# Structural Issues in Linkages - PowerPoint PPT Presentation

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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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#### Presentation Transcript

R. Lindeke, Ph. D.

ME 3230 Kinematic & Mechatronics

Me 3230

### Topics Of Interest

• Grashof’s Law

• Motion Limits for “Slider Cranks”

• Practical Considerations

• Revolute Joints

• Prismatic Joints

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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)

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### Consequences of Failure of Grashof’s inequality:

s can’t rotate! s+l-p>q

s can’t rotate! s+q-p<l

See text for proof of Grashof’s theorem!

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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

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• 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

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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

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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!

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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

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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)

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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

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