Machine Kinematics
solved, then drawn
Depth

Series

A field says what an essay is about. A series follows one idea essay by essay — from the question that introduces it to the one that assumes all the others.
The identification Jacobian of a four-bar, read by protractor. One row for every number the instrument reads and one column for every parameter that might be wrong. Each cell is the derivative of that reading with respect to that parameter, drawn to the right of its centre line when positive and to the left when negative, with the largest entry in the whole matrix at 4.09e-1. 14 rows against 4 columns: far more equations than unknowns, which is what makes an identification a least-squares problem rather than a solve, and what makes the question of which combinations of columns cancel a real one. These are the same derivatives the tolerance field computes one at a time — the same matrix read down instead of across.

Parameter

  1. 1 A dimension is a measurement
  2. 2 The matrix a calibration inverts
  3. 2 The direction no protractor can see
  4. 3 The coordinates the site already had
  5. 3 A ruler and a protractor
  6. +27 more
32 essays · metrology A ratio that is not a number. The output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.398 to 0.333 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.

Misconception

  1. 1 The ratio that is not a number
  2. 2 Two things called jamming
  3. 3 Exactly right, and unbuildable
  4. 4 The solver was refusing a quarter of the sweep
  5. 5 The number on the box
  6. +14 more
19 essays · wrong Three bars and four bars. On the left, two bars to a common point: three links, three joints, and Grübler gives 3(3−1) − 2(3) = 0. The Jacobian agrees — two free coordinates, rank 2, nothing left over — and the shape cannot change without a bar changing length. On the right, one more bar and one more joint gives mobility 1, and the whole of this site follows from that difference. The triangle is why bridges are triangulated and the quadrilateral is why machines are not.

Mobility

  1. 1 What decides whether it moves
  2. 2 Counting and measuring mobility
  3. 2 What each joint takes away
  4. 3 The mechanism Grübler says cannot move
  5. 4 A roller is not a slider
  6. +13 more
18 essays · constraint The involute, unwound. Hold a string taut against a circle and unwind it: the end traces this curve. Two properties follow immediately and between them they are the whole of gear geometry. The string is always tangent to the base circle, and it is always perpendicular to the curve. So the normal to an involute at any point is a tangent to its base circle — measured on the generated points here, to 2.3e-5 — and when two involutes touch, the common normal is a line tangent to both base circles, which does not move as the gears turn.

Tooth

  1. 1 Why a tooth is an involute
  2. 2 What happens in a mesh
  3. 2 Undercutting, and the seventeen-tooth rule
  4. 2 Epicyclic ratios, two ways
  5. 3 Moving the cutter out
  6. +13 more
18 essays · gears A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding.

Cam

  1. 1 Prescribing motion
  2. 2 The law that costs least is not the smoothest
  3. 2 Stopping thirty times a second
  4. 3 The cam that cannot be cut
  5. 4 The cam is not the valve
  6. +12 more
17 essays · cams Five points on one coupler. The same four-bar, with a tracing point rigidly attached to the coupler at five different places. Each curve is a sextic — degree six — and moving the attachment point a little changes it a great deal. That sensitivity is the reason coupler-curve synthesis was done with atlases of printed curves for most of the twentieth century: there is no simple inverse, so the practical method was to look one up.

Coupler

  1. 1 What a coupler point draws
  2. 2 The straight-line problem
  3. 2 Where the coupler is turning
  4. 3 Peaucellier and the exact answer
  5. 4 Where a hinge pin can go
  6. +12 more
17 essays · curves A four-bar at 60°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 0.0e+0 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 66.9°.

Fourbar

  1. 1 Four bars and four pins
  2. 2 Grashof, predicted and then swept
  3. 2 The transmission angle
  4. 2 The slider-crank
  5. 3 The return stroke is quicker
  6. +12 more
17 essays · linkages Kutzbach's count against the measured mobility. Five closed loops of revolute joints. The count is 6(L − 1) − 5j, a statement about how many links and joints there are; the measurement is the number of joints minus the rank of the loop's screw system, which knows only where the axes point. They disagree for four of the five, and the one they agree on is the generic seven-joint loop — so the formula is not broken, it is blind to the special geometry that makes the other four work. The universal joint is counted at -2 degrees of freedom and is in every car built.

Spatial

  1. 1 Six freedoms, not three
  2. 2 The joint that is not constant velocity
  3. 2 Sarrus, and the straight line that is exact
  4. 3 Bennett, and the condition that moves it
  5. 4 Every motion is a screw
  6. +12 more
17 essays · spatial Three prescribed positions of a rigid body. The whole of the design problem, before any mechanism exists. A body has to occupy these three positions — each one a place and an angle, three numbers — and what carries it between them is not yet decided. A forward analysis starts from link lengths and finds the motion. This starts from the motion, and the lengths are what has to be found. The marked points are the poles: any planar displacement is a rotation about one point, so each pair of poses has one, and the arcs show the turn each represents through the body's own origin. A pole is a property of the displacement and not a mechanism — nothing has been chosen yet. 1 of the 3 poles lies outside this frame and is not drawn; near-parallel displacements push their pole a long way off.

Synthesis

  1. 1 The problem the other way round
  2. 2 Three positions, and a circumcentre
  3. 3 What the fourth position costs
  4. 3 Three linkages, one curve
  5. 3 Where a pin becomes a slide
  6. +12 more
17 essays · synthesis The output is a band, not an angle. The rocker's angle through one turn of the crank, for a four-bar whose four lengths are each specified to ±0.01. The line is the nominal mechanism; the band is where the output of an actual one lies, found by building all sixteen extreme combinations of the four lengths at every crank angle and solving each. The band is not a constant width: it is 0.73° at its widest, near 30°, and 0.36° at its narrowest — a factor of 2.0. Which of those a designer is told depends entirely on where the mechanism was measured.

Tolerance

  1. 1 A length is a range
  2. 2 Two routes to a sensitivity
  3. 3 The four lengths do not matter equally
  4. 4 Worst case and the square root
  5. 5 Where a stack-up stops working
  6. +12 more
17 essays · practice Two circles, four answers, two of them nowhere. A four-bar with its crank held at 52° is two circles: the coupler pin is 3.5 from the crank pin and 3 from the far ground pivot. Two quadratics in two unknowns, so Bézout's number is four — and the tracker finds two. The other two paths run off to infinity, and they do so for every pair of circles ever drawn: two circles meet the line at infinity in the same two points, and those are what the fourth and third answers are.

Algebra

  1. 1 Two circles, four answers
  2. 2 Following a root from a problem already solved
  3. 2 The paths that leave
  4. 2 The count that does not move
  5. 3 Twenty-eight, not forty
  6. +11 more
16 essays · algebra a crank rocker with a post: the closest pair at one position. The same four-bar with a post bolted to the frame, just clear of the coupler's path. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.2561 here, between rocker · post. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

Body

  1. 1 A link that takes up room
  2. 2 A gap is a number
  3. 2 A shape with a dent in it
  4. 3 A gap with corners in it
  5. 3 A sweep that missed nothing
  6. +11 more
16 essays · bodies What the machine draws, against where the polynomial vanishes. Two objects, found two ways. The thin line is the set where x^4 + 2x^2y^2 + y^4 − 1.2x^2 + 1.2y^2 is zero, walked over a grid with no mechanism involved. The marks are where the compiled machine's tracing point went, one per converged solve, over the 147 positions of its working arc. The machine's constraint set never mentions the polynomial, so evaluating it at each traced point is an independent check: the worst value over the whole arc is 3.7e-13. The arc is 1.30 radians of the driving angle and not the whole turn, and past that arc it draws something else.

Compute

  1. 1 A demand that is an equation
  2. 2 Every curve is a sum of cosines
  3. 2 A circle costs one term
  4. 3 Four bars that add two angles
  5. 3 A parallelogram carries an angle, and only so far
  6. +11 more
16 essays · computing The coupler's motion at 66°. Every point drawn as a stub is a point of the coupler's own plane, and the stub is that point's velocity — solved, not sketched. They all point different ways and they are all consistent with one statement: at this instant the whole plane is turning about a single point, the pole, marked with a cross. It is off this frame at 1.5 coupler lengths from the crank pin, which happens whenever the coupler is close to translating. The arrow through it is the pole's own velocity, which is a quantity about the motion rather than about any point of it, and half of everything in this field follows from its direction and its size. positioned by solving, not by drawing.

Curvature

  1. 1 The mechanism drops out
  2. 2 Every point has a centre
  3. 2 The circle of points going straight
  4. 3 The other pole
  5. 3 A construction with no arithmetic in it
  6. +11 more
16 essays · curvature Every way of stopping, on the same four questions. Six mechanisms that all turn a continuous input into an output that moves and then waits. Index is how far the output steps. Moving is the fraction of the input's turn the output is actually going for; the rest is dwell. From rest says whether the output starts and stops at zero velocity, and acceleration whether its acceleration is a number at all. Every entry is computed from the mechanism's own library, which matters for two of them: a Geneva's moving fraction is (n − 2)/2n and not 1/n, and its entry rate is zero in closed form rather than to the accuracy of a sampled sweep. The three rows whose acceleration is not a number are not badly made — they are mechanisms whose output velocity has a step, and no tolerance improves that.

Intermittent

  1. 1 The mechanism that waits
  2. 2 A joint that works one way
  3. 2 The resolution is the pitch
  4. 3 The arc that is concentric with the pivot
  5. 3 The gear with its teeth cut away
  6. +11 more
16 essays · timing One contact, and the point the normal has to pass through. Two wheels on fixed centres, turning in the ratio 24 : 36, with one flank of each drawn. The contact is found by solving n·(v₁ − v₂) = 0 along the first flank — the two velocities are formed from the two rotations and subtracted, and nothing in that calculation knows where the pitch point is. The pitch point, marked with a cross, is computed separately as the one place where the two bodies' material points have the same velocity. The common normal misses it by 8.44e-15 of a millimetre, which is the law of gearing arriving as a measurement rather than as an assumption. positioned by solving, not by drawing.

Meshing

  1. 1 The second shape is not a choice
  2. 2 Any shape has a partner
  3. 2 Rolling at one point only
  4. 3 The tool is the definition
  5. 3 The shape that does not mind where the shafts are
  6. +11 more
16 essays · meshing Three legs, one platform. A 3-RRR planar parallel mechanism at (0.20, -0.15) turned 11.5°, elbows up/up/up. The three actuator angles were computed one leg at a time and independently, which is what makes this direction cheap. The dashed lines extend each leg's second link: those are the three forces the legs can transmit to the platform, and the mechanism is controllable exactly while they stay independent. Here they miss one another by 0.651, and the smallest singular value of the three is 0.9550. At this position the platform can be turned through 206° in all before a leg runs out of reach.

Parallel

  1. 1 The easy problem and the hard one change places
  2. 2 Locked, and still moving
  3. 2 One command, six answers
  4. 2 Six legs and a square root
  5. 3 Why the platform stays flat
  6. +11 more
16 essays · parallel A rolling wheel, where it was driven to. The mechanism at a configuration nothing wrote down: it was reached by integrating permitted velocities from the start of the trail, and there is no equation here whose root it is. The barred line at each wheel is the direction that wheel forbids — the subject of the whole field, and the one thing a photograph of a car cannot show. The constraint residual along the drawn history is 0.0e+0.

Rolling

  1. 1 A constraint that takes nothing away
  2. 2 One character apart
  3. 2 The motion left over by going nowhere
  4. 3 How many wiggles
  5. 3 Parking is an exponent
  6. +11 more
16 essays · rolling elbow arm at a posture. elbow arm, drawn from 6 joint values through a product of 6 exponentials — no equation is solved anywhere in this picture, because an open chain has none to solve. The thin lines are the joint axes at this configuration, which are also the columns of the arm's Jacobian: the tool's velocity is a sum of turns about exactly those lines. Here the smallest singular value of that Jacobian is 0.3674 and the largest is 2.407, so the arm is comfortably away from a configuration where a direction of motion is lost. Drag θ₂ shoulder.

Serial

  1. 1 The chain that does not close
  2. 2 Four numbers or a screw
  3. 2 Where the hand can go
  4. 3 Two routes to a Jacobian
  5. 3 Eight ways to hold the same tool
  6. +11 more
16 essays · serial One routine, six strand systems. Every row is the same function: a list of bodies, each with a sense, handed to a routine that returns the tangent runs between them, the arcs on them, and the total. Nothing in it knows what a belt is, what a tackle is or what a tendon is. The right-hand column is what each row was checked against — a textbook formula, an integer, a convex hull's perimeter, a second route to the same length — and it is the reason the middle column can stay the same all the way down. positioned by solving, not by drawing.

Strand

  1. 1 A member with no length of its own
  2. 2 Where a strand leaves a body
  3. 2 The wraps add up to a turn
  4. 3 A ratio that is a derivative of a length
  5. 3 The strand that is slack
  6. +11 more
16 essays · strands

All essays