Hysteretic Damping
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## 1 Introduction to Hysteretic Damping[1]

Experiments on the damping that occurs in solid materials and structures which have been subjected to cyclic stressing have shown the damping force to be independent of frequency. This internal, or material, damping is referred to as hysteretic damping.

The viscous damping force $LaTeX: c\dot{x}$ is dependent on the frequency of oscillation. Hysteretic damping is not dependent on frequency so $LaTeX: c\dot{x}$ is not an adequate model. Hysteretic damping requires the damping force $LaTeX: c\dot{x}$ to be divided by the frequency of oscillation $LaTeX: \omega_{n}$.

## 2 Hysteretic Damping: Equation of Motion

The equation of motion is therefore

 $LaTeX: m\ddot{x}+\left(\frac{c}{\omega_n}\right)\dot{x}+kx=0$

Structures under harmonic forcing experience stress that leads the strain by a constant angle, $LaTeX: \alpha$. For a harmonic strain, $LaTeX: \epsilon=\epsilon_{0}\mbox{ sin }\nu t$, where ν is the forcing frequency. The induced stress is

 $LaTeX: \sigma=\sigma_{0}\mbox{ sin}\left(\nu t+\alpha \right)$

Hence

 LaTeX: \begin{alignat}{2}\sigma & =\sigma_{0} \mbox{ cos} \left(\alpha\right) \mbox{ sin}\left(\nu t\right)+\sigma_{0} \mbox{ sin} \left(\alpha\right) \mbox{ cos}\left(\nu t\right) \\ & =\sigma_{0} \mbox{ cos} \left(\alpha \right) \mbox{ sin}\left(\nu t\right)+\sigma_{0} \mbox{ sin} \left(\alpha\right) \mbox{ sin}\left(\nu t+\frac{\pi}{2}\right) \end{alignat}

The first component of stress is in phase with the strain $LaTeX: \epsilon$; the second component is in quadrature with $LaTeX: \epsilon$ and $LaTeX: \frac{\pi}{2}$ ahead. Replacing $LaTeX: \frac{\pi}{2}$ with $LaTeX: j=\sqrt{-1}$ leads to,

 $LaTeX: \sigma=\sigma_{0} \mbox{ cos}\left(\alpha\right) \mbox{ sin}\left(\nu t \right)+j\sigma_{0}\mbox{ sin}\left( \alpha \right) \mbox{ sin}\left(\nu t \right)$

### 2.1 Hysteretic Damping: Loss Factor

A complex modulus E* is formulated, where

 LaTeX: \begin{alignat}{2}E^{*} & =\frac{\sigma}{\epsilon}=\frac{\sigma_{0}}{e_{0}} \mbox{ cos} \left(\alpha \right)+j\frac{\sigma_{0}}{\epsilon_{0}} \mbox{ sin}\left(\alpha \right) \\ & = E^{'}+jE^{''} \end{alignat}

where

$LaTeX: E^{'}$ is the in-phase or storage modulus, and
$LaTeX: E^{''}$ is the quadrature or loss modulus.

The loss factor $LaTeX: \eta$, which is a measure of the hysteretic damping in a structure, is equal to $LaTeX: \frac{E^{''}}{E^{'}}$, that is, $LaTeX: \mbox{tan}\alpha$. Typically the stiffness of a structure cannot be separated from its hysteretic damping, so these quantities are considered as a single coefficient. The complex stiffness $LaTeX: k^{*}$ is given by

 $LaTeX: k^{*}=k\left(1+j\eta\right)$,

where

$LaTeX: k$ is the static stiffness and
$LaTeX: \eta$ the hysteretic damping loss factor.

### 2.2 Hysteretic Damping: Equation of Free Motion for a single DOF system

Figure 1: Hysteretic Damping for single DOF system

The equation of free motion for a single DOF system with hysteretic damping is therefore

 $LaTeX: m\ddot{x}+k^{*}x=0$

Figure 1 shows a single DOF model with hysteretic damping of coefficient $LaTeX: c_{H}$.

The equation of motion is

 $LaTeX: m\ddot{x}+\left(\frac{c_{H}}{\omega}\right)\dot{x}+kx=0$

Now if $LaTeX: x=Xe^{j\omega t}$,

 $LaTeX: \dot{x}=j\omega x$ and $LaTeX: \left(\frac{c_{H}}{\omega}\right)\dot{x}=jc_{H}x$

Reformulating the equation of motion it becomes

 $LaTeX: m\ddot{x}+\left(k+jc_{H}\right)x=0$

Since

 $LaTeX: k+jc_{H}=k\left(1+\frac{jc_{H}}{k}\right)=k\left(1+j\eta\right)=k^{*}$

we can write

 $LaTeX: m\ddot{x}+k^{*}x=0$

That is, the combined effect of the elastic and hysteretic resistance to motion can be represented as a complex stiffness, $LaTeX: k^{*}$.

## 3 Hysteretic Damping Loss Factors[2]

A range of values of η for some common engineering materials is given below. For more detailed information on meterial damping mechanisms and loss factors.

Table 1: Loss Factor for select materials
Material Loss Factor (η)

Aluminum-pure

0.00002-0.002

Aluminum alloy-dural

0.0004-0.001

Steel

0.001-0.008

0.008-0.014

Cast iron

0.003-0.03

Manganese copper alloy

0.05-0.1

Rubber-natural

0.1-0.3

Rubber-hard

1.0

Glass

0.0006-0.002

Concrete

0.01-0.06

## 4 Energy dissipated by Hysteretic Damping[3]

The energy dissipated per cycle by a force F acting on a system with hysteretic damping is $LaTeX: \int F dx$, where

 $LaTeX: F=k^{*}x=k\left(1+j\eta\right)x$,

and x is the displacement.

For harmonic motion $LaTeX: x=X\mbox{ sin } \omega t$, so

 LaTeX: \begin{alignat}F & = kX\mbox{ sin}\left(\omega t\right)+j\eta kX\mbox{ sin}\left(\omega t\right) \\ & = kX\mbox{ sin}\left(\omega t\right)+\eta kX \mbox{ cos}\left(\omega t\right)\end{alignat}. A

Now

 $LaTeX: \mbox{sin}\left(\omega t\right)=\frac{x}{X}$ B.1

 $LaTeX: \mbox{cos}\left(\omega t\right)=\frac{\sqrt{\left(X^2-x^2\right)}}{X}$ B.2

Substituting the Eqn. (B.1) and Eqn. (B.2) into Eqn. (A)

 $LaTeX: F=kx \pm \eta k \sqrt{\left(X^2-x^2\right)}$. C

Eqn. (C) is an ellipse as shown in Figure 2. The energy dissipated is area of this ellipse.

 TODO Find image

Performing the integration of Eqn. (C) we get the total energy dissipated

 $LaTeX: \int F dx & =\int_{0}^{x} \left(kx \pm \eta k \sqrt{\left(X^2-x^2\right)}\right) dx = \pi X^2 \eta k$.

## 5 Notes

• Beards, C. F. 1995 Engineering Vibration Analysis with Applications to Control Systems. ISBN 034063183X
• Lazan, B. J. 1968 Damping of Materials and Members in Structural Mechanics.

### 5.1 References

1. Beards, pp. 41-43
2. Lazan
3. Beards, pp. 43-45