workbook theory of machines and …1.2. modelling of machines and mechanisms the completion of a...

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WORKBOOK

THEORY OF MACHINES AND MECHANISM

LUBLIN 2014

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Projekt współfinansowany ze środków Unii Europejskiej w ramach Europejskiego Funduszu

Społecznego

Author: Łukasz Jedliński

Desktop publishing: Łukasz Jedliński Technical editor: Łukasz Jedliński

Figures: Łukasz Jedliński

Cover and graphic design: Łukasz Jedliński

All rights reserved.

No part of this publication may be scanned, photocopied, copied or distributed in any

form, electronic, mechanical, photocopying, recording or otherwise, including the placing or

distributing in digital form on the Internet or in local area networks,

without the prior written permission of the copyright owner.

Publikacja współfinansowana ze środków Unii Europejskiej w ramach Europejskiego Funduszu

Społecznego w ramach projektu Inżynier z gwarancją jakości – dostosowanie oferty Politechniki Lubelskiej

do wymagań europejskiego rynku pracy

© Copyright by

Łukasz Jedliński, Lublin University of Technology

Lublin 2014

First edition

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TABLE OF CONTENTS

1. INTRODUCTION TO THE MOTION SIMULATION APPLICATION IN THE NX 8.5 PROGRAMME......................................................................................................................... 4

1.1. Introduction ......................................................................................................................... 4

1.2. Modelling of machines and mechanisms .............................................................................. 4

1.3. Example .............................................................................................................................. 4

1.4. Reference .......................................................................................................................... 15

2. THE KINEMATIC ANALYSIS OF A FOUR BAR LINKAGE USING THE ANALYTICAL

AND NUMERICAL METHODS ............................................................................................ 16

2.1. The aim of the experiment ................................................................................................. 16

2.2. Theoretical information ..................................................................................................... 16

2.2.1. Analytical method .......................................................................................................... 16

2.3. The course of the study ...................................................................................................... 17

2.4. Reference .......................................................................................................................... 18

3. DETERMINING VELOCITIES AND ACCELERATIONS OF THE PARTICULAR PLANAR

MECHANISMS USING THE PLOT AND NUMERICAL METHOD .................................... 19


3.2. Diagram method ................................................................................................................ 19

3.2.1. Example ......................................................................................................................... 19

3.3. Course of the study ............................................................................................................ 25

3.4. Reference .......................................................................................................................... 26

4. KINEMATICS OF FIXED-AXIS GEARS. SETTING GEAR RATIO, DETERMINING ROTATIONAL SPEED AND ROTATIONAL DIRECTION OF AN OUTPUT SHAFT ........ 27


4.2. Kinematic analysis of a fixed-axis gear .............................................................................. 27


4.4. Reference ........................................................................ Błąd! Nie zdefiniowano zakładki.

5. KINEMATICS OF PLANETARY GEARS. SETTING GEAR RATIO, DETERMINING ROTATIONAL SPEED AND ROTATIONAL DIRECTION OF AN OUTPUT SHAFT ........ 29


5.2. Kinematic analysis of a planetary gear ............................................................................... 29


5.4. Reference ........................................................................ Błąd! Nie zdefiniowano zakładki.

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1. INTRODUCTION TO THE MOTION SIMULATION APPLICATION IN

THE NX 8.5 PROGRAMME

1.1. Introduction

Due to the limited volume of hours dedicated to the subject and, above all, the volume of the

study, the presented information concerning the NX program will contain only the minimum

information necessary for the completion of the course will be presented.

1.2. Modelling of machines and mechanisms

The completion of a simulation model is a process consisting of stages preformed in a particular

order. For a simple mechanism and a basic analysis, after a model has been loaded into the Motion

Simulation application for the first time, the following steps should be taken:

1. Create a new simulation - Simulation command.

2. Specify the non-deformable links on the basis of the loaded model - Link command.

3. Define the motion relations between the links and choose the driving link(s) - Joint and Coupler

commands.

4. Solve the simulation model - Solution command.

1.3. Example

Conduct a simulation model, which makes it possible to carry on kinematic analysis of the

machine, on the basis of the solid model demonstrated in the fig. 1.1. As it is adopted in the machine

construction, the motion of parts should be consistent with their function.

The angle of Shaft_2's rotation during a single rotation of a crankshaft should be examined.

Additionally, one has to measure the angle between the axes of Shaft_2 and Shaft_3, as well as prepare

a graph illustrating the rotational speed of Shaft_3.

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Fig. 1.1. Simplified solid model of the engine and drivetrain

Solution

Developing simulation

The solid model of the machine is being loaded to the program, after the Open command and

the Assembly_engine.prt file have been chosen. Then, the Motion Simulation application has to be

launched by clicking on the Start button . The next step is to create a new

simulation. In the Environment window, the type of the analysis has to be defined as Kinematics. After

clicking OK, the Motion Joint Wizard window is opened. Here, the establishment of links and

kinematic pairs can be done automatically, on the basis of the parts and relations determined in the

assembly. Unfortunately, in case of the more complex machines, the final effect is frequently

disappointing; therefore, one should choose the Cancel command. The creation of the new simulation,

the default name of which is motion_1, changed the status of the toolbars related to this application to

active, thus, enabling the further development of the simulation model.

Defining links

The next stage is to define links. Parts that do not move against each other comprise one link,

which the program describes as Link. The first link, named Piston, will consist of two parts: Piston and

Piston_pin, which should be indicated in the Link window in the Link Objects field. It can be done

by indicating with the cursor in the workspace or in the Assembly Navigator. Here, the first method is

used. To facilitate the selection of the solid models, Cylinder can be hidden by clicking the left mouse

button on the checkbox (fig. 1.2). Mass Properties Option is kept together with the Automatic default

option, while the name in the Name field has to be changed to Piston.

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Fig. 1.2. Creating the Piston link

Other links are created analogously to the Piston link. According to the table 1.1., Cylinder is a

fixed link (base) and does not have to be defined as Link.

Table 1.1. Parts comprising the links

Link name Names of the parts comprising the link

Connecting_rod Connecting_rod

Crank_helical_gear_1 Crankshaft_part_1, Crankshaft_part_2, Crank_pin, Helical_gear_1

Gear_halical_2_bevel_1 Helical_gear_2, Shaft_1,Bevel_gear_1

Bevel_gear_2_Yoke_1 Bevel_gear_2, Shaft_2, Yoke (connected with Shaft_2)

Yoke_2 Yoke (connected with Shaft_3), Shaft_3

After all the links have been defined, the Links node should include six links in the structure of the

movement simulation (fig. 1.3).

Establishing kinematic pairs

When the reaction forces are not the aim of the study, the user has considerable freedom in the

choice of the type of kinematic pairs for the same links. In case the rigid system is obtained (Gruebler

Count< 0), it is sufficient to remember about the choice of the Dynamics analysis type. However, in

this example, the choice of the connection types aims at achieving Gruebler Count = 0, which means

that the machine's motion is unequivocally determined and there are no passive constraints.

Fig. 1.3. List of all the created links

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The first kinematic pair is formed by the Piston link and the fixed link. Since Cylinder has not

been defined as Link, it will not be pointed as the second link. The situation for other links is similar.

The movement possibilities of the Piston link will be limited by the Slider connection. To

appropriately determine three positions, mark the edges of the piston in the Action field of the Joint

command , as shown in the fig. 1.4. The direction of the Z axis, which determines the movement

possibility and is set in the Specify Vector option, is essential and should comply with the piston axis.

Fig. 1.4. Defining the Slider connection of the Piston link

Subsequently, the Spherical pair between the Piston and Connecting_rod links has to be

established. It does not matter which of the links will be placed as the second one in the Base field.

Here, it has been assumed that Connecting_rod is the second link. Fig. 1.5. shows how to set this pair.

Establishing the beginning of the connection coordinate system has to be carried out using the

Between Two Points method, and of the points themselves using Arc/Ellipse/Sphere Center. While

marking the passive link, the fact that the cursor is placed on this link is not important.

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Fig. 1.5. Determining the Spherical pair between the Piston and Connecting_rod links

The Cylindrical pair will be created between the Connecting_rod and Crank_helical_gear_1

links. To determine the first link, click on the edge of the pin. The Z axis of the coordinate system of

the pair should be situated as shown on fig. 1.6. The second link (Connecting_rod) should be marked

when the Select Link option in the Base field is active (fig. 1.6).

Crank_helical_gear_1 link requires the enforcement of the Revolute connection, so that it could

revolve on its axis (fig. 1.7). Moreover, an engine, which propels the entire machine, will be created

on the Driver page. The angular velocity is 90 degrees/s (constant).

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Fig. 1.6. Establishing the Cylindrical pair between the Connecting_rod and Crank_helical_gear_1 links

Fig. 1.7. Defining the Revolute connection with the engine of the Crank_helical_gear_1 link

Two Revolute connections for the Gear_halical_2_bevel_1 (fig. 1.8) i Bevel_gear_2_Yoke_1 (fig.

1.9) links should be established analogously.

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Fig. 1.8. Establishing the Revolute connection for the Gear_halical_2_bevel_1 link

Fig. 1.9. Establishing the Revolute connection for the Bevel_gear_2_Yoke_1 link

Links Bevel_gear_2_Yoke_1 and Yoke_2 require the Universal type of connection. Firstly, choose

(Select Link) the Bevel_gear_2_Yoke_1 link; then, determine the position of the beginning of the

coordinate system (Specify Origin) using Between Two Points method and of the points themselves

using the Arc/Ellipse/Sphere Center method (fig. 1.10). Axis direction (Specify Vector) is determined

through marking the edges of the hole. In case of the second link, after it has been chosen, only the

direction of the X axis has to be specified (Specify Vector) by marking the edge shown in the fig. 1.10.

For the Universal symbol to become clearly visible, the scale has to be changed with Display Scale to

5.

The last connection from the Joint command will be given to Yoke_2 link. One has to choose

Parallel type and click on the edge of the shaft, as shown in the fig. 1.11a.

After all the connections have been defined with the Joint command, the motion simulations

structure should include eight kinematic pairs (fig. 1.11b).

At this point, the model of the machine lacks gears. To change that, one should choose the Gear

command . While defining the cylindrical gear, one has to mark the Revolute Joint J004 connection

in the First Joint field and the Revolute Joint J005 in the

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Fig. 1.10. Defining the Universal connection between the Bevel_gear_2_Yoke_1 and Yoke_2 links

Second Joint field. The gear ratio will be determined on the basis of the number of teeth on the

gears, thus, in the Settings field, one has to enter the value 20/31 in the Ratio line (fig. 1.12).

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Fig. 1.11. Defining the Parallel connection of the Yoke_2 link a), and the motion simulation structure b)

The analogous procedure is to be undertaken to define the bevel gear (fig. 1.13):

First joint - Revolute Joint J005

Second Joint - Revolute Joint J006

Ratio = 20/26

Fig. 1.12. Defining the cylindrical gear with the Gear command

Fig. 1.13. Defining the bevel gear with the Gear command

Developing new solution

The access to the results and the machine animation option is possible after the created model has

been solved. To achieve that, one has to choose the Solution command and set the parameters of the

solution, as shown in the fig. 1.14.

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Fig. 1.14. Solution window with the solution parameters

With the duration of the simulation set to be 4s and the rotational speed of the engine being 90

degrees/s, the crankshaft will make one full rotation during the animation

Obtaining results from the simulation model

The rotation angle of Shaft_2 can be measured using the Measure command . The type of the

measurement has to be changed to Angle. In the First Set field, mark the wall of the yoke (fig. 1.15). It

will lead to the creation of the first vector, which is perpendicular to the wall. In the Second Set field,

where the cylindrical wall of the Shaft_2 is marked, the second vector, which overlaps with the axis of

the shaft, is generated. Both vectors are parallel and have the same direction, thus, the angle will be

measured starting from 0. To obtain the result, one has to launch the machine motion simulation using

the Animation command and choose the option Measure in the command's window. Shaft_2

rotated by 178,66 .

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Fig. 1.15. Measuring the rotation angle of Shaft_2

The angle between the axes of Shaft_2 and Shaft_3 will be determined by the Simple Angle

command . The axes of the shafts are marked in the command's window (fig. 1.16).

Fig. 1.16. Measuring the angle between the axes of Shaft_2 and Shaft_3

The last task is to prepare the rotational speed graph for Shaft_3. Do this using the Graphing

command . In the Objects field, indicate the J008 connection and define the type of the represented

data as Velocity against the RZ object in the Request option (fig. 1.17).

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Fig. 1.17. Creating the rotational speed graph for the Shaft_3 link

Return to the model's view after choosing the Return to Model command .

1.4. Reference

1. Help and Training -NX 8.5.

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2. THE KINEMATIC ANALYSIS OF A FOUR BAR LINKAGE USING

THE ANALYTICAL AND NUMERICAL METHODS

2.1. The aim of the experiment

Acquiring practical skills as regards the analysis of displacement, velocity and acceleration of the

four bar mechanisms using the analytical and numerical methods.

2.2. Theoretical information

2.2.1. Analytical method

In the typical analysis of mechanism, the kinematics of the driving link is known. Instead, one

looks for information on the behaviour of other elements. For the known lengths of links L1, L2, L3, L4

and angle θ2 (fig. 2.1), the remaining angles and the segment |BD| equal [1]:

221

2

2

2

1 cos2 LLLLBD (2.1)

43

22

4

2

3

2arccos

LL

BDLL (2.2)

coscos

sinsin2

42231

4223

LLLL

LLarctg

(2.3)

coscos

sinsin2

31422

3224

LLLL

LLarctg

(2.4)

Fig. 2.1. Quadrilateral linkage mechanism with marked link lengths and angles [1]

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For the known angular velocity of the driving link 2, the velocities of other links can be

calculated from the formulas [1]:

sin

sin

3

24223

L

L (2.5)

sin

sin

4

23224

L

L (2.6)

In general, when the driving link’s acceleration is 2, the accelerations of other links are illustrated

by the relations [1]:

343

343

2

34

2

4422

2

242223

sin

coscossin

L

LLLL (2.7)

344

3

2

3344

2

3322

2

232224

sin

coscossin

L

LLLL (2.8)

2.3. The course of the study

1. On the basis of the given formulas, calculate the displacement, velocities and accelerations of the

rocker. The crank movement parameters are determined by the instructor.

2. For the same conditions, carry out a movement simulation of a crank-and-rocker mechanism. To

do that:

load the Four_bar_mechanism.prt assembly file,

change the application to Motion Simulation ,

define links using the Link command and name them according to fig. 2.2,

Fig. 2.2. Solid model of the quadrilateral linkage

define four Revolute pairs using the Joint command . In the rotational pair, consisting of

Base and Crank, create the engine,

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solve the simulation model with the Solution command ,

using the Graphing command , prepare graphs and, on their basis, acquire the data on the

kinematic parameters of the rocker motion,

moreover, determine the smallest and the largest angle between Coupler and Rocker using the

Measure command .

3. Prepare the report on the study, which should include:

table with results, analogous to the tab. 2.1

graphs of the motion parameters calculated on the basis of the formulas,

charts showing the difference between motion parameters obtained with analytical and

simulation methods,

conclusions drawn from the study, including the interpretation of the possible differences in

results,

answers to the questions: what information is gained from the knowledge of the angle ; what is

the crank's position for the min. and max.?

Table 2.1. Kinematic parameters of the quadrilateral linkage crank motion

Angular

position of the crank θ2

[ ]

Angular

position of the rocker – formulas θ4a

[ ]

Angular

position of the rocker –simulation

θ4b [ ]

Difference

= θ4a -

θ4b

[ ]

Angular

velocity of the rocker -

formulas

4a [ /s]

Angular

velocity of the rocker - simulation

4b [ /s]

Difference

= 4a -

4b

[ /s]

Angular

accel. of the rocker –

formulas 4a

[ /s2]

Angular

accel. of the rocker –

simulation 4b

[ /s2]

Difference

= 4a -

4b [ /s2]

0

1

2

3

4

…

356

357

358

359

360

2.4. Reference

1. Myszka D. H.: Machines & mechanism. Applied kinematic analysis. Prentice Hall, 4 edition, 2011.

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3. DETERMINING VELOCITIES AND ACCELERATIONS OF THE

PARTICULAR PLANAR MECHANISMS USING THE PLOT AND

NUMERICAL METHOD


Acquiring practical skills as regards the analysis of the velocity and acceleration of the planar

lever mechanisms using the plot and numerical methods.

3.2. Diagram method

Graphic methods, such as the diagram method, are currently used mainly for the verification of

results obtained with the use of other methods and for the study of kinematic analysis of mechanisms.

It results from the characteristics of these methods, such as simplified verification of the obtained

results, clarity of solution and the fact that the obtained results cover only one position of the links. If

the position or dimensions of the elements are different, the calculations have to be carried out from

the beginning.

Acquiring information on the velocities and accelerations of the links with the use of the plot

method involves graphic solving of vector equations as well as, in some cases, algebraic equations.

3.2.1. Example

Calculate the velocity and acceleration of the points C2 (part of link 2) and E of the mechanism

presented in fig. 3.1 using the diagram method. Dimensions and positions of the links are known, as is

the fact that link 1 rotates with the constant rotational speed:

n1 = 120 rev/min,

lAB = 86 mm, lBC = 180 mm, lCD = 115 mm, lDE = 70 mm, lBD = 218,45 mm,

1 = 130 , 2 = 14,17 , 3 = 7,33 , 4 = 114

Solution

Determining velocities

Links 1 and 3 perform rotational motion, whereas link 2 moves in planar motion. Firstly, calculate

the angular velocity of the driving link:

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rad/s 430

120

30

11

n

and the linear velocity of the point B:

m/s 0807,1086,041 ABB lv

Obviously, in a rotational pair, the velocities of the connected ends of the elements are equal,

thus: vB1 = vB2 = vB.

Since there is not enough information on the movement of links 2 and 3, in order to specify the

velocity of the point C2, the velocity of link 3 in the point B3 has to be determined first. It should be

mentioned that no part of link 3 is located in that point. Still, it is assumed that a point, which is rigidly

connected with link 3, is situated in this position. The movement of the point B3 will be considered

complex motion. The rise velocity is vB and the relative velocity is vBB3, thus the vB3 velocity equals:

CD

BB

AB

B

BD

B vvv||

33

Due to the construction of link 2, which includes the slider, the distance of the points B and B3 in

relation to the line determined by the points C and D is always constant during the movement of the

mechanism. It means that the direction of the relative velocity vBB3 is parallel to the segment CD

(hence, single underlining in the formula). The direction of the velocity vB is perpendicular to the

segment AB; additionally, the velocity value is known, therefore, double underlining was used.

Fig. 3.1. Kinematic diagram of the mechanism

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Direction of the velocity vB3is also known. Because link 3 moves in rotational motion, its velocity

is perpendicular to the segment BD. To determine the vB3 velocity from the plot, the

mm

1

s

m 01,0v graduation scale has been adopted.

The length of the velocity vector vB3, read from the fig. 3.2, is 175,18 mm, thus, the velocity value

is:

s

m 7518,101,018,17533 vBB vv

and

s

m 0205,101,005,10233 vBBBB vv

hence the angular velocity of link 3 equals:

rad/s 0192,821845,0

7518,133

BD

B

l

v

The velocity of the point E can be calculated from the relation:

s

m 56134,007,00192,83 DEE lv

The remaining issue is the calculation of the velocity of the point C2. The equation on the velocity

of this point in the complex motion can be designed analogously to the equation for the point B3:

CD

CC

CD

CC vvv||

3232

where:

3Cv – rise velocity,

32CCv – relative velocity.

The value of the velocity vC3 is calculated from the formula:

m/s 9222,0115,00192,833 CDC lv

There is not enough data to graphically solve the equation on the velocity of the point C2. It

requires one more relation, which may be established once the planar motion of link 2 is regarded as

the combination of the translational and rotational motions. Then, the connection is formed between

the velocities of the points of the element:

BC

BC

AB

BC vvv 2222

The length of the velocity vector vC2 is 137,52 mm, thus, the velocity value is:

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m/s 3752,101,052,13722 vCC vv

and

m/s 0204,101,001,1023232 vCCCC vv

m/s 4430,101,030,1442222 vBCBC vv

Fig. 3.2. Velocity plot

Determining accelerations

The order of determining the accelerations is identical to the one for velocities, since it results

from the structure of mechanism and the available data.

The driving link rotates with the constant angular velocity, therefore, the only present acceleration

is the normal one, which, in the point B, equals:

222

1 m/s 5806,13086,04AB

n

BB laa

To determine the acceleration in the point B3, two equations have to be used. In the first one, the

movement of the point B3 is treated as a complex motion:

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CD

c

BB

CD

t

BB

n

BB

AB

BBBBB aaaaaaa 13

||

130

13

||

11313

where:

1Ba – rise velocity,

13BBa – relative velocity.

The normal acceleration n

BBa 13 equals zero, for the path of the point B3 is rectilinear in relation to

the point B1 during the relative motion. Coriolis acceleration is calculated from the relation:

2

13313 m/s 3671,160205,10192,822 BB

c

BB va

and the manner of determining the direction is shown in the acceleration plot (fig. 3.3). The

second equation for the point B3 results from the rotational motion of link 3:

BD

t

B

BD

n

BB aaa 3

||

33

while:

2

22

33 m/s 0513,14

21845,0

7518,1

BD

Bn

Bl

va

The adopted acceleration graduation scale equals mm

1

s

m 5,0

2a . On the basis of the length

of vectors on the plot, the following values are calculated:

2

33 m/s 6525,225,0305,45aBB aa

2

33 m/s 7678,175,05357,35a

t

B

t

B aa

Determining the acceleration of the point C2 requires two equations as well. The first one

describes the relation of the point C2 against C3 during the complex motion:

CD

c

CC

CD

t

CC

n

CC

CD

t

C

CD

n

CCCCC aaaaaaaa 32

||

320

323

||

33232

where:

2

22

33

s

m 3952,7

115,0

9222,0

CD

Cn

Cl

va

2

333

s

m 3557,9115,0

218450

7678,17

,l

l

ala CD

BD

t

BCD

t

C

232332s

m 3656,160204,10192,822 CC

c

CC va

Considering the planar motion of link 2 as a sum of translational and rotational motion leads to

obtaining of the second equation:

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BC

t

BC

BC

n

BC

AB

BBCBC aaaaaa 22

||

22

||

22222

where:

2

22

2222

s

m 5680,11

18,0

443,1

BC

BCn

BCl

va

hence, the acceleration of the point C2 equals:

2

22 m/s 5656,275,01312,55aCC aa

The final calculation will be the acceleration of the point E:

2

222

2

3

22

22

s

m 2576,707,0

218450

7678,17

07,0

5613,0

,

ll

a

l

vaaa DE

BD

t

B

DE

Et

E

n

EE

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Fig. 3.3. Acceleration plot

3.3. Course of the study

1. Calculate the velocities and accelerations of the chosen points of the mechanism using the diagram

method. The instructor determines the movement parameters.

2. For the same conditions, carry out a movement simulation of the mechanism. To do that:

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load the Mechanism_X.prt file. A sample 2D mechanism model is shown in fig. 3.4,

Fig. 3.4. Mechanism model made of 2D elements

change the application to Motion Simulation ,

define links using the Link command and name them,

establish the Revolute and Slider pairs using the Joint command . Create the engine in the

driving link of the rotational pair,

solve the simulation model with the Solution command ,

using the Graphing command , prepare graphs and, on their basis, acquire the data on the

kinematic parameters of the motion.

3. Prepare the report on the study, which should include:

velocity and acceleration plots together with the necessary calculations,

graphs on the motion parameters of the simulations,

conclusions drawn from the study, including the interpretation of the possible differences in

results.

3.4. Reference

1. Miller S.: Teoria maszyn i mechanizmów. Analiza układów kinematycznych. Oficyna Wydawnicza Politechniki Wrocławskiej 1996.

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4. KINEMATICS OF FIXED-AXIS GEARS. SETTING GEAR RATIO,

DETERMINING ROTATIONAL SPEED AND ROTATIONAL

DIRECTION OF AN OUTPUT SHAFT


Acquiring skills as regards the setting of ratio for fixed-axis gears, calculating the angular velocity

and determining the rotational direction of the output shaft.

4.2. Kinematic analysis of a fixed-axis gear

The gear ratio of fixed axis gearing is relatively easy to determine using kinematic or geometric

dependencies. Cylindrical gears can additionally be described by the gear ratio sign. If the driving and

driven gears rotate in the same direction, the gear ratio sign is positive. The sign is negative if the

gears rotate in opposite directions. Gears with non-parallel planes of gear rotation

(e.g. bevel gears and worm gears) do not have any fixed gear ratio sign, therefore it is always assumed

to be positive.

The gear ratio of a cylindrical internal gear is defined as:

(4.1)

gdzie:

1 is the angular velocity of the driving gear (primary),

2 is the angular velocity of the driven gear (secondary),

z1 is the number of teeth of the driving gear,

z2 is the number of teeth of the driven gear,

while the gear ratio of external gears is expressed as:

(4.2)

Cylindrical gears can come with two types of gearing: series and parallel gears. The gear ratio of a

series gear depends on parameters of the first and last gears, rather than on the number of teeth on the

intermediate gears. The gear ratio of the gear shown in Fig. 4.1 is:

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(4.3)

Fig. 4.1. Design of a series gear Fig. 4.2. Design of a parallel gear

In parallel gearing systems, the total gear ratio is the product of gear ratios of individual teeth. The

total gear ratio of the gear shown in Fig. 4.2 is:

(4.4)

Gears where rotational velocity on the input shaft is lower than that on the output shaft are known

as reduction gears (|i| > 1). In contrast, gears which require a higher rotational velocity on the output

shaft are known as multiplicator gears (|i| < 1).


1. Calculate the gear ratio, direction and velocity of the output shaft using the analytical method.

2. Analogous calculations should be performed for the simulation model.

3. Prepare the report on the study.

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5. KINEMATICS OF PLANETARY GEARS. SETTING GEAR RATIO,

DETERMINING ROTATIONAL SPEED AND ROTATIONAL

DIRECTION OF AN OUTPUT SHAFT


Acquiring skills as regards the setting of ratio for planetary gears, calculating the angular velocity

and determining the rotational direction of the output shaft.

5.2. Kinematic analysis of a planetary gear

Contrary to fixed axis gearing where the gears only rotate, planetary gears have planet gears that

perform plane motion. Planet gears can rotate around their own axis and, additionally, their axes can

perform rotational motion, too. This makes determination of their gear ratio difficult. One of the

methods for gear ratio determination is the Willis method. This method consists in changing the

reference system from the wheelcase to the cage. This is done by assigning the angular velocity of the

cage yet in opposite direction to all elements.

The Willis formula for determination of the gear ratio of planetary gears (Fig. 5.1) is:

(5.1)

where:

– denotes the gear ratio between the driving gear 1 and the driven gear 3 at fixed planetary

cage,

With some other gear unit made stationary, the above formula is rewritten such that the cage

becomes fixed. This can easily be done by exchanging the gear ratio signs in accordance with the

dependence:

(5.2)

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Fig. 5.1. Design of a planetary gear


1. Calculate the gear ratio, direction and velocity of the output shaft using the analytical method.

2. Analogous calculations should be performed for the simulation model.

3. Prepare the report on the study.

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