Technical Report: Coefficients Representing Damping Characteristics (2)

Free vibration is vibration that occurs "without any external force applied." Without any external force, it would remain stationary indefinitely, but if initial conditions such as an initial displacement and initial velocity are applied, it will begin to vibrate. For example, consider the spring-mass model shown in Figure 4. If you initially pull on a mass m to impart a displacement (initial displacement) to a spring k, and then suddenly release it, it will begin to vibrate; this is free vibration.

_ωd is somewhat smaller than _ω0, but in real-world vibration systems, the value of ζ is small, so _ωd is close to _ω0. As can be seen from equation (14), the behavior of a damped vibration system is determined by the initial conditions and the damping ratio ζ. Figure 5 shows the response for different damping ratios ζ when the initial velocity is 0 and the initial displacement is 1, and it can be seen that the behavior differs greatly depending on the damping ratio ζ.

Note that when ζ ≥ 1, the calculation cannot be performed using equation (14), and a different equation is required. While that calculation formula is omitted here, Figure 5 shows the response for comparison. Incidentally, the state where ζ = 1 is called critical damping, ζ > 1 is called overdamping, and 1 > ζ > 0 is called underdamping. In overdamping and critical damping, the motion is damped without oscillation. In Figure 5, ζ = 1 (critical damping) is emphasized for clarity, but this merely indicates the boundary between oscillation and non-oscillation, and does not imply that critical damping is particularly important.

The response for ζ = 0.707 (= 1/√2) is also shown, which is an important value in the following steady-state oscillation. Furthermore, although there is some overshoot (undershoot), this value is often used in control systems and other applications where response time is important, as it results in the shortest settling time (the time it takes for the response to converge to within 5% of the target value).

Next, let's consider the case where a continuous external force is applied to a free-vibrating system.

Vibration caused by an external force is called forced vibration. However, vibration in a state where the external force is a sine wave and a sufficient amount of time has elapsed since the force was applied (steady state) is called steady vibration. In contrast, the process from the application of the external force to reaching the steady state is called the transient state, which will be explained in the next section.

As shown in Figure 6, consider a model in which an excitation force F cos ωt is applied to a one-degree-of-freedom vibration system.

First, let's review the symbols and formulas.

Figure 6: Forced Excitation Vibration Model

To summarize the characteristics of amplitude magnification:

ζ < 1/√2 At that time, ω _r = √1-2 ζ ² ω _o It has one resonance point with a resonant frequency of [frequency]. ω _r teeth ζ The larger the value, the lower the value, but a low damping system, i.e. ζ When it is small (approximately) ζ < 0.05) ω _r ≒ ω ₀ In other words, it resonates at its natural frequency. ζ ≥ 1/√2 Then it won't resonate.

The amplitude magnification is maximized at the resonant frequency.

img-dampingfactor2-17 （22）

This becomes equal to the Q-factor. When ζ is small, i.e., when the resonance is sharp, the Q-factor is often used.

At frequencies below the resonance point, the amplitude magnification asymptotically approaches 1.
At frequencies higher than the resonance point, the amplitude multiplier asymptotically approaches 1/(ω/ω ₀) ², i.e., a slope of -40 dB/decade.

Equation (19) represents the phase difference between the excitation force and the steady-state vibration. Graphing this gives us Figure 8.

What we will learn from this is:

In the frequency range where the excitation force frequency is lower than _ω0, the phase lag of the steady-state vibration asymptotically approaches 0 degrees, meaning it vibrates with a phase slightly delayed from the excitation force.

In the frequency range above _ω0, it asymptotically approaches 180 degrees, meaning it oscillates with a phase close to the opposite phase to the excitation force.

At ω = _ω0, the oscillation is delayed by 90 degrees, or 1/4 of a period.

When ζ is small, the phase changes abruptly around _ω0, and as ζ increases, the change becomes more gradual.

In the previous section, we considered a state where a sufficient amount of time had elapsed since an external force was applied to a steady state. Next, let's consider the transient state, that is, the state from the time an external force is applied until a steady state is reached.

Considering the vibration system in Figure 6, its equation of motion is given by equation (24). However, for simplicity, we will assume the external force is F sin ω t, and the initial condition is complete rest, meaning the initial displacement and initial velocity are zero.

img-dampingfactor2-20 （24）

The first term of equation (25) is the free vibration component, which decays over time until only the steady-state vibration component (the second term) remains. Figures 9, 1 to 4, show this process graphically. Here, with ζ = 0.01, the changes in the transient response are shown when ω/ _ω0 is varied to 0.1, 0.9, 1.0, and 2.0.

When ω/ _ω0 is small, free vibrations are simply superimposed on steady vibrations, and these free vibrations decay over time, transitioning to steady vibrations.

As ω/ _ω0 approaches 1, that is, as the excitation frequency approaches the natural vibration frequency, the amplitude increases and a humming sound is produced.

ω/ _ω₀ = 1, that is, when the excitation frequency matches the natural vibration frequency, the amplitude increases almost proportionally to time, reaching a very large amplitude, i.e., a resonance state.

When ω/ω ₀ > 1, the amplitude decreases, but the waveform becomes more complex.

Here, we used a small value of ζ = 0.01 to clearly illustrate the transient state. However, a larger ζ value results in faster deceleration of the free vibration and a smaller amplitude of the steady-state vibration. This amplitude is shown in Figure 7. Conversely, a smaller ζ value results in a prolonged transient state and a longer period of instability. The amplitude of the steady-state vibration also increases, and particularly at frequencies around ω/ _ω₀ = 1, even small vibrations will gradually increase in amplitude over time, growing into very large vibrations. (Note that Figures 9-1 to 9-4 have different vertical axis scales.)