Читать книгу Vibroacoustic Simulation - Alexander Peiffer - Страница 18

The last case is a transition between both systems. There is only one root s=−ω0, and the solution in Equation (1.4) becomes:

(1.18)

This solution does not provide enough constants to fulfil the initial conditions, so that we need an extra term te−ω0t:

(1.19)

Introducing the initial conditions again, the constants are:

(1.20)

(1.21)

Critically damped systems can be of practical relevance, because the motion returns to rest in the shortest possible time, which is useful if periodic motion shall be prevented. In contrast to the overdamped oscillator the equilibrium is reached as can be seen in Figure 1.4.

Figure 1.4 Motion of the critically damped oscillator. Source: Alexander Peiffer.

Let us summarize some facts and observations about free damped oscillators:

1 Oscillation occurs only if the system is underdamped.

2 ωd is always less than ω0.

3 The motion will decay.

4 The frequency ωd and the decay rate are properties of the system and independent from the initial conditions.

5 The amplitude of the damped oscillator is u^(t)=u^0e−βt with β=ζω0. β is called the decay rate of the damped oscillator.

The decay rate is related to the decay time τ. This is the time interval where the amplitude decreases to e−1 of the initial amplitude. Thus, the decay time is:

(1.22)