Testing

• Kinematics is the study of object motion independent of forces
• Kinetics is the study of forces causing or caused by motion
• Movement-induced stresses are accounted for after a motion analysis for a mechanism is completed
• Links are the building blocks of mechanisms and are also known as kinematic links
• Links can be classified based on the number of nodes in their body: binary link (2 nodes), ternary link (3 nodes), quarternary link (4 nodes), and so on
• When two or more links are joined together at their nodes, their group is called a linkage
• If a linkage can exhibit controlled output motion, it is called a kinematic chain
• If at least one link of a kinematic chain is fixed, grounded, or secured to a reference frame, it is called a mechanism
• A rigid space potato suspended in outer space has 6 degrees of freedom: 3 translational axes (x, y, z) and 3 rotational axes (x, y, z)
• Each arbitrary point on the potato has a different number of real degrees of freedom: A (3 DOF), B (2 DOF), C (1 DOF), and other points (0 DOF)
• Joints or kinematic pairs are formed when two links connect while allowing some motion
• Joint's degree of freedom is the number of motions that must be controlled to fully define the joint
• Lower Pairs (Full Joints) have area contact between links and include:

  • Revolute (R) Joint: restricts motion to rotation on one axis (θ) with 1 DOF
  • Prismatic (P) Joint: restricts motion to translation on a single axis (x) with 1 DOF
  • Helical (H) Joint: exhibits translation along and rotation on one axis, but only has 1 DOF as the motion types are not independent
• Cylindrical (C) Joint:

  • Allows independent translation (x) and rotation (θ) in one axis
  • Has 2 degrees of freedom
  • Examples include player rods in a foosball table and sliding latch locks on doors
• Spherical (S) Joint:

  • Known as the ball-and-socket joint
  • Allows rotation in all three axes (θ, ϕ, γ)
  • Has 3 degrees of freedom
  • Examples include shoulder and hip joints
• Planar (PL) Joint:

  • Permits general plane motion on a level surface (x, y, θ)
  • Has 3 degrees of freedom
  • An example is a mouse on a mousepad
• Universal (U) Joint:

  • Allows transmission of rotation (θ) between two shafts at an angle (ϕ)
  • Can transmit rotation even when shafts are bent
  • Has 2 degrees of freedom
  • Modern automobiles use double U-joints to counteract eccentricity
• Pin-in-Slot:

  • Joint where a pin moves along a sliding path and acts as a revolute joint
  • Example: sliding pin lock on doors
• Gear (G) Pairs:

  • Meshed gear pairs transmit rotation
  • Purpose is to transmit rotation or translation
  • Has 2 degrees of freedom
• Cam Pairs:

  • Consists of a cam as input and a follower that moves accordingly
  • Can convert rotation into rectilinear motion or intermittent rotation in another axis
  • Has 2 degrees of freedom
• Wheel:

  • Forms a joint at its contact point on a surface
  • Can roll along the surface with no slip or unintentionally skid
  • Has 2 degrees of freedom
  • If constrained to only roll or slide, its degrees of freedom is 1
• Wrapping Pairs:

  • Joints consisting of flexible mechanical elements
  • Can have multiple point or line contacts at a time
  • Include pulleys, belts, roller chains, and others
• Kinematic Pairs by Enclosure:

  • Two main ways to keep a kinematic pair from decoupling: force-closed and form-closed
  • Force-closed joints require external force to ensure coupling
  • Form-closed joints use geometry to constrain the joint
• Kinematic Pairs by Constraint:

  • Constraints limit the number of degrees of freedom a joint can exhibit
  • Completely constrained joints are restricted to only one motion
  • Partially constrained joints require external force to limit motion
  • Incompletely constrained joints are not restricted to one motion
• Grübler’s Formula:

  • Formula to calculate the degree of freedom of mechanisms
  • DOF = m(N - 1 - J) + Σfi
  • Mechanism will always have a DOF of 1 or higher
• Order of Joints:

  • Joint may connect 3 or more links, becoming an nth ordered joint
  • Order of joint is N - 1
  • Order of joint is also the individual DOF of the complex joint
• Grübler’s Blindspots:

  • Grübler’s formula may have miscalculations due to ignoring geometry
  • Grübler’s Paradox can occur where mechanisms with zero DOF can still move due to geometric constraints
• Having three legs of equal length predisposes the chain to act like a parallelogram linkage
• This anomaly covers many spatial mechanisms, such as the Bennett linkage
• Inversions of a mechanism involve changing the grounded link
• A four-bar linkage consists of four links forming a loop connected by R-joints
• The four links in a four-bar linkage are labeled as:

  • Link 1: ground
  • Link 2: input
  • Link 3: coupler
  • Link 4: output
• Grashof's Criterion classifies four-bar linkages based on the relative lengths of the links
• Class I Four-Bar Linkage:

  • Shortest link + Longest link < Remaining links
  • Also known as Grashof linkages
  • Have at least one link that can undergo continuous motion
  • Four possible inversions
• Class II Four-Bar Linkage:

  • Shortest link + Longest link > Remaining links
  • Also called triple rockers
  • None of the unground links can complete a rotation
• Class III Four-Bar Linkage:

  • Shortest link + Longest link = Remaining links
  • Have change points where the mechanism can transition to different configurations
  • Not desirable due to uncertainty in operations
• The crank-slider mechanism consists of four linkages with three revolute joints and one prismatic joint
• Inversions of the crank-slider mechanism include:

  • Crank-slider/rocker-slider
  • Inversion 2: link 2 is fixed
  • Inversion 3: link 3 is fixed
  • Inversion 4: link 4 is fixed
• The double slider is obtained when a crank-slider's link 2 is replaced by another slider
• Inversions of the double slider include:

  • Scotch Yoke mechanism
  • Oldham coupling