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Contents
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 is dependent on the frequency of oscillation. Hysteretic damping is not dependent on frequency so is not an adequate model. Hysteretic damping requires the damping force to be divided by the frequency of oscillation .
2 Hysteretic Damping: Equation of Motion
The equation of motion is therefore
Structures under harmonic forcing experience stress that leads the strain by a constant angle, . For a harmonic strain, , where ν is the forcing frequency. The induced stress is
Hence
The first component of stress is in phase with the strain ; the second component is in quadrature with and ahead. Replacing with leads to,
2.1 Hysteretic Damping: Loss Factor
A complex modulus E^{*} is formulated, where
where
 is the inphase or storage modulus, and
 is the quadrature or loss modulus.
The loss factor , which is a measure of the hysteretic damping in a structure, is equal to , that is, . 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 is given by
, 
where
 is the static stiffness and
 the hysteretic damping loss factor.
2.2 Hysteretic Damping: Equation of Free Motion for a single DOF system
The equation of free motion for a single DOF system with hysteretic damping is therefore
Figure 1 shows a single DOF model with hysteretic damping of coefficient .
The equation of motion is
Now if ,
and 
Reformulating the equation of motion it becomes
Since
we can write
That is, the combined effect of the elastic and hysteretic resistance to motion can be represented as a complex stiffness, .
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.
Material  Loss Factor (η) 

Aluminumpure 
0.000020.002 
Aluminum alloydural 
0.00040.001 
Steel 
0.0010.008 
Lead 
0.0080.014 
Cast iron 
0.0030.03 
Manganese copper alloy 
0.050.1 
Rubbernatural 
0.10.3 
Rubberhard 
1.0 
Glass 
0.00060.002 
Concrete 
0.010.06 
4 Energy dissipated by Hysteretic Damping^{[3]}
The energy dissipated per cycle by a force F acting on a system with hysteretic damping is , where
, 
and x is the displacement.
For harmonic motion , so
.  A 
Now
B.1 
B.2 
Substituting the Eqn. (B.1) and Eqn. (B.2) into Eqn. (A)
.  C 
Eqn. (C) is an ellipse as shown in Figure 2. The energy dissipated is area of this ellipse.
Performing the integration of Eqn. (C) we get the total energy dissipated
. 
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.