Performance rate

The rate of performance is a formula that is used in physics .

The principle of performance of mechanics

The power law of mechanics is a generalization of the energy law. It states that the sum of all power acting on a system at any point in time is equal to the temporal change in the kinetic energy of the system:

${\ displaystyle {\ frac {d} {dt}} E_ {kin} = \ sum {P_ {i}}}$ .

The services affecting a system are always composed of conservative and non-conservative services:

${\ displaystyle {\ frac {d} {dt}} E_ {kin} = \ sum {P_ {i, K}} + \ sum {P_ {i, NK}}}$ .

Non-conservative performances are, for example, the performance of frictional or damping forces from which energy is dissipated .

Special case of only conservative services

If the sum of the non-conservative benefits is identical to zero, i.e. H. only conservative services have an effect on a system, then one can quickly show that the rate of performance now changes into the rate of energy . You get it now

${\ displaystyle {\ frac {d} {dt}} E_ {kin} = \ sum {P_ {i, K}} + 0}$

and due to the definition of the (temporal change in) potential energy , the energy law of mechanics now follows directly ${\ displaystyle {\ frac {d} {dt}} E_ {pot} = - \ sum {P_ {i, K}}}$

${\ displaystyle {\ frac {d} {dt}} E_ {kin} + {\ frac {d} {dt}} E_ {pot} = 0}$

and after integration according to the time the known notation

${\ displaystyle \ int \ left ({\ frac {d} {dt}} E_ {kin} \ right) dt + \ int \ left ({\ frac {d} {dt}} E_ {pot} \ right) dt = \ int 0dt}$

${\ displaystyle \ Rightarrow E_ {kin} + E_ {pot} = const.}$

The power of a force

The power of a (vectorial) force is defined as follows: ${\ displaystyle {\ vec {F}}}$

${\ displaystyle P_ {F} = {\ vec {F}} \ ast {\ vec {v_ {F}}}}$

with the speed of the force application point. So you remember: "Power is power multiplied by speed". ${\ displaystyle {\ vec {v_ {F}}}}$

If the force acts exactly in the direction of the speed of its point of application along a coordinate axis, i.e. if both vectors have exactly one and the same non-zero component, then the scalar product of the two vectors is simplified to the product of the two scalar quantities (amount of force times amount the speed). Recognizing this simplifies many calculations considerably, as you can save yourself the cumbersome handling of vector components.

The achievement of a moment

The power of a (vectorial) moment results as ${\ displaystyle {\ vec {M}}}$

${\ displaystyle P_ {M} = {\ vec {M}} \ ast {\ vec {\ omega}}}$

with the angular velocity of the torque application point. This can be traced back to the performance of a force if you keep the following in mind: Since you can decompose a moment M into a product of force F and lever arm r, (here non-vectorial!), And at the same time the speed of a point Equal to a rotating body is the angular velocity of the rotation of the body multiplied by the distance r from the center of rotation,, follows the claim for the power of a moment. Here, too, applies again, only in a different representation: "Power is power times speed". ${\ displaystyle {\ vec {\ omega}}}$ ${\ displaystyle M = r \ cdot F}$ ${\ displaystyle v = \ omega \ cdot r}$

Equation of motion and scope

After cutting free the system, calculating all unknown forces and moments as well as the speeds of the respective force application points, the equation of motion for an unknown degree of freedom can be formed from the performance set . It does not matter whether the degree of freedom is an angle or a coordinate, it just has to appear in the expression . To do this, the individual kinetic energies of the system, consisting of translational and rotational movements, are added up depending on the one degree of freedom and the term obtained is derived from the time. It is important that the system may only have exactly one degree of freedom if you want to treat it with the performance rate. In the same way as the energy law, note that with the power law, a system with more than one degree of freedom can no longer be treated in this way, because then it would not be clearly defined how the energies are distributed among the individual degrees of freedom. If there is more than one degree of freedom, the Lagrange formalism is chosen for setting up the equations of motion. ${\ displaystyle {\ frac {d} {dt}} E_ {kin}}$