Technical Report: Coefficient Representing Damping Characteristics 2

Free vibration refers to vibration in a state where "no external force is applied." In this state, it remains at rest indefinitely, but when initial conditions such as initial displacement or initial velocity are provided, it begins to vibrate. For example, considering the spring-mass model shown in Figure 4, if we initially pull the mass m to give it a displacement (initial displacement) in the spring k and then suddenly release it, it will start to vibrate, which is free vibration.

ω _d becomes somewhat smaller compared to ω ₀, but in real vibrating systems, the value of ζ is small, so ω _d is close to ω ₀. As can be seen in equation (14), the behavior of a damped vibrating system is determined by the initial conditions and the damping ratio ζ. Figure 5 shows the response due to differences in the damping ratio ζ when the initial velocity is 0 and the initial displacement is 1, and it can be seen that the behavior varies significantly depending on the damping ratio ζ.

In addition, when ζ ≧ 1, it cannot be calculated using equation (14) and relies on a different equation. The calculation formula is omitted here, but for comparison, the response is shown in Figure 5. Incidentally, the state of ζ = 1 is called critical damping, ζ > 1 is over-damping, and 1 > ζ > 0 is under-damping. In both over-damping 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.

Also, the response for ζ = 0.707 (= 1/√2) is shown, which is an important value in the following steady-state oscillations. Although there may be some overshoot (or undershoot), this value is often used in control systems where response time is emphasized, as it represents the shortest settling time (the time it takes for the response to converge within 5% of the target value).

Next, we consider the case where a continuous external force is applied to a free oscillating system.

Forced vibrations occur when an external force is applied, and if the external force is a sine wave, the vibrations that occur after a sufficient amount of time has passed since the force was applied (steady state) are called steady-state vibrations. In contrast, the period from when the external force is applied until the system reaches steady state is referred to as the transient state, which will be explained in the next section.

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

First, I will review the symbols and expressions.

Summarizing the characteristics of amplitude magnification:

When ζ ＜ 1/√2, there is one resonance point with the resonance frequency ω _r = √1-2 ζ ² ω _o . The resonance frequency ω _r decreases as ζ increases, but in the case of a low-damping system, that is when ζ is small (approximately ζ＜ 0.05), ω _r ≒ ω ₀, meaning it resonates at the natural frequency. There is no resonance when ζ ≧ 1/√2.

The amplitude ratio is maximized at the resonance frequency.

img-dampingfactor2-17 （22）

Next, it becomes equal to the Q value. When ζ is small, that is, when the resonance is sharp, it is often treated with the Q value.

At frequencies lower than the resonance point, the amplitude ratio approaches 1.
At frequencies higher than the resonance point, the amplitude ratio approaches 1/(ω/ω ₀) ², which means it approaches a slope of −40 dB/decade.

Equation (19) represents the phase difference between the excitation force and the steady-state vibration. When this is graphed, it results in Figure 8.

What will be understood from now on;

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

In the frequency range higher than ω ₀, it approaches 180 deg, meaning it vibrates in a phase close to being out of phase with the excitation force.

ω ＝ ω ₀ is delayed by 90 deg, that is, oscillates with a phase shift of 1/4 period.

When ζ is small, the phase changes rapidly near ω ₀, and as ζ increases, the change becomes gradual.

In the previous section, we considered the state after a sufficient amount of time has passed since the external force was applied in the steady oscillation. Next, we will examine the state from the moment the external force is applied until it reaches a steady state, that is, the transient state.

Considering the vibration system in Figure 6, its equation of motion becomes Equation (24). Here, for clarity, we assume the external force to be F sin ω t, and we consider the initial conditions to be at rest, meaning both 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 and eventually becomes only the steady-state vibration component of the second term. This behavior is illustrated in Figures 9-1 to 4. Here, with ζ = 0.01, the changes in the transient response are shown for ω/ω ₀ values of 0.1, 0.9, 1.0, and 2.0.

When ω/ω ₀ is small, the steady-state vibration is simply superimposed on the free vibration, and the free vibration decays over time and transitions to the steady-state vibration.

As ω/ω ₀ approaches 1, that is, as the excitation frequency approaches the natural frequency, it can be seen that the amplitude increases and a hum is generated.

ω/ω ₀ = 1 which means that when the excitation frequency matches the natural frequency, the amplitude increases almost proportionally with time, reaching a very large amplitude, thus entering a resonant state.

ω/ω ₀＞ 1 results in a decrease in amplitude, but presents a complex waveform.

In this case, to clearly illustrate the transient state, we set ζ = 0.01 as a small value, but when ζ is large, free vibrations converge quickly, and the amplitude of steady-state vibrations also becomes small. The amplitude is as shown in Figure 7. Conversely, when ζ is small, the transient state takes a long time to settle, resulting in an unstable state that lasts longer. Additionally, the amplitude of steady-state vibrations increases, and especially near the frequency of ω/ω ₀ = 1, even if the initial vibrations are small, the amplitude gradually increases over time, leading to very large vibrations. (Note that Figures 9-1 to 4 have different vertical axis scales.)