Chap 11

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Synthesis of LinkagesSynthesis of LinkagesChap-11Chap-11

Theory of Machines and Mechanisms Theory of Machines and Mechanisms 33rdrd edd edd

J.J. Uicker, G.R. Pennock, J.E. ShigleyJ.J. Uicker, G.R. Pennock, J.E. Shigley

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Synthesis Synthesis The process of contriving a scheme or a method of accomplishing a given purpose. It is the process of prescribing the sizes, shapes material compositions, and arrangements of parts so that the resulting machine will perform the prescribed task.Kinematic Synthesis:The design or the creation of mechanism to yield a desired set of motion characteristics.

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1. Type Synthesis: It refers to the kind of mechanism selected; which may be a:• Linkage • A gear sytem

• Belts and pulleys or

• A cam system

This phase of design problem involves • Manufacturing processes• Materials• Safety

• Space and • economics

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1. Number Synthesis: It deals with the number of links and joints or

pairs that are required to obtain a certain mobility.

1. Dimensional Synthesis:Determining the dimensions of the individual

links.

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1. Function Generation: It is a design requirement to cause an output member to

rotate, oscillate, or reciprocate according to a specified function of time or input motion. This is called function generation e.g.:

Synthesis of a four-bar linkage to generate y=f(x)Where x = the motion (crank angle) of the input crank andy = the motion of the output rocker.1. Path Generation: It refers to the problem in which a coupler point is to generate

a path having a prescribed shape. e.g: circular arc, elliptical, or straight line.

1. Body Guidance: Here we are interested in moving an object from one position

to another. i.e. translation or a combination of translation and rotation. e.g: scoops & bulldozer blades must be moved through a series of prescribed positions.

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Two-Position Synthesis of Slider-Two-Position Synthesis of Slider-Crank MechanismCrank Mechanism

• The limiting positions are found by drawing arcs of radius r3-r2 and r3+r2.

• All points on the coupler of the isosceles slider crank (r3= r2) generate elliptical path.• Else the path generated the points on coupler are not elliptic

but are always symmetric about O2B.

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Two-Position Synthesis of Two-Position Synthesis of Slider-Crank MechanismSlider-Crank Mechanism

• Changing the offset e results in obtaining certain results. e.g. • the stroke B

1B

2 > 2r

2 (always)

• crank angle required to execute forward stroke is different from return stroke. (quick return mechanism for slower working stroke)

General or Offset slider crank Mechanism.The limiting positions B

1B

2 are found in the

same manner

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Two-Position Synthesis of Crank Two-Position Synthesis of Crank & Rocker Mechanism& Rocker Mechanism

• Crank executes angle ψ ------- rocker moves angle ø• Crank executes angle 360o-ψ ----- rocker forward angle ø

• The limiting positions are found by drawing arcs of radius r3-r2 and r3+r2. (same as for slider crank)• Note that the crank &

rocker form a single straight line at the extreme positions.

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Synthesis problem:• Given: angle ø and time ratio Q (or α )• To find: Dimensions or geometry

Time ratio:

Q=time of advance stroketime of return stroke

Q=180 + α180 - α

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Crank-Rocker Mechanism with Crank-Rocker Mechanism with Optimum Transmission angleOptimum Transmission angle

Brodell and Soni’s synthesis of crank-rocker: for Q = 1 and γmin = 180o - γmax

Where γ = transmission angle

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The results of these equations are plotted as a design chart.•γ>30o ----- for good quality motion. •γ should be even larger if higher speeds are involved.

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The resulting linkage should always be checked to ensure that crank-2 is capable of rotating through a complete circle.

Hall and Soni’s synthesis of crank-rocker:When Q ǂ 1 the solution for optimum transmission angle is more difficult.

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Three Position SynthesisThree Position Synthesis

In obtaining the inversions, great care must be taken to ensure that rotation is made:•in the correct direction and •with the correct angles

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Three Position SynthesisThree Position SynthesisProblem:Synthesize a linkage in which it is desired to determine the dimensions of a linkage in which the output lever is to occupy three specified positions corresponding to three given positions of the input lever.

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Given: θ2, ψ12, ψ23, ψ13, ø12, ø23, ø13

To find: θ4, r4

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Four Position Synthesis; Four Position Synthesis; Point-Position ReductionPoint-Position Reduction

• The linkage is made symmetrical about the frame centerline O2O4.• This makes two of the A’ points coincident.• Thus resulting in 3 equivalent A’ points through which a

circle is drawn as in 3-position synthesis.Problem:Let us synthesis a linkage to generate the function y=log x

over the interval 10 ≤ x ≤ 60 using an input crank range of 120o and an output range of 90o. The angle ψ is evaluated for the four design positions from the equation ψ = ax + b and from the boundary condition

ψ = 0 when x = 10 and ψ = 120o when x = 60Similarly ø is evaluated from equation ø = cy + d (with 0 ≤ ø ≤ 90o)

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•Only the length of the input rocker O2A can be specified.• The distance O2O4 is

dependant upon the values of ψ and ø.

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Precision Position; Structural Precision Position; Structural Error; Chebychev SpacingError; Chebychev Spacing

• The synthesis of the preceding sections are of the function generation type. i.e. we are trying to find the dimensions of a linkage with the input / output relation y = f(x).• In general a mechanism has only a limited number

of design parameters. i.e. a few link lengths starting values of input & output angles a few more.• Therefore, except for very special cases, there is no

exact solution over the entire range of travel.

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Precision Position; Structural Error; Precision Position; Structural Error; Chebychev SpacingChebychev Spacing

Precession Positions:The two, three and four positions of the Linkage are called

Precession Positions.• The implicit assumption is that if the design fits the

specifications at these few positions, then it will probably deviate only slightly from the desired function between the precision positions.

Structural Error:The theoretical difference between the function produced by

the synthesized linkage and the function originally prescribed.• The structural error in a four-bar linkage solution can be

less than 4%.• Structural error usually exists even if There is no graphical error No mechanical error due to imperfect manufacturing.

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• The amount of structural error in the solution can be affected by the choice of the precession positions.• One of the problem of linkage design is the selection of a

set of precession positions which minimize the structural error.

Chebychev Spacing:A good trial for the spacing of precession positionsFor n precision positions in the range xo ≤ x ≤ xn+1

Chebychev Spacing according to Freudenstein and Sandor

Example:Devise a linkage to generate a function y = x0.8

Over the range 1 ≤ x ≤ 3 using 3-precesion positions

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Graphical approach:Construct a circle of diameter equal to the range Δx. i.e.:

Δx = xn+1 – x0

Inscribe a regular polygon with sides = 2nWith 1st side symmetric about x-axis.

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• It is a good approximation of precision positions that reduces the structural error in design.

Branch defect:It refers to a possible completed design that meet all the

prescribed requirements at each of the precision positions, much which cannot be moved continuously between these positions without being taken apart and reassembled.

Order defect:It refers to a developed linkage that can reach all of the

precision positions, but not in the desired order.

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The Overlay MethodThe Overlay Method

Synthesis of a function generator by this technique is the• easiest.• quickest• poor accuracy• can employ as many precision positions as

are desired

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Example:Design a function generator to solve the equation

y = x0.8 , 1 ≤ x ≤ 3Take six precision positions with uniform spacing of the

output rocker.

Position x Ψ,deg y Φ,deg

1 1 0 1 0

2 1.366 22.0 1.284 14.2

3 1.756 45.4 1.568 28.4

4 2.16 69.5 1.852 42.6

5 2.58 94.8 2.136 56.8

6 3.02 121.0 2.420 71.0

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Coupler-Curve SynthesisCoupler-Curve Synthesis

Synthesising a linkage to generate a path:• choose upto six precision positions along

the path.• for a successful synthesis the tracing point

will pass through each precision position.•However, because of branch or order

defects, the final result may or may not approximate the desired path.

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Coupler-Curve Synthesis Coupler-Curve Synthesis (2-positions)(2-positions)

Note: C1 & C3 are especially selected so that the mid normal c13 passes through O4

< C1O4C3 = < A1O4A3

∆ C3A3O4 ≡ ∆ C1A1O4

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• In this case it is possible to make two point reduction.• Choose two pairs out of these positions for reduction

purposes.

Figure 11.19

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Figure 11.20

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Figure 11.21

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Cognate Linkages; The Roberts-Cognate Linkages; The Roberts-Chebychev TheoremChebychev Theorem

An unusual property of the planar four-bar linkage: •There is not one but three four bar linkages that generate the same coupler curve. (Roberts and Chebychev)

Figure 11.22

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Cognate Linkages; Cognate Linkages; Cayley DiagramCayley Diagram

• Unfasten O1, O2 & O3 • Pull all 3 apart until a straight line is formed by the crank,

coupler & follower of each linkage.

Figure 11.23

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• If tracing point P is on the straight line AB or its extensions all 3 linkages are compressed into a single straight line.Locate O

3 on extension of O

1O

2 in ratio AB : BP

Construct parallelogramsa)O

1A

1PA

b)O2B

2PB

c)O3C

1PC

2

Figure 11.24

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Hartenberg & Denavit •ω9 = ω2 = ω7

•ω10 = ω3 = ω5

•ω8 = ω4 = ω6

Figure 11.22

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Bloch’s Method of SynthesisBloch’s Method of Synthesis

• Replace the links of a four-bar linkage by position vectors.

Figure 11.25

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FREUDENSTEIN'S FREUDENSTEIN'S EQUATIONEQUATION

Freudenstein offers the following suggestions, helpful in synthesizing such function generators:

• The total swing angles of the input & output members should be less than 120o.

• Avoid the generation of symmetric functions such as y = x2 in the range -1 ≤ x ≤ 1.

• Avoid the generation of functions having abrupt changes in slope.

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Analytical Synthesis using Analytical Synthesis using Complex AlgebraComplex Algebra

• Write complex Algebra equations describing the final linkage in each precision position.

• Magnitudes of these complex vectors do not change but angles do change from one position to another.

• By writing equations at several precision positions, obtain a set of simultaneous equations to be solved for the unknown magnitudes & angles.

Example:Design a mechanical strip chart recorder with the final design

concept shown in figure. The signal to be recorded is available as a shaft rotation having a range of 0 ≤ ø ≤ 90o clockwise. This rotation is to be converted into a straight line motion of the recorder pen over a range of 0 ≤ s ≤ 4 in to the right with a linear relationship between changes of ø and s.

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• Before we finish we should evaluate the quality of our solution by analyzing the designed linkage.

• Using equations of Chapter # 2.• The location of coupler point was found for 20 equally

spaced crank increments.• There is structural error;

the coupler curve of the recording pen tip is not exactly straight. The displacement increments are not perfectly linear over the range

of travel• However, the solution is quite good

The deviation from straight line is less than .020 in or 0.5% of the travel.

• The structural error vanishes at the 3 precision positions.• The design may be improved slightly by using additional

precision positions.

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Synthesis of Dwell MechanismsSynthesis of Dwell MechanismsHrones and Nelson Atlas

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Synthesis of Dwell MechanismsSynthesis of Dwell MechanismsOverlay for use with the Hrones and Nelson AtlasIt reveals immediately •The radius of curvature of the segment•The location of pivot point D.•The swing angle of the connecting link

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Synthesis of Dwell MechanismsSynthesis of Dwell Mechanisms

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Intermittent Rotary MotionIntermittent Rotary MotionThe Geneva Wheel or Maltese Cross:•A cam like substance that provides intermittent rotary motion. •Widely used in both low & high-speed machinery e.g:

• Stop to prevent over winding of Watches• Indexing of a spindle, turret, or worktable• Motion picture projectors to provide intermittent advance of the

film.

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Intermittent Rotary MotionIntermittent Rotary MotionThe six slot Geneva Mechanism:•The centerlines of the slot & cam are perpendicular at engagement &

disengagement.•One revolution of the crank rotates a fraction of the Geneva wheel.•The amount of wheel rotation depends on the no. of slots.•The circular segment effectively locks the wheel during disengagement.

Design of Geneva Mechanism:•The design is initiated by specifying:

Crank radius Roller diameter Number of slots (3-12)

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Intermittent Rotary MotionIntermittent Rotary Motion

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Intermittent Rotary MotionIntermittent Rotary Motion• It is desirous to reduce the wheel acceleration wheel acceleration > inertia forces > wear on the sides of the slot

• Use a curved slot (reduces acceleration but increases deceleration thus wear of one side of slot)

• Use the Hrones –Nelson atlas to synthesis a 4-bar linkage (During the driving period the roller should have a low value of

acceleration)

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The Inverse Geneva Wheel:• The wheel & crank

rotates in the same direction.

• Requires less space

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Problem 11.3Problem 11.3

Determine a set of link lengths for a slider crank linkage such that the stroke is 16 in and the time ratio is 1.25.

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Problem 11.14Problem 11.14The figure shows a function-generator linkage

in which the motion of rocker 2 corresponds to x and the motion of rocker 4 to the function y = sinx (0 ≤ x ≤ π/2). Use four position points and Chebychev spacing and synthesize a linkage to generate the function.

Chap 11

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