Hysteretic Damping

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Hysteretic Damping
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Viscous Damping Single DOF
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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

Lead

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

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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