Pole plan

Two discs (gray) are held by an articulated fixed bearing A and two monovalent bearings B and C. The disks are connected to one another via a sliding sleeve S and a roller bearing R. The main poles are marked in red and the secondary pole in blue.

The pole plan is a graphic method in statics (mainly structural engineering) to examine a system of supported and interconnected panes - as in the picture - for its stability. The bodies are called disks here because this is obvious with plane problems; however, they can actually have any three-dimensional shape. The pole plan makes statements about whether the system is static (fixed and immovable), kinematic (movable) or partially kinematic (in its parts or parts of it can be moved).

First, the system is checked for static certainty . If it is under-determined, i.e. inadequately stored, further storage is added until the system is stored in a certain or indefinite way. In this system, the pole plan is created with the rules mentioned below, with the main poles and secondary poles being constructed in the system. The main pole Π _{i of} a disk i is its pivot point (red in the picture) and each disk has exactly one of them. The connection points of two disks i and j define secondary poles (i; j) around which the adjacent disks - provided they are free - can rotate.

If contradictions arise in the search for the main pole of a disc, then this disc cannot be moved and its main pole cannot be located. If there are contradictions in the search for the secondary pole (i; j) of two disks i and j, disks i and j are firmly connected to one another and can be viewed as one disk. When a definitive statement has been made about all main and secondary poles of a system (whether or not they have been found without contradiction), the plan is considered complete. If it is not possible to determine all poles graphically, the poles can also be calculated using pole face equations. If one or more main poles are found (in the picture there are two), then the system is kinematic or partially kinematic and unsuitable as a supporting structure. The system is stable only if there are contradictions in all main poles in the construction of the pole plan.

Rules for creating the pole plan

The slices are numbered with i = 1, 2, 3, ... and each slice is assigned a main pole Π _i . The connections of disks are represented by secondary poles and denoted by (i; j) for two disks i and j connected to one another. The pole plan is drawn up with the following rules, which should be explained using the picture shown at the beginning:

A firmly clamped disc is immovable and has no main pole. All articulated connections to this disc become corresponding bearings and are treated as such. In the picture, the articulated fixed bearing A (gray) can be seen as such a washer.
An articulated fixed bearing is the main pole of the adjacent discs. In the picture, the main pole Π _{1 of} the first disk is in bearing A.
The main pole of a disk resting on a displaceable bearing lies on the straight line which is perpendicular to the direction of movement of this bearing and which leads through the bearing. In the picture, the distance from bearing B to the main pole Π _{2 of} the second disk is perpendicular to the direction of movement of bearing B.
A moment joint forms a secondary pole of the adjacent discs. If the fixed bearing A were the third disk in the picture, then its connection with the first disk would be a moment joint and the secondary pole (1; 3) would be at the point of this joint.
The secondary pole of two disks connected by a normal or shear force joint lies on the straight line which is perpendicular to the direction of movement of the bearing and which leads through the bearing. In the picture, the distance from the secondary pole (1; 2) to the roller bearing R and the sliding sleeve S is perpendicular to the directions of movement of the joints R and S.
The main poles of two disks and their common secondary pole lie on a straight line. In the picture, the secondary pole (1; 2) is on the straight line Π ₁ -Π _{2 connecting} the main poles of disks one and two (which is why the system can be moved).
The secondary poles of three disks lie on a common straight line.

Possible outcomes

If all main and secondary poles are found, the system is considered to be kinematic (movable) and is therefore not stable. If there are contradictions in the pole plan (e.g. a main pole has already been found, but bearing or joint conditions clearly indicate that it must also be in a different location), then that means

if there is a contradiction in the main pole : the disk is fixed (connected to the earth disk, quasi part of the earth disk) or

if there is a contradiction in the secondary pole : the two disks connected by hinges are firmly and rigidly attached to each other (but not necessarily to the earth disk) and thus act as one disk, even if there is no direct material connection between them. The new double disc now has only one main pole that has to be determined.

A system with contradictions in all main poles is firm. A system in which all the main poles can be found without contradiction is movable (kinematic or unstable). If some panes are fixed and others can be moved, the system is partially kinematic.

Pole plane equations

The above-mentioned rules for the construction of the pole plan define points or straight lines in the plane in which or on which the main and secondary poles (must) lie and which can be represented in the form of equations by means of analytical geometry . These equations lead to a linear system of equations that can advantageously be written down in matrix form . The linear algebra knows Lösbarkeitskriterien can be decided according to which, if the system is zero, one or infinitely many solutions.

If no solution exists, this means that the determining equations for (at least) one point are contradictory.
1. If the point is a secondary pole (i; j), then the disks i and j lying in it are firmly connected and can be treated as one disk. The pole plan and / or the equations are set up with this new double disc and solved again.
2. If the point is a main pole Π _i , then the disk i is immovable. If the pane is the only one remaining - possibly after the panes have been folded according to the previous measure - then the pole plan is complete and the system is fixed. Otherwise - if there are several remaining disks - the pole plan and / or the equations are set up anew and solved again as if the disk i were firmly clamped.
If there is exactly one solution, this means that all slices in the system can be moved.
An infinite number of solutions means that at least one pane can be moved.

example

The creation of the pole plan should be reproduced for the system of two panes shown in the picture above. First of all, the static determinacy has to be checked. In the camps, two reactions act in A, one in B and C, and two in each of the joints R and S. Thus there are 2 + 1 + 1 + 2 + 2 = 8 unknown reactions 2 * 3 = 6 equations in the form of two force and one moment equilibrium condition per disc: The system is statically indeterminate. The pole plan should now be used to check whether kinematics still occur.

According to the second rule, the main pole of the first discs is placed in bearing A.
According to the third rule, the main pole of the second disk must lie on the perpendicular to the direction of movement of the bearing B-Π ₂ , which passes through the bearing B.
According to the same rule, the main pole of the second disc must also lie on the perpendicular to the direction of movement of the bearing C-Π ₂ , which passes through the bearing C. The intersection of the straight lines B-Π ₂ and C-Π ₂ marks the main pole Π _{2 of} the second disk.
According to the fifth rule, the secondary pole (1; 2), which connects the first and second discs, lies on the perpendicular R- (1; 2) to the direction of movement of the bearing, which runs through the roller bearing R.
According to the same rule, the secondary pole (1; 2) also lies on the perpendicular S- (1; 2) to the direction of movement of the joint, which runs through the sliding sleeve S. The secondary pole (1; 2) lies at the intersection of the two perpendiculars R- (1; 2) and S- (1; 2).
According to the sixth rule, the main poles of the first and second discs and the secondary pole (1; 2) must lie on a straight line: That is the case.

As a result it can be determined: The two main poles of the disks are to be fixed without contradiction, which is why the considered system is kinematic.

An identical result is obtained with pole plane equations. Point A is at the coordinates (A _x , A _y ) and points B, C, R and S are at the coordinates from the table.

Point	A.	B.	C.	R.	S.
x coordinate	−1	0	2	1	0.5
y coordinate	7th	0	0	4th	2.5

Let the slope of the sliding direction of bearing B be −2, so that the vertical has a slope of 1/2. Then follows:

According to rule two, the main pole of the first disc must be in bearing A:

{\ displaystyle {\ begin {pmatrix} \ Pi _ {1x} \\\ Pi _ {1y} \ end {pmatrix}} = {\ begin {pmatrix} A_ {x} \\ A_ {y} \ end {pmatrix }} = {\ begin {pmatrix} -1 \\ 7 \ end {pmatrix}} \ ,.}

According to rule three, the main pole of the second disk must be parallel to the y-axis through point C: Π _2x = C _x = 2.

Furthermore, according to rule three, the main pole of the second disc must be perpendicular to the direction of movement of bearing B through point B:

{\ displaystyle {\ frac {\ Pi _ {2y}} {\ Pi _ {2x}}} = {\ frac {\ Pi _ {2y}} {2}} = {\ frac {1} {2}} \; \ rightarrow \ quad \ Pi _ {2y} = 1 \; \ rightarrow \ quad {\ begin {pmatrix} \ Pi _ {2x} \\\ Pi _ {2y} \ end {pmatrix}} = {\ begin {pmatrix} 2 \\ 1 \ end {pmatrix}} \ ,.}

According to the fifth rule, the secondary pole (1; 2) has the same x-coordinate as the sliding sleeve S because it can be moved in the x-direction, and the same y-coordinate as the roller bearing R because it can be moved in the y-direction:

{\ displaystyle (1; 2) = {\ begin {pmatrix} 0.5 \\ 4 \ end {pmatrix}} \ ,.}

It remains to be checked whether the secondary pole (1; ₂ ) lies on the straight line between the two main poles (Π ₁ Π ₂ ) according to the sixth rule . This is the case, because the secondary pole is exactly in the middle between the two main poles: ${\ displaystyle (1; 2) = {\ begin {pmatrix} 0.5 \\ 4 \ end {pmatrix}} = {\ frac {1} {2}} \ left [{\ begin {pmatrix} -1 \ \ 7 \ end {pmatrix}} + {\ begin {pmatrix} 2 \\ 1 \ end {pmatrix}} \ right]}$

All main poles and secondary poles are thus determined and the system can therefore be moved in all parts.

Free body images of the panes from the system of two panes presented at the beginning.

One of the following measures could help:

Move bearing B or C so that Π _{2 is} no longer on the straight line Π ₁ - (1; 2).
Change the sliding direction of bearing B so that Π _{2 is} no longer on the straight line Π ₁ - (1; 2).
Shift or twist joints R or S so that the secondary pole (1; 2) no longer lies on the straight line connecting the two main poles.

One of these measures would have the effect that the secondary pole (1; 2) could no longer be fixed without contradiction, which is why both disks would now be firmly connected to one another and treated like one disk. According to the second rule, the main pole Π of this new double disk would have to lie in bearing A and, according to the third rule, at the intersection of the straight lines Π-B and Π-C, which is impossible: the modified system would be fixed.

The calculation of the bearing reactions based on the free-body images of the two panes in the adjacent figure confirms the mobility. An externally acting moment M _{A is} assumed to be the cause of the bearing reactions in bearing A. The reaction moments M _{R, S} in the joints R and S are designated in the same way . The equilibrium of moments in the two main poles provides two equations for the two joint reactions R and S in the joints of the same name and the reaction moments there:

{\ displaystyle {\ begin {array} {rcl} \ sum M_ {i \ Pi _ {2}} & = & - 3R + 1,5S-M_ {R} -M_ {S} = 0 \\\ sum M_ {iA} & = & M_ {A} -3R + 1,5S + M_ {R} + M_ {S} = M_ {A} + 2M_ {R} + 2M_ {S} = 0 \,. \ end {array} }}

If the joints are not supposed to (or cannot) absorb moments, M _A = 0 must be ensured: the system turns out to be susceptible to rotations.

literature

Rolf Mahnken: Textbook of Technical Mechanics . Basics and Applications. 2nd Edition. tape 1 : Rigid body statics. Springer Vieweg, Berlin, Heidelberg 2016, ISBN 978-3-662-52784-9 .
Christian Spura: Technical Mechanics 1. Stereostatics . 1st edition. Springer Vieweg, Wiesbaden 2016, ISBN 978-3-658-14984-0 .

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