rigidity Condition of rigidity:

+ [f] = 0,01m – Shaft assemble with cylindrical gears

+ [f] = 0,005m – Shaft assemble with bevel gears

+ [] = 0,01rad - Thrust bearing ( ổ bi đỡ )

+ [] = 0,05rad- Ball bearings ( ổ bi lòng cầu )

+ [] = 0,001rad – Friction bearing ( ổ bi trượt )

In machine manufacturing, we can be selected : [f] =(0,0002 ÷ 0,0003)I

where, l: Pin distance.

F

f

f ≤ [ ]

≤ 

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b Torsional deflection

Basic criterial of Torsional deflection of shafts:

[ ]: allowable angel of twist ( góc xoắn cho phép ), rad

G: shear modulus module ( trượt đàn hồi ) G=8,3.104,MPa

J0: moment of inertia in torsional ( moment quán tính ) (J0= /32)

l: length of twist ( chiều dài đoạn trục xoắn ), mm

where,

2 Calculating rigidity.

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3 Calculating oscillation.

a Cause

- Off centerline generate external

forced generate vibration ( dao

động )

b Damage

- Making additional stress ( ứng suất phụ )

=> Effecting on strength

- Resonance ( vùng cộng hưởng )

=> Breaking shaft

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c Criteria of calculation

- Calculating amplitude ≤ [ ]

- Determining resonance ( vùng cộng hưởng )

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d Oscillation problems

- According the off-centerline m

=

= 48

where,

( ) =

f = = (natural frequency) → ∞ ( )

The rigidly of shaft (TLTK)

l/2 l/2

Flt

I Overview

II Basic criteria calculation of

shaft

III Fundamentals of calculation

and design

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Fundamentals of calculation and design

Notice:

1 Analyzing acting force in 2 plane (coordinates)

2 Calculating support reaction at pins on shaft

3 Drawing moment diagram (From left to right)

4 Determine diameter at critical cross-section

Bending Fa =

Torque Ft = ∗ = ∗

Moment

Sequence of calculation

O

y

z x

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1 Calculating force acting

a Specifying gears parameter

- Determining the pitch diameter by formula :

+ Spur gear : d = mz1 (mm)

+ Helical gear: d = mnz1/cosβ (mm)

+ Bevel gear: d = mmz1(mm)

where,

m – module

z – Number of teeth on gear

β – Helix angle

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b Obtaining the acting Force

- Radial force Fr:

+ Chain: (N)

α

 20sin(1/ 2)

r

F k F k P zpn

- Tangential force on coupling Ft: Ft= 2T/Dkn(N)

where,

+ Dkn– coupling diameter (đường kính khớp nối) (mm)

+ T – twist torque (N.mm)

The radial acting force on shaft Frdue to misalignment can be calculated by

1 Calculation acting force

 (0.2 0.3) 

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2 Analysis the acting force

a Force acting diagram

- The Figure describes force acting diagram of transmission

x

+

Ft1

Fr1

x

+

X

Fr2

Ft2

z

Fkn

Fxy

Fxx

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b Drawing moment diagram

3 Determining diameter of shaft

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+ L1= L2= L3= 125mm

+ Ft1= 500N, Fr1= 182N

+ Ft2= 800N, Fr2= 291N

+ T = 50000 N.mm

Reaction force at pin A and D:

+ RAY= 24.3 N, RDY= 133.3 N

+ RDX= 700 N, RAX= 600 N

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Exercise

Q 1 Given the shaft of transmission system, Torque on shaft T = 100000 Nmm.

Helical gears, mn= 2 mm and helix gear β = 12o The number of teeth on gearing:

Z1= 22 and Z2= 67 The yield stress of shaft material is [σF] = 60 MPa.

Calculating the force acting on shaft?

β



1

cos

44,98

n

m Z

d

mm

1

2

4446

t

T

tan 1655 cos

β

2

cos

n

m Z

d

2

1459,9

t

T

cos

β



2 2

2

t

T

F

d Fr2 Ft2tan α Fa2 0

Spur gears

Helical gears

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Allowable stress = 50

Q 2

Exercise

a.Determining the reaction support at pin B and C

b.Drawing moment diagram , , T

c Determining diameter at cross-section

Tangential force = 1000 and radial force = 364

Tangential force = 2500 and radial force = 910

O

y

z

x

The reactive force at pin B and C:

= − ∗ + − ∗ + = 0

= − + + − = 0

 = + − = 1674,3 + 364 − 910 = ,

 = − + + = −3400 + 1000 + 2500 =

2.a

20-Nov-23

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Exercise

Q 2

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2.b

Q 2

Exercise

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2.b

Q 2

Exercise

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Diameter at danger cross-section C:

Equivalent moment at C:

= 136489 + 375000 + 0,75 ∗ 200000

=

Diameter of shaft at cross-section C:

For assembling bearing at C we choose dc= 45 mm

2c

Q 2

Exercise

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Given the working shaft in picture 3

The helical gear parameter

= 240 , = 5000 , = 1885 ,

= 1340

The spur gear parameter

= 120 , = 10000 , = 3640

Length

a Determining the reaction support at pin B and C b.Drawing moment diagram , , T

c Determining the diameter of shaft at danger cross-section C

Q 3

Exercise

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Determining the reaction support at pin B and C

+ The moment equation at B in Y direction :

+ The moment equation at B in X direction :

Q 3

Exercise

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Ma1

4.b

Q 3

Exercise

In yoz plane: Fror Ma

In xoz plane: only Ft

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+ Equivalent moment at C:

đ = + + 0.75

= 440100 + 225000 + 0,75 ∗ 600000 =

Diameter of shaft at cross-section C:

≥ đ

0.1 ≥ .

For assembling gear at C, we choose: = ( )

Bonus:

+ Equivalent moment at B:

đ = + + 0.75

= 436764 + 1200000 + 0.75 ∗ 600000 =

Diameter of shaft at cross-section C::

≥ đ 0.1 = ,

For assembling bearing at B, we choose: = ( )

Q 3

Exercise

The bevel gear 1: = 200 , 1= 1000 , 1= 163 , 1= 325 ,

and helical gear Z2 2= 160 , 2= 1250 , 2= 465 , 2= 486

Distance 1= 100 , 2= 200 , 3= 100

The allowable bending stress [ ] = 60

a Determine the reaction support at pin A and D.

b Drawing the moment diagram Mx, Myand T.

c Determining the diameter of shaft at danger cross-section B.

Q 4

Exercise

Q 4

Exercise

Tiêu đề	Chapter 10: Shaft
Tác giả	TS. Phan Công Bình
Trường học	Hochiminh City University Of Technology And Education
Chuyên ngành	Theory Of Machine And Machine Design
Thể loại	chapter
Năm xuất bản	2023
Thành phố	Hochiminh City

Định dạng
Số trang	18
Dung lượng	1,19 MB