Mechanisms Final Report

Kuwait UniversityCollege of Engineering & PetroleumDepartment of Mechanical Engineering

Engineering Fundamental Laboratory ME 372

Experiment Title:

Mechanisms Report 1

Submission Date:

Sunday 26-10-2008

Group members:

Fawaz Ahmed Alshammeri 203112780Mishary Fowzi Alsinny 204112440Ahmad Abdullah Al-Ahmad 204215860Ali Abdullah Al-Basri 205115632

Table of Content

DescriptionPages

Table of Content

2

Lists of Figures & Tables

3-4

Abstract5

Starting Chapter 1

6

Experimental Background7-12

Objectives of Experiment13

Starting Chapter 2

14

Theoretical Background15-18

Starting Chapter 3

19

Experimental apparatus and procedure20-21

Starting Chapter 4

22

Results23-28

Sample Calculations29-31

Figures32-37

Discussion38-39

Sources of error40

Starting Chapter 541

Conclusion42

References43

Lists of Figures & Tables

DescriptionSequence

Four Bar Linkage Mechanism.Figure.1.1

The Possible Types of Four Bar LinkageFigure.1.2

External Geneva Mechanism in starting positionFigure.1.3

Geneva Stop.Figure.1.4

Quick Return MechanismFigure.1.5

The Quick Return Mechanism in the Shaping MachineFigure.1.6

Slider Crank MechanismFigure.1.8

Four-Bar Linkage NomenclatureFigure.2.1

A Way to Solve Four-Bar LinkageFigure.2.2

Geneva StopFigure.2.3

Time ratio in the Whitworth quick-return mechanismFigure.2.4

Slider-Crank MechanismFigure.2.6

The displacement Curve for the four bar mechanism(Case. I)Fig.4.1

The displacement curve for the four bar mechanism (Case. IIFig.4.2

The displacement diagram of the Geneva mechanismFig.4.3

The displacement diagram of the Whitworth Quick-ReturnFig.4.4

The velocity diagram of the Whitworth Quick-ReturnFig.4.5

The acceleration diagram of the Whitworth Quick-ReturnFig.4.6

The displacement diagram of the slider crankFig.4.7

The velocity diagram of the slider crankFig.4.8

The Acceleration diagram of the slider crankFig.4.9

The displacement diagram of the slider crankFig.4.10

The velocity diagram of the slider crankFig.4.11

The Acceleration diagram of the slider crankFig.4.12

Link measurementsTable.1

The input angle and the corresponding output angles in each caseTable.2

The input and output angles for Geneva mechanismTable.3

The crank angle and the displacement of the Whitworth Quick-Return mechanismTable.4

Measurements tableTable.5

The crank angle and the displacement of the slider , Velocity and acceleration (theoretical Data)Table.6

The crank angle and the displacement of the slider , Velocity and acceleration (practical Data ).Table.7

The transmission angle (of each case for each input angleTable.8

Abstract There are several kinds of mechanisms that will be studied during the Engineering Fundamental Laboratory (ME-372) lectures and many experiments will illustrate these mechanisms. The main objective of these experiments is to learn more about each type of mechanisms, motion function and properties. The data obtained in the laboratory have been analyzed using mathematical equations and graphical methods. These data will represent the motion, velocity and acceleration of each type of mechanism. Finally, the objectives mentioned above are followed and fully achieved.

There are two types of mechanisms that will be studied. Firstly, Reciprocating Motion Mechanisms which included Slider Crank Mechanism and Whitworth Quick Return Motion Mechanism. The second type is called Linkage Mechanisms which included Four Bar Linkage Mechanism and Geneva Stop Mechanism.

Chapter 1

1. Experimental Background:

1. a) Four Bar Linkage:

Mechanical linkages are a series of rigid links connected with joints to form a closed chain. Each link will have two or more joints, and the joints will have various degrees of freedom to allow motion between the links. A linkage is called a mechanism if two or more links are movable with respect to a fixed link. Mechanical linkages are usually designed to take an input and produce a different output, altering the motion, velocity, acceleration, and applying mechanical advantage.

The Four Bar-Linkage is the simplest possible closed-loop mechanism, and has numerous uses in industry and for simple devices found in automobiles, toys, etc. The device gets its name from its four distinct links (or bars), as shown in Figure.1. Link 1 is the ground link (sometimes called the frame or fixed link), and is assumed to be motionless. Links 2 and 4 each rotate relative to the ground link about fixed pivots (A0 and B0). Link 3 is called the coupler link, and is the only link that can trace paths of arbitrary shape (because it is not rotating about a fixed pivot).

Figure.1.1

Usually one of the "grounded links" (link 2 or 4) serves as the input link, which is the link which may either be turned by hand, or perhaps driven by an electric motor or a hydraulic or pneumatic cylinder. If link 2 is the input link, then link 4 is called the follower link, because its rotation merely follows the motion as determined by the input and coupler link motion. If link 2 is the input link and its possible range of motion is unlimited, it is called a crank, and the linkage is called a crank-rocker. Crank-rockers are very useful because the input link can be rotated continuously while a point on its coupler traces a closed complex curve.

Grashof's law is applied to pinned linkages and states; the sum of the shortest and longest link of a planar four bar linkage cannot be greater than the sum of remaining two linkages if there is to be continuous relative motion between the links. Figure.2. shows the possible types of pinned, four-bar linkages, where our experiment was done on the Crank-rocker one.Figure.1.2

1. b) The Geneva Stop:

The Geneva mechanism is a timing device. It is used in many counting instruments and in other applications where an intermittent rotary motion is required. Essentially, the Geneva mechanism consists of a rotating disk with a pin and another rotating disk with slots (usually four) into which the pin slides as shown in Figure.3.Figure.1.3

The Geneva mechanism was originally invented by a watch maker. The watch maker only put a limited number of slots in one of the rotating disks so that the system could only go through so many rotations. This prevented the spring on the watch from being wound too tight, thus giving the mechanism its other name, the Geneva Stop. The Geneva Stop was incorporated into many of the first film projectors used in theaters.

In Optimum Design of Mechanical Elements, the Geneva mechanism is used to provide an intermittent motion of the conveyor belt of a "film recording marching." There are several weak points in the Geneva mechanism. For instance, for each rotation of the Geneva (slotted) gear the drive shaft must make one complete rotation. Thus for very high speeds, the drive shaft may start to vibrate. Another problem is wear, which is centralized at the drive pin. Finally, the designer has no control over the acceleration the Geneva mechanism will produce. Also, the Geneva mechanism will always go through a small backlash, which stops the slotted gear. This backlash prevents controlled exact motion.

An interesting example of intermittent gearing is the Geneva Wheel shown in Figure 2. In this case the driven wheel, B, makes one fourth of turn for one turn of the driver, A the pin, a, working in the slots. b, causing the motion of B. The circular portion of the driver, coming in contact with the corresponding hollow circular parts of the driven wheel, retains it in position when the pin or tooth a is out of action. The wheel A is cut away near the pin a as shown, to provide clearance for wheel B in its motion.

Figure.1.4

If one of the slots is closed, A can only move through part of the revolution in either direction before pin a strikes the closed slot and thus stops the motion. The device in this modified from was used in watches, music boxes, etc., to prevent over winding. From this application it received the name Geneva Stop. Arranged as a stop, wheel A is secured to the spring shaft, and B turns on the axis of the spring barrel. The number of slots or interval units in B depends upon the desired number of turns for the spring shaft.

1. c) The Whitworth Quick Return:

A quick return mechanism such as the one seen opposite is used where there is a need to convert rotary motion into reciprocating motion. As the disc rotates the black slide moves forwards and backwards. Many machines have this type of mechanism and in college's workshop the best example is the shaping machine.

Figure.1.5

The shaping machine is used to machine flat metal surfaces especially where a large amount of metal has to be removed. Other machines such as milling machines are much more expensive and are more suited to removing smaller amounts of metal, very accurately. The reciprocating motion of the mechanism inside the shaping machine can be seen in the diagram. As the disc rotates the top of the machine moves forwards and backwards, pushing a cutting tool. The cutting tool removes the metal from work which is carefully bolted down.

Figure.1.6

1. d) The Slider Crank:A slider-crank mechanism is a four-bar linkage with 3 revolute joints and one prismatic joint. This mechanism converts rotary motion into reciprocating linear motion, or vice versa. The components of a slider-crank are shown in the figure below. Notice that the ground link is the base of the machine that connects the axis of rotation of the crank with the surface upon which the slider moves.

Figure.1.8

This mechanism is composed of three important parts; the crank which is the rotating disc, the slider which slides inside the tube and the connecting rod which joins the parts together. As the slider moves to the right the connecting rod pushes the wheel round for the first 180 degrees of wheel rotation. When the slider begins to move back into the tube, the connecting rod pulls the wheelround to complete the rotation. One of the best examples of a crank and slider mechanism is a steam train. Steam pressure powers the slider mechanism as the connecting rod pushes and pulls the wheel round. The cylinder of an internal combustion engine is another example of a crank and slider mechanism.

1.2: Objectives of Experiment:

- To study the function and performance of each coupling.- Analytical derivation of its kinematics.- Experimental verification of analytical results.

Chapter 2

1. Theoretical Background:

1. a) The Four-Bar Linkage:

In order to analyze the motion of the four-bar linkage, geometrical analysis should be carried out. See Figure.11.

Figure.2.1

Given

------------------------------(1)

------------------------------(2)Equations 1 and 2 are nonlinear equation. In order to solve them, numerical methods or the following procedures should be used:A line z -connects the points A, O2- splits the mechanism into two triangles and by using Cosine Law we can solve the system. See Figure.12.

------------------------------(3)

------------------------------(4)

------------------------------(5)

------------------------------(6)

------------------------------(7)

------------------------------(8)

------------------------------(9)

Figure.2.2

In order to find the angular velocities in the mechanism, Equations 1 and 2 should be differentiated with respect to time:

----------------------(10)

----------------------(11)

2.b)The Geneva Stop:

Figure.2.3

Given

------------------------------(12)

------------------------------(13)

------------------------------(14)

------------------------------(15)

2.c)The Whitworth Quick-Return Mechanism:The following equations show how to find the time ratio (TR) in the whitworth quick-return mechanism.

Figure.2.4

-------------------------------------------------------(16)

-------------------------------------------------------(17)

(18)

2.d) The Slider-Crank Mechanism:The main idea of analyzing motion of the slider-crank mechanism is by using geometrical analysis. The following equations represent the motion analysis:

Figure.2.5

Given

----------------------------(19)

----------------------------(20)In order to analyze the velocity

----------------------------(21)

----------------------------(22)

----------(23)

(24)

Chapter 3

3. Experimental apparatus and procedure:

3.a) The Four-Bar Linkage:1. Set the input angle at 0 and record the corresponding output angle.2. Rotate the input crank in steps of 15 until it turns one complete revolution. At each step, record the reading on the output disk.3. Plot a graph of the output angle against the input angle.4. Calculate the values of z and for each step.5. Repeat steps 1-4 for different lengths.

3.b) The Geneva Stop:1. Set both scales to zero, then rotate the input rotor in steps of 5 from 0 to 360 and record the output angle for each position.2. Plot a graph of output angle from 0 to 360 against the input angle.

3.c) The Whitworth Quick-Return Mechanism:1. Determine the output response for rotation of the input shaft. Plot slider displacement against input shaft angle (0 to 360 in steps of 15).2. From the graph using software get the 6th degree displacement equation and differentiate it to get the velocity and acceleration relations .3. Plot the velocity and acceleration relations .

3.d) The Slider-Crank Mechanism:1. Set the crank to 0. Record the reading on the slider scale.2. Move the crank through 15 steps until it turns one complete revolution. At each step, record the reading on the slider scale.3. Plot a graph of the displacement .4. From the graph get the 6th order polynomial and differentiate it twice to get the velocity and acceleration relations .5. Plot the velocity and the acceleration relations .6. Measure the crank radius and the length of the connection link. Calculate the theoretical values of slider displacement, velocity and acceleration from theoretical background for a range of values of crank angle. Plot the theoretical results on the same axes as the experimental curves ( or separated ) .7. Compare the displacement, velocity and acceleration curves of the slider crank mechanism producing a pure sinusoidal motion. For unit angular velocity,

----------------------------------------------(25)

----------------------------------------------(26)

----------------------------------------------(27)Calculate these functions for a range of crank angular positions, and plot the points on the experimental curves.

Chapter 4

4.1 Results:

a) Four Bar Linkage:

CasesFixed frame link length (mm)Crank length (mm)Coupler link length (mm)Output link length (mm)

case 12005022.320.1

case22005022.310

Table.1

Input AngelsOutput Angles

In DegreesIn radIn Degrees ( case 1 )In rad ( case 1 )In Degrees ( case 2 )In rad ( case 2 )

00.0002273.960209.53.655

150.2622253.925202.03.524

300.5232223.873194.03.384

450.7852183.803186.03.245

601.0472133.716175.03.053

751.3082083.620165.02.878

901.5702023.524158.02.756

1051.8321983.454155.02.704

1202.0931953.402156.02.721

1352.3551943.384160.02.791

1502.6171943.384166.02.896

1652.8781963.419172.53.009

1803.1401993.471181.03.157

1953.4022033.541188.53.288

2103.6632073.611196.03.419

2253.9252113.681203.03.541

2404.1872143.733209.03.646

2554.4482183.803214.03.733

2704.7102223.873218.03.803

2854.9722243.908220.03.838

3005.2332263.942221.03.855

3155.4952283.977220.03.838

3305.7572283.977218.03.803

3456.0182283.977214.03.733

3606.2802273.960209.03.646

Table.2

b) Geneva Mechanism:

Input angle in DegreesInput angle in radOutput angle in Degreesoutput angle in rad

00.0001.00.017

50.0872.00.035

100.1742.80.048

150.2623.20.056

200.3494.20.073

250.4366.00.105

300.5237.00.122

350.61110.50.183

400.69813.50.236

450.78517.60.307

500.87221.00.366

550.95926.00.454

601.04729.50.515

651.13434.00.593

701.22137.50.654

751.30842.90.748

801.39646.50.811

851.48349.50.864

901.57052.00.907

951.65754.50.951

1001.74456.00.977

1051.83257.51.003

1101.91958.51.021

1152.00659.01.029

1202.09359.01.029

1252.18159.01.029

1302.26859.01.029

1352.35559.01.029

1402.44259.01.029

1452.52959.01.029

1502.61759.01.029

1552.70459.01.029

1602.79159.01.029

1652.87859.01.029

1702.96659.01.029

1753.05359.01.029

1803.14059.01.029

1853.22759.01.029

1903.31459.01.029

1953.40259.01.029

2003.48959.01.029

2053.57659.01.029

2103.66359.01.029

2153.75159.01.029

2203.83859.01.029

2253.92559.01.029

2304.01259.01.029

2354.09959.01.029

2404.18759.01.029

2454.27459.01.029

2504.36159.01.029

2554.44859.01.029

2604.53659.01.029

2654.62359.01.029

2704.71059.01.029

2754.79759.01.029

2804.88459.01.029

2854.97259.01.029

2905.05959.01.029

2955.14659.01.029

3005.23359.01.029

3055.32159.01.029

3105.40859.01.029

3155.49559.01.029

3205.58259.01.029

3255.66959.01.029

3305.75759.01.029

3355.84459.01.029

3405.93159.01.029

3456.01859.01.029

3506.10659.01.029

3556.19359.01.029

3606.28059.01.029

Table.3

c) Whitworth Quick-Return mechanism:

Crank angle in degreeCrank angle in radDisplacement (mm)Velocity (mm/s)Acceleration (mm/s2)

00.0009.5000.328-0.018

150.26210.5000.323-0.018

300.52311.5000.318-0.018

450.78511.9000.313-0.018

601.04710.8000.309-0.018

751.3087.2000.304-0.017

901.5703.8000.300-0.017

1051.8322.2000.295-0.017

1202.0931.4000.290-0.016

1352.3551.0000.286-0.016

1502.6170.7000.281-0.015

1652.8780.7000.277-0.014

1803.1400.8000.273-0.014

1953.4020.8000.268-0.013

2103.6631.1000.264-0.012

2253.9251.5000.260-0.011

2404.1872.0000.255-0.010

2554.4482.1000.251-0.009

2704.7103.3000.247-0.008

2854.9724.1000.243-0.007

3005.2335.1000.239-0.006

3155.4956.1000.235-0.004

3305.7577.3000.230-0.003

3456.0188.4000.226-0.002

3606.2809.5000.2220.000

Table.4

d) Slider crank mechanism:

crank length (mm)50

connecting rod length (mm)150

Table.5


00.0000.0000.0000.037

150.2622.2600.0100.037

300.5238.7740.0190.036

450.78518.7940.0290.036

601.04731.2230.0390.036

751.30844.7990.0490.036

901.57058.2940.0590.036

1051.83270.6750.0680.036

1202.09381.2120.0780.036

1352.35589.4900.0880.035

1502.61795.3610.0980.035

1652.87898.8420.1080.035

1803.140100.0000.1180.035

1953.40298.8700.1290.035

2103.66395.4180.1390.035

2253.92589.5760.1490.035

2404.18781.3270.1590.034

2554.44870.8160.1690.034

2704.71058.4530.1800.034

2854.97244.9660.1900.034

3005.23331.3840.2010.034

3155.49518.9330.2110.034

3305.7578.8760.2210.034

3456.0182.3140.2320.033

Table.6


150.26210.0000.0170.005

300.52311.4000.0180.005

450.78512.7000.0190.005

601.04713.2000.0200.005

751.30814.3000.0220.005

901.57014.5000.0230.005

1051.83214.3000.0240.005

1202.09313.6000.0250.005

1352.35512.6000.0260.004

1502.61711.3000.0280.004

1652.87810.0000.0290.004

1803.1408.7000.0300.004

1953.4027.4000.0310.004

2103.6636.4000.0320.004

2253.9255.5000.0330.004

2404.1874.9000.0340.004

2554.4484.6000.0350.004

2704.7104.5000.0360.004

2854.9724.6000.0370.004

3005.2334.9000.0380.004

3155.4955.5000.0390.004

3305.7576.3000.0400.004

3456.0187.4000.0410.004

3606.2808.7000.0420.004

Table.7

4.2) Sample Calculations:


Same procedure is doing for the each case.

Input Angle in degreesCase ICase II

00.7160220.583152

151.2005711.525147

300.9694841.097895

450.8340.899667

601.2463031.611598

750.7422430.644098

901.1248111.384347

1051.0729281.288828

1200.7770920.720609

1351.2565651.631209

1500.8130110.795587

1651.0280231.206221

1801.1616831.452589

1950.7242850.602715

2101.2301261.580854

2250.9106140.987213

2400.9222941.009374

2551.2250141.571179

2700.7213310.595751

2851.170151.468327

3001.0164561.184898

3150.8228960.81569

3301.2554181.629013

3450.7691350.703515

3601.0837471.308722

Table.8


We fit the curve to the sixth order polynomial more accurate than fifth order .

Linear model Polynomial of sixth order :y = 3E-12x6 - 4E-09x5 + 2E-06x4 - 0.0004x3 + 0.0323x2 - 0.5152x + 3.0297


Selecting a sixth order polynomial regression for the displacement data yields the following equation:

y = -5E-13x6 + 7E-10x5 - 3E-07x4 + 8E-05x3 - 0.0091x2 + 0.3275x + 8.7242

Differentiating the above equation gives the following velocity equation:

= -5E-21x5 + 3E-16x4 - 6E-12x3 + 7E-08x2 - 0.0003x + 0.3275

Differentiating the above equation gives the following acceleration equation:

= 1E-15x4 - 2E-11x3 + 1E-07x2 + 4E-11x - 0.0182


Selecting a sixth order polynomial regression for the displacement data yields the following equation:

y = 3E-13x6 - 4E-10x5 + 2E-07x4 - 4E-05x3 + 0.0025x2 + 0.0153x + 9.4024

Differentiating the above equation gives the following velocity equation:

y' = 3E-21x5 - 2E-16x4 + 4E-12x3 - 4E-08x2 + 9E-05x + 0.0153

Differentiating the above equation gives the following acceleration equation:y'' = 8E-19x4 - 4E-14x3 + 7E-10x2 - 4E-06x + 0.005

The following equations were also used in plotting the graphs:a) Displacement:

b) Velocity:

c) Acceleration:


Case .I:

Fig.4.1

Case.II:

Fig.4.2:


Fig.4.3


Fig.4.4

Fig.4.5

Fig.4.6


-The graphs of the Theoretical data

Fig.4.7

Fig.4.8

Fig.4.9

-The graphs of the practical data

Fig.4.10

Fig.4.11

Fig.4.12

4.4) Discussion:


Here four different cases were analyzed, and in each case the displacement behavior is being studied as follows: ()

Case I - Maximum coupler length and maximum output length (Fig.4.1):As shown from table (4.2) & figure (4.1), there are small difference between the fit curve and the data points. The curve motion of the output angles vs. input angle take shape of sinusoidal wave. The maximum output angle=170o at the beginning and the end of motion. The minimum output angle= 137o at input angle= 135o. That means link (4) oscillates between 170o and 137o as link (2) rotates one complete revolution. It can be seen from (Fig.4.1) that the out output angle changes rapidly with the input angle when reaching their extreme values, whereas the response of the output angle varies uniformly with the input angle between its extreme values.

Case II - Maximum coupler length and minimum output length (Fig.4.2):In (Fig.4.2), the output angle is first set to be equal to 152 o, and then it reaches a minimum value of 98 o at an input angle of 105o. When the input angle is equal to 285, the output angle reaches a maximum value of 163o, and then it decreases back again to a value of 152o at an input angle of 360o. In this case, it can be seen from (Fig.4.4) that the amplitude of oscillation is equal 3.5 degrees. This illustrates the behavior of the four bar linkage when the length of the output link is minimum, in which case it oscillates with a constant amplitude.


For the Geneva mechanism, the output angle was plotted against the input angle (Fig.4.3). The output angle increases with the increase of the input angle until it a constant value of 59o starting from an input angle of 15o. This behavior illustrates the intermitted motion of the Geneva mechanism. The displacement function was obtained by a sixth order polynomial regression.


For the Whitworth Quick-Return mechanism, the displacement versus the crank angle was plotted (Fig.4.4). The displacement starts with a value of 9.5 mm, and then it goes up to reach a value of 11.900 mm at a crank angle of 45 o. After that, the displacement decreases in a uniform manner to reach value (0.800 mm) at the end angle equal 195 o . As expected form the above analysis, the sudden drop in the displacement is due to the quick-return nature of the Whitworth mechanism.In a similar way of the slider crank , The velocity and acceleration functions were obtained and plotted against the crank angle. In (Fig.4.5 and Fig.4.6), the velocity is initially at a value of 0.328 mm/s, it then decreases rapidly to reach a minimum value of 0.222 cm/s at a crank angle of 360 o. For the acceleration diagram (Fig.4.6), the acceleration stars with a value of -0.018 mm/s2, and it increases to a value of 0.0 mm/s2 at a crank angle of 360 o.


In the analysis of the slider crank mechanism, first the displacement diagram (Fig.4.7) and (Fig.4.10) was obtained by plotting the displacement of the slider versus the crank angle. But one of them is obtained practically and the other one obtained theoretically . In the practical diagram, the displacement curve was found by fitting the displacement data with a sixth order polynomial. The initial value is 10 mm when the crank angle is set at 0, and increases to reach a maximum value of 14.300 mm when the crank angle reaches 75o. The displacement curve then decreases back to its initial value when the crank angle is at 165o. At a crank angle of 225o, the displacement reaches a minimum value of 5.500 mm, and increases back again to reach the initial value (10 mm) when the crank angle is at 360 o.The velocity was determined by differentiating the displacement function, and the velocity diagram was obtained by plotting the velocity against the crank angle (Fig.4.11). The velocity starts with a value of 0.017 mm/s when the crank angle is at 0, and then reaches a maximum value of 0.42 mm/s at a crank angle of 360o.To get the acceleration function, the velocity function was differentiated; the obtained function was then plotted against the crank angle as shown in (Fig.4.12). Initially the acceleration has a value of 0.005 mm/s2, this value decreases to its minimum value .004 mm/s2) when the crank angle is equal to 360o. 4.6 Sources of error :a. Human error sourcehuman error according to the wrong readings or plugging a wrong values on the equations .b. Device accuracy The device have a specific accuracy and resolution In our case :The accuracy of the length measuring device ( ruler ) is .5 mmAnd the devices resolution = .5 o c. Device defects .

Chapter 5

5.1) Conclusion:

After covering the discussed experiments many conclusions and statements are reported and learned. All the functions and properties of commonly used mechanisms are well studied. The importance of the mechanisms is shown clearly and their most common types also. Each type has its own output motion and is used according to the needed application.

5.2) References

http://en.wikipedia.org http://www.sportdevices.com/shockabsorber/ http://www.technologystudent.com/cams/crank1.htm http://www.google.com

44

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