Force system

The force system (also force system or force / force group ) is a term from mechanics that describes a system of mechanical interactions between bodies . When analyzing force systems, it can be a matter of calculating unknown forces in the system, converting the force system into a simpler, equivalent one, which facilitates further analyzes, or determining the accelerations of the bodies involved caused by the forces. The analysis of general systems of forces is not possible without the concept of torque .

In mechanical engineering and structural engineering , knowledge of the loads is essential for dimensioning objects. For the design of chassis, for example, the force systems in the various movement states are calculated using multi-body simulation , which then allows the dynamic properties and loads on the chassis to be assessed.

Internal, external and reaction forces

Fig. 1: Cut-out (red) of a car driving on a slope (gray) (blue)

According to the principle actio and reaction, forces are always interactions between bodies, in Fig. 1 between slope and car. If one body K ₁ exerts a force F on the other K ₂ , then the other also does this on the former:

${\ displaystyle {\ stackrel {K_ {1}} {\ square}} \, {\ stackrel {F} {\ leftarrow}} \ quad {\ stackrel {F} {\ rightarrow}} \, {\ stackrel {K_ {2}} {\ square}}}$

Both forces have the same line of action and are oppositely equal. Forces can be exerted on contact or non-contact by mass attraction and magnetism . The forces between bodies belonging to the system are called internal forces and all together cancel each other out. In addition, there are interactions between bodies that belong to the system and those that do not belong to the system. These are externally attacking or external forces and they always only occur once in the system, for example as a constraining force or an impressed force.

Bodies can change their state of motion and / or deform through the action of force. To determine the forces, the body is cut free from its surroundings . The incision must always be made on the original system and be closed as shown in Fig. 1.

Definition, site plan and power plan

Fig. 2: a: Site plan for a car driving uphill (gray) with external forces (red) and dimensions (blue), b: Force plan

Since forces and torques cause a system of forces is KS only by giving all ( n ) forces along with their force application points clearly defined: ${\ displaystyle {\ vec {F}} _ {1, ..., n}}$ ${\ displaystyle {\ vec {r}} _ {1, ..., n}}$

{\ displaystyle KS: = \ {({\ vec {F}} _ {i}, {\ vec {r}} _ {i}), i = 1, \ ldots, n \}}

In the true-to-scale site plan , the forces are drawn in as they act in the system, see Fig. 2. The site plan corresponds to a true-to-scale free body diagram of the system. In the force plan , the forces are graphically combined into a force polygon using a scale with the dimension length per force unit . The resulting force F _R determined in the force plan is transferred to the site plan. Because this is pointing up the slope in the picture, the car will accelerate.

Basic tasks of statics

The basic tasks of statics are the reduction to a resulting force and a resulting moment, determination of the equilibrium of force systems and the decomposition of a force.

Reduction is about simplifying a system of forces arithmetically or graphically. It turns out that every system of forces in its effect on a rigid body is equivalent to a resulting force with a certain point of application and a resulting moment:

{\ displaystyle \ sum _ {i = 1} ^ {n} {\ vec {F}} _ {i} = {\ vec {F}} _ {R} \ quad {\ text {and}} \ quad \ sum _ {i = 1} ^ {n} ({\ vec {r}} _ {i} - {\ vec {c}}) \ times {\ vec {F}} _ {i} = {\ vec { M}} _ {cR}.}

The resulting force and moment together form the dyname .

A body is in equilibrium when it remains at rest or maintains its state of motion, which places conditions on the system of forces acting on it: In an equilibrium system, the resulting force and the resulting moment disappear. In the past this was called the first and second law of the statics of rigid bodies . A force system that keeps a body in equilibrium is an "equilibrium system" or an "equilibrium group." Such systems do not change the balance or the effect of the force system on a rigid body and may be added to or removed from a force system accordingly.

The decomposition of forces is the opposite of determining the resultant. A force is broken down into components, the resultant of which is the force itself. In practice, it is broken down into perpendicular components in pairs, which analytically amounts to representing the force vector with respect to an orthonormal basis .

Fig. 3: Both forces (red) have the same meaning for equilibrium, not so for deformation (blue)

Forces may only be added, subtracted or shifted in thought for the analysis of the effect on rigid bodies or equilibrium considerations on deformable bodies. If the cutting reactions and the deformation of bodies due to the applied forces are of interest, then the forces must not be changed, see Fig. 3. Only the decomposition into components or the combination of forces acting at a point to form a resultant, see below, is always permitted .

Static equivalence

Two systems of forces that act on a rigid body are statically equivalent if they produce the same performance for any rigid body movements of the force application points . The services of a force and a moment are: ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle {\ vec {M}}}$

{\ displaystyle L_ {F}: = {\ vec {F}} \ cdot {\ vec {v}} \ quad {\ text {or}} \ quad L_ {M}: = {\ vec {M}} \ cdot {\ vec {\ omega}}.}

This is the speed of the point of application of force on the rigid body and the angular speed of the rigid body. The idea behind this definition is that if a body at rest starts moving somehow, for example because a bearing fails, the force systems acting on it are statically equivalent if each system enters the body with the same amount of energy per unit of time (power) for every such hypothetical movement brings in. The assumption of small deformations of a body, which is widespread in technology, implies that its movements are at least initially rigid body movements. ${\ displaystyle {\ vec {v}}}$ ${\ displaystyle {\ vec {\ omega}}}$

It turns out: two systems of forces are statically equivalent if and only if their resulting force and their resulting moment are equal with respect to any reference point.

Analytically this means that the systems of forces

{\ displaystyle KS ^ {a}: = \ {({\ vec {F}} _ {i} ^ {a}, {\ vec {r}} _ {i} ^ {a}), i = 1, \ ldots, n \} \ quad {\ text {and}} \ quad KS ^ {b}: = \ {({\ vec {F}} _ {i} ^ {b}, {\ vec {r}} _ {i} ^ {b}), i = 1, \ ldots, m \}}

are statically equivalent if and only if the following applies for each reference point : ${\ displaystyle {\ vec {c}}}$

{\ displaystyle \ sum _ {i = 1} ^ {n} {\ vec {F}} _ {i} ^ {a} = \ sum _ {i = 1} ^ {m} {\ vec {F}} _ {i} ^ {b} \ quad {\ text {and}} \ quad \ sum _ {i = 1} ^ {n} ({\ vec {r}} _ {i} ^ {a} - {\ vec {c}}) \ times {\ vec {F}} _ {i} ^ {a} = \ sum _ {i = 1} ^ {m} ({\ vec {r}} _ {i} ^ { b} - {\ vec {c}}) \ times {\ vec {F}} _ {i} ^ {b}.}

Earlier this was called the third law of the statics of rigid bodies .

Processing of force systems

With the illustrative procedures presented in this section, force systems can be graphically converted into other force systems that are easier to understand.

Single force

A single force is the simplest possible system of forces, which consists of a force with magnitude and direction and the point of application of force. The point of application and the direction of the force determine its line of action , on which the force can be shifted without changing its effect on a rigid body. The force is thus a bound vector .

Parallelogram of forces

Fig. 4: Parallelogram of forces from two forces (green and blue) and their resultant (red)

Two forces acting at the same point can be replaced by a resultant, see Fig. 4. The forces form two sides of a parallelogram (yellowish) and the resulting force is the diagonal of the parallelogram (red). This graphical construction corresponds analytically to the addition of the vectors representing the forces .

The opposite approach is taken when a force is broken down into components that act parallel to a selected vector space basis and that add up to the force itself. The components of the force are then analytically the components of their vector with respect to the chosen basis.

Unequal, parallel forces

Even two forces working on different but parallel lines of action, which are not oppositely equal, can be replaced by a resultant in their effect on a rigid body, see Fig. 5.

Fig. 5: Construction of the resultant (c, green) for two unequal, parallel forces (a, black) with the help of a zero force (b, blue)

For this purpose, an equilibrium group consisting of two mutually canceling, oppositely equal forces with an identical line of action is added (blue in Figure 5b). The resultant from the original forces and one of the added forces each form a central system of forces (red) and can be replaced by the resultant at the intersection of their lines of action (green in Fig. 5c).

Couple of forces and torque

Fig. 6: Two pairs of forces (blue and black) correspond to a resulting pair of forces (red)

A pair of forces is a system of forces that consists of two oppositely equal forces F , whose lines of action according to the scheme

{\ displaystyle F \ uparrow \; \ longleftarrow a \ longrightarrow \; \ downarrow F}

run parallel at a distance a . The sum of the two forces is indeed the zero vector , but the forces cause a moment M that works perpendicular to the plane defined by the two lines of action and is equal in terms of the product of the distance between the lines of action and one of the forces:

M = a * F .

A couple of forces is equivalent in its effect on a rigid body at this moment. The pair of forces cannot be replaced by a resulting force. The couple strives to turn a body and is therefore not a system of equilibrium.

As with forces, as indicated in Fig. 6, a resulting pair of forces can also be constructed by determining the resultant (red) from the partners (black and blue), which form the resulting pair of forces. This fact is called the moment theorem or first Varignon movement by Pierre de Varignon .

A couple of forces can be shifted freely in space without changing its effect on a rigid body, see Fig. 7.

Fig. 7: A pair of forces (black) can be shifted in parallel anywhere in space.

For this purpose, an equilibrium group (blue in 7b) is applied to the couple of forces (black in 7a, b). The resultants (red in 7b) can be shifted along their lines of action (dashed in 7b, c). Decomposition of the resultant into an equilibrium group (blue in 7c) and the shifted pair of forces (black in 7cd) shows that the force systems in Fig. 7a and d are equivalent. With the pair of forces, the corresponding torque can also be freely shifted, which is also called the second Varignon theorem. The point of application can be freely chosen for torques, they are free vectors .

Displacement or misalignment moment

A force can be shifted along its line of action without changing its effect on a rigid body. With the parallel displacement perpendicular to the line of action, an offset or offset moment arises, see Fig. 8.

Fig. 8: A single force (a, black) is equivalent to an offset force (c, green) and a dislocation moment (c, red).

By adding an equilibrium system of force and oppositely equal, extinguishing force (blue in Fig. 8b), an equivalent system is created from the displaced force (green in Fig. 8c) with a dislocation moment (red in Fig. 8c).

Power screw or winder

With the help of the offset torque , a system of forces consisting of a force and an arbitrary moment can be converted into a statically equivalent system of forces consisting of a force that is equal in amount, parallel shifted, and a moment that is parallel to the force, see Fig. 9. ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle {\ vec {M}}}$ ${\ displaystyle {\ vec {F}} _ {\ parallel}}$ ${\ displaystyle {\ vec {M}} _ {\ parallel}}$

Fig. 9: Any force and moment becomes an offset force and a parallel moment. A detailed description is given in the text.

For this purpose, the moment is converted into a force couple (red in 9b), which is broken down into a force couple with force components perpendicular to the force (blue) and parallel to the force (green). From the latter pair of forces, together with the force, a force system results from two parallel forces (green in 9c), from which the resultant is constructed using the means shown above . In terms of amount, this is equal to the original force but shifted in parallel. The pair of forces with the forces that are too perpendicular create a moment that is parallel to (blue in 9e). The system of the force displaced in parallel and the moment parallel to it is statically equivalent to the original system of forces and is called a power winder or power screw , which is an important term in screw theory . ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle {\ vec {F}} _ {\ parallel}}$ ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle {\ vec {F}}}$

Special systems of forces

General and spatial systems of forces

In the general system of forces, the forces work on arbitrary lines of action, which do not all have to meet at one point. General systems of forces are therefore generally not central. They can be spatial or flat, in balance or not.

In spatial force systems, the forces work in any spatial direction. Spatial force systems are generally not flat. But they can be general or central and in balance or not.

Level system of forces

In the plane system of forces, the lines of action of the forces are all in a common plane. Level systems of forces can be graphically processed in a particularly clear manner, using the methods presented above. If there are many forces, the rope corner method can be used to determine the resultant.

Central system of forces

A central force system consists of forces whose lines of action all through a point P go. The parallelogram of forces is a simple central system of forces. The resulting force is the sum of the forces and its effect on a rigid body is equivalent to the central force system. There is no torque with respect to point P. The inner forces that cancel each other out in pairs form particularly simple central force systems with vanishing resultants, which can be added or removed at any time and anywhere without affecting the body in any way. The central system of forces is an equilibrium system when the resulting force disappears.

Equilibrium system

A system of forces is in equilibrium when both the resulting force and the resulting moment vanish in a reference point. Equivalent in rigid body mechanics are the statements that no resultants act or the system of forces is statically equivalent to a system in which no forces act.

The resultant is determined graphically by shifting the forces along their lines of action in the reference point, possibly with the introduction of a dislocation moment. This creates a resulting force and a resulting moment at the reference point, and if both are the zero vector, then the system is in equilibrium. The analysis can often be simplified by choosing the right reference point.

Analytically, six linearly independent equilibrium conditions are derived for the components of the resulting force ( force equilibrium ) and the resulting moment ( moment equilibrium ) with respect to a point : ${\ displaystyle {\ vec {s}}}$

{\ displaystyle {\ vec {F}} _ {R}: = \ sum _ {i = 1} ^ {n} {\ vec {F}} _ {i} = {\ vec {0}} \ quad { \ text {and}} \ quad {\ vec {M}} _ {sR}: = \ sum _ {i = 1} ^ {n} ({\ vec {r}} _ {i} - {\ vec { s}}) \ times {\ vec {F}} _ {i} = {\ vec {0}}.}

If the forces and moments are in equilibrium with respect to one point , then their moment result vanishes with respect to every other reference point as well . Because with the above equilibrium conditions it follows: ${\ displaystyle {\ vec {s}}}$ ${\ displaystyle {\ vec {c}}}$

{\ displaystyle {\ begin {aligned} {\ vec {M}} _ {cR}: = & \ sum _ {i = 1} ^ {n} ({\ vec {r}} _ {i} - {\ vec {c}}) \ times {\ vec {F}} _ {i} = \ sum _ {i = 1} ^ {n} ({\ vec {r}} _ {i} - {\ vec {s }} + {\ vec {s}} - {\ vec {c}}) \ times {\ vec {F}} _ {i} \\ = & \ sum _ {i = 1} ^ {n} ({ \ vec {r}} _ {i} - {\ vec {s}}) \ times {\ vec {F}} _ {i} + ({\ vec {s}} - {\ vec {c}}) \ times \ sum _ {i = 1} ^ {n} {\ vec {F}} _ {i} = {\ vec {M}} _ {sR} + ({\ vec {s}} - {\ vec {c}}) \ times {\ vec {F}} _ {R} = {\ vec {0}}. \ end {aligned}}}

A system of equilibrium can be added to another system of forces without disturbing its equilibrium. If each of its subsystems is in equilibrium, then the entire system is also in equilibrium.

Application in rigid body mechanics

Every system of forces can be replaced in its effect on a rigid body by a resulting force and a resulting torque acting in its center of mass . The law “ force equals mass times acceleration ” and Euler's gyroscopic equations then provide six differential equations with which the movement of the system is determined, see multi-body simulation .

The theory of stability deals with the question of whether a small change in position of the body is stimulated or dampened by the external force system, which is of interest for the hull of ships , for example , see picture. The force system consisting of buoyancy A and weight G tries to bring the dinghy back into an upright position: the hull is dimensionally stable .

Application in strength theory

In strength theory, we are interested in the loads on the body that the force system acts on. Force application points must not be moved on deformable bodies. From the equilibrium conditions in statically determined supported bodies, as in the picture, possibly still unknown support reactions (green) can be calculated from the known impressed forces . Then all external forces are available and internal cutting reactions can be determined at any point. Material assumptions, of which linear elasticity is the simplest, allow conclusions to be drawn about the stress on the body at the location of the cutting reactions and thus provide a basis for dimensioning the body.

Individual evidence

^ ^A ^b H. Egerer: Engineering Mechanics . Textbook of technical mechanics in mainly graphic treatment. tape 1 . Springer, Berlin, Heidelberg 1919, ISBN 978-3-662-32061-7 , pp. 124 ( google.de [accessed on January 3, 2017]).
↑ MB Sayir, J. Dual, S. Kaufmann: Engineering Mechanics 1 . Basics and statics. Springer, 2008, ISBN 978-3-8351-0018-3 ( springer.com [accessed December 30, 2016]).
↑ C. Hartsuijker, JW Welleman: Engineering Mechanics . Volume 1: Equilibrium, ISBN 978-1-4020-4120-4 , pp. 64 ( google.com ).

literature

Alfred Böge: Vieweg manual mechanical engineering . Basics and applications of mechanical engineering. 18th edition. Vieweg-Verlag, Wiesbaden 2008, ISBN 978-3-8348-0110-4 .
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 .

[egerer-1] A ^b H. Egerer: Engineering Mechanics . Textbook of technical mechanics in mainly graphic treatment. tape 1 . Springer, Berlin, Heidelberg 1919, ISBN 978-3-662-32061-7 , pp. 124 ( google.de [accessed on January 3, 2017]).

[2] MB Sayir, J. Dual, S. Kaufmann: Engineering Mechanics 1 . Basics and statics. Springer, 2008, ISBN 978-3-8351-0018-3 ( springer.com [accessed December 30, 2016]).

[3] C. Hartsuijker, JW Welleman: Engineering Mechanics . Volume 1: Equilibrium, ISBN 978-1-4020-4120-4 , pp. 64 ( google.com ).