chapter 3 v2008 v3 color noprint

Copyright 2008 Department of Mechanical Engineering - University of Massachusetts Lowell Chapter 3 - 1

Mechanical Engineering - 22.321 Design of Machinery

Chapter

GRAPHICAL LINKAGE SYNTHESIS



3.0 Introduction

Engineering Design involves

1. Synthesis2. Analysis



3.1 Synthesis Qualitative Synthesis creation of potential solutions

in the absence of a well defined algorithm which configures or predicts the solution.

Real design problems involve more unknowns than equations. Need to do some type of qualitative

judgment. Use: Drawings, Simulations,

Paper Models Qualitative design by successive

analysis. Type Synthesis refers to the definition of the proper

type of mechanism best suited to the problem (form of qualitative analysis)



Design is essentially an exercise in trade-offs

Quantitative Analysis (a.k.a. Analytical Analysis) Generation of one or more solutions of a particular type and for which a synthesis algorithm is defined. Set of Eqns

But in general: # Eqns < # VariablesBest done in some computer code.

Dimensional AnalysisDetermination of linkage sizes required to accomplish the desired motions.



3.2 Function, Path and Motion Generation

Function Generation the correlation of an input motion with an output motion in a mechanism.

e.g. gun aiming systems (mechanical analog, computer) and computer controlled servos

Path Generation control of a point in the plane so that it follows some prescribed path.

Typically uses at least four bars. Motion Generation control of a line in the

plane so that it follows some prescribed set of sequential positions.

e.g. Bucket on a bulldozer.



Have a potential solution evaluate its quality

Toggle Position is a position s.t. two of the moving links are collinear.

Only undesirable if it prevents the mechanism from moving from one desired position to another.

3.3 Limiting Conditions



A toggle may be desirable for locking.



Transmission Angle the angle between the output link and the coupler

Taken as the absolute value of the acute angle at the intersection of two links

Varies continuously throughout the motion


Mechanical Engineering - 22.321 Design of Machinery T2 induces an axial force in Link 3. At Point D, the linkage has forces normal to and radially inline with

Link 4. Radial component only increases friction at pivot O4. Tangential (normal to Link 4) produces torque.

= 90o is optimal. In design, keep > 40o

To promote smooth running and good force transmission.



3.4 Dimensional Analysis

The determination of the proportions (lengths) of links necessary to accomplish the desired motions.



Example 3-1



Example 3-2



Example 3-3



Example 3-4



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Example 3-5



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Example 3-6



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Example 3-7



Example 3-8



3.5 Quick Return Mechanisms Desire work done in forward stroke and quick return in

back stroke (return). See Fig 3-12 (next page) Fig 3-4a shows a linkage w/ equal time in forward and

return strokes.Why?

Because the center of the crank is inline with the chord of the rockers extreme points.



ANIMATE ME

Example 3-9



Time in forward and return strokes is different. TR = Time Ratio = /

Recall that + = 360o Use a construction angle , where

= 180o - = 180o - To synthesize the linkage

See Example 3-9 (Fourbar crank-rocker)

This design works well with TR s down to 1:1.15



ANIMATE ME



Sixbar Quick Return Good for larger TR s up to 1 : 2.

Strategy1. First design a fourbar drag link mechanism

which has the desired TR between its driver crank and its driven crank (dragged crank)

2. Then add a dyad (Twobar) output stage driven by the dragged crank.

See Example 3-10



Example 3-10



ANIMATE ME



Crank Slider Quick Return

Capable of larger TRs than fourbar quick-return linkage.

Easy to synthesize. Simply move O4 until desired TR is achieved.

Whitworth Mechanism behaves as a double crank with both pivots making full rotations.

Crank-Shaper Mechanism Driving crank is shortest link Crank-Rocker linkage

See Fig 3-14



Crank-Shaper Mechanism Driving crank is shortest link Crank-Rocker linkage

ANIMATE ME



3.6 Coupler CurvesCoupler Undergoes complex motion Coupler curve is always closed

Fourbar Coupler CurvesCusp Sharp point on the curve associated w/ zero

velocity (but acceleration 0).Crunode Creates a figure-eight shaped curve which

contains a double point at the crossover. Two slopes, i.e. velocities.

The Hrones and Nelson Atlas of Fourbar Coupler Curves is a good reference. (On reserve in the library BE GENTLE WITH IT PLEASE!) Better to use online version



ANIMATE ME



3.7 Cognates Consider the problem:

A good solution to a linkage synthesis problem is found to satisfy path-generation constraints

BUT linkage has fixed pivots in inappropriate locations for attachment

or Linkage is non-Grashof when Grashof is required.

Use Cognate A linkage of different geometry, which generates the same coupler curve

See Figs. 3-24 and 3-25



Chebychev discovered that any fourbar coupler curve can be duplicated w/ a geared fivebar linkage.

Useful to avoid toggle lock-up.See Fig 3-28.

The fivebar is constructed by simply drawing Link 6Link 2, Link 7Link 4, Link 5A1P and Link 8B1P

A three-gear set is required to couple links 5 and 8 with a ratio of +1 (i.e. Dia5 = Dia8 and have same direction of rotation)



3.8 Straight Line Mechanisms A common application of coupler curves is for

the generation of approximate straight lines. See Fig 3-29 a d

Note: Hoeckens and Chebychev are cognates of one another

To generate an exact straight line requires more than four links.

The geared fivebar linkage can generate an exact straight line.



Peaucellier discovered an exact straight-line mechanism of eight bars. (Fig 3-29e) Links 5, 6, 7 and 8 form a rhombus. Links 3 and 4 are of equal length. When O2 = O4, Point C generates an arc of

infinite radius.By moving O2 left or right and only changing the length of Link 1: Point C will generate true circle arcs with

radii much greater than any one of the link lengths.



3.9 Dwell MechanismsDwell = zero output motion for some

nonzero input motion.

Dwells are often achieved by using cams and followers (but w/ cost!)

Can also achieve dwells with pure linkages of only links and pin joints (at a lower cost than cams)



Single Dwell Linkages

Results in a sixbar linkage First find a fourbar linkage,

then add a dyad Design a fourbar with a coupler curve that

contains an approximate circular arc portion (good to use coupler curve atlas)

See Example 3-11



Example 3-11


Mechanical Engineering - 22.321 Design of MachineryDouble Dwell Linkages

Can use a Fourbar coupler curve to create a double-dwell output motion.

Approach 1 Similar to Example 3-11 Need a coupler curve which has two approximate circle

arcs of the same radius but with different centers

Both of these arcs are either concave or convex. Link 5 of length equal to the radius of the arcs so that it

and Link 6 will remain nearly stationary at each of the arc centers while the coupler traverses arcs of its path.

Multiple dwells can be created by higher-order linkages which possess coupler curves with multiple, approximate circle arcs.



Approach 2 Use a coupler curve with two approximate straight-line

segments of appropriate duration Attach a pivoted slider block (Link 5) to coupler path

point Link 6 will slide on Link 5 Choose pivot point O6 at the intersection of the

extensions of the straight line segments While Link 5 is moving along the straight line path, Link 6

will not experience any angular motion

Note: Approximate Straight Line allows jitter in Link 6

(see Example 3-12)



3.10 Other Useful LinkagesConstant Velocity Piston Motion

Fourbar slider crank linkage is one of the most frequently used in machinery

Used in IC engines Consider a situation where a constant velocity is

needed, e.g. piston pump for metering fluids which needs to be constant during the delivery stroke



Large Angular Excursion Rocker Motion It is often desired to obtain a rocking motion through a

large scale w/ continuous rotary input In Grashof linkage, rocker motion is limited to 120o to

keep transmission angles > 30o

To get a larger oscillation and good transmission angles requires six links



Remote Center Circular Motion Used when rotary motion is needed but the center of

that rotation is not available to mount the fixed pivot of the crank

The linkage will generate circular motion in the air

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