ME154-02 graphical linkage synthesis 202002 01 PDF

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Graphical Linkage Synthesis

San José State University | S. J. Lee | 2020 Feb 01

Linkage synthesis 

Linkage synthesis is the determination of a solution to a linkage design problem.

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Dimensional synthesis of a linkage is the determination of the lengths of the links necessary to accomplish required motion.

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Linkage synthesis may involve qualitative approaches, quantitative methods, or a combination of both.

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Graphical linkage synthesis is often the simplest and fastest approach to dimensional synthesis for four-bar linkages having up to three design positions.

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Analytical and/or numerical methods are usually needed for linkages with many links and/or many design positions.

San José State University | S. J. Lee | 2020 Feb 01

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Generating outcomes 

Motion generation is the control of a rigid link in a reference frame such that it assumes some prescribed set of orientations and locations. Planar motion generation applies to a the orientation and location of a line segment in a plane.

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Path generation is the control of a point such that it follows a prescribed path.

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Function generation is a prescribed relationship between an input motion and an output motion, often with respect to time.

San José State University | S. J. Lee | 2020 Feb 01

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Generating outcomes – what do these highlight?

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Transmission angle 

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The transmission angle, defined as the (acute) angle between the output link and the coupler, provides an indication of how well force and velocity are transmitted through the linkage. Transmission angle varies as the linkage goes though its range of motion, and is meaningful only if the input and output links are pivoted to ground. Only the tangential force component promotes rotation of the output link, so a large transmission angle is desirable. (40º conventionally “good enough”, and 90º is the ideal best.)

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Mechanical advantage 

Mechanical advantage compares the ratio of output force to input force, and many machines are designed to achieve favorable mechanical advantage. Fout Fin

Image from Design of Machinery, 5th ed. by R. L. Norton, © 2012, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Toggle positions 

A toggle position is a limiting condition for the motion of a linkage, wherein there occurs colinearity between two of the moving links and motion may be self-impeded.

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Linkage synthesis generates paths and trajectories, but inspection in some form (e.g. animation, prototype model or software program) is usually needed to check for toggle positions.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Design example with toggle mechanism

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Also seen in many folding tables, piano benches, and similar applications.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Two-position graphical linkage synthesis

San José State University | S. J. Lee | 2020 Feb 01

Two-position synthesis – Example 1 

Design a four-bar Grashof crankrocker to give 45º of rocker rotation from a constant speed motor input.

1.

Place O4 conveniently from the extreme positions B1 and B2 for the specified 45º.

2.

Place O2 conveniently somewhere on the extension of chord B1B2. Bisect the chord B1B2, and draw a circle of the same radius about O2. Assign O2A1 as Link 2.

3.

4.

Place A1 and A2 at the circle’s intersection with the extended B1B2. Assign A1B1 as Link 3.

5.

Verify Grashof condition, and iterate (e.g. relocate O2 or O4) if non-Grashof.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Two-position synthesis – Example 2 

Design a four-bar linkage to move link CD from position C1D1 to C2D2.

1.

Draw construction lines C1C2 and D1D2.

2.

Place O4 at the intersection of the perpendicular bisectors of C1C2 and D1D2. This is called a “rotopole”.

3.

Place B1 and B2 equidistant from O4 along O4C1 and O4C2, respectively.

4.

Place O2 conveniently somewhere on the extension of chord B1B2.

5.

Bisect the chord B1B2, and draw a circle of the same radius about O2. Assign O2A1 as Link 2.

6.

Place A1 and A2 at the circle’s intersection with the extended B1B2. Assign A1B1 as Link 3.

7.

Verify Grashof condition, and iterate if non-Grashof.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Two-position synthesis – Example 3 

Design a four-bar linkage to move link CD from position C1D1 to C2D2.

1.

Place O2 “conveniently” on the perpendicular bisector of chord C1C2.

2.

3.

Place O4 “conveniently” on the perpendicular bisector of chord D1D2. Assign O2C1 as Link 2.

4.

Assign C1D1 as Link 3.

5. 6.

Assign O4D1 as Link 4. Check for toggle positions.

7.

Evaluate transmission angles.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Two-position synthesis – Example 4 

1. 2. 3. 4.

5.

6.

Given a four-bar linkage to move link CD from position C1D1 to C2D2, add a dyad to limit the motion of the linkage entirely betweeen the two design positions. Place a point B1 conveniently somewhere on Link 2. Use as circular arc about O2 to find the corresponding point B2. Place O6 conveniently somewhere on the extension of chord B1B2. Bisect the chord B1B2, and draw a circle of the same radius about O6. Assign O6A1 as Link 6. Place A1 and A2 at the circle’s intersection with the extended B1B2. Assign A1B1 as Link 5. Verify Grashof condition, and iterate if non-Grashof.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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San José State University | S. J. Lee | 2020 Feb 01

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Non-uniqueness of driver dyads

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

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Graphical linkage synthesis using CAD O4

A1 B1

B2 Q2 P2

P1

O2

A2

Link 1 (ground) includes O2 and O4 . Link 2 (crank) is O2A . Link 3 (coupler) is AB . Link 4 (rocker) is O4B .

Q1

Position 2 Position 1

San José State University | S. J. Lee | 2020 Feb 01

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Three-position graphical linkage synthesis

San José State University | S. J. Lee | 2020 Feb 01

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Three-position synthesis – Example 1 

Design a four-bar linkage to move link CD from position C1D1 to C2D2 to C3D3.

1.

Draw construction lines C1C2, C2C3, D1D2, and D2D3.

2.

Place O2 at the intersection of the perpendicular bisectors of C1C2 and C2C3. Place O4 at the intersection of the perpendicular bisectors of D1D2 and D2D3.

3.

5.

Assign O2C1 as Link 2, C1D1 as Link 3, O4D1 and as Link 4. Check for toggle positions.

6.

Evaluate transmission angles.

4.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Three-position synthesis – Example 2 

Design a four-bar linkage having no toggle positions, that is able to move link CD from position C1D1 to C2D2 to C3D3.

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The given conditions as shown would result in toggle positions because colinearity happens to occur in position 1 and position 2. Using different moving pivots (E and F) other than C and D can still achieve the required design positions, as long as E and F have a rigid relationship to C and D. Different moving pivots changes the lengths of links in the four-bar linkage and further adds design flexibility for locating the fixed pivots O2 and O4. Not only can alternate pivots avoid toggle positions, but they can simplify driving dyad connection and also improve transmission angles.

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Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Using alternate moving pivots

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Path generation and coupler curves

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Coupler curves  

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A coupler in a linkage in general has complex motion and provides the greatest variety of paths that can be traced. Varying with the location of coupler point P with respect to connected nodes (the moving pivots of a four-bar mechanism, for example), there can be an infinite number of paths generated. Two special features that occur in some coupler paths are: A cusp is a sharp point on the curve that has instantaneously zero velocity. A crunode is a point in a multi-loop curve that has two different velocities depending on which portion of the curve is traced.

San José State University | S. J. Lee | 2020 Feb 01

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San José State University | S. J. Lee | 2020 Feb 01

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Examples of coupler curves

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Coupler points

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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San José State University | S. J. Lee | 2020 Feb 01

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Partial “atlas” of coupler curves

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Examples of straight-line mechanisms

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Prescribed timing

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Quick-return mechanisms 

When the end positions of a rocker are driven with a constant-speed crank connected at unequal angles, the forward and return strokes occur at different speed.

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The time ratio TR is the ratio of angles  and , which sum to 360º and are proportional to the fractional time for each stroke.

TR  

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 

As with symmetric timing, the couplerplus-crank length must match each of the end positions of the rocker. Transmission angle puts a practical limit on extreme differences in angle.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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Quick-return graphical linkage synthesis example 

1. 2.

Design a four-bar linkage to swing a rocker 45º with a time ratio of 1:1.25. Calculate the angles  and . Draw construction lines conveniently through end positions B1 and B2, and intersect the lines with the following angle at a grounded pivot O2.   180    180  

3.

Establish compatible coupler & crank lengths connecting O2 and the points B1 and B2.

4.

Check Grashof condition and transmission angle. Iterate if necessary.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

TR 

 

    360 

360  1  TR

San José State University | S. J. Lee | 2020 Feb 01

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Dwell mechanisms 

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A dwell is zero output motion for some nonzero input motion. One obvious approach would be to use a cam. How?

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Simple linkages of links and pin joints can also achieve dwells, often with lower cost and higher reliability.

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Synthesizing a dwell is based on “pseudo-arcs” along the curve traced by a selected point on the coupler, and are thus usually only approximate dwells. An extensive coupler curve atlas is generally needed.

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Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

San José State University | S. J. Lee | 2020 Feb 01

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San José State University | S. J. Lee | 2020 Feb 01

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Dwell linkage synthesis example 1.

2.

3.

4.

The dwell is based on a special pivot point D that is at the center of a “pseudo arc” along the coupler curve. Another point E is found on the perpendicular bisector of the pseudo arc, sharing the same Link 5 as D but at a different extreme distance on the coupler curve. A pivot O6 is placed perpendicular to DE, and its distance from DE establishes the angle swept by output Link 6. Output Link 6 pivots about O6, but during a portion of the motion, Link 6 is approximately stationary even though Link 5 continues to move along the coupler curve.

Reference: Design of Machinery, 3rd ed. by R. L. Norton, © 2004, McGraw-Hill

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