Gangpolbahn

Animated crank arm :
red: velocity vectors;
Yellow point: momentary pole ;
Blue: detent pole track of the middle bar;
Green: gang pole path of the middle rod

In the case of a plane rigid body movement , the momentary center is the point in space in which the speed of the particle located there disappears and in which the movement is momentarily represented as a pure rotation. The (body-fixed) gang pole trajectory ( English : moving centrode ) determines the position of the momentary poles as a function of time in the body-fixed reference system , see picture. The gang pole trajectory is the curve on which all particles of the rigid body lie that ever come to a standstill in the movement at the momentary pole.

The following applies to the gang pole path when moving in the xy plane and rotating around the z axis:

{\ displaystyle {\ begin {array} {rcl} x_ {M} & = & {\ frac {1} {\ omega (t)}} [\ sin (\ varphi (t)) {\ dot {s}} _ {x} (t) - \ cos (\ varphi (t)) {\ dot {s}} _ {y} (t)] \\ y_ {M} & = & {\ frac {1} {\ omega (t)}} [\ cos (\ varphi (t)) {\ dot {s}} _ {x} (t) + \ sin (\ varphi (t)) {\ dot {s}} _ {y} (t)] \ end {array}}}

The index "M" refers to the momentary pole, (x, y) are the coordinates in the plane, the speeds in the x or y direction of the reference point around which the rigid body rotates and is the angle of rotation whose time derivative is the angular speed ω around the z-axis. If ω = 0, then there is a translation and the instantaneous pole and the point assigned to it on the gang pole path is not defined. The gang pole track rolls smoothly on the locking pole track . ${\ displaystyle ({\ dot {s}} _ {x}, {\ dot {s}} _ {y})}$ ${\ displaystyle \ varphi}$

The gang pole track is interesting in the kinematics of vehicles, transmission technology, robotics and also prosthetics.

Gang pole path in the complex number plane

Latch level (yellow) with latching coordinates (black) and aisle level (sky blue) with aisle coordinates (blue)

The instantaneous center of gravity and the gang pole trajectory are only defined in the case of plane movements and therefore the plane rigid body movement can be modeled as a movement of the complex plane of numbers. The fixed image space is the locking plane , which represents the space of our perception and which contains the locking coordinate system and the locking pole path. The moving archetype space is the corridor level , which contains the rigid body resting in it, the corridor coordinate system and the corridor pole path. All particles of the rigid body move synchronously with the aisle level. Based on the spatial Eulerian and the material Lagrangian point of view , the coordinates in the rest plane are designated as spatial and with lowercase letters and the coordinates in the corridor plane as material and with uppercase letters, see picture.

Each point in the complex number plane corresponds to a complex number. The translation of a point is modeled with the addition of another number and the rotation around the origin with the product with the complex number , where the angle of rotation, e is Euler's number and i is the imaginary unit . ${\ displaystyle e ^ {\ mathrm {i} \ varphi}}$ ${\ displaystyle \ varphi}$

The motion function χ (Z, t) and speed of a particle Z can then be in the locking plane as ${\ displaystyle {\ dot {\ chi}} (Z, t)}$

{\ displaystyle \ chi (Z, t) = s (t) + e ^ {\ mathrm {i} \ varphi (t)} Z \ quad \ rightarrow \ quad {\ dot {\ chi}} (Z, t) = {\ dot {s}} (t) + \ mathrm {i} \ omega (t) e ^ {\ mathrm {i} \ varphi (t)} Z}

to be written. The point s (t) denotes a moving point of reference, in which the origin of the gear coordinate system, and the rotational speed ω is obtained from the time derivative of the angle of rotation: . The momentary pole stands still in the locking plane, so that ${\ displaystyle \ omega = {\ dot {\ varphi}}}$

{\ displaystyle {\ dot {s}} (t) + \ mathrm {i} \ omega (t) e ^ {\ mathrm {i} \ varphi (t)} M (t) = 0 \ quad \ rightarrow \ quad M (t) = \ mathrm {i} {\ frac {e ^ {- \ mathrm {i} \ varphi (t)}} {\ omega (t)}} {\ dot {s}} (t)}

follows. The material points M, the real and imaginary parts of which were specified at the beginning, designate the gangway pole path in the gangway plane. All particles of the body that ever stand still in the momentary pole lie on the gang pole orbit. The latching pole track, on the other hand, supplies spatial points in the latching plane that are at some point the momentary pole.

Unrolling the gang pole track on the locking pole track

When the rigid body moves, the gang pole track rolls smoothly on the latching pole track. The latching and gang pole tracks are parameterized with the time t. Because this is now known, the time parameter is not stated for the sake of clarity. The pole change speed on the detent pole track is in the detent plane: ${\ displaystyle m = s + {\ frac {\ mathrm {i}} {\ omega}} {\ dot {s}}}$

{\ displaystyle {\ dot {m}} = {\ dot {s}} - {\ frac {\ mathrm {i}} {\ omega ^ {2}}} {\ dot {\ omega}} {\ dot { s}} + {\ frac {\ mathrm {i}} {\ omega}} {\ ddot {s}} \ ,.}

The pole change speed on the gear pole path is transferred to the latching plane with the movement function : ${\ displaystyle {\ dot {M}}}$ ${\ displaystyle \ chi (M, t) = s + Me ^ {\ mathrm {i} \ varphi}}$

{\ displaystyle {\ begin {array} {rcl} {\ dfrac {\ mathrm {d}} {\ mathrm {d} t}} \ chi (M (t), t) & = & {\ dfrac {\ partial } {\ partial M}} \ chi (M, t) {\ dot {M}} + \ overbrace {{\ dfrac {\ partial} {\ partial t}} \ chi (M, t)} ^ {= v | _ {M = const} = 0} = {\ dot {M}} e ^ {\ mathrm {i} \ varphi} \\ [2ex] & = & {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ left (\ mathrm {i} {\ frac {e ^ {- \ mathrm {i} \ varphi}} {\ omega}} {\ dot {s}} \ right) e ^ {\ mathrm {i} \ varphi} = \ left (\ mathrm {i} (- \ mathrm {i} \ omega) {\ frac {e ^ {- \ mathrm {i} \ varphi}} {\ omega}} {\ dot {s}} - \ mathrm {i} {\ frac {e ^ {- \ mathrm {i} \ varphi}} {\ omega ^ {2}}} {\ dot {\ omega}} {\ dot {s }} + \ mathrm {i} {\ frac {e ^ {- \ mathrm {i} \ varphi}} {\ omega}} {\ ddot {s}} \ right) e ^ {\ mathrm {i} \ varphi } = {\ dot {m}} \ end {array}}}

In the latching plane, the pole change speeds on the latching pole track and gear pole track are the same in amount and direction, which is why the two curves roll over one another without sliding.

example

Detent pole and gear pole path for a right-angled cross slide (animation 1.6 M)

The green-colored gang pole trajectory of the system given on the right in the picture should be calculated. The center point S of the cross slide moves with the constant angular velocity Ω on the circular path (not shown) with the radius R around the origin. In the complex number plane, the reference point is and the point running around the reference point on the cross slide rotates with the opposite angular velocity ω = -Ω. This results for the momentary pole: ${\ displaystyle s = Re ^ {\ mathrm {i} \ Omega t}}$

{\ displaystyle M = {\ frac {\ mathrm {i} e ^ {- \ mathrm {i} \ varphi (t)}} {\ omega (t)}} {\ dot {s}} (t) = { \ frac {\ mathrm {i} e ^ {- \ mathrm {i} (- \ Omega t)}} {- \ Omega}} \ mathrm {i} \ Omega Re ^ {\ mathrm {i} \ Omega t} = Re ^ {2 \ mathrm {i} \ Omega t}}

d. H. the radius of the green-colored gear pole curve is equal to half the length of the bold black modeled cross slide and thus equal to the radius of the circular path of point S. The radius of the blue-colored latching pole curve is twice the radius of the gear pole curve and thus identical to the length of the cross slide.

Point-by-point determination of the gear pole curve

Determination of the gear pole curve of a revolving double crank

Nowadays, CAD systems are often used to simulate rigid body systems that enable the point-by-point construction of paths. CAD systems thus enable the easy implementation of the drawing sequence calculation method with little mathematical knowledge and at the same time this method allows the user to easily check the results

As can be seen from the picture of the animated crank arm (at the top), the momentary pole is always on the perpendicular to the velocity vector of a particle of the rigid body and thus on the normal to the path of the particle. The momentary pole is therefore at the intersection of the normal to the paths of 2 particles. The gang pole path results from connecting the instantaneous poles.

In four-link articulated gears such. B. With double cranks, the gang pole path of the coupling shown in black in the picture on the right can be determined point by point, since the path of the joints of the coupling each form an arc with the frame joint as the center (compare picture at the top) and thus the normals with the direction of the gray Cranks and swing arms are identical. In the case of four-link articulated gears, the instantaneous poles are therefore always at the intersection of the gray cranks or rockers or at the intersection of their purple extensions.

literature

M. Husty: Kinematics and Robotics . Springer, 2012, ISBN 978-3-642-63822-0 .

K. Luck, K.-H. Modler: Gear Technology: Analysis Synthesis Optimization . Springer, 1990, ISBN 978-3-211-82147-3 .

G. Bär: Level kinematics . Script for the lecture. Institute for Geometry, TU Dresden ( tu-dresden.de [PDF; accessed on April 1, 2015] Contains further literature recommendations ).

Web link

Commons : Rast- und Gangpolbahnen - Collection of pictures, videos and audio files

Textbook of technical mechanics - dynamics: a descriptive introduction to the Google book search