Full text of DOI:10.1061/(ASCE)0733-9399(2000)126:11(1189)

EQUIVALENT
V
ISCOUS
DAMPING FOR A
BILINEAR
HYSTERETIC
O
SCILLATOR
1
2
By C. K. Reddy and R. Pratap
A
BSTRACT: A bilinear hysteretic model is commonly used to study elastoplastic structures. In this paper, a
damped, bilinear hysteretic oscillator is studied under harmonic loading. We show the existence of an equivalent
viscous damping for small values of a loading parameter such that the associated linear structure and the
hysteretic structure have the same frequency response curves. We use the Kryloff-Bogoliuboff method of av-
eraging to find the equivalent viscous damping as a function of the steady state amplitude. We present a model
of a bilinear elastic oscillator which captures the steady-state dynamics of the hysteretic oscillator for low values
of the loading parameter. We also study the nature of the dependence of the equivalent viscous damping on the
kinematic hardening parameter.
INTRODUCTION
EQUATION OF MOTION
A bilinear hysteretic model is used to study the response of
various elastoplastic structures. For fundamental analysis and
understanding, a single-degree-of-freedom model often suf-
fices. Much of the literature on studies of elastoplastic struc-
tures is based on this model. There have been numerous stud-
ies on the forced response of elastoplastic structures
The equation of motion of a forced, damped bilinear hys-
teretic oscillator is written as
m y¨ ϩ c y˙ ϩ F(y, y˙ ) = P cos ⍀t
(1)
F is the hysteretic restoring force, c represents the damping
coefficient, and y is the displacement. P is the amplitude of
the forcing with frequency ⍀. The nondimensional equation
of motion is given by
(
Tanabashi 1956; Caughey 1960; Iwan 1964; Ballio 1970; Pra-
tap et al. 1994; Savi and Pacheco 1997). It is often convenient
to study the steady-state dynamics of an elastoplastic oscillator
by considering an equivalent linear structure (Jacobsen 1960;
Berg 1965; Jennings 1968; Iwan and Gates 1979). Jennings
x¨ ϩ c x˙ ϩ f(x, x˙ ) = p cos ␻␶
0
(2)
(
1968) compares the various ways in which an equivalent lin-
2
0
ear structure can be defined to get frequency and amplitude
matching. He considers an undamped, elasto-perfectly-plastic
oscillator under harmonic excitation. Iwan and Gates (1979)
compare the accuracy of the various methods for defining
equivalent linear systems by considering a damped bilinear
hysteretic oscillator subjected to earthquake loading and har-
monic excitation.
where x = y/y
c/ km; ␻ = ⍀/␻
͙
0
; ␶ = ␻
0
t; ␻
= k/m; f = F/F
0
; p = P/F
0
; c =
0
0
. The force-deflection diagram is shown in
Fig. 1. The nondimensional restoring force f is not a single-
valued function of the nondimensional displacement x. It de-
pends upon x as well as the sign of x˙ . For the plastic flow
rule, we have assumed kinematic hardening. This essentially
means that the total range of displacement in the elastic phase
remains the same, irrespective of the net plastic displacement
(Mendelson 1968). The parameter ␩ represents the kinematic
hardening parameter; ␩ = 0 would mean a perfectly plastic
case, and ␩ = 1 a perfectly elastic case.
The resonant amplitude method, dynamic stiffness method,
and dynamic mass method are some of the methods used for
defining equivalent linear systems. In all the above methods,
the frequency shift exhibited by hysteretic systems cannot be
taken care of by defining the equivalent viscous damping ceq
alone. In this paper, we address the issue of whether the
steady-state dynamics of a hysteretic oscillator can be com-
pletely captured by an equivalent linear system by just defining
a new equivalent viscous damping. We analyze the response
of a damped single-degree-of-freedom oscillator, having bilin-
ear hysteresis, to harmonic forcing. For small values of the
forcing amplitude and damping, it is shown that the steady-
state response of the hysteretic oscillator is represented com-
pletely in terms of an equivalent linear structure by finding an
equivalent viscous damping. The bilinear hysteretic oscillator
and the bilinear elastic oscillator both show soft resonance
behavior. We use this behavior to get a better frequency and
amplitude match by studying a model of the bilinear nonhys-
teretic oscillator. The functional relationship between the non-
dimensional equivalent viscous damping defined in the model
and the kinematic hardening parameter is also studied.
2
2
2
APPROXIMATE FREQUENCY RESPONSE EQUATION
Here we obtain an approximate response of the hysteretic
oscillator to the applied harmonic load by using the method
of averaging on the harmonic response of the oscillator, ig-
noring hysteresis. We first assume the solution of the form
x(␶) = x
s
cos(␻␶ ϩ ␾)
(3)
1
Grad. Student, Dept. of Mech. Engrg., Indian Inst. of Sci., Bangalore
5
5
60012, India. E-mail: konda@mecheng.iisc.ernet.in
Asst. Prof., Dept. of Mech. Engrg., Indian Inst. of Sci., Bangalore
60012, India. E-mail: pratap@mecheng.iisc.ernet.in
2
Note. Associate Editor: Dimitrios Karamanlidis. Discussion open until
April 1, 2001. To extend the closing date one month, a written request
must be filed with the ASCE Manager of Journals. The manuscript for
this paper was submitted for review and possible publication on May 17,
1
1
999. This paper is part of the Journal of Engineering Mechanics, Vol.
26, No. 11, November, 2000. ᭧ASCE, ISSN 0733-9399/00/0011-1189–
1196/$8.00 ϩ $.50 per page. Paper No. 20980.
FIG. 1. Force Deflection Diagram for Bilinear Hysteresis
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FIG. 2. Comparison of Approximate Solution (Solid Curve) against Numerical Solution (Dotted-Dashed Curve)
FIG. 3. Typical Frequency Response Curve
2
where x
method of slowly varying parameters, and for ␻ Ϸ 1, we get
see Appendix I for details of the calculation)
s
and ␾ are slowly varying parameters of ␶. Using the
␻ = G(x )
(5)
s
2
where G(x
s
) represents the functional dependence of ␻ on x .
s
(
Fig. 2 shows a comparison between the exact (numerical) so-
lution and the approximate solution. It is seen that the method
of averaging approximates the exact solution quite well, even
for values of ␻ away from unity.
2
0
2
0
2
2
2
2
(2␤ Ϫ c
) Ϯ
͙
(2␤ Ϫ c
) Ϫ 4(␣ ϩ ␤ Ϫ ␥ Ϫ 2c
0
␣)
2
␻
=
2
(
4)
EQUIVALENT VISCOUS DAMPING
The variables ␥, ␤, and ␣ are functions of the amplitude of
2
motion x
s
(see Appendix I). Hence, for constant c
0
, p, and ␩ ,
Fig. 3 shows a typical frequency response curve for a bilin-
ear hysteretic oscillator, which exhibits soft resonance. To find
(4) can be written as
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2
2
0
FIG. 4. Region I Giving Values of p and c for Which Curves for ␩ ʦ [0, 1) Fall within Curve for ␩ ؍ 1
0
FIG. 5. Frequency Response Curves for Values of p and c outside Region I
the extent of frequency shift, we can find out the difference
values of c
has a slope equal to 2. Fig. 5 shows the case for values of p
and c outside Region I. The above observations lead to the
0
, the line separating Region I is almost linear and
2
between the frequency ␻
␩
␩
2
for a given ␩ ʦ [0, 1) and ␻
= 1 for different values of x . A positive value of (␻
) = ⌬␻ will mean that the curve for a given ␩ falls inside
1
for
Ϫ
2
s
␩
2
0
2
␻
1
following propositions.
2
the curve for ␩ = 1, and a negative value will mean that the
Proposition 1. In all steady-state motions of the form x
s
2
curve falls outside the curve for ␩ = 1. We can find the values
cos(␻␶ ϩ ␾) involving plastic cycles, for small values of c
0
2
2
of p and c
0
for which the frequency response curves for all ␩
and for (p/c
0
) < 2, there exists c
e
> 0 for all ␩ ʦ [0, 1), where
2
2
ʦ [0, 1) fall within the curve for ␩ = 1. See Fig. 4. For all
p and c within Region I, we get frequency response curves
c
e
= c
e
(x
s
, ␩ ), such that the following linear equation
0
2
2
for all ␩ ʦ [0, 1) within the curve for ␩ = 1. For small
x¨ ϩ (c
0
e
ϩ c ) x˙ ϩ x = p cos ␻␶
(6)
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2
e 0
FIG. 6. Frequency Response Curves for Different Values of ␩ (Dotted-Dashed Curve) and c (Solid Curve) for Same p and c
2
2
2
and (2) are equivalent, that is, the frequency response curves
␥ Ϫ (1 Ϫ ␻ )
c
e
=
Ϫ c
0
(7)
(8)
ͱ
2
are the same for a given p and c
0
. We define c as the equiv-
e
␻
alent viscous damping.
2
Substituting for ␻ from (5), we have
Proof: The proof is based upon the nature of the frequency
2
response curves for different values of ␩ ʦ [0, 1) for values
2
␥
2
Ϫ (1 Ϫ G(x
s
))
of (p/c ) < 2 (Fig. 6). It is seen that all curves fall within the
0
c
e
=
Ϫ c
0
ͱ
2
s
G(x )
curve for ␩ = 1. We also know that the frequency response
curves of the associated linear system given by (6) with in-
2
As ␩ decreases, the range of values that c
because the amount of plastic damping increases.
e
takes increases
2
creasing values of c
e
lie within the curve for ␩ = 1 (or c
e
=
0). Since the hysteretic curves and the curves for increasing
constant values of c start and end at different points respec-
e
BILINEAR ELASTIC OSCILLATOR AND EQUIVALENT
VISCOUS DAMPING
tively, the intersection of these curves is always transversal,
except for the case discussed in the next proposition. This
2
means that every point on the curve with a particular ␩ is
The frequency response curves of both the bilinear nonhys-
teretic (elastic) oscillator and the bilinear hysteretic oscillator
show a leftward shift. This behavior is typical of soft springs
intersected by a curve with a particular constant value of c
Since there is a continuous one-to-one mapping between the
e
.
curves with different constant values of c
e
, every point on a
such that the am-
(
Nayfeh and Mook 1979). Because the linear system does not
2
curve with a particular ␩ has a unique c
e
exhibit soft spring behavior, we do not generally get frequency
match when the amplitudes are matched by defining the equiv-
alent viscous damping only. However, since the bilinear hys-
teretic and the nonhysteretic oscillator show the same behavior
as far as frequency response curves are concerned (for low
loading parameters), we can expect a better frequency match
when the amplitudes are matched in the equivalent bilinear
nonhysteretic oscillator.
plitude is the same at the same frequency. This shows that the
2
frequency response for a particular ␩ can be exactly repre-
sented by (3) with a variable c
Proposition 2. For small values of c
e
that depends on x
and (p/c
for a particular ␩ ʦ [0, 1), such that the
frequency response curve with c is tangent to the frequency
response curve with ␩ for a given p and c
Proof: To prove Proposition 2, it is enough to note that if a
particular curve with a constant c intersects a curve for a
s
.
0
0
< 2, there
2
exists a constant c
e
e
2
0
.
Hysteretic damping comes into play when the oscillator
goes into the plastic regime (͉x͉ > 1 in the nondimensional
model). So, the only way this can be captured in the bilinear
nonhysteretic oscillator is by introducing additional damping
e
2
particular ␩ , it does so in at most two points. Now there is a
continuous one-to-one mapping between the curve with a con-
2
stant c
e
that intersects with the curve for ␩ and one with a
2
in the branch with slope ␩ . The equivalent viscous damping,
different value of c
e
that does not intersect. This implies that
which would be a function of the loading parameter p and
there exists one curve between these two curves that is tangent
viscous damping c
same resonant amplitude for a given value of ␩ . The value
of c together with p decides the values of ␻, for which we
0
, would be just that value which gives the
2
to the ␩ curve (Fig. 6).
2
0
APPROXIMATE EXPRESSION FOR THE EQUIVALENT
VISCOUS DAMPING
have ͉x͉ > 1. Since we are introducing damping partly, the
range of these values of ␻ remains unaffected, which is not
the case when we introduce additional damping throughout.
The bilinear nonhysteretic model studied is described by the
following equations:
In case of the linear oscillator given by (6), we can express
c
e
as a function of x and ␻ as
s
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FIG. 7. Frequency Response Curves of Bilinear Hysteretic Oscillator (Solid Curve) and Bilinear Nonhysteretic Oscillator (Dotted-
Dashed Curve) with Corresponding Value of c for Resonant Amplitude Match
a
2
FIG. 8. Variation of X against c
x¨ ϩ d( x˙ ) ϩ s(x) = p cos ␻t
a
for Given Values of p and ␩ for Bilinear Nonhysteretic Oscillator
(9)
and
1
x > 0
where
sign(x) =
ͭ
Ϫ1 x < 0
c
0
x˙
͉x͉ Յ 1
The value of c , which gives the same resonant amplitude for
a given ␩ , is defined as the new nondimensional equivalent
for a given p and c . Fig. 7 shows that
we get a resonant frequency match when the resonant ampli-
tudes are matched.
a
d( x˙ ) =
2
ͭ
(c
0
ϩ c
a
) x˙ ͉x͉ > 1
viscous damping C
e
0
x
͉x͉ Յ 1
(x Ϫ 1) ϩ sign(x) ͉x͉ > 1
s(x) =
2
ͭ
␩
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FIG. 9. Logarithmic Plot of X ؊ 1 against c
a
for Bilinear Nonhysteretic Oscillator
2
FIG. 10. Comparison of Approximate Relation between C
e
and ␩ with Values Obtained Numerically
2
2
FUNCTIONAL RELATIONSHIP BETWEEN C
UNDAMPED CASE
e
AND ␩ :
4(1 Ϫ ␩ )
X =
(10)
2
4
(1 Ϫ ␩ ) Ϫ ␲p
Consider an undamped (c
a relationship between the equivalent viscous damping C
the kinematic hardening parameter ␩ . For the undamped hys-
0
= 0) hysteretic oscillator to seek
and
If we can get a relation between the resonant amplitude X
and c for the bilinear nonhysteretic oscillator, for a range of
values of ␩ and some given p, then equating this relation with
(10) would give us the desired relationship between C and
will tend to unity.
e
a
2
2
teretic oscillator, we have a simple relation between the res-
e
2
2
onant amplitude X and ␩ for a given p (Caughey 1960).
␩ . It is clear that as c
a
takes large values, X
s
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See Fig. 8, which suggests an exponential relationship between
where x
a solution of the form
s
and ␾ are constants. For nonzero but small ␮, assume
2
X and c
unity on the right hand side of the probable relation between
X and c . A logarithmic plot of X Ϫ 1 versus c gives a straight
line (Fig. 9).
a
for a given ␩ and p. This would mean that we have
x(␶) = x
s
cos(␻␶ ϩ ␾)
(15)
a
a
where x
s
and ␾ are slowly varying parameters of ␶. Therefore
It is clear that the relation between X and c
of chosen values of ␩ is of the form
a
for the range
the velocity is given by
x˙ (␶) = Ϫ␻x sin ␪ ϩ x˙
where ␪ = ␻␶ ϩ ␾ for simplicity. Now, from (14), we get
x˙ (␶) = Ϫ␻x sin(␪)
Comparing (16) and (17) we have
2
˙
s
s
cos ␪ Ϫ x
s
␾ sin ␪
(16)
A
a
X = Bc
ϩ 1
(11)
2
The values of ␩ are chosen away from the value that gives
2
s
(17)
(18)
(19)
unbounded resonance (␩ = 1 Ϫ ␲p/4) for a given p. Also the
values of c start away from zero to circumvent the difficulty
a
of finding the resonant amplitude numerically. A is the slope
of the straight line obtained from the logarithmic plot, and B
is equal to the exponential of the vertical intercept. A and B
˙
cos ␪ Ϫ x ␾ sin ␪ = 0
x˙
s
s
2
Similarly
x¨ (␶) = Ϫ␻ x
depend upon ␩ for a given p. It has been observed that the
2
variation of A and B with respect to ␩ is very slight. A and
2
˙
sin ␪ Ϫ ␻x ␾ cos ␪
s
cos ␪ Ϫ ␻x˙
s
s
B are calculated using average values of slope and the vertical
2
intercept over the range of values of ␩ used. A = Ϫ0.8 and
The method of averaging (Caughey 1960) is used to find the
B = 0.5142 in our case for p = 0.3. Equating (10) and (11),
frequency ␻ as a function of the steady-state amplitude x
Ϸ 1, as
s
, for
and noting that the value of c
amplitude is C , we have
a
that gives the same resonant
␻
e
2
0
2
0
2
2
2
2
(
2␤ Ϫ c
) Ϯ
͙
(2␤ Ϫ c
) Ϫ 4(␣ ϩ ␤ Ϫ ␥ Ϫ 2c
0
␣)
1/A
2
␻
=
␲
p
2
C
e
=
(12)
ͩ
2
ͪ
(20)
B(4(1 Ϫ ␩ ) Ϫ ␲p)
For a bilinear hysteretic oscillator, ␤ and ␣ can be found to
be
It can be seen from Fig. 10 that the approximate relationship
given by (12) compares well with the values obtained numer-
ically.
␮
2
␣
= Ϫ [sin ␪*]
(21a)
The relation in (12) is a quick way to obtain a measure of
the dissipation in a structure that shows hysteresis. Engineers
are primarily concerned with the resonant amplitude and res-
onant frequency of a structure. The preceding discussion
shows that it is possible to study the steady-state dynamics of
a somewhat involved model of a hysteretic structure by study-
ing a simpler nonhysteretic model with equivalent damping.
␲
1
␮
␤
=
␮␪* ϩ (1 Ϫ ␮)␲ Ϫ sin 2␪* ; ␥ = p/x
s
(21b,c)
ͫ
ͬ
␲
2
Ϫ1
where ␪* = cos [1 Ϫ (2/x )].
s
ACKNOWLEDGMENT
CONCLUSIONS
This work was supported by the Department of Science and Technol-
ogy, Govt. of India, through the grant no. HR/OY/E-13/98.
It is shown that for certain values of the loading parameter
and viscous damping that is present throughout, the frequency
response of a single-degree-of-freedom damped bilinear hys-
teretic oscillator under harmonic loading can be represented
exactly by the frequency response of a linear oscillator under
the same loading by introducing a variable equivalent viscous
damping. An approximate implicit relation between the equiv-
alent viscous damping and the steady-state amplitude is de-
rived using the Kryloff-Bogoliuboff method of averaging. The
APPENDIX II. REFERENCES
Ballio, G. (1970). ‘‘On the dynamic behavior of an elastoplastic oscillator
(experimental and theoretical investigation).’’ Meccanica, 5, 87–97.
Berg, G. V. (1965). ‘‘A study of the earthquake response of inelastic
systems.’’ Proc., 34th Convention of Struct. Engrs. Assoc. of California,
Coronado, 63–67.
Caughey, T. K. (1960). ‘‘Sinusoidal excitation of a system with bilinear
hysteresis.’’ J. Appl. Mech., 27, 640–643.
Iwan, W. D. (1964). ‘‘The dynamic response of the one degree of freedom
bilinear hysteretic system.’’ Proc., 3rd World Conf. on Earthquake
Engrg., 783–796.
Iwan, W. D., and Gates, N. C. (1979). ‘‘Estimating earthquake response
of simple hysteretic systems.’’ J. Engrg. Mech. Div., ASCE, 105(3),
391–405.
Jacobsen, L. S. (1960). ‘‘Damping in composite structures.’’ Proc., 3rd
World Conf. on Earthquake Engrg., Vol. II, 1029–1044.
Jennings, P. C. (1968). ‘‘Equivalent viscous damping for yielding struc-
tures.’’ J. Engrg. Mech. Div., ASCE, 94, 103–116.
Mendelson, A. (1968). Plasticity, McMillan, New York.
Nayfeh, A. H., and Mook, D. T. (1979). Non-linear oscillations, Wiley,
New York.
Pratap, R., Mukheejee, S., and Moon, F. C. (1994). ‘‘Dynamic behavior
of a bilinear hysteretic elastoplastic oscillator, Part II: Oscillations un-
der periodic impulsive loading.’’ J. Sound and Vibration, 172(3), 339–
‘
‘soft’’ type of resonance exhibited by the damped bilinear
hysteretic oscillator and the bilinear nonhysteretic oscillator is
used to suggest a model of the bilinear nonhysteretic oscillator
with extra damping introduced to capture the steady-state dy-
namics of the hysteretic oscillator. Resonant amplitudes are
matched, and the corresponding extra damping is termed the
equivalent viscous damping. It is observed that the resonant
frequencies are nearly the same when the resonant amplitudes
are matched. An approximate relation between the equivalent
viscous damping in the case of the bilinear nonhysteretic os-
cillator and the kinematic hardening parameter is derived for
a given value of the loading parameter.
APPENDIX I. APPROXIMATE FREQUENCY
RESPONSE EQUATION
3
58.
Savi, M. A., and Pacheco, P. M. C. L. (1997). ‘‘Non-linear dynamics of
an elastoplastic oscillator with kinematic and isotropic hardening.’’ J.
Sound and Vibration, 207(2), 207–226.
2
Let ␮ = 1 Ϫ ␩ . Hence, for the case ␮ = 0, we have
Tanabashi, R. (1956). ‘‘Studies on nonlinear vibration of structures sub-
jected to destructive earthquakes.’’ Proc., World Conf. on Earthquake
Engrg., University of California, Berkeley, Calif., 6-1–6-7.
x(␶) = x
s
cos(␻␶ ϩ ␾)
sin(␻␶ ϩ ␾)
(13)
(14)
x˙ (␶) = Ϫ␻x
s
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APPENDIX III. NOTATION
k = stiffness of elastic branch of hysteretic oscillator;
m = mass of hysteretic oscillator;
P = forcing amplitude;
The following symbols are used in this paper:
p = nondimensional forcing amplitude;
t = time;
X = resonant amplitude;
C
e
= equivalent viscous damping in case of bilinear nonhyster-
etic oscillator;
c = coefficient of viscous damping;
x = nondimensional displacement;
c
a
= extra damping in case of bilinear nonhysteretic oscillator;
= equivalent viscous damping in case of linear oscillator;
= nondimensional coefficient of viscous damping;
x = steady state amplitude;
s
c
e
c
0
y = displacement of hysteretic oscillator;
2
F = restoring force for bilinear hysteretic oscillator;
f = nondimensional restoring force for bilinear hysteretic os-
cillator;
␩ = kinematic hardening parameter;
␶ = nondimensional time;
␾ = slowly varying phase;
G = function defining relation between frequency ratio and
steady state amplitude;
⍀ = frequency of harmonic forcing; and
␻ = nondimensional frequency.
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