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critical viscous-damping. There are additional expressions for the period and frequency

       (1.7)

      where f₀ is the natural frequency and T₀ the oscillation period. Equations (1.1)–(1.3) can now be written as

       (1.8)

       (1.9)

(1.10)

      The problem falls into three cases:

       ζ > 1 overdamped

       ζ < 1 underdamped

       ζ = 1 critically damped.

The first case leads to two real roots, and no oscillation is possible. The second case gives two complex roots, which means that (damped) oscillation occurs. The third case is a transition case between the two other. Subsections 1.1.2–1.1.4 deal with each case in detail.

      1.1.2 The Overdamped Oscillator (ζ > 1)

      Both roots in Equation (1.10) are real, distinct and negative. The motion is called overdamped because introducing this into Equation (1.4) gives a sum of decaying exponential functions:

       (1.11)

The movement of such a system is illustrated in Figure 1.2. Using the above solution and applying the initial conditions u₀ and vx0 we get for B_i:

       (1.12)

      Figure 1.2 Decaying components of the overdamped oscillator. Source: Alexander Peiffer.

      1.1.3 The Underdamped Oscillator (ζ < 1)

      Here, the roots are complex conjugates and the solution of Equation (1.10) becomes:

       (1.13)

       (1.14)

      The motion is oscillatory with a frequency that is lower than in the undamped configuration:

       (1.15)

Introducing the initial conditions u₀ and vx0 at t = 0 the solution for the initial amplitude u^0 and phase ϕ₀ reads as:

       (1.16)

       (1.17)

The damped oscillatory motion is illustrated in Figure 1.3. It shows a decreasing motion that never approaches the equilibrium.

      Figure 1.3 Damped, sinusoidal motion of the underdamped oscillator. Source: Alexander Peiffer.

      1.1.4 The Critically Damped Oscillator (ζ = 1)

      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

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