Damped harmonic oscillator with initial conditions a) and external force excitation b). Source: Alexander Peiffer.
1.1.1 Homogeneous Solutions
Without external excitation as shown in Figure 1.1 a) the motion depends on the initial conditions at time t = 0 with the displacement u(0)=u0 and velocity vx(0)=vx0. The damping is supposed to be viscous, thus proportional to the velocity Fxv=−cvu˙. The equation of motion
is a homogeneous second order equation with a solution of the form u=Aest. Entering this into Equation (1.1) leads to the characteristic equation
with the two solutions
Hence,
with B1 and B2 depending on the initial conditions. The root in Equation (1.3) is zero when cv equals 4mks. This specific value is called the critical viscous damping damping
We use the following definitions:
ω0 is the natural angular frequency, ζ is ratio of the viscous-damping to the critical viscous-damping. There are additional expressions for the period and frequency
where f0 is the natural frequency and T0 the oscillation period. Equations (1.1)–(1.3) can now be written as
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:
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 Bi:
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:
