Chapter 8 Gear Train.pdf

Microsoft PowerPoint Chapter 8 GEAR TRAIN 28/10/2021 1 TS Phan Công Bình binhpc@hcmute edu vn binhpc tpm@gmail com Theoretical contents I Overview II Classification III Transmission Ratio IV Problem 1[.]

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

I Overview

II Classification

III Transmission Ratio

IV Problem

28-Oct-21

TS Phan Công Bình binhpc@hcmute.edu.vn binhpc.tpm@gmail.com 28-Oct-21

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The gears are matching together in series or parallel

Transmitting and distributing motion

Increasing or decreasing rotation speed

Large ratio and far centre distance

1 Overview

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2 Classification.

Gear drives system

(Planar, universal) Gear drives system

(Planar, universal)

Open Differential Open Differential Planetary differentialdifferentialClose Close

Combination

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TS Phan Công Bình binhpc@hcmute.edu.vn binhpc.tpm@gmail.com 28-Oct-21 28-Oct-21

Centerlines of all gears are fixed

a Simple gear drive system (hệ thường)

2 Classification.

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Epicyclic gear system has at least 1 gear’s centerline motion

b Epicyclic gear system (hệ ngoại luân)

2 Classification.

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Degree of freedom:

W = 3𝑛 − 2𝑝 − 𝑝4 = 3.3 − 2.3 − 2 = 1

Z1 C

Z’2

Z3

Z2

C Z2

Z1

Z3 Z’2

b Epicyclic gear system

Planetary gear (hành tinh)is at least 1 fixed center gear

The gear that the centerline is in the main axis of the system is so-called center gear

2 Classification.

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Open differential: (Vi sai hở) All center gears are moveable

2 Classification.

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Degree of freedom:

W = 3𝑛 − 2𝑝 − 𝑝4 = 3.4 − 2.4 − 2 = 2

Z1 C

Z’2

Z3

Z2

Open differential All center gears are moveable

C

𝒁𝒃

𝒁𝒂

Z1

Z’2

C

Z2

Z3

Za

Zb

2 Classification.

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Close differential (vi sai kín) 2 in 3 kinematic links (C, Z1, Z3) are engaged

each other

Z1

C

Z’2

Z3

Z’4

Z2

Z’1

Z’3

Z4

Degree of freedom:

W = 3𝑛 − 2𝑝 − 𝑝4 = 3.5 − 2.5 − 4 = 1

Z1

Z’2

C

Z2

Z3

Z4

Z’3

Z’4

Z5

2 Classification.

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Simple gearing coupling with a epicyclic (planetary, open differential, close

differential)

C Z’2

Z2

Z’3

Zd

Z3

Z1

Zb

Za

c Combination gear system

2 Classification.

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(+): Inside engage (𝜔 ↑↑ 𝜔) (−): Outside engage (𝜔 ↑↓ 𝜔)

• If 𝑢 < 0 gear 1 and n gear are opposite direction

• If 𝑢 > 0 gear 1 and n gear are same direction

For Universalgear (bevel gear) Determining (+) (-) in reality

a Simple gear system

Ratio of 1 gearing

Ratio of simple gear system

3 Transmission ratio

ω

 1 1  2

12

u

ω



n

Z n

u

(−1)kPlanar where,kis number of outside engage

±

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Z1

Z2 Z’2

Z3

Z’3

Z4

Planary

Universal

Given:

Z1 = 20 Z2 = 40 Z’2= 20

Z3 = 50 Z’3= 40 Z4 = 20 Calculating u13?

Z1

Z’2

C

Z2

Z3

Z4

Z’3

Exercise

a Simple gear system

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• If arm Cis stationary, angular velocity of each link will be −𝜔

b Epicyclic system

theangular velocitiesof

the various gears in the

system is taken by

transforming the arm C

into stationary observer

𝐾ℎâ𝑢 1 𝑍1 𝜔 − 𝜔

𝐾ℎâ𝑢 2 (𝑍2 & 𝑍′2) 𝜔 − 𝜔

Kℎâ𝑢 3 (𝑍3) 𝜔− 𝜔

𝐾ℎâ𝑢 4 (𝑪): 0

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In absolute system 𝑢 = −1 = 5

0

b Epicyclic system

Planetary gear

Z1

C

Z’2

Z3

Z2

Given: Z1=20; Z2=60;

Z’2=30; Z3=50

13/

3 1 c C



1

c



(Arm C and Z1are oppositedirection)

In reference frame C

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Z1

C

Z’2

Z3

Z2

Given: Z1=20; Z2=60; Z’2=30; Z3=50

ω =8 rad/s ω =1 rad/s

Z and C are the opposite direction Calculating:

ω =? rad/s and direction Z3?

Open differential

b Epicyclic system

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In reference frame C

(Z3 and Z1are the same direction)

Open differential

b Epicyclic system

1 13/

3

c C c







13

u



In absolute system

1 3 4 5 c

Assuming direction Z1+

Or C

-𝑢 / = 𝑢 where,

  

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Z1

C

Z’2

Z3

Z’4

Z2

Z’1

Z’3

Z4

Given: Z1=20; Z2=60; Z’2=20; Z3=40 Z’1=20; Z’4=40; Z4=20; Z’3=40 ω =8 rad/s

Calculating ω =? rad/s

Close differential

In reference frame C 𝑢 / =

Z . Z Z′ = 6

ω − ω = 6 (ω − ω ) =>ω = Close differential: 𝑢 = = −1 Z’4Z’1 Z’3Z

=>ω =+ ;nên ω = + = + rad/s

b Epicyclic system

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𝑈 = 𝑈 𝑈

𝑈 =

𝑈 / =ω − ω

ω − ω = 1 −

ω ω

𝑈 = (−1)𝑍 𝑍

𝑍′ 𝑍 =

−𝑍 𝑍′

Determining the sign ( + or -)

following the ‘’k’’ indication

𝑈 = −𝑍𝑍

Simple gear Planetary gear

c Combination

Z1

Z2

Z4

Z3

Z’2

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Problem Solution Method

1 Determine kind of gear train

Planar: based on pair of external gearing k

Space: based on realistic schematic diagram

2 Epicyclic gear train

1

13/

3

C







  2 3

'

1 2

Z

Z Z

Planetary:ω = 0

Direction (the same or opposite)

Given 2 of 3 speed C, Z1, Z3

Close differentialClose differential equation

3 Combination Solve the Epicyclic

Determine the remained variable in Simple train Note: The notation must be based on the realistic schematic diagram

Open differential

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Q 1.

𝐴𝑙𝑙 𝑔𝑒𝑎𝑟 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑚𝑜𝑑ule

𝐻𝑒𝑟𝑒, Z = 2Z , Z = 20, Z = 40, Z′ = 35, Z = 70 teeth

Gear Z has n = 120 𝑟𝑝𝑚 , n = 70 𝑟𝑝𝑚 ,

𝑍 𝑎𝑛𝑑 𝑙𝑖𝑛𝑘 𝐶 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛

D𝐞𝐭𝐞𝐫𝐦𝐢𝐧𝐞 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑛 𝑎𝑛𝑑 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑍

Problem

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Solution Q1.

Problem









1 13/

3

'

1 3

3 (1) 4

C C

C

n n u

n n Z Z

Z Z

n

Epicyclic gear

Simple gear     

 1  

2 (2) 2

ab

a b

u

n

n n

 

 3 2 3 (1) & (2)

4

a C

n

Assumption C +

or Za+

(120) 3(70)

4

Z3rotates at 37,5 rpm and the same direction Arm C

Conclusion

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Q 2.

a.Degree of freedom W of gear?

b.Direction of gear?

c.Rotation speed n3of gear Z3?

d.Rotation speed nbof gear Zb?

Given: Z1=20; Z2=80; Z’2=40; Z3=40

Z = 2Z , n = 400 𝑟𝑝𝑚 , n = 200 𝑟𝑝𝑚

Z and C are opposite direction of rotation

Calculate:

Z1

Z’2

C

Z2

Z3

Za

Zb

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Solution 2.

Assume Z1+

C-Za+

Z1

Z’2

C

Z2

Z3

Za

Zb

Problem



1 3

4

C C C C

u

n n n

    

2 (2) 2

ab

a

u

n

5 2 (1) & (2)

4

a

n n n

  3

5(400) (200) 2 4

300 / n

v p

Z3rotates at 300 rpm and opposite direction Z1(or the same Arm C)

Conclusion:

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Q 3.

a.Degree of freedom W of gear?

b.Rotation speed n3of gear Z3?

c.Rotation speed n5of gear Z5?

Z1= Z’2= Z’3= 50, Z2= Z3= 25,

Z4= 100, Z’4= 20, Z5= 40

Rotation speed n1= 800 (rpm) Calculating:

Problem

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Q 4. Z

a= 25, Zb= 50, 𝑍 = 25, 𝑍 = 50, 𝑍′ = 15, 𝑍 = 30

Gear Zaand arm C turn same direction

Rotation speed na= 200 (rpm), nc= 100 (rpm)

Determine the direction and rotation speed n3of gear Z3?

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Q5. Z

1= Z5= Z4= 30, Z2=60, Z3= 80, Z6=140

n1= 960 rpm Determine

a) DOF of the given gear train?

b) nCand rotating direction of Arm C

(compared to direction of Z1)?

IV Problem

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

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Tiêu đề	Gear Train
Tác giả	TS. Phan Công Bình
Trường học	Hochiminh City University Of Technology
Thể loại	theoretical contents
Năm xuất bản	2021
Thành phố	Ho Chi Minh City

Định dạng
Số trang	9
Dung lượng	1,66 MB