Technique for Prescribing Topology of Linkages: Degrees of Freedom and Force Balancing, Lecture notes of Kinematics

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THE KINEMATICS (^) AND (^) VIBRATION OF PLANAR (^) LINKAGE (^) MECHANISMS

Abstract

This (^) thesis (^) reports (^) an investigation (^) into three (^) problems encountered in^ the^ design^ of linkage^ mechanisms, namely kinematic (^) synthesis, balancing (^) of inertia forces (^) and vibration analysis. A (^) general (^) method of (^) synthesizing (^) planar linkages (^) with (^) pin and sliding joints^ using (^) an Optimization^ approach has^ been investigated. (^) A (^) concise but (^) easily interpreted (^) technique for (^) prescribing the topology (^) of linkages formed by (^) connecting pairs of links^ together^ has^ been^ developed.^ The^ displacement analysis (^) of (^) a linkage^ is^ achieved (^) using a direct^ method which is (^) considerably faster (^) than (^) alternative techniques. A (^) non- linear (^) optimization (^) algorithm has been (^) modified to (^) cater for non-linear constraints (^) such as transmission^ angle. These techniques have^ been (^) incorporated into (^) a computer program. Two (^) case-studies (^) of using the (^) program are given. The first (^) is (^) the (^) synthesis (^) of a (^) six-bar linkage (^) for (^) a motorcycle rear (^) suspension (^) such that^ a constant centre distance^ is

maintained between^ the^ chain-wheels as the^ suspension deflects. The (^) second (^) concerns the (^) modification of two linkages, (^) containing eight (^) and ten^ links^ respectively, to^ give (^) an improved^ knitting action for^ a warp-knitting (^) machine. Operating linkages^ at high^ speeds can result in^ rapidly varying forces^ acting (^) on the^ frame^ due^ to^ the^ mass of the^ moving links. (^) A (^) procedure to determine (^) suitable counterweights to balance (^) these forces (^) has been (^) developed. (^) Since (^) adding the counterweights (^) may double^ the^ total^ mass of the^ linkage,^ the links (^) should have (^) minimum mass. If (^) the (^) mass (^) of a link^ is^ reduced too far,^ the link^ may vibrate (^) and so detrimentally^ affect the^ performance of the linkage. (^) Accordingly (^) the final (^) part reports an investigation into (^) the forced (^) vibration, assuming (^) stability, of a 'Uniform, pin-jointed, binary^ link.^ The^ equations of motion are derived and stability boundaries^ determined.^ The^ theoretical^ predictions are (^) compared with (^) experimental results from^ the^ coupler of (^) a four-bar (^) linkage. e

UNIVERSITY (^) OF (^) NEWCASTLE (^) UPON (^) TYNE

THE KINEMATICS AND VIBRATION

0FPLANARLINKAGEMECHANISMS,

by

K. OLDHAM,^ M. A., (^) C. Eng., (^) M. I. Mech. (^) E., A. T. I.

A (^) Thesis (^) submitted for (^) the (^) Degree (^) of Doctor (^) of Philosophy (^) in

the Faculty (^) of Applied (^) Science, (^) University (^) of Newcastle (^) upon Tyne.

November 1977

P-R-EF-A,CE

The (^) work reported in^ this thesis (^) was (^) carried out under three^ Science^ Research^ Council^ grants^ and^ a Department (^) of Industry^ contract. The^ original intention

was to^ devote^ it^ to^ an^ investigation^ into^ vibrations^ in

planar linkage^ mechanisms.^ However,^ the^ Science^ Research Council (^) visiting (^) panel felt^ that^ the^ needs of industry should be^ ascertained^ to^ ensure industrial^ relevance. As (^) a result, three (^) problems encountered in^ the design^ of

planar linkage^ mechanisms^ have^ been^ investigated,^ namely kinematic (^) synthesis, balancing^ of inertia^ forces^ and vibration analysis. The kinematic^ synthesis of planar linkages^ containing pin and^ sliding^ joints^ using^ an^ optimization^ approach^ has been investigated.^ A (^) concise but^ easily interpreted technique has^ been^ developed^ for^ prescribing the^ topology of linkages^ formed^ by^ connecting^ pairs^ of^ links^ together. The displacement^ analysis of (^) a linkage^ is^ achieved using a direct^ method^ which^ is^ considerably^ faster^ than alternative techniques.^ A^ non-linear^ optimization algorithm has^ been^ modified^ to^ cater^ for^ non-linear constraints such^ as^ transmission^ angle^ and^ loop^ closure. These techniques^ have^ been^ incorporated^ into^ a (^) computer program. Two (^) case studies of (^) using the^ program (^) are given. The first^ is^ the^ synthesis of (^) a six-bar linkage^ for^ a motorcycle rear^ suspension.^ The^ linkage^ maintains^ a constant centre^ distance^ between^ the^ chainwheels^ as^ the suspension deflects.^ The^ second^ concerns^ the^ modification of two^ linkages,^ containing^ eight^ and ten^ links^ respectively, to (^) give an improved^ knitting^ action for^ a warp-knitting machine. The^ discussion^ following^ these^ case studies differs from^ the^ usual form^ in^ that it^ compares different techniques (^) and (^) methods (^) rather than^ theory^ and (^) experiment. operating linkages^ at high^ speeds (^) can result in rapidly varying^ forces^ acting^ on^ the^ frame^ due^ to^ the mass of^ the^ moving^ links.^ Accordingly^ the^ second (^) part contains the^ theory^ and^ a procedure to^ determine^ suitable counterweights to^ balance^ these^ forces.^ Since^ adding the

C0NTENTS

Paqe

GENERAL (^) INTRODUCTION 1.1 (^) Application (^) of Linkage Mechanisms 1- 1.2 (^) Survey (^) of Design Factors 1-

A. KINEMATIC SYNTHESIS

2. (^) SCOPE 2.1 (^) Requirement 2- 2.2 (^) Review (^) of Current (^) Methods 2- 2.3 (^) Choice (^) of Approach 2-

SELECTION 3.1 (^) Terminology (^) and Conventions (^) 3- 3.2 (^) Degrees (^) of Freedom 3- 3.3 (^) Kinematic (^) Chains (^) and Planar 3- Linkages

TOPOLOGY

4.1 (^) Specification 4- 4.2 (^) Survey 4- 4.3 (^) Loops 4- 4.4 (^) Undetermined (^) Dyads 4- 4.5 (^) Arcs (^) and Arrays 4-

KINEMATICS 5.1 (^) Approach (^) 5- 5.2 (^) Theory 5- 5.2.1 (^) Undetermined dyad (^) with 5- three (^) revolut e joints 5.2.2 (^) Undetermined dyad (^) with 5- two (^) revolute joints^ and one (^) prismatic joint 5.2.3 (^) Undetermined dyad (^) with 5- one revolute joint^ and two (^) prismatic joints 5.3 (^) Comparisons 5-

6. OPTIMIZATION

6.1 (^) Variables (^) and Parame ters 6- 6.2 (^) Objective 6-

B. FORCE (^) BALANCING

VIBRATION (^) ANALYSIS

Paqe

14. FOUR-BAR (^) LINKAGE 14.1 (^) Choice (^) of Linkage 14- 14.2 (^) Theory 14- 14.3 (^) Experimental (^) Set-up (^) and Procedure 14- 14.4 Results (^) and Discussion 14- 14.5 (^) Conclusions 14-

D. FURTHER^ WORK

15. (^) FURTHER WORK 15.1: (^) Kinematic (^) Synthesis 15- 15.2 (^) Balancing 15- 15.3 (^) Vibration 15-

APPENDICES

A. REFERENCES

B. PSALM^ INPUT^ DATA Bl (^) Title B- B2 Topological^ Data Set (^) - LINKS B- B3 Optimization (^) Data Set (^) - METHOD (^) B- B4 (^) Constraint (^) Data Set (^) - LIMITS B- B5 Complete Sets (^) of Data (^) B- C. INTERMEDIATE ARRAYS D. COIAPUTER-AIDED^ SYNTHESIS^ OF^ LINKAGES -A MOTORCYCLE^ DESIGN^ STUDY

I. GENERALANTR0DUCT10N

1.1 (^) 'APPLICATIONS OF LINKAGE MECHANISMS Linkage (^) mechanisms have^ a long^ history.^ For example, a (^) crank-rocker linkage^ is^ illustrated^ in^ a compendium of^ agriculture and rural engineering dated (^1313) AD (See^ Nolle {1.11 (^) Linkage (^) mechanisms are in widespread use today,^ both^ in^ the^ home^ and in^ industry. Examples (^) range in^ complexity from^ slider-crank mechanisms used in^ conventional^ umbrellas and a cream-making attachment for^ a domestic^ food-mixer^ to^ an (^) eleven-bar linkage (^) used in (^) shoe machinery '11.2a)^ and (^) a fourteen- bar (^) circuit-breaker (^) mechanism {1.31. Linkage (^) mechanisms are indispensable^ where an operation'depends (^) on the^ relative motion between^ two^ or more (^) machine elements (^) undergoing simultaneous but different (^) movements (^) at high (^) speed. For (^) example, Kestell {1.2b} (^) describes (^) a (^) machine for (^) stitching a (^) sole to (^) the

welt (^) of (^) a shoe at a rate of (^1000) stitches/min. In^ this

machine, the^ combined movements of several (^) elements, e. g. the (^) awl, needle (^) and looper,^ are required to form (^) a stitch, so linkage^ mechanisms are (^) used to^ control them.^ The^ awl and (^) needle mechanisms are crank-driven (^) whereas the^ looper mechanism is^ cam-driven. Cams (^) and followers^ are (^) sometimes regarded as alternatives to^ linkages,^ e. g. Eder^ {1-4}^ , but^ there are (^) circumstances, (^) where cams (^) are used, that^ require a linkage (^) as well. In (^) such cases, one or more of the following (^) conditions (^) applies (^) : a) there^ is^ insufficient^ space to^ accommodate the cam close to^ where the^ output motion is required, (^) e. g. the^ valve-gear for^ an internal combustion (^) engine, b) (^) the (^) output (^) motion is (^) adjustable (^) e. g. Kestell fl. (^) 2c} illustrates (^) a (^) cam-driven, thread-

• (^) References (^) will be found (^) in (^) Appendix A.

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measure mechanism^ in^ which^ the^ length^ of thread (^) paid out depends^ on the^ thickness^ of the (^) work being^ stitched,

C) the^ amplitude of^ the^ output^ motion^ is different from^ that (^) of the (^) cam follower;^ the output amplitude^ may^ be^ larger,^ as^ in^ the^ case of a draw^ mechanism^ for^ a^ weft-knitting machine (one^ version^ of^ which^ is^ given^ by Wignall '11.5})^ in^ which the^ output amplitude is (^) adjustable and may be^ a maximum of 42 in. compared with a^ cam^ throw^ of^9 in.,^ or^ smaller, as in^ a^ copying^ machine^ where a^ negative magnification, to^ reduce errors, is^ obtained using a^ pantograph mechanism,

d) (^) two (^) cams are (^) used to (^) generate a two-dimensional

output motion, the^ linkage^ being^ required^ to combine the^ motions,^ e. g.^ Bloom'{1.6)^ and^ Wray et al.^ 11.71^ where one^ component^ of^ the^ output is adjustable^ to^ provide different^ spacings^ of^ the loops in^ a looped^ pile fabric^ machine, e) a^ positive drive^ is^ required^ from^ a^ single^ cam profile using swinging followers^ e. g.^ Jackowski and Dubil'fl.^ 8}^ , an early example^ of^ its^ use^ is illustrated by^ Willkomm'{1.91. There is^ a continual demand^ made on industrial designers for^ machines that (^) will run (^) faster and yet be simpler to^ manufacture.^ If^ cams^ are^ used^ at^ higher speeds, greater attention^ must^ be^ paid^ to^ producing^ a profile that^ gives^ the^ desired^ dynamic^ performance^ as this is^ significantly (^) affected by^ local^ profile errors. In (^) addition, if^ a (^) spring-loaded follower^ is^ used, high spring forces^ will be^ necessary to^ maintain contact and these forces^ will cause high^ contact stresses and possible surface^ failure.^ A^ closed-track^ cam^ with^ a single roller follower^ suffers from^ impact^ and^ skidding, causing noise and^ wear,^ when^ the^ roller^ changes^ from one side of the^ track^ to^ the^ other.^ To^ overcome^ these

Material (^) Density t/m^3

Specific Modulus MJ/kg

Specific Strength kJ/kg

Cast iron^ 7.2 5 16 40 BS 1452,^ grade 17

SG iron,^ annealed (^7 1 24 ) BS 2789,^ type SNG 27/^.

Mild (^) steel 7.8 26.5 50

Low (^) alloy (^) steel- 7.8 (^) 26.5 125

. Aluminium.^ casting^.^ 2.7^ 25.8^80

Magnesium (^) casting, (^1 8 25 ) heat-treated (^).

Titanium (^) alloy, (^4 5 25) 2CO DTD 5173.

Carbon fibre^ reinforced plastic, unidirectional, (^1 5 115 ) 50% high^ modulus. polyacrylonitrile based

Table 1.1.^ Representative Values (^) of Specific Modulus (^) and Strength (^) of Various Materials.

measures of^ the^ stiffness-weight^ and^ strength-weight ratios respectively.^ Cast^ iron^ is^ easy^ to^ machine and has^ good^ damping^ properties.^ Aluminium^ alloy links (^) should form flat^ laminae^ {1.111^ but^ the low density (^) means that thicker (^) sections can be (^) used for a given mass. Titanium^ has^ been^ used for^ the guide-bars of^ a^ warp-knitting machine but^ has^ now^ been replaced by^ 'Grafil',^ a carbon fibre^ reinforced plastic (CFRP)'{1.23}.^ Partly^ as a result of^ this change, the^ machine^ speed was increased^ by^ 10%.^ A 50% improvement in (^) the (^) speed (^) of a narrow-fabric loom was obtained^ by^ using CFRP^ instead^ of^ steel for^ the heddle frames'{1.211.^ Although CFRP (^) offers the twin advantage (^) of high^ specific (^) modulus and strength, close co-operation between^ the^ manufacturer and the^ machine designer is^ required to (^) obtain the best (^) results since the directional (^) properties of the (^) material depend^ on the lay^ of the^ carbon fibres. If the (^) stiffness of the links^ is^ too low,^ the linkage (^) may (^) start to (^) resonate at various (^) speeds resulting in^ material fatigue^ with possibly an^ incorrect output (^) motion and noise (^) radiation from^ other parts of the (^) machine. If the (^) mechanism is^ driven^ by^ a cam running (^) at constant speed, it^ is^ possible^ to^ design^ the cam (^) profile to^ take^ into^ account the^ effect of^ mechanism flexibility (^) on the (^) output (^) motion. Dudley {1. proposed the^ use^ of^ polynomial curves^ and^ Kanzaki^ and Itao {1.251^ have^ investigated^ the (^) use of (^) such cams for positioning the^ typehead^ in^ high-speed^ teleprinters. However, this (^) approach is^ not (^) applicable to (^) a crank- driven linkage,^ so, in (^) this (^) case, the designer^ will

want to^ know^ how^ light^ the^ links^ can be^ to^ avoid these

problems (^) within the^ envisaged speed range. The (^) output (^) motion can (^) also be^ affected by^ clearance at the^ joints.^ For^ example, in^ a^ plain revolute^ joint', the (^) pin (^) may move from^ one side of the^ bush^ to^ the^ other

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which, again, is^ affected by^ the^ tolerances^ on^ the link dimensions. (^) There (^) are five (^) assembly methods from (^) which to (^) choose (^) :

a) (^) random assembly with total^ interchangeability

• this^ may^ require^ tight^ tolerances^ on^ the link (^) dimensions, b) (^) random assembly (^) with limited interchangeability based (^) on (^) probability theory
• it^ is^ assumed^ that^ the^ variations^ in^ each component due^ to^ manufacturing errors are independent; (^) as a (^) result there is^ only a small probability that^ the^ maximum variation^ will occur (^) simultaneously on (^) several links^ and^ that the (^) effects (^) of all of the (^) variations will be additive, C) (^) random assembly (^) with a compensating member
• the^ mechanism^ is^ assembled^ without^ the compensating member which is^ then^ machined^ to suit the^ combined (^) variations in^ the^ other links, d) (^) random assembly (^) with an adjustable link
• this^ is^ similar^ to^ (c)^ except^ that^ the^ link may be^ re-adjusted incorrectly^ at^ any time, e) (^) selective assembly
• for^ this^ to^ be^ feasible,^ there^ must^ be^ a sufficient number of each component^ from^ which to (^) select and the (^) number of components in^ each assembly should be^ small^ so^ that^ their interaction (^) can be^ determined. Thus (^) the (^) choice of the (^) tolerances is^ an important decision (^) and, to (^) make the (^) correct choice, the designer must know^ the^ sensitivity of the^ output motion from^ the mechanism to^ variations in^ the^ link^ lengths.^ It^ is desirable (^) that the (^) method for^ determining^ this^ shall be the (^) same as that (^) used originally for^ selecting the^ link lengths.

Further development (^) of design (^) methods is (^) both possible (^) and desirable.^ Contacts^ with industry have (^) shown that (^) machine designers (^) would appreciate assistance (^) with the^ determination^ of the link^ lengths and the^ anticipation (^) of linkage^ vibration. Reflecting these (^) requirements, the (^) thesis is (^) divided into (^) three parts (^) entitled Kinematic^ Synthesis,^ Force^ Balancing and Vibration^ Analysis^ respectively.

. 'Suwznary

After (^) noting some applications of linkage (^) mechanisms, the conditions (^) where linkages^ are (^) necessary are listed.^ Their relationship (^) with, and their^ advantages over, (^) cam-driven mechanisms (^) are then^ considered. This^ is^ followed^ by^ a survey of factors^ to^ be^ taken^ into^ account in^ the^ design^ of linkage mechanisms. These^ factors^ include^ material selection, force balancing (^) and vibration in (^) addition to (^) clearance and friction (^) at the joints.^ As (^) well (^) as determining (^) the (^) nominal dimensions (^) of the links, (^) the designer^ must decide (^) on suitable tolerances for manufacture (^) and assembly. In^ conclusion, the^ need for^ further work in^ kinematic^ synthesis, force^ balancing^ and (^) vibration is emphasized.

synthesizing linkages^ to^ generate^ coupler curves. They (^) can be^ divided^ into^ two (^) groups depending^ on the relationship between^ the^ number of points used^ to define (^) the desired (^) motion and the (^) number of unknown parameters, (^) such as link^ lengths,^ in^ the^ mechanism. Precision (^) point methods depend^ on the (^) number of points being^ less^ than^ or (^) equal to^ the^ number of parameters. The^ resulting mechanism will produce the desired^ motion exactly (^) at those (^) points but^ there is (^) no control over intermediate (^) positions. Optimization (^) methods are (^) used (^) where the (^) number of points is^ greater than^ the^ number^ of parameters. In this (^) case, there^ is^ no guarantee that^ the^ resulting mechanism (^) will pass through^ any of the^ specified points exactly but^ there^ is^ more control^ over^ the general shape of the^ motion curve. Thompson*{2. concludes that^ this^ is^ the^ best^ type^ for^ kinematic synthesis. Kestell'{2.41^ also followed^ this^ approach, using (^) graphical methods, by^ 'plotting^ the^ trend'.^ Of the (^) graphical (^) and analytical methods, only the analytical (^) ones (^) offer the^ promise of general application, but,^ to^ obtain the^ full^ benefit^ for linkages (^) with more than (^) a few links, (^) they (^) require the use of (^) a computer. There^ have^ been^ three^ attempts to develop (^) computer-based methods applicable to (^) virtually all planar linkages. The first,^ called KOGEOP (KOppel-GEtriebe- OPtimierung), has^ been^ developed^ by^ Dresig (^) et al 12.5 (^) - 2. ý)^ for both (^) kinematic (^) synthesis and dynamic optimization (^) using a BESM-6^ computer. No^ indication of the^ size of the^ program (I^ reserve this^ spelling for computer (^) programs) is^ given, but^ it^ must be considerable (^) since the^ user has^ the^ choice of (^) eight optimization (^) routines. The (^) second, called KINSYN, has^ been^ developed by (^) Kaufman {2.8,2.91.^ The (^) original version used a precision point approach and (^) relied on^ the^ user^ being familiar (^) with Burmester theory. It (^) was designed^ for^ an

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IBM (^1130) computer with 8K^ words of core storage and a (^) storage tube^ display.^ Later^ it^ grew (^) until, by 1973, (^) it (^) consisted of (^) well over 15,000 (^) cards Of Fortran (^) and Assembler language (^) code. Since this is difficult (^) to (^) manage, a new version is being (^) developed to (^) make use (^) of a computer system with a (^) refresh-type graphical display^ (which^ is^ considerably (^) more expensive than (^) a storage tube display). The (^) most (^) ambitious attempt is^ that by^ Bona (^) et al {2.10). (^) This (^) consists of a (^) suite of (^27) programs (one of which is^ based^ on KINSYN).^ The^ scope is considerable, including,^ in^ addition to^ linkage^ synthesis and analysis, (^) cam and (^) gear design,^ stress and tolerance analysis, (^) automatic drafting^ and preparation of NC^ tapes. The intention^ was also to include^ a mechanism data^ bank but, (^) although this is (^) considered to be (^) feasible, it has been (^) discontinued (^) because (^) of lack (^) of storage space and

limited (^) availability (^) of suitable computer terminals. The package (^) uses (^) an IBM^1130 computer with 32K^ words of core storage (^) and (^) one million words (^) of disc^ storage (^) with an IBM (^2250) refresh-type graphic display. These (^) attempts, (^) which cater for^ only (^) planar linkages, all (^) require a large^ amount of computer (^) storage and two use refresh-type (^) graphic displays.^ This^ combination of facilities (^) is (^) not (^) generally available to (^) machine designers (^) in (^) this (^) country. The (^) use (^) of analogue, rather than digital,^ computers has (^) also been (^) advocated for (^) the kinematic (^) synthesis of planar linkages.^ Keller'f2^.^111 usdd an analogue computer to^ simulate a slider-crank mechanism and a four- bar (^) linkage. (^) Brat (^) and Vaclavik'f2.12) (^) synthesized a four-bar (^) linkage for (^) a crane (^) and Rees Jones'{2. synthesized (^) a (^) six-bar linkage^ for^ a power (^) press. There are two^ main (^) advantages. The^ first^ is^ speed of computation. Secondly,^ the^ user-can control the linkage (^) parameters directly (^) and (^) so determine (^) the (^) effect