Traveling pole chain

In the case of a plane rigid body movement , the Bereis pole chain consists of spatial points in which a derivative of the movement of a particle located there with respect to time disappears. In the case of a plane rigid body movement in which the rigid body also rotates, there is always a point at which the speed of the particle in it is zero. This point is the momentary pole . The point at which the acceleration disappears is the acceleration pole . R. Bereis recognized "useful tools" for the investigation of rigid body movements also in those spatial points in which the higher derivatives disappear after time. According to this, the momentary center is the pole of the first order, because the speed is the first derivative of the movement with respect to time. The second order pole is the acceleration pole in which the acceleration disappears. In general, the nth derivative of the motion with respect to time in the nth pole is zero. The totality of these poles forms the Bereis pole chain .

Definition in the complex number plane

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

The momentary center is only defined for plane movements and therefore the rigid body movement is 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 and the corridor coordinate system. 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 ${\displaystyle {\dot {\chi }}(Z,t)}$

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

The time parameter t was not given for the sake of clarity. The point S 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: . This is constant: . ${\displaystyle \omega ={\dot {\varphi }}}$ ${\displaystyle {\dot {\omega }}={\ddot {\varphi }}=0}$

The first pole, the instantaneous pole, is the point p ₁ at which the velocity is zero:

${\displaystyle {\dot {z}}={\dot {s}}+\mathrm {i} \omega (z-s)=0\quad \rightarrow \quad p_{1}=s-{\frac {\dot {s}}{\mathrm {i} \omega }}=s-{\frac {\dot {s}}{(\mathrm {i} \omega )^{1}}}}$

The second pole, the acceleration pole, is the point p ₂ where the acceleration vanishes:

${\displaystyle {\ddot {z}}={\ddot {s}}+\mathrm {i} \omega ({\dot {z}}-{\dot {s}})={\ddot {s}}+(\mathrm {i} \omega )^{2}(z-s)=0\quad \rightarrow \quad p_{2}=s-{\frac {1}{(\mathrm {i} \omega )^{2}}}{\ddot {s}}\,.}$

n-th time derivative s ⁽ⁿ⁾ at reference point s and n-th pole p _n

The higher derivatives are determined in the same way:

${\displaystyle {\begin{array}{rclclcl}z^{(3)}&=&s^{(3)}+\mathrm {i} \omega ({\ddot {z}}-{\ddot {s}})&=&s^{(3)}+(\mathrm {i} \omega )^{3}(z-s)=0&\rightarrow &p_{3}=s-{\frac {1}{(\mathrm {i} \omega )^{3}}}s^{(3)}\\z^{(4)}&=&s^{(4)}+\mathrm {i} \omega (z^{(3)}-s^{(3)})&=&s^{(4)}+(\mathrm {i} \omega )^{4}(z-s)=0&\rightarrow &p_{4}=s-{\frac {1}{(\mathrm {i} \omega )^{4}}}s^{(4)}\\&\vdots &&\vdots &&\vdots &\\z^{(n)}&=&s^{(n)}+\mathrm {i} \omega (z^{(n-1)}-s^{(n-1)})&=&s^{(n)}+(\mathrm {i} \omega )^{n}(z-s)=0&\rightarrow &p_{n}=s-{\frac {1}{(\mathrm {i} \omega )^{n}}}s^{(n)}\,.\end{array}}}$

The sequence (p ₁ , p ₂ , p ₃ , ...) is the Bereis pole chain. The identity carries with it ${\displaystyle \mathrm {i} =e^{\mathrm {i} {\frac {\pi }{2}}}}$

${\displaystyle p_{n}=s-e^{-\mathrm {i} {\frac {n\pi }{2}}}{\frac {s^{(n)}}{\omega ^{n}}}=s+e^{\mathrm {i} \left(\pi -{\frac {n\pi }{2}}\right)}{\frac {s^{(n)}}{\omega ^{n}}}}$

on the construction in the picture.

Pole chain and time derivative

n-th time derivative z ⁽ⁿ⁾ at point z and n-th pole p _n

By adding a zero, the nth time derivative of the movement can advantageously be written with the nth pole:

${\displaystyle {\begin{array}{rcl}z^{(n)}&=&s^{(n)}+(\mathrm {i} \omega )^{n}(z-p_{n}+p_{n}-s)=\underbrace {s^{(n)}+(\mathrm {i} \omega )^{n}(p_{n}-s)} _{=p_{n}^{(n)}=0}+(\mathrm {i} \omega )^{n}(z-p_{n})\\\rightarrow z^{(n)}&=&(\mathrm {i} \omega )^{n}(z-p_{n})=e^{\mathrm {i} {\frac {n\pi }{2}}}\omega ^{n}(z-p_{n})\,.\end{array}}}$

At constant angular velocity, the n-th time derivative of the movement with the ω ⁿ -fold direction vector to the n-th pole in the counterclockwise direction includes the n-fold right angle, see figure.

example

Detent pole and gear pole path for a cross slide rotating on a circular path and rotating in opposite directions (animation 544 kB)

The Bereis pole chain of the system animated in the picture is to be calculated. The center point s of the cross-slide moves with the constant angular velocity Ω on the circular path of radius R around the origin: . The cross slide rotates around its center point at an angular velocity ω = -Ω in the opposite direction. Accordingly, the motion function and the speed are: ${\displaystyle s=Re^{\mathrm {i} \Omega t}}$

${\displaystyle {\begin{array}{rcl}z&=&Re^{\mathrm {i} \Omega t}+e^{-\mathrm {i} \Omega t}Z\\\rightarrow {\dot {z}}&=&\mathrm {i} \Omega Re^{\mathrm {i} \Omega t}-\mathrm {i} \Omega e^{-\mathrm {i} \Omega t}Z=\mathrm {i} \Omega Re^{\mathrm {i} \Omega t}-\mathrm {i} \Omega (z-Re^{\mathrm {i} \Omega t})=2\mathrm {i} \Omega Re^{\mathrm {i} \Omega t}-\mathrm {i} \Omega z\,.\end{array}}}$

The pole chain is thus calculated:

${\displaystyle {\begin{array}{rclcl}z^{(1)}&=&2\mathrm {i} \Omega Re^{\mathrm {i} \Omega t}-\mathrm {i} \Omega z=0&\rightarrow &p_{1}=2Re^{\mathrm {i} \Omega t}\\z^{(2)}&=&2(\mathrm {i} \Omega )^{2}Re^{\mathrm {i} \Omega t}-\mathrm {i} \Omega z^{(1)}=2(\mathrm {i} \Omega )^{2}Re^{\mathrm {i} \Omega t}-\mathrm {i} \Omega [2\mathrm {i} \Omega Re^{\mathrm {i} \Omega t}-\mathrm {i} \Omega z]\\&=&(\mathrm {i} \Omega )^{2}z=0&\rightarrow &p_{2}=0\\z^{(3)}&=&(\mathrm {i} \Omega )^{2}z^{(1)}=0&\rightarrow &p_{3}=p_{1}\,.\end{array}}}$

The fourth pole would then be at the origin again. The poles jump back and forth between the origin and the momentary pole:

${\displaystyle p_{n}={\begin{cases}0\quad &{\text{falls n gerade}}\\2Re^{\mathrm {i} \Omega t}\quad &{\text{falls n ungerade}}\,.\end{cases}}}$

The n-th time derivative of the movement at a location z can be given quickly with these poles:

${\displaystyle z^{(n)}=e^{\mathrm {i} {\frac {n\pi }{2}}}(-\Omega )^{n}(z-p_{n})={\begin{cases}e^{\mathrm {i} {\frac {n\pi }{2}}}\Omega ^{n}z\quad &{\text{falls n gerade}}\\-e^{\mathrm {i} {\frac {n\pi }{2}}}\Omega ^{n}(z-2Re^{\mathrm {i} \Omega t})\quad &{\text{falls n ungerade}}\,.\end{cases}}}$

For the center point of the cross slide is calculated

${\displaystyle {\begin{array}{rcl}z^{(n)}&=&{\begin{cases}e^{\mathrm {i} {\frac {n\pi }{2}}}\Omega ^{n}Re^{\mathrm {i} \Omega t}=(\mathrm {i} \Omega )^{n}Re^{\mathrm {i} \Omega t}\quad &{\text{falls n gerade}}\\-e^{\mathrm {i} {\frac {n\pi }{2}}}\Omega ^{n}(Re^{\mathrm {i} \Omega t}-2Re^{\mathrm {i} \Omega t})=e^{\mathrm {i} {\frac {n\pi }{2}}}\Omega ^{n}Re^{\mathrm {i} \Omega t}=(\mathrm {i} \Omega )^{n}Re^{\mathrm {i} \Omega t}\quad &{\text{falls n ungerade}}\,.\end{cases}}\\&=&(\mathrm {i} \Omega )^{n}Re^{\mathrm {i} \Omega t}\,,\end{array}}}$

as expected.

Individual evidence

1. ↑ R. Bereis: "The far pole position of the plane movement", in the publications of the Technical University of Vienna, issue 11, 1953

literature

• 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 ).