Hysteretic Damping Modelling by Nonlinear Kelvin-Voigt Model

1
PUBLISHING HOUSE
OF THE ROMANIAN ACADEMY
PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A,

Volume 3, Number 3/2002.
HYSTERETIC DAMPING MODELLING BY NONLINEAR KELVIN-VOIGT MODEL
Dinu BRATOSIN, Tudor SIRETEANU

Institute of Solid Mechanics - Romanian Academy,
Calea Victoriei 125, 71102 Bucharest
Corresponding author: Dinu BRATOSIN: e-mail: bratosin@acad.ro
ABSTRACT. This paper present a nonlinear Kelvin-Voigt model (NKV model) with the stiffness and
damping characteristics as function in term of displacements. The behaviour of this model for harmonic imput was verified by means of the resonant column experimental data.
Key word: nonlinear dynamics, soils dynamics, damping modelling
1. INTRODUCTION
The strong dependence of the soils dynamic
properties on strain or stress level produced by external loads is very well known. In the previous
author's papers [1], [2], [3] this nonlinear behaviour was modeled assuming that the geological
materials are nonlinear viscoelastic materials. The
dynamic model obtained was built upon two dynamic nonlinear functions one for material
strength modeling and another including material
damping, both in terms of strain level caused by
external loading conditions and both functions being completely determined from resonant column
test data.
The resonant column system can be considered
as a one degree-of-freedom system that is made up
of a single mass (the vibration device) supported
by a spring and a damper represented by the
specimen [3]. But, due to the mechanical properties of the specimen materials both spring and
damper have non-linear characteristics and thus the
entire system is a non-linear one [4].
In linear dynamics a usual description of a solid
single-degree-of-freedom behaviour is given by the
Kelvin-Voigt model consisting of a spring (with a
stiffness k) and a dashpot (with a viscosity c) connected in parallel. The governing equation of this
system for torsional harmonic vibrations (usually
resonant column system excitation) is:
J 0 && + c & + k = M 0 sin t
__________________________
Recommended by Radu P.VOINEA
Member of the Romanian Academy
(1.1)
where is the system's displacement (rotation, in

this case), J0 is the moment of inertia of the vibrator, M 0 are the amplitude and the pulsation of the
harmonic external imput.
In the non-linear case, due to the mechanical
properties of the specimen materials both spring
and dashpot characteristics become non-linear
functions in terms of deformation (or rotation)
level [4], [5]. The most expected form of the governing equation for non-linear behaviour of a single- degree-of-freedom system is:
J 0 && + c() & + k() = M 0 sin t
(1.2)
with the analogic model from fig.1.1.

The purpose of this paper is to verify this nonlinear forms of the Kelvin-Voigt model and the
capabilities of this model to modelling the hysteresis loops.
Mt = M0 sin t
m ; J0
k = k ()
c = c()
Fig.1.1 Non-linear Kelvin-Voigt model
2. NONLINEAR MATERIAL FUNCTION

In order to model the non-linear material behaviour, in [1], [2], and [3] a non-linear viscoelastic
constitutive law for dynamic response of soils was
presented. This model describes the nonlinearity
by the dependence of the material mechanical parameters: shear modulus G and damping ratio D in
terms of shear strain invariant : G = G ( ) ,

D = D ( ) , or twisting angle : G = G ( ) ,
D = D ( ) .
As an example, in fig.2.1 such non-linear material functions obtained from resonant column test
performed upon clay sample are given.
Fig.2.1 Dynamic material functions
Using the same method that describes the nonlinearity by strain dependence of the material parameters, we assume that the damper viscosity c
and the spring stiffness k are functions in terms of
rotation [4], [5] (fig.2.2):
c ( ) = 2 J 0 0 D ( ) [Nms]
k ( ) =
Ip
h
G ( )
(2.3)
[Nm]
where 0 is the system undamped natural pulsation, Ip = 4 /32 is the polar moment of the specimen and (, h) are the diameter and height of the
cylindrical specimen. For the linear systems, the
undamped natural pulsation 0 is defined in terms
of spring stiffness k : 0 = k / J 0 . In this
case, the spring stiffness is a function, and we de-
fine the undamped natural pulsation in terms of

initial
value
of
stiffness
function
k (0) : 0 =
k (0) / J 0 .
3. VALIDATION OF THE NON-LINEAR

FORM
Eq. (1.1) can be numerical solved [4], [11], [12]
and the computed results can be compared with the
measured resonant test results. Thus, by using the
change of variable = 0 t and by introducing a
new "time" function [4]:
( ) = (t ) = ( / 0 )
(3.1)
one obtains from eq.(1.2) a dimensionless form of

the non-linear equation of motion:
A nonlinear Kelvin-Voigt model for soils
Fig.2.2 Dynamic non-linear characteristics
+ C ( ) + K ( ) = sin
(3.2)
where the superscript accent denotes the time derivative with respect to , and:
C () = C() =
K () = K () =
=
M0
J0
2
0
c()
= 2 D ()
J 00
k ()
k () G ()
=
=
= G n ( ) (3.3)
2
J 0 0
k (0 ) G (0 )
=
M0
k (0 )
= st
For a given normalized amplitude and relative

pulsation , the non-linear equation (3.2) can be
numerically solved and a solution of the form
= (; , ) can be obtained in n points [4],
[11]. After that, dropping the transitory part of the
solution and keeping only the stationary part, the
amplitude of rotation 0 becomes:
0 = (t ) = ()
(3.4)
The same rotation 0 can be obtained directly

form resonant column output:
0 =
A
ra 2
(3.5)
where A is the measured accelerometer value, ra is

the distance from the axis of rotation to the accelerometer axis (ra = 0.03175m for Drnevich resonant column) and = 2f r is the pulsation of the
vibrator device under resonant frequency f r.
The comparison of the values obtained from
the steady-state solution of the non-linear single
degree-of-freedom system with experimental data
can give a pertinent information about the model
validity. The results of such evaluation, given in
fig.3.1, show a good behaviour for the NKV
model.
Fig.3.1 Model validation
4. HYSTERETIC DAMPING MODELLING

A great number of laboratory tests on soils
shows that the cyclic stress-strain curves are high
nonlinear and constitute a closed hysteresis loops.
These testing results seem to indicate that damping
properties are especially of hysteretic type, and
not viscous as those corresponding to the KelvinVoigt model.
All of the resonant column determinations of
the damping capacity for a certain strain level are
based on the equivalence between the hysteretic
damping of the soil specimen and the viscous
damping for a uniform viscoelastic specimen of the
same mass, density and dimensions as the soil
specimen [2]. As a result of this rheo-hysteretic
hypothesis, the Kelvin-Voigt model is able to describe the dissipated energy of the specimenvibrator system and the verifying method presented
in capther 3 demonstrated a good agreement between model and experimental data.
Several methods use for damping evaluation
of the experimental registered hysteretic loops and
determine the damping ratio as:
D=
1 W
4 W
(4.1)
where W is the maximum stored energy and W is

the energy loose per cycle represented by the area
enclosed inside the hysteresis loop (fig.4.1). By
another methods the hysteresis loop is obtained
from the skeleton curve by applying the Masing
rule (the superior and inferior branches are obtained from the skeleton curve by multiplying by a
factor two in both directions). [9], [10].
Fig.4.1 Hysteretic damping definition
Using the same single degree-of-freedom system from fig.1.1 there is a possibility to compare
the resonant column experimental damping values
Hysteretic damping modelling
with the values obtained from non-linear KelvinVoigt model and Masing model.
We mention that in the ordinary resonant column test the damping evaluation uses a different
method the magnification factor method based
on measuring both current and acceleration at two
different frequencies - from resonant frequency f r
and at 2 f r .
To verify the capabilities of the non-linear Kelvin-Voigt equation (1.2) to modelling a hysteresis
loop one can use a inverse strategy starting from
the given dynamic material functions c = c ( )
and k = k ( ) included in the restoring force
Q = Q , & one can built the hysteresis loops for a
( )
certain levels 0 and then the damping ratio value

D for level 0 can be obtained from eq. (4.1). This
value can be compared with the experimental va lues at level 0 : D = D ( ) = 0
Fig.4.2 NKV hysteresis loop
Thus, for nonlinear Kelvin-Voigt model from

fig.1.1 the restoring force is:
( )
( )
Q , & = Qel () + Qdam , & =

= k() + c() &
(4.2)
where Q el ( ) = k ( ) is the backbone curve or

skeleton curve.
For a certain amount of the excitation
M = M 0 sin t the response rotation (after the
dropping the transitory part) has the form:
= 0 cos t
(4.3)
& = 0 sin t
(4.4)
and, then:
Therefore, by eliminating the time t between this

two equations, (4.3) and (4.4), result:
& = 20
(4.5)
and the restoring force (4.2) becomes:

Q() = k() c() 20
(4.6)
where the sign "+" is for the superior branch of the

hysteresis loop and the sign "-" for the inferior
branch (fig.4.2).
Fig.4..3 Masing hystersis loop
For comparison, the Masing hysteresis loop is

given in fig.4.3 for the same tested clay, at the
same amplitude level 0 = 2 .076 % built using
the same skeleton curve Q el ( ) = k ( ) . The
superior and inferior branches are obtained usig the
Masing rule starting from the skeleton curve and
multiplying by a factor two in both Q and directions.
As can see in these figures the geometrical aspect of these hysteresis loops are different. But, the
damping value is directly connected with the loop
area and not with its form. Fortunately, the loop
area differences are not so obvious. This can be
proved by computing the damping ratio for different amplitude 0 and for each kind of hysteresis
loop. The results of such calculus together with the
corresponding D experimental values are given in
fig.4.4.
Fig.4.4 NKV and Masing damping modelling
4. CONCLUDING REMARKS
The non-linear viscoelasticity can be used as

starting point for building a dynamic model for
soils behaviour.
The non-linear dynamic characteristics for
damping and stiffness can be obtained as an
extension in the non-linear domain of the corresponding linear constants.
The non-linear Kelvin-Voigt model provides a
good agreement with the experimental resonant column data.
The non-linear Kelvin-Voigt model is able to
model the damping characteristics of the hysteretic type materials.
2.
3.
4.
5.
6.
7.
8.
9.
ACKNOWLEDGEMENTS
10.
This work was supported by the CNCSIS

Grant no. 7014/2001-2002.
11.
12.
REFERENCES
1.
BRATOSIN, D., Nonlinear Relationships between Stress

and Strain Invariants of Soils, Revue Roumaine des Sciences Technique, srie de Mchanique Applique , 6,
1985.
13.
BRATOSIN, D., Nonlinear Viscoelastic Model for Soils,

Rev.Roum.Sci.Tech.-Mec.Appl,, 1, 1986.
BRATOSIN, D., A Dynamic Constitutive Equation for
Soils, Rev.Roum.Sci.Tech.-Mec.Appl,, 5, 1993.
BRATOSIN D., SIRETEANU T., BLA C., Equivalent
Linear Model for Nonlinear Behaviour of Soils,
Rev.Roum.Sci.Tech.-Mec.Appl,, 4, 1998.
Bratosin D., Structural damping in geological materials,
Proceedings of the Romanian Academy, 2, 1-2/2001.
DRNEVICH V.P., HARDIN B.O., SHIPPY D.J.,
Modulus and Damping of Soils by Resonant-Column
Method, in Dynamic Geotechnical Testing, American
Society for Testing and Materials, 1978.
HARDIN B.O., Suggested Methods of Tests for Shear
Modulus and Damping of Soils by the Resonant Column,
American Society for Testing and Materials, 1970.
HARDIN B., DRNEVICH V.P., Shear Modulus and
Damping in Soils: Design Equations and Curves,
Journ.Soil Mech.Found.Eng.Div., July 1972.
IDRISS I.M., DOBRY R., SINGH R.D., Nonlinear Behavior of Soft Clays during Cyclic Loading,
Journ.Geotech.Eng.Div., Dec., 1978.
ISHIHARA K., Soil Behaviour in Earthquake Geotechnics, Clarendon Press, Oxford, 1996.
LEVY S., WILKINSON J.P.D., The Component Element
Method in Dynamics, McGraw-Hill Book Company,
1976.
PRESS W.H., FLANNERY B.P., TEUKOLSKY S.A.,
VETTERLING W.T., Numerical Recipes, The art of Scientific Computing, Cambridge University Press, 1990.
* * * * * Drnevich Long-Tor Resonant Column Apparatus, Operating Manual, Soil Dynamics Instruments
Inc., 1979.
Received August 13, 2002

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1

PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A,

HYSTERETIC DAMPING MODELLING BY NONLINEAR KELVIN-VOIGT MODEL

Dinu BRATOSIN, Tudor SIRETEANU

where is the system's displacement (rotation, in

with the analogic model from fig.1.1.

Fig.1.1 Non-linear Kelvin-Voigt model

Dinu BRATOSIN, Tudor SIRETEANU

2. NONLINEAR MATERIAL FUNCTION

terms of shear strain invariant : G = G ( ) ,

Fig.2.1 Dynamic material functions

fine the undamped natural pulsation in terms of

3. VALIDATION OF THE NON-LINEAR

one obtains from eq.(1.2) a dimensionless form of

A nonlinear Kelvin-Voigt model for soils

Fig.2.2 Dynamic non-linear characteristics

For a given normalized amplitude and relative

The same rotation 0 can be obtained directly

where A is the measured accelerometer value, ra is

Dinu BRATOSIN, Tudor SIRETEANU

Fig.3.1 Model validation

4. HYSTERETIC DAMPING MODELLING

where W is the maximum stored energy and W is

Fig.4.1 Hysteretic damping definition

Hysteretic damping modelling

certain levels 0 and then the damping ratio value

Fig.4.2 NKV hysteresis loop

Thus, for nonlinear Kelvin-Voigt model from

Q , & = Qel () + Qdam , & =

where Q el ( ) = k ( ) is the backbone curve or

Therefore, by eliminating the time t between this

and the restoring force (4.2) becomes:

where the sign "+" is for the superior branch of the

Fig.4..3 Masing hystersis loop

For comparison, the Masing hysteresis loop is

Dinu BRATOSIN, Tudor SIRETEANU

Fig.4.4 NKV and Masing damping modelling

The non-linear viscoelasticity can be used as

This work was supported by the CNCSIS

BRATOSIN, D., Nonlinear Relationships between Stress

BRATOSIN, D., Nonlinear Viscoelastic Model for Soils,

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