Mathematical pendulum

Pendel (reibungsfrei)
pendulum (frictionless)

a mathematical pendulum is an idealization of a material pendulum. It is a fundamental model to understand of hunting oscillations.

Mathematical pendulums are characterized by two substantial characteristics:

  • It does not prevail a friction in any form, thus neither to flow resistance, nor internal friction in thread and point of suspension
  • the entire mass of the pendulum is concentrated in only one point. The thread is regarded as massless, the mass distribution of the pendulum body is represented by its center of mass.

In practice one can approximate a mathematical pendulum, if the used thread is as long and thinly as possible, the pendulum body as small and heavy as possible. If one regards the oscillations in a too not expanded time interval, one can likewise neglect the friction usually.

A typical characteristic with the oscillation of a mathematical pendulum is that the frequency depends only on the length of the pendulum, not however from initial deflection or the attached mass.

mathematical description

the general motion equation for the description of such a pendulum is (similar to the law of Newton)

<math> m \ cdot l \ cdot \ ddot {\ varphi} = - m \ cdot g \ sin \ left (\ varphi \ right) </math>

whereby <math> \ {} /math< ddot> for the second derivative after the time stands. It concerns thus a differential equation.

g is  acceleration due to gravity,  l the thread length and m  the mass of the pendulum.

With small scan angles <math> \ varphi< /math> (<math> \ leq< /math> ) one knows 5° the small angle approximation

<math> \ sin (\ varphi) \ approx \ varphi </math>

use, so that itself the motion equation too

<math> \ ddot {\ varphi} \ approx - \ frac {g} {l} \ varphi< /math>

simplified. This describes a harmonious oscillation. One receives from it two from each other independent solutions

<math> \ varphi_1 (t) = \ varphi_ {\ rm max} \ cos \ left (\ sqrt {\ frac {g} {l}} \ cdot t \ right) </math>
<math> \ varphi_2 (t) = \ varphi_ {\ rm max} \ sin \ left (\ sqrt {\ frac {g} {l}} \ cdot t \ right) </math>

the two a harmonious oscillation with the period duration

<math> T=2 \ pi \ {\ frac {l sqrt} {g}}< /math>

represent. Refrained from the porportionality factor <math> 2 \ pi </math> this connection can be deduced also with the Buckingham Π-theorem. The only kosinusförmige solution results, if one moves and then releases pendulums at the beginning, without exercising a Kraft. The only sinusoidal solution results, if one gives the resting pendulum thrust. If one moves the pendulum and if it schubst at the beginning, then a mixture of the two solutions results:

<math> \ varphi (t) = A \ varphi_1 (t) + b \ varphi_2 (t)< /math>

material pendulums

there genuine pendulums than to be ever more infinitesimally expenditure-steered, behave them in reality nonlinearly. The general differential equation is elementarily not solvable and requires knowledge of elliptical integrals. Thus the general solution in a row can be developed:

<math> T (\ varphi) = 2 \ pi \ sqrt {\ frac {l} {g}} \ left (1+ \ left (\ frac {1} {2} \ right) ^2 \ sin^2 \ left (\ frac {\ varphi} {2} \ right) + \ left (\ frac {1 \ cdot 3} {2 \ cdot 4} \ right) ^2 \ sin^4 \ left (\ frac {\ varphi} {2} \ right) +… \ right)< /math>

In addition the absorption is larger by friction losses with a genuine pendulum than zero, so that deflections decrease approximately exponentially with the time.

 

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