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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 l.jpg

Structural Issues in Linkages

R. Lindeke, Ph. D.

ME 3230 Kinematic & Mechatronics

Me 3230

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Topics Of Interest

  • Grashof’s Law

  • Motion Limits for “Slider Cranks”

  • Interference in Linkages

  • Mechanical Advantage

  • Practical Considerations

    • Revolute Joints

    • Prismatic Joints

Me 3230

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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

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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

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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

Me 3230

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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

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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

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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

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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

Me 3230

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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?

Me 3230

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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

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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

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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.

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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

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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

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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

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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”

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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

Me 3230

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