When I see a website unconsciously (Sorry, this is Japanese site), I found the following writeup.

There are similar name in damping, such as damping coefficient \(c\), damping constant \(h\), hysteretic damping ratio \(h\).

The only description I found is a book seismic ground response analysis¹⁾, in which it is wrhtten that

\(\hspace{20mm}\)\(h=\cfrac{1}{4\pi}\cfrac{\Delta W}{W}\)

but I cannot understand the meaning of the equation. Where does \(4\pi\) comes from? Is \(2\pi\) or \(\pi\), which has the same dimension, is OK or not?

Since my book is listed up there, I found this site. I am not responsible on his writeup. But I think there are many engineers who have the same question. This is the reason why I write this topic

At first, equation of motion of one-degree of freedom system is written as

\(\hspace{20mm}\)\(m\ddot{u}+c\dot{u}+ku=f(t)\) (1)

\(c\) which is multiplied to the velocity is damping coefficient. Since the product of \(c\) and velocity becomes force, \(c\) is a quantily with dimension

Let's consider the case that \(f(t)=0\), i.e., the case with no external load or free vibration problem. Then vibration terminates quicker as damping ratio \(c\) becomes large. Finally, when damping coefficient becomes

\(\hspace{20mm}\) \(c=2\sqrt{mk}=c_c\) (2)

where

is called damping constant. Here, it is noted, in the case of spirng, coefficient and constant are same meaning such as spring coefficient and spring constant, but in the case of damping, damping coefficient and damping constant are different meaning. \(c=c_c\) is called critical damping, then \(\beta\) is a ratio of damping coefficient to critical damping constant. Since Eq. (3) is ratio, it is also called as damping ratio or critical damping ratio.

Instead of \(\beta\), \(h\) is also used for damping ratio. If you are interested in elastic material, knowledge above may be sufficient.

In the elastic system above, energy is absorved by the velocity dependent term. On the other hand, in the nonlinear material, energy is absorved by the hysteretic behavior of stress-strain relationships. This is also damping.

Stress-strain behavior of the nonlinear material, stress-strai relasionships looks as

Here, light blue color is absorved energy by hysteretic behavior. This shape is called as sprindle shape. The area is denoted as \(\Delta W\). On the other hand, red hatched area is stran energy or elastic energy and is denoted by \(W\). The ratio \(\Delta W/W\) is a definition of damping ratio. Howerver, it is more conveninent so that the damping ratio has the same meaning with 1 degree of freedom system. We should define damping ratio as

\(\hspace{20mm}\)\(h=\cfrac{1}{4\pi}\cfrac{S}{W}\)(4)

At the beginning of this section, he has a question "is \(2\pi\) or \(\pi\) is OK?", but it is not good.

I do not see a textbook in which these explanation is written in detail kindly. It is writtend in a book "Vibration theory of building" by Prof. Tajimi²⁾ in detail. According to this book,

One more important concept is written in the above Prof. Tajimi's book. He wrote that, in the case of 1 degree of freedom system, damping ratio depends on mass and damping ratio, therefore, it is defined as damping characteristic of a vibration system. On the other hand, damping ratio in Eq. (4) is an inherent constant of material when it is derived from nonlinear stress-strain relationships. Howerver, this fact is not well recognized in general.

This is very iportant feature in the seismic response analysis of ground. However, this is not well recognized even in SHAKE, which is shown in Complex modulus in SHAKE in detail.

1) Yoshida, N.: Seismic ground response analysis, Springer, 365pp., 2014
2) Tajimi, H.: Vibration theory of building, Corona Publishing, 213pp., 1965