Lab 2- Graphical Linkage Synthesis

8/11/2019 Lab 2- Graphical Linkage Synthesis

1/16

1

3. Graphical Linkage Synthesis

3.1 Types of Motion

Function Generation: correlation of output motion with input motion

Example: Scotch Yoke.Input: RotationOutput: Sinusoidal Function

Path Generation: control of a point on a mechanism such that it follows a prescribed path.

Example: piston engine

Control Points: TDC, BDC

Motion Generation: Control of a line on a mechanism such that in assumes a set of sequentialpositions

Example: Windshield Wiper (Mercedes)

- Vertical in middle of windshield

- Angled at edges of windshield- Does not move in a circular arc!

TDC

BDC


2/16

2

3.2 Graphical Dimensional Synthesis

Given: a set of prescribed positions & link angles, find the dimensions (link lengths & pivotpositions) to produce this motion.

Example 3-1: 2 Position synthesis with rotational displacement (windshield wiper)

Find a linkage which produces 60of rocker rotation with equal time forward and back using aconstant-speed motor. (windshield wiper mechanism)

Given:2 desired output anglesFind:Link lengths, fixed and moving pivot locations

1.

Draw link O4B in extreme positions B1, B2so that 60is swept out.

Note: link 4 is already defined at this point, as is the position of one fixed pivot. Now we must find

a way to drive the rocker through 60. The simplest way is to push point B back and forthhorizontally.

2.

Draw chord B1B2.3.

Draw a circle having as its center the midpoint of B1B2and radius extending out to B2.4.

Extend B1B2out to a convenient point.5.

Select a point O2on B1B2for the other fixed pivot.6. Copy the circle drawn in Step 3 so that its center is at O2.7.

Label the intersections of circle and B1B2A1and A2.

8.

Measure length of coupler from A1B1or A2B2.9.

Measure rocker length O4B1, crank length O2A1.10.

Find Grashof condition. If non-Grashof redo steps 3-8 with O2further from O4.

Note: Two free choices

1.

length of O4B2.

length of O2B

This implies an infinite number of possible solutions.


3/16

3

Example 3-1


4/16

4

Example 3-2: 2 position synthesis with complex displacement (bottle capper)

For this example we wish to move a line on the rocker through two specified locations.

1.

Draw link CD in its desired positions C1D1and C2D22. Draw lines from point C1to C2and D1to D23. Bisect lines C1C2 and D1D2 and extend perpendicular bisectors in convenient directions.

Their intersection is the rotopole

Note: it takes three points to define a circular arc. The three points may be on the arc, or two pointsmay be on the arc with the remaining point at the center. In this case we require two circular arcswith a common center: the arcs swept out as C1moves to C2and D1to D2. The two perpendicularbisectors we just drew are possible center locations for arcs which pass through C 1C2and D1D2,respectively. Their intersection is thus a common center.

4.

Select a convenient radius and draw an arc about O4to intersect O4C1and O4C25. Label the intersection points B1and B2.6. Do steps 2 through 8 of Example 3-1

Note: Free choices

1. position of joint B2. length of O2B

this problem reduced to Example 3-1 once the correct rocker link geometry was found.

What happens if C1D1and C2D2are parallel to each other?


5/16

5

Example 3-2


6/16

6

Example 3-3: 2 positions with complex coupler output

Here we wish to move a line on the couplerthrough two positions. This is useful in situations whererocker output is not up to the job (e.g. parallel movement.)

1.

Draw link CD in two desired positions (coupler link)2.

Bisect lines C1D1and C2D2and draw perpendicular bisectors.3.

Select convenient points on bisectors for O2, O4.4.

Connect O2with C1: link 25.

Connect O4with D1: link 4

Note: this time our arc centerpoints are separated since we are defining two links instead of just one.Since this fourbar has two rocker links, we must construct a driver dyad to drive it.

6.

Select a point on link 2, call it B1.7.

Draw arc around O2to intersect B1and O2C2. Call the intersection B2.8.

Rest is same as example 3-1.

Free choices:

1.

length of link 22.

length of link 43.

position of joint B4.

length of O6B

Note: there is a problem with this linkage, in that it probably cannot move smoothly betweenpositions 1 and 2. Linkage design is an iterative process!


7/16

7

Example 3-3


8/16

8

Example 3-4: 3 positions with complex coupler output

Here we wish to synthesize a linkage which will move the coupler through 3 positions.

1.

Draw link CD in its three desired positions.

2.

Draw construction lines between C1C2, C2C3, etc.3.

Bisect lines C1C2 and C2C3 and extend perpendicular bisectors until they intersect. Label theintersection O2.

Note that in this case we have three points which define an arc about O2. The intersections of thebisectors is the center of the arc.

4.

Bisect lines D1D2and D2D3and extend perpendicular bisectors until they intersect. Label theintersection O4.

5.

Connect O2with C1and call it link 2. Connect O4with D1and call it link 4.6.

Construct a driver dyad to drive the linkage between its two extreme positions.


9/16

9

Example 3-5: 3 positions with alternate moving pivots

In some cases we wish a line on the coupler which is distinct from the line between the two pivotsto move through a specified path. This might happen in the case of a windshield wiper mechanism,where we do not want to locate pivot points in the middle of the windshield.

1.

Draw line CD in the three desired positions.2.

Define new attachment points E and F on coupler link. These points have a fixedrelationship with CD. Draw E1F1on link 1. Draw a cirele of radius E1C1about C2. Draw acircle of radius E1D1about C2. The intersection of these circles is point E2. Use the sameprocedure to find F2, E3and F3.

Explanation: Essentially we are using similar triangles to transfer the relationship between C1D1andE1to C2D2and E2. The first circle we drew makes the distance E2C2the same as the distance E1C1.The second circle makes the distance E2D2 the same as E1D1. Since the distance E1C1=E2C2, thethree sides of the triangles E1C1D1and E2C2D2are the same, thus, similar triangles.

3.

Draw construction lines E1E2and E2E3. Repeat for F.4.

Draw perpendicular bisectors of E1E2, E2E3, F1F2and F2F3.5.

Label intersections O2and O4.6.

Connect E with O2and F with O4to make links 2 and 4, respectively.7.

Construct a driver dyad to drive the linkage.

Note: free choices

1.

Locations of E and F on coupler link

There is a problem with this construction! The pin at O4appears to interfere with link 2. It would

be nice if there were a procedure to specify the locations of the fixed pins and find the locations ofthe moving pins on the coupler (points E and F.)


10/16

10

Example 3-5


11/16

11

Example 3-6: 3 position synthesis with specified fixed pivots

This problem is called an inversion problem, that is, we will pretend a moving link is fixed, andperform the analysis from there.

1.

Draw line CD in its three desired positions. Draw ground link O2O4in its position. For theremainder of the analysis we will rename O2O4G1H1.

2. Take the relationship between line C2D2and O2O4and transfer it to C1D1. Call the new lineG2H2.

3. Take the relationship between line C3D3and O2O4and transfer it to C1D1. Call the new lineG3H3.

Note: what we are doing here is pretending that C1D1is afixedlink and that the ground link movesaround it. This is the so-called inversion part of the analysis. The three positions that themoving ground link takes bear the same relationship with fixed C1D1 as those between thefixed ground link and moving CD.

4. Draw construction lines G1G2and G2G3. Repeat for H.5. Draw perpendicular bisectors of G1G2, G2G3, H1H2and H2H3.6. Label intersections E1and F1. E1F1is the coupler link in position 1.7. Connect E with O2and F with O4to make links 2 and 4, respectively.8. Construct a driver dyad to drive the linkage.

To find the layout of the rockers and coupler in positions 2 and 3, we must repeat steps 2 and 3,transferring the relationships to lines C2D2and C3D3, respectively.


12/16

12

Example 3-6


13/16

13

Quick-Return MechanismsIn the preceding examples, we didnt worry about how long it took for the linkage to travel its path.In the case of the windshield wiper mechanism, for example, it is probably best for the linkage tospend an equal time pushing the wipers to the left as to the right. But for some mechanisms, timingis critical. Consider the egg lifter mechanism shown above. The purpose of this mechanism is tolift an egg (or other delicate object) from one conveyor belt to another. We wish to lift the eggslowly and gently, so as not to damage it, but the return stroke (when the egg is not on the platform)

should be quick, so as not to waste time.

To see how we might accomplish this, consider the cartoon above. As you can see, the rockerspends and equal time moving forwards as backwards, because the crank sweeps out 180 for eachmotion. To change the timing, we need to tailor the angle of sweep to our needs.

Egg

180oforward

180oback


14/16

14

By simply lowering the fixed crank pivot, we change the parts of the cranks rotation that are spentmoving forward or backward. In the figure above, the crank spends 200 of rotation moving therocker forward and 160 of rotation moving the rocker backward. Let us call the forward angle and the return angle . Now define the time ratioas

In most cases, the time ratio will be specified, and we must construct the linkage to accomplish thismotion. Finally, well use the construction angle to help design our linkage.

200oforward

160oback


15/16

15

Example 3-7

2-position quick-return mechanism

Problem Statement: design a driver dyad which sweeps out 40 of rocker rotation and has a timeratio of 1.25:1

| |

1. Draw output link O4B in desired position2. Draw a construction line through B1at a convenient angle

3. Draw a line through B2at an angle with the first line4. Label the intersection O25. Measure lengths O2B1and O2B2

Note:

Solve for L2and L3:

( )

( )

6. Measure angle


16/16

16

Example 3-7

Lab 2- Graphical Linkage Synthesis

Documents

Transcript of Lab 2- Graphical Linkage Synthesis