Lublin, dnia 18

Transkrypt

Lublin, dnia 18

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WORKBOOK
THEORY OF MACHINES AND MECHANISM
LUBLIN 2014
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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
Projekt współfinansowany ze środków Unii Europejskiej w ramach Europejskiego Funduszu
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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
Gear_halical_2_bevel_1
Bevel_gear_2_Yoke_1
Yoke_2
Crankshaft_part_1, Crankshaft_part_2, Crank_pin, Helical_gear_1
Helical_gear_2, Shaft_1,Bevel_gear_1
Bevel_gear_2, Shaft_2, Yoke (connected with Shaft_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
rotated by 178,66 .
and choose the option Measure in the command's window. Shaft_2
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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]:
L12
BD
arccos
3
2arctg
4
2arctg
L22
2L1L2 cos
L23
2
L24 BD
2 L3 L4
2
(2.1)
(2.2)
L2 sin 2 L4 sin
L3 L2 cos 2 L4 cos
(2.3)
L2 sin 2 L3 sin
L2 cos 2 L4 L1 L3 cos
(2.4)
L1
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
calculated from the formulas [1]:
3
2
L2 sin 4
L3 sin
4
2
L2 sin 3
L4 sin
In general, when the driving link’s acceleration is
by the relations [1]:
L sin
2 2
2
4
3
L sin
2 2
4
2
3
2
2 2
L cos 2
L3 sin
the velocities of other links can be
2
(2.5)
2
(2.6)
the accelerations of other links are illustrated
2
4 4
L
4
2
3 3
L cos
4
3
(2.7)
4
2
2 2
L cos 2
L4 sin
2,
2,
3
2
3 4
L cos
3
4
3
2
3 3
L
(2.8)
4
3
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
Base and Crank, create the engine,
. In the rotational pair, consisting of
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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
Angular
Angular Difference Angular
Angular Difference Angular
Angular Difference
position of position of position of
= θ4a - velocity of velocity of
= 4a - accel. of the accel. of the
= 4a 2
the crank θ2 the rocker – the rocker –
the rocker - the rocker rocker –
rocker –
θ4b
4b
4b [ /s ]
formulas θ4a simulation
formulas
simulation
[]
formulas
simulation
[]
[ /s]
4a
4b
[]
θ4b [ ]
[ /s2]
[ /s2]
4a [ /s]
4b [ /s]
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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n1
30
1
120
30
4
rad/s
and the linear velocity of the point B:
vB
l
1 AB
4
0,086 1,0807 m/s
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:
vB 3
BD
vB
AB
vB 3B
||CD
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
v
m 1
s mm graduation scale has been adopted.
0,01
The length of the velocity vector vB3, read from the fig. 3.2, is 175,18 mm, thus, the velocity value
is:
vB 3
vB 3
vB 3 B
vB 3 B
m
s
175,18 0,01 1,7518
v
and
m
s
102,05 0,01 1,0205
v
hence the angular velocity of link 3 equals:
3
vB 3
lBD
1,7518
0,21845
8,0192 rad/s
The velocity of the point E can be calculated from the relation:
vE
l
3 DE
8,0192 0,07 0,56134
m
s
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:
vC 2
vC 3 vC 2C 3
CD
||CD
where:
vC 3
– rise velocity,
vC 2C 3
– relative velocity.
The value of the velocity vC3 is calculated from the formula:
vC 3
l
3 CD
8,0192 0,115 0,9222 m/s
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:
vC 2
vB 2 vC 2 B 2
AB
BC
The length of the velocity vector vC2 is 137,52 mm, thus, the velocity value is:
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vC 2
vC 2
137,52 0,01 1,3752 m/s
v
and
vC 2C 3
vC 2 B 2
vC 2C 3
vC 2 B 2
v
v
102,01 0,01 1,0204 m/s
144,30 0,01 1,4430 m/s
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:
aB
aBn
2
1 AB
l
4
2
0,086 13,5806 m/s2
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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aB3
a B1 a Bn3 B1 a Bt 3 B1 a Bc 3 B1
a B1 a B 3 B1
0
|| AB
||CD
CD
where:
aB1 – rise velocity,
aB3B1 – relative velocity.
n
The normal acceleration aB3B1 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:
aBc 3B1
2 8,0192 1,0205 16,3671 m/s2
2 3vB3B1
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:
aB 3
aBn3 aBt 3
||BD
BD
while:
vB2 3
lBD
aBn 3
1,75182
14,0513 m/s2
0,21845
The adopted acceleration graduation scale equals
a
0,5
m 1
s 2 mm . On the basis of the length
of vectors on the plot, the following values are calculated:
aB 3
aB 3
aBt 3
aBt 3
a
a
45,305 0,5 22,6525 m/s2
35,5357 0,5 17,7678 m/s2
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:
aC 2
aC 3
aC 2C 3
aCn3 aCt 3 aCn2C 3 aCt 2C 3 aCc 2C 3
||CD
CD
0
||CD
CD
where:
aCn 3
aCt 3
aCc 2C 3
3lCD
vC2 3
lCD
0,92222
0,115
7,3952
m
s2
aBt 3
lCD
lBD
2 3vC 2C 3
17,7678
m
0,115 9,3557 2
0,21845
s
m
2 8,0192 1,0204 16,3656 2
s
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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aC 2
aB 2
aC 2 B 2
aB 2 aCn2 B 2 aCt 2 B 2
|| AB
BC
|| BC
where:
aCn 2 B 2
vC2 2 B 2
lBC
1,4432
0,18
11,5680
m
s2
hence, the acceleration of the point C2 equals:
aC 2
aC 2
55,1312 0,5 27,5656 m/s2
a
The final calculation will be the acceleration of the point E:
aE
a
n 2
E
0,56132
0,07
a
2
t 2
E
vE2
l DE
2
17,7678
0,07
0,21845
aBt 3
lDE
lBD
2
7,2576
2
m
s2
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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
driving link of the rotational pair,
 solve the simulation model with the Solution command
. Create the engine in the
,
 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),
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,
2
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.
Społecznego

Lublin, dnia 18

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