Analytical Modeling of Kinematic Linkages, Part 2

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Analytical Modeling of Kinematic Linkages, Part 2

• ME 3230
• R. Lindeke, Ph.D.

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Topics

• Slider Crank Modeling
• Inverse Slider Crank System Modeling
• RPRP 4 vectors
• RRPP 3 vectors
• Compound Mechanisms

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General Approach to Slider Crank Linkages

• rP r1 r4 r2r3
• Just like 4-Bar mechanisms
• Must be satisfied throughout the entire motion of
  the linkage
• r1 has fixed angle but variable length
• ?4 ?1 ?/2 rad
• (both constants!)

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Looking at Solutions

• Constants in Slider Crank
• Thus, Depending on which of the 3 remaining terms
  is the Input to the system we will solve the
  component equations for ?2 and/or ?3 and/or r1 in
  terms of the one which is the input

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Velocity and Acceleration

• Again, we need to take the derivatives (1st and
  2nd w.r.t time) of the solutions from our
  position equations (for 2 of the 3 values in
  terms of the 3rd value)
• When we work the solutions, we must specify an
  assembly mode value (?) to match the solution we
  desire!
• Finally, Since we will encounter Sqrt functions
  too, we must recognize that negative values
  indicate geometries that cant be assembled
  (remembering the A-B-C rules we encountered
  earlier)
• Tables 5.4 and 5.5 include summaries of equations
  for Slider-Crank with angular or Linear inputs

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Try One? The venerable 350CI V8

• Solve with ?2 as input
• Currently ?2 is at 55?
• Crank is turning at 8300 RPM (869.17 rad/s)
• Determine the velocity and acceleration of X1

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Starting Point Finding ?1 and Magnitude of X2
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Solutions Positions

• Drawing equations from Table 5.4
• r1 7.1319
• ?3 .7340 rad (42.06?)

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Solution Velocity
The velocity of the piston is 347.35/s downward
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Solution Acceleration
Crank Acceleration is constant!
Piston is accelerating downward at 1.44
million/s2!
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Looking at the Inverse Slider-Crank

• These are Mechanisms (seen in Backhoes, Boom
  Trucks, etc) where linear actuators are used to
  move the linkage and this Slider is not direct
  connected to an end of the mechanism as in a
  Slider-Crank
• Analytically they consist of revolute to slider
  joint connected by a link

Again we have 4 links driving Pt. P Solution is
found by the link closure eqns rP r2
r1r3r4
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Positional Models

• r1 (base vector) constant in direction
  magnitude
• r2 and r4 constant in magnitude (directions
  change)
• r3 variable in both magnitude and direction
• Therefore r1, r2, r4, ?1 ?4 constants

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Solutions

• Again, we need to solve models for position,
  velocity and acceleration depending on the input
  variable one of ?2, ?3, or r3
• These models are summarized in Table 5.6 (?2 as
  input)
• Table 5.7 when ?3 is the input value
• Table 5.8 when r3 is the input value

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RPRP Mechanisms

• A common mechanism using it is the Rapson slide
  used in steering gear for ships
• Closure Equations are again 4-bar
• The constants are r1, r2, ?1 and ?4

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Models
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Solution follows as before

• Select which of r3, r4, ?2 or ?3 should be chosen
  as input and with 3 equations we can solve the
  system
• Practically since ?2 and ?3 are directly related
  only one makes sense as a choice.
• This last fact typically means that only one
  angle or one length is chosen as the input
• When Angular Input is used the trajectory
  equations are summarized in Table 5.9
• Linear Input r4 are summarized in Table 5.10
• Linear Input r3 are summarized in Table 5.11
• Example Problem 5.7

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The RRPP Linkage

• A common application is the Scotch Yoke mechanism
• One finds them in places where we index or move
  items using a rotary input drive.
• These mechanisms are typically 3-vector sets

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Solution

• In this mechanism r2, ?1 and ?3 are constants
• Additionally where ? is also
  constant
• Typically ?2 is chosen for input but r1 or r3 can
  also be chosen
• Solution for Trajectory is found in Tables
• 5.12 if ?2 is input
• 5.13 if either r1 or r3 is chosen
• See Problem 5.18 as an example

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Elliptic Trammel (RRPP) and Oldham Mechanisms
(RPPR)

• Both are 3 link mechanisms as seen above as
  mechanisms and models

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Elliptic Trammel (RRPP) and Oldham Mechanisms
(RPPR)

• Elliptic Trammel
• These mechanisms are used to generate elliptical
  coupler curves for function generation
• Analytically
• Constants are r3, ?1, ?2 (and ?1 ?2 ?)
• Variables are ?3, r1, and r2 (one must be known)
• Trajectory Models are seen in Tables 5.14
  (angular input) or 5.15 (r1 input)
• See Problem 5.5 as an example

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Elliptic Trammel (RRPP) and Oldham Mechanisms
(RPPR)

• An Oldham coupler is a method to transfer torque
  between two parallel but not collinear shafts. It
  has three discs, one coupled to the input, one
  coupled to the output, and a middle disc that is
  joined to the first two by tongue and groove. The
  tongue and groove on one side is perpendicular to
  the tongue and groove on the other. Often springs
  are used to reduce backlash of the mechanism. The
  coupler is much more compact than, for example,
  two universal joints. Wikipedia Definition
• Originally developed for paddlewheel steamers but
  now being used in Compressors, Disc Brake
  Schemes, etc.

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Elliptic Trammel (RRPP) and Oldham Mechanisms
(RPPR)

• Oldham Mechanism
• R1 and ?1 are constants
• Additional model (beyond vector state
  components) ?4 ?2 ? (? is constant)
• Variable (one must be input) ?2, ?4, r2 r4
• Using the 3 equations we can solve the mechanism!
• Table 5.16 includes trajectory eqns. For ?s as
  input
• Table 5.17 includes trajectory eqns. When rs as
  inputs

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Addressing Multiple loops

• Using Systematic Loop equations with universal
  reference frame and vectors drawn from joint to
  joint
• Treat sliders as two vectors one in slider
  direction, one normal to the slider path

After sketching the mechanism to near-scale
determine known angles and lengths and variable
angles and lengths
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Following up on Compound Mechanisms

• Write all separate loop paths as state
  equations (x-components and Y-components)
• Write any auxiliary equations (typically angles
  or perhaps lengths)
• Call n the total number of equations for the
  mechanism 2loops Aux. Eqns.
• Determine the DOF of the mechanism (f)
• The mechanism can be uniquely solved if
• n f of Unknown lengths or angles

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Following up on Compound Mechanisms

• These position solution involve non-linearity's
  (transcendental trigonometric equations in
  the unknowns)
• Solution requires guess and iterate techniques
  (Newton-Rapson methods) or
• Separation of mechanism into simpler loops or
• Separation into a series of 2-Equation/2-Unknown
  systems
• For Velocities, take derivatives of the position
  equations (these are linear in the unknowns!)
• So they can be directly solved using linear
  solvers (matrix methods based on Gaussian
  elimination)
• For Acceleration take derivatives of the velocity
  equations while involved and extended, they are
  also linear in the unknowns
• So they can be directly solved using linear
  solvers (matrix methods based on Gaussian
  elimination)