Technical Report: Coefficient Representing Damping Characteristics 1

Vibrations are an unavoidable phenomenon when machines are in operation, and they can not only be uncomfortable but also cause failures, potentially leading to the destruction of the machine in severe cases. Addressing vibrations after they occur can be labor-intensive and costly, so it is important to conduct thorough considerations regarding vibrations during the design of machines and to implement measures to prevent or reduce them. Furthermore, vibration control is essential in structures such as buildings and bridges.

There are four main methods to consider for vibration control in machines and structures.

This paper focuses on 'damping' and explains the methods for calculating coefficients that represent the characteristics of damping, such as the damping ratio, damping rate, and Q factor, as well as the effects of these coefficients in the phenomenon of vibration.

The coefficients that represent the damping characteristics such as damping vibrations and damping materials include damping ratio (damping factor), logarithmic decrement, loss coefficient,Qvalue, etc. The definitions and physical meanings of each coefficient will be explained later, but here we will first explain how to calculate these coefficients.

In general, the amplitude of a damped free vibration waveform decreases exponentially, as shown in Figure 1. Taking the logarithm of the ratio of adjacent amplitudes results in a constant value. This natural logarithm of the ratio of adjacent amplitudes is called the logarithmic decay rate and 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.

Let the n-th amplitude at time t _n be a _n, and similarly, let the amplitudes for n + 1, ..., n + m be a _{n + 1}, ..., a _{n + m}. The logarithmic decay rate δ is defined by the following equation.

（1）

If a single cycle is insufficient for accuracy, calculations will be performed over multiple cycles.

  （2）

Therefore,

  （3）

You can calculate the damping ratio (damping factor) and loss coefficient.

  （4）

  （5）

Using the Hilbert transform function of the FFT analyzer, you can obtain the logarithmic decay rate δ and the decay ratio ζ.

By importing the decay waveform from Figure 1 as a time axis waveform into the FFT analyzer, performing the Hilbert transform to convert it into an amplitude envelope, and displaying the Y-axis in dB, you will obtain a downward sloping straight line as shown in Figure 2.

Using the cursor, if you find ΔX and ΔY between the two points of this straight line section, the logarithmic decay rate δ and the decay ratio ζ can be calculated by the following formula.

If the vibration frequency is f _d, then,

  （6）

  （7）

( f _d is obtained from the frequency spectrum. Also, some FFT analyzers can automatically calculate ζ.)

As mentioned later, when the damping ratio ζ is small, the vibrating system exhibits resonance characteristics, and the damping ratio, loss coefficient, or Q value can be calculated from the frequency/amplitude characteristics near the resonance frequency.

In the frequency/amplitude characteristics as shown in Figure 3, the Q value, loss coefficient η, and damping ratio ζ can be determined from the peak frequency f ₀ and the frequency width Δf at the point that is 3 dB below the peak value using the following formula.

  （8）

  （9）

  （10）

An amplitude of −3 dB means that the energy is reduced to 1/2, so this calculation method is called the half-value width method.

[Note]

The half-value 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 in Figure 3, the attenuation ratio and attenuation coefficient can be determined using the half-value width method. (For the derivation of Equation (10), refer to [Supplement] "Derivation of the Half-Value Width Method Calculation Formula")

Damping ratio and logarithmic decrement are used as evaluation indices or design parameters for the vibration characteristics of machines and structures, while the loss factor is often used as an evaluation index for damping materials. Additionally, the Q value is used to evaluate the characteristics of electrical and mechanical resonance.

When determining the damping coefficient of materials with a large loss factor, methods such as the decrement method based on the amount of damping over one second or the impedance method are used. The performance measurement of damping materials is explained in detail below.