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Structural Issues in Linkages. R. Lindeke, Ph. D. ME 3230 Kinematic & Mechatronics. Topics Of Interest. Grashof’s Law Motion Limits for “Slider Cranks” Interference in Linkages Mechanical Advantage Practical Considerations Revolute Joints Prismatic Joints. Grashof’ Law.

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Structural Issues in Linkages

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Structural Issues in Linkages

R. Lindeke, Ph. D.

ME 3230 Kinematic & Mechatronics

Me 3230

Topics Of Interest

  • Grashof’s Law

  • Motion Limits for “Slider Cranks”

  • Interference in Linkages

  • Mechanical Advantage

  • Practical Considerations

    • Revolute Joints

    • Prismatic Joints

Me 3230

Grashof’ Law

  • The fundamental 4-bar linkage design law: s + l < p + q

    • Here, s is the shortest link

    • l is the longest link

    • p and q are the other two links

  • This law states that for there to be continuous relative motion between any 2 links, this inequality must be true (Grashof Type 1 linkages)

Me 3230

Consequences of Failure of Grashof’s inequality:

Links can’t connect! s+l+p<q

s can’t rotate! s+l-p>q

Links can’t connect! s+q+p<l

s can’t rotate! s+q-p<l

See text for proof of Grashof’s theorem!

Me 3230

Type 1 or 2 Grashof Linkages

  • Type 1: the inequality holds

    • We state that these linkages (type 1) have two joints that perform complete (360) rotation – and they are located at either end of the shortest link

  • Type 2: the inequality is not held

    • They have no fully rotating joints

    • All 4 joints oscillate between limits

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Addition Type 1 Nomenclature

  • Base frame is the fixed link

  • Two members connected to base by revolute joints are Turning Links

  • Member jointed to both turning links is the Coupler

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Type 1 Mechanisms:

  • s is connected to the base (a, b) – this is a crank rocker

  • s is the base link (c) – this is double crank or “drag-line” mechanism

  • s is the coupler link – this is a double rocker where the coupler can perform a complete rotation relative to the base

Me 3230

Topological Interference – So it should rotate shouldn’t it?

  • This point is really one of construction – making sure that the design will actually operate

  • Topological interference is a fundamental property of the structure – it can’t be eliminated by “reshaping” links

  • When we are assembling a Linkage – there is a “right way” to assure that the various links will not run through each other

  • Motion will be transferred to rotating link – using shafts (perhaps) so this issue also addresses how the shafts are connected to the linkage too

Me 3230

A Simple Crank rocker – as a structure

Notice the Positioning of Coupler vs. the turning links (a). In this arrangement we can bring in/take out torques with shafts thru the base link (b). And the crank can completely rotate without striking the shaft or coupler!

Me 3230

Setup for Drag– Link Mechanisms

  • Here the “drive Shafts” must be connected directly to the turning links

  • In the drag line, the base link has become a pair of fixed bearings and the link is essentially “turned inside out”

  • This is a must otherwise the coupler must pass thru the base or shafts thus becoming locked up

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Lets try one:

  • Problem 1.35 where you are to select from a set of 8 links (2”; 3”; 4”; 7”; 9.5”; 13” and 9”) From this set choose 4 links to build a mechanism that can be driven by a continuous rotational motor? Identify each link by appropriate name.

  • What type of mechanism results?

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Motion Limits for a Slider Crank

  • There are 2 rules that must be held for freedom of motion – full rotation

    • b>a in mechanism

    • b - a > c

  • Where a is length of crank

  • Where b is length of coupler

  • Where c is the distance from ground pivot to slider pin

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Design Considerations

  • Consider Slider Crank Half first A-B-C: does it meet motion criteria?

    • BC>AB (yes)

    • BC-AB>c (yes)

  • Consider C-BD-EF- ‘AF’ as a crank rocker

  • EF must be the crank – we will let upper link (A-B-D) “rock” to move the Slider at C

  • Positioning of E is along the locus of E (from DE length) -- upper sketch

  • Use Grashof calculation for the shortest link of Crank-rocker to establish true limits for E (lower sketch)

Me 3230

Mechanical Advantage in a Mechanism

  • MA is the ration of the output torque to the input torque of a mechanism

  • This ration is directly proportional to Sin() (coupler to driven turner) and inversely proportional to Sin() (coupler to driver turner)

  • When  is 0 or 180 (position A-B1 and A-B2 in the figure) a small input torque delivers a large (infinite) output torque – the rocker is said to be in Toggle

  • When  is small MA is very low – this transmission angle should never be designed to work at angles of less than about 45 . If MA is too small, only a small amount of friction can lock up the mechanism

Me 3230

Mechanical Advantage in a Mechanism

  • Considering rin & rout as level arms of the input and output shafts (and loads), then:

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Try One?

  • A crank-rocker linkage has a 100-mm frame, a 25-mm crank, a 90-mm coupler, and a 75-mm rocker. Draw the linkage and find the maximum and minimum values of the transmission angle. Locate both toggle positions and record the corresponding crank angles and transmission angles.

Me 3230

Green Lines At “Gamma” limits – Crank at 0 or 180

 is about 53.1 and 98.1 respectively

Purple Lines at “Toggle” Crank at about 226 and 40 (rocker back and forward respectively)

 is about 90.9 and 59.1 respectively

Me 3230

Practical Design Issues: Revolute Joints

  • Lubrication of the Bearing Surfaces in rotation

    • Hydrodynamic lubrication occurs under conditions of unidirectional rotation under speed

      • A lubricating film, carrying the bearing load, is established between the bearing surfaces and only lubricant viscous friction (low friction) results

      • additionally no metal to metal contact is present and no running wear is observed (only startup and stopping wear)

      • Lubricant can by pumped in to assist in establishing HD lubrication

      • HD lubrication is seen in internal combustion engines for crankshaft support bearings and connecting rod/crankshaft bearings

Me 3230

Practical Design Issues: Revolute Joints

  • Lubrication of the Bearing Surfaces in rotation

    • Hydrostatic lubrication is a system where lubricant is pumped in to the bearing gap under elevated pressure to carry the bearing loads

      • It can be used even if rotational speed is low or even reverses

      • Used in main bearings in large turbo-generator sets

      • Fits need to be made to tight tolerances so oil will not leak out in operation or idle activities

Me 3230

Practical Design Issues: Revolute Joints

  • Can use “Grease Bearings” in slow or reversing motion

  • Solid Contact bearing (teflon bearings) and dissimilar metal bearing can be used – like babbitt’s metal

    • 90% tin 10% copper

    • 89% tin 7% antimony 4% copper

    • 80% lead 15% antimony 5% tin

  • Also can use roller, ball or pin bearings as contactor for revolute joint “systems”

Me 3230

Practical Design Issues: Prismatic Joints

  • Jamming of the slider is purely a design issue!

  • This problem is a function of the friction of the slider, the applied force and its direction

    • If the angle of the ‘coupler to slider’ is less than the Friction angle: f = tan-1m (mis coefficient of friction) the slider will jam

  • Sliders loaded by offset forcing loads will also Jam if the width of the slider b < 2 mawhere a is the offset between the slider and the forcing load

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Practical Design Issues: Prismatic Joints

  • Since Jamming is a function of slider friction, using means to reduce friction is the most effective way to reduce problems

  • The use of rolling contact joints is an effect means to reduce the friction

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