Technical Report: Coefficient representing damping characteristics 1

Vibration is an unavoidable phenomenon when machinery is in operation, but it is not only unpleasant, but can also cause malfunctions, and in severe cases, even lead to the destruction of the machine. Since taking countermeasures after vibration occurs is time-consuming and costly, it is important to thoroughly consider vibration in advance when designing machinery and to implement measures to prevent or reduce vibration. Furthermore, vibration countermeasures are essential for structures such as buildings and bridges.

There are four main types of methods for counteracting vibrations in machinery and structures.

This paper focuses on "damping" and explains how to determine coefficients that represent the characteristics of damping, such as the damping ratio, damping rate, and Q value, as well as the effects these coefficients have on the phenomenon of vibration.

The coefficients used to describe the damping characteristics of damped vibrations and vibration-damping materials include: Damping ratio (damping factor), logarithmic damping rate, loss coefficient,QThere are values, etc. We will explain the definitions and physical meanings of each coefficient later, but here we will first explain how to find these coefficients.

Generally, the amplitude of a damped free vibration waveform decreases exponentially, as shown in Figure 1. Therefore, taking the logarithm of the ratio of adjacent amplitudes always yields a constant value. This natural logarithm of the ratio of adjacent amplitudes is called the logarithmic decay rate, and it is widely used as an easily understandable coefficient to represent damping characteristics. Physical characteristic values such as the damping ratio and loss coefficient can be calculated from the logarithmic decay rate.

If an is the nth amplitude at time tn, and similarly an+1, ..., an+m are the n+1, ..., an+m amplitudes, then the logarithmic decay rate δ is defined by the following equation:

（1）

If the accuracy is insufficient with only one period, we will perform calculations using multiple periods.

  （2）

Therefore,

  （3）

This allows you to determine the damping ratio (damping factor) and loss coefficient.

  （4）

  （5）

The Hilbert transform function of an FFT analyzer can be used to determine the logarithmic decay rate δ and the decay ratio ζ.

When the decay waveform in Figure 1 is imported into an FFT analyzer as a time-domain waveform, a Hilbert transform is performed to convert it into an amplitude envelope, and the Y-axis is displayed in dB, a downward-sloping straight line like that shown in Figure 2 is obtained.

By using the Δ cursor to find ΔX and ΔY between two points on this straight line, the logarithmic decay rate δ and decay ratio ζ can be calculated using the following equations.

If the vibration frequency is fd,

  （6）

  （7）

(fd is determined from the frequency spectrum. Some FFT analyzers can also automatically calculate ζ.)

As will be explained later, when the damping ratio ζ is small, the vibration system exhibits resonant characteristics, and the damping ratio, loss coefficient, and Q value can be calculated from the frequency/amplitude characteristics near the resonant frequency.

In the frequency/amplitude characteristics shown in Figure 3, the Q value, loss factor η, and attenuation ratio ζ can be determined from the peak amplitude frequency f0 and the frequency width Δf at a point 3 dB below the peak value using the following equation.

  （8）

  （9）

  （10）

Since this method reduces the amplitude by -3 dB, or the energy by half, it is called the half-width method.

【Note】

The half-width method can only be used to determine the attenuation ratio when the value of ζ is small (at most 0.1 or less). In other words, when the frequency/amplitude characteristics show a clear unimodal characteristic as shown in Figure 3, the attenuation ratio and attenuation coefficient can be determined using the half-width method. (For the derivation of equation (10), please refer to the [Supplement] "Derivation of the Half-Width Method Calculation Formula".)

Damping ratios and logarithmic damping rates are used as evaluation indices and design parameters for the vibration characteristics of machinery and structures, while loss coefficients are commonly used as evaluation indices for damping materials. The Q value is used to evaluate the characteristics of electrical systems and mechanical resonances.

When determining the damping coefficient of materials with a large loss factor, methods such as the damping rate method, which is calculated from the amount of damping per second, or the impedance method are used. Performance measurement of vibration-damping materials is explained in detail below.