Aperiodic borderline case
The aperiodic borderline case describes a damping state of a harmonic oscillator . It is the smallest damping at which the deflection without overshoot, i.e. H. a change of direction that tends towards equilibrium if it is released from a deflected state without an initial speed. The approach to equilibrium takes place in a very short time. If the oscillator has an initial speed, a zero crossing can occur in the aperiodic borderline case. If the damping is even greater, one speaks of an overperiodic fall or creep fall .
The aperiodic limit case corresponds to a Lehr's damping of D = 1 or a quality factor of Q = 0.5.
Linear damped harmonic oscillator
The equation of motion of a damped oscillating mass is:
with the deflection x , the damping constant d , the mass m and the spring constant k .
Usually one identifies as the undamped natural angular frequency of the harmonic oscillator and as the decay constant , so that the following form results for the equation of motion of a damped harmonic oscillator:
This equation can be solved with the exponential approach . The result is the characteristic equation :
With the solution:
The aperiodic borderline case results for, since the discriminant of this equation then becomes 0. Therefore, the oscillator does not oscillate periodically, but rather returns to the rest position in a minimal time.
It then holds and the general solution for the case of a double zero has the following form:
If the oscillator is released at the point in time zero at the point with the speed zero, then and applies , so that the following special solution results:
If, on the other hand, the transducer is "kicked" at the point with the velocity at the point in time zero , the solution is:
If both initial conditions are specified, these solutions can also be superimposed linearly, so that, overall , the solution for the initial data
reads.