Resulting power

Resultant:
a) the site plan with forces F ₁ and F ₂ and the resultant F _R
b) force corner to determine the resultant
c) alternative to b): the parallelogram of forces
d) force couple - resultant equal to zero, but torque not equal to zero

The resultant force (short resultant or resultant ) is in the mechanics , the vector sum of forces , line loads , surface loads and volume forces at one physical system attack on the same or at different points. In the case of only one single force, the resultant is identical to this force. In the case of exactly two non- parallel individual forces, it is given by the diagonal of the associated force parallelogram .

If all forces act at the same point, the system reacts as if only the resulting force were acting at this point. If the forces act at different points, the center of mass of the system reacts as if only the resulting force was acting on it (principle of the center of gravity ). In the event that the resulting force is zero, the center of mass does not move at all or maintains its linear, uniform movement. Regardless of this, however, the forces can exert a torque that influences the rotating movement of the system. The best known example of this is the couple of forces . Here the resulting force is zero.

If the system is a rigid body and the lines of action of the individual forces intersect at a point, then the resulting force, if it acts at this point of intersection, has the same effect on the body in every respect as all individual forces combined (see static equivalence ) .

General procedure for determining the resultant

The resultant is determined by vector addition : with
${\ displaystyle \ mathbf {R} = \ left [\ sum _ {i} ^ {n} \ mathbf {F} _ {i} \ right] + \ left [\ sum _ {i} ^ {m} \ int _ {x} \ mathbf {q} _ {i} (x) \ mathrm {d} x \ right] + \ left [\ sum _ {i} ^ {o} \ iint _ {A} {\ boldsymbol {\ sigma}} _ {i} (x, y) \ mathrm {d} x \ mathrm {d} y \ right] + \ left [\ sum _ {i} ^ {p} \ iiint _ {V} {\ varvec symbol {\ gamma}} _ {i} (x, y, z) \ mathrm {d} x \ mathrm {d} y \ mathrm {d} z \ right] \ quad {\ textrm {with}} \ quad {n , m, o, p} \ in \ mathbb {N} _ {0}}$

${\ displaystyle \ mathbf {R}}$ the resultant
${\ displaystyle \ mathbf {F} _ {i}}$ Single force i
${\ displaystyle \ mathbf {q} _ {i} (x)}$ the line load i
${\ displaystyle {\ boldsymbol {\ sigma}} _ {i} (x, y)}$ the traction vector i
${\ displaystyle {\ boldsymbol {\ gamma}} _ {i} (x, y, z)}$ the volume force i

Procedure for determining the resultant of individual forces

The resultant is determined by vector addition : There are various methods for this.
${\ displaystyle \ mathbf {R} = \ sum _ {i} ^ {n} \ mathbf {F} _ {i} \ quad {\ textrm {with}} \ quad n \ in \ mathbb {N} _ {0 }}$

Analytical procedure

The resultant is determined analytically from the following conditions:

The components of the resultant with respect to a Cartesian coordinate system are equal to the sum of the components of the individual forces and
the components of the moment of the resultant in relation to any point are equal to the sum of the components of the moments of the individual forces.

If the vector sum of the individual forces disappears, the resulting force is zero. This is e.g. B. the case with a couple of forces (Figure d.); a single moment remains , whereby the law of leverage applies.

Graphic process

The force corner (Figure b.) Or the parallelogram of forces (Figure c.) Are used to graphically determine the resultant of two forces .

The three-force method is used to determine the resultant or to determine a third, unknown force if two of three forces are known. The resultant with two or more forces can be z. B. determine with the help of the rope corner method.

The four-forces method according to Karl Culmann serves as the Cremona diagram for graphic determination of the resulting bar or rod forces, for example in the design of trusses .

Individual evidence

↑ Dankert, Dankert: Technische Mechanik , Springer, 7th edition, 2013, p. 20.
↑ Böge: Technische Mechanik , Springer, 31st edition, p. 38.
^ Gross, Hauger, Schröder, Wall: Technische Mechanik - Statik , Springer, 11th edition, 2011, p. 50.
↑ Böge (Ed.): Handbuch Maschinenbau , Springer, 21st edition, 2013, p. B12f.
↑ Mahir Sayir, Jürg Dual, Stephan Kaufmann, Edoardo Mazza: Engineering Mechanics 1: Fundamentals and static . Springer-Verlag, 2015, ISBN 978-3-658-10047-6 ( google.at [accessed December 7, 2019]).

[1] Dankert, Dankert: Technische Mechanik , Springer, 7th edition, 2013, p. 20.

[2] Böge: Technische Mechanik , Springer, 31st edition, p. 38.

[3] Gross, Hauger, Schröder, Wall: Technische Mechanik - Statik , Springer, 11th edition, 2011, p. 50.

[4] Böge (Ed.): Handbuch Maschinenbau , Springer, 21st edition, 2013, p. B12f.

[5] Mahir Sayir, Jürg Dual, Stephan Kaufmann, Edoardo Mazza: Engineering Mechanics 1: Fundamentals and static . Springer-Verlag, 2015, ISBN 978-3-658-10047-6 ( google.at [accessed December 7, 2019]).