ANALYTICAL MODEL OF VISCOELASTIC
FLUID DAMPERS
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By Nicos Makris, ~ M. C. Constantinou, 2 Associate Member, ASCE, and
G. F. Dargush 3
ABSTRACT: An analytical approximate constitutive relation is derived for a form
of fluid damper, which exhibits viscoelastic behavior. The damper is used for vibration isolation of piping systems and industrial equipment, as well as for vibration
and seismic isolation of building structures. The damper consists of an open pot,
filled with highly viscous fluid, and a piston that moves within the fluid. The
analytical solution relates the damping constant of the unit at vanishingly small
frequencies with the material constants of the fluid used in the damper and the
geometric characteristics of the damper. With the determination of the damping
constant, a macroscopic model may be constructed that describes the damper behavior over a large frequency range. The other parameters of the macroscopic
model are identical to those in the corresponding constitutive relation of the damper
fluid. The results of the analytical solution are in very good agreement with experimental data and with numerical solutions based on boundary-element analysis.
INTRODUCTION
Viscous dampers consisting of a moving piston i m m e r s e d in highly viscous
fluid (Fig. 1) have found wide application in the shock and vibration isolation
of industrial machines, e q u i p m e n t , p i p e w o r k systems, and buildings. Furthermore, they have been recently used in c o m b i n e d vibration and seismic
isolation of buildings (Huffman 1985; Makris and Constantinou 1991, 1992).
Modeling the behavior of the viscous d a m p e r s is an increasingly important
problem because of their wide range of applicability. A t t e m p t s have b e e n
made in modeling viscous d a m p e r s either as simple linear viscous elements
( " P i p e w o r k " 1986) or by models of the classical theory of viscoelasticity
(Schwann et al. 1988). The writers (Makris and Constantinou 1991) proposed a fractional derivative m o d e l for these devices and showed that the
model is capable of describing their behavior with very good accuracy. The
model has the following form:
co -a.(r
-
a dqP(l)
P(t) +
dl q
=
dt
...................................
(1)
in which P = force applied at the piston; u = piston displacement; Co =
zero-frequency damping coefficient; h = relaxation fractional time; and q
= order of derivative in the range (0, 1). It was anticipated and indeed
verified (Makris and Constantinou 1991) that values for the o r d e r of time
derivative and relaxation fractional time of the force-displacement m o d e l
for the d a m p e r unit are almost the same as the corresponding values in the
1Asst. Prof., Dept. of Civ. Engrg. and Geological Sci., Univ. of Notre Dame,
Notre Dame, IN 46556-0767.
2Assoc. Prof., Dept. of Civ. Engrg., State Univ. of New York, Buffalo, NY 14260.
3Res. Assoc. Prof., Dept. of Civ. Engrg., State Univ. of New York, Buffalo, NY.
Note. Discussion open until April 1, 1994. 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 February 18,
1992. This paper is part of the Journal of Structural Engineering, Vol. 119, No. 11,
November, 1993. 9
ISSN 0733-9445/93/0011-3310/$1.00 + $.15 per page.
Paper No. 3476.
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I
@
1_
-"
]
I
(a)
(b)
FIG. 1. Geometry of: (a) Typical Viscous Damper; (b) Idealized Damper
stress-strain relation for the fluid used within the damper (a form of silicon
gel).
In principle it is possible to determine the time history of force needed
to produce any displacement history of the damper piston by integrating
the set of governing equations. This set includes the fluid constitutive relation, along with the dynamic equilibrium and continuity equations. Such
an approach is impractical for dynamic analysis of isolated structures. However, it is very useful for determining the mechanical properties of dampers,
from which macroscopic models can be developed, like the one presented
by Makris and Constantinou (1991).
In the present paper an analytical solution is presented that determines
the stresses and force acting on the piston under steady-state harmonic
motion. The solution is based on simplifying assumptions that are physically
motivated and describe well the behavior of the damper at low-frequency
motion. The solution is used in the derivation of an analytic expression for
the zero-frequency damping constant Co. With the determination of constant
Co, the macroscopic fractional derivative model describing the damper behavior, (1), may be constructed by assuming the other two constants of the
model to be identical to those in the corresponding constitutive relation of
the damper fluid. Thus, the analytical solution eliminates the need for extensive testing of the devices.
The values of damping constant Co predicted by the approximate analytical solution are found to compare well with experimental results and
results of a rigorous boundary-element analysis. Furthermore, the complete
macroscopic model of the damper is found to predict well the experimentally
determined mechanical properties and response of the damper under harmonic and transient motion. Thus, the macroscopic model is quite suitable
for use as part of the design process.
ASSUMPTIONS IN APPROXIMATE SOLUTION
The solution derived herein is for the vertical motion of the piston. Vertical motion occurs not only in the vertical mode of vibration but also in
the rocking mode, which is the fundamental mode of vibration of springviscous damper isolated structures.
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Fig. l(a) shows the geometry of a typical viscous damper. The outer
surface of the piston features a smooth top portion and a lower portion
containing ribs. The ribs serve the purpose of preventing slippage of fluid
during motion of piston. The bottom of the piston is hollow and contains
a large number of inner pipes of small diameter (usually 15 to 20 mm
diameter). This arrangement allows for penetration of the fluid into the
pipes and ensures full contact of the fluid with the bottom surface of the
piston.
In this simplified mathematical model, the piston is assumed to be cylindrical with radius
rlenle
rl =
+ ruHu
nl
..........................................
(2)
which represents the weighted average radius of the actual piston. Fig. l(b)
shows the idealized damper in its deformed position where it has undergone
vertical displacement equal to amplitude u0. During this motion, stresses at
the side and the bottom of the idealized piston result in the force needed
to maintain the imposed motion. These two components of the piston force
are evaluated separately by assuming that: (1) The fluid between the lateral
faces of the piston and housing is subjected to pure shearing; and (2) the
fluid between the bottom of the piston and the housing is subjected to
compression-extension.
The vertical motion of the piston is harmonic and described by
U
=
n o d it~
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(3)
where to = frequency of the motion. The force required to maintain this
motion is evaluated under steady-state conditions so that
P = Poe ~ '
..................................................
(4)
in which Po = complex amplitude of force. Under steady-state conditions,
the exponential time-dependent terms drop out and the analysis involves
only amplitudes.
FORCE ACTING ON LATERAL SURFACE OF PISTON
It is assumed that the vertical displacement profile uz of the fluid between
the lateral piston surface and the housing is parabolic and that the radial
displacement of the fluid is zero. Furthermore, it is assumed that this displacement pattern prevails over the entire height of the piston. Thus
uz(r)
= uof(r)
= uo(etr 2 + fSr + 3')
............................
(5)
where constants et, 1~, and 3' are determined from the boundary conditions
and volume conservation. That is
uz(r,)
= -Uo
u~(r2) = 0
uo
!
..............................................
.................................................
2~rrf(r) dr -
~rr~uo
= 0 .................................
(6a)
(6b)
(7)
The third condition is based on the assumption of fluid incompressibility.
3312
J. Struct. Eng. 1993.119:3310-3325.
Eqs. (6)-(7) form the following system of algebraic equations with ct, [3,
and ~/as unknowns.
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I t
2
r22
(r 4 -
rx
re
1t([3)a
1
2
r 4 ) ~ ( r23 - r 3)(r 2 - r 2
=
O)1
. ...............
(8)
r2
The stress-strain relation of the fluid in the frequency domain is
"r,z(Oa) = G*(to)~/r~(to) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(9)
where "rr~and Yrz = amplitudes of the harmonic shear stress and strain; and
G*(to) = complex frequency-dependent shear modulus
itotx
G*(co) =
. .................
(10)
where I~ = fluid zero-shear rate viscosity. Eqs. (9) and (10) are derived
from the time-domain fractional derivative model by application of Fourier
transform (Makris and Constantinou 1991).
Using (4) and the assumption of zero radial displacement, the following
equation is derived for the shear stress:
rrz = G*uo d f
dr
..............................................
(11)
The amplitude of force exerted by the fluid on the lateral face of the
piston is evaluated from the stresses Vrz at r = rl after multiplication by the
surface area of the idealized piston
PoL = 2~rrlH1G*uo df(rO
dr
....................................
(12)
FORCE ACTING ON BOTTOM SURFACE OF PISTON
The bottom surface of the piston is imposing a compression-extension
type of deformation. The fluid displaced from the moving piston tends to
move radially. The solution to this two dimensional problem is approximated
by a one-dimensional model. It is assumed that only the fluid within a
truncated cone is primarily affected by the motion of the piston. This cone
is limited by the base of the damper and the bottom of the piston. Its axis
of symmetry is the vertical axis z [Fig. l(b)]. This simplified model has been
extensively used in deriving approximate expressions for the dynamic stiffness and radiation damping of foundations (Gazetas 1984; Wolf 1988).
Under this simplifying assumption, dynamic equilibrium of a horizontal
slice of the cone along the vertical direction takes the form
_ _ , t) + 2 z % z ( z , t) = pz 2 aeuz(z, t)
Z20Tzz(Z
......................
OZ
82t
(13)
where u~(z, t) = vertical displacement of the slice at depth z and time t;
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J. Struct. Eng. 1993.119:3310-3325.
and P = fluid mass density. Under harmonic, steady-state conditions, (13)
reduces to an ordinary differential equation in terms now of the amplitudes
of the stress ,r= and displacement Uz
z2&zz(Z) +
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dz
2z%z(z)
+ Oto2Z2Uz(Z)
= 0
.......................
(14)
The deformation pattern of the slice is reminiscent of biaxial stretching
flow (Bird 1987). In this type of flow, the normal stress difference %z - "r,r
is related to the rate of normal strain ~ through the elongation viscosity "q.
At low rate of strain (~ < 10 -2 s-1), the elongation viscosity -q approaches
the value of 3tx. This result is valid not only for Newtonian fluids (Trouton
viscosity), but also for non-Newtonian viscoelastic fluids (Bird 1987). At
low strain rates, the elastic component of the viscoelastic fluid vanishes and
the fluid tends to behave as a Newtonian fluid. Accordingly, at low strain
rates and for a harmonic input motion of frequency to, the difference of
normal stresses is given by
Tzz(Z )
-- Trr(Z ) =
3G* a(~vu.,z, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Oz
(15)
where 3G* = elongation modulus.
At this point, the radial stresses within the cone have to be approximated.
This is achieved by balancing the tractions exerted on the portion of the
fluid located above the truncated cone. Fig. 2 illustrates a cross section of
this fluid portion (areas A B C D and EFGH) and the shear stresses acting
on its inner and outer boundaries for a downward motion of the piston. It
is assumed that for vertical equilibrium only the illustrated stresses contribute.
Under these simplifying assumptions, the total vertical force, Fv, that is
transmitted to the truncated cone from the fluid above is
Fv = 2~rHiG*uo
df(rl)
dr
r,
df(r2)]
r2---~r J
........................
(16)
This vertical force is balanced by the vertical component of the resultant of
normal stresses acting on the inclined surface of the upper fluid portion.
The radial component Fr of the resultant of those normal stresses is
...................................
Fr = F~tan O = F ~ - r 2
-
(17)
r I
where 0 = angle between vertical and normal to the inclined surface directions.
Substituting Fv (16), into (17) yields
F~ = 2~rH1G*uo
rl
df(rl)
dr
-
-
-
-
r2
df(r2) 1 - -H2
dr J r2
rl
. . . . . . . . . . . . . . . . . .
(18)
-
The radial stress within the cone is, of course, a function of the radial
coordinate r. Nevertheless a representative average value may be obtained
by dividing the total radial force Fr by the lateral area A = 2 " r r r l H 2. Accordingly, the radial stress may be approximated by
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A
]3
H
E
l
U0
l
H1
C-
~
II
s I'I', ~ I'I'I'I' l
2
l
I
FIG. 2. Schematic Representation of Stresses Acting on Fluid within Damper
Fr
~" = -~
H,
G*uo
=
-
r~(r~
-
r[ df(rO r= df(r2,|~]
~,)
~
d---;-
-
dr
............
(19)
I
where the minus sign indicates compression. Substitution of %, into (15)
results in
%z(z) = 3G* Ou~(z)
Oz
G*uo
H1
[ d/(r,)
rl(r-~- r,) rl dr
df(r2)]
re--~r ]
"'" (20)
Returning to the governing equation (14) and using (20), the following
equation is derived in terms of only the vertical fluid displacement:
duz(z)
p~02
2
HA
z 2 d2uz(z)
d2-----~ - + 2z dz + ~ zZuz(Z) = 5 u~ rl(r ~ 3315
J. Struct. Eng. 1993.119:3310-3325.
r,)
dr(r,)
[_r' dr
[
df(r2)]
-
r2
dr
...............................................
(21)
J
A solution of the homogeneous part of (21) is
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uztt(z) = A1 1_ e(_ikz ) Jr- A2 1 e(ikz)
Z
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(22)
.
Z
where
P ~1/2
k = to \~G-g ]
............................................
(23)
is the complex wave n u m b e r and A1, A2 are integration constants to be
determined from boundary conditions. A particular solution of (21) is
uze(z) = A o -
1
.............................................
(24)
Z
where
G*
Ao = 2 - - Uo
DO)2
H1
~ dr(r1)
- rl rl(F2 -- q ) [ - dr
r2
-
df(r2)] . . . . . . . . . . . . . . . .
dr _]
(25)
The boundary conditions are uz(z 0 = uo and uz(z2) = 0 where zl and z2
are the coordinates of the bottom of the piston and of the bottom surface
of the housing, respectively [Fig. l(b)]. Coordinates Zl and z2 are, respectively, given by
7,1
~-
//2
- - rl ,
r2 - rl
Z 2 = Z 1 ~- n 2
...........................
(26)
Application of the boundary conditions results in the following constants in
(22):
A1 = UoZl - A0(1 - e -ik~I2) e ikz2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2i sin(kH2)
(27)
A2 =
(28)
UoZl - A0(1 - e i~/42) e_ikz2
2i sin(kH2)
...........................
The solution of (21) is the superposition of the homogeneous, (22), and
particular, (24), solutions. The substitution of the values of the integration
constants into (22) produces
zl sin[k(z2 - z)]
Ao {sin[k(z2 - z)] + sin[k(z - zl)]
z
sin(kH2)
z
sin(kH2)
..........................................................
Uz(Z)
=
Uo
-
1
}
(29)
The second term in (29) corresponds to the contribution of the particular
solution. It vanishes for both values z = Zl and z = z2 and can be neglected
within the accuracy of this approximate method.
With this last approximation, the normal stress in the fluid is determined
from (20)
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r~z(Z) -
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I
G ~ Uo
3G'u~ z {~ sin[k(z2- z)] + k cos[k(z2- z)]}
sin(kH2) zl
r,(r2-- r,) Jr'
-
dr
-
-
dr(r2)] ......................
r2 dr J
(30)
The stresses on the piston are given by (30) at z = Za and with the negative
sign replaced by positive sign. Accordingly, the amplitude of force that the
fluid exerts on the bottom circular surface of the piston is given by
[ 1
cos(kH2)]
Po8 = ~rr~G*uo 3 ~ + k sin(kH2)J
H1
[ dr(r1)
+ r,(rz -~ r,) rl - dr
-
r2
dr
jj
For harmonic input motion, the predicted total force on the piston is
P(t) = (Pot + PoB)ei~ 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(32)
where PoL and P0B are given by (12) and (31).
MACROSCOPIC MODEL OF DAMPER
The macroscopic model of the damper undergoing vertical motion is the
fractional derivative model of (1). The model parameters )t, q, and Co are
normally determined by fitting experimental data from tests on dampers.
Herein, parameter h and q are assumed to be same as those of the constitutive relation of the fluid and constant Co is determined from the approximate analytical solution. In this respect, prediction of the behavior of the
damper is accomplished without the need for testing of the damper. Rather,
only testing of the fluid is needed.
For the fractional derivative Maxwell model (1), the damping coefficient
is given by
qcos(Vl]
c(to)
=
..........................
1+
+
(33)
cos
The presented analysis, valid for small frequencies, is used to calculate C(to).
Then at the limit of to tending to zero, C(to) converges to the value Co,
which is the zero-frequency damping constant.
In harmonic motion of frequency to and amplitude uo, the damping coefficient is given by
WD
C(to) - r176 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
where Wn = energy dissipated per cycle and is given by
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J. Struct. Eng. 1993.119:3310-3325.
(34)
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147o =
[du(t)]
(35)
J 0 Re[P(t)]Re L---~-- j dt
where Re = the real part of a complex quantity; T = period of one cycle
(T = 2~r/oJ), and P(t) and u(t) are given by (32) and (3), respectively. The
real part of force P(t) is of the form
Re[P(t)] = Pc cos m t -
P, sin o~t . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(36)
where Pc and P, = force components independent of time. The real part
of the velocity of the piston is
Integration of (35) yields
wo
=
...............................................
(38)
In the limit of small frequencies, the expression for Ps, after a lengthy
algebra, reduces to
Ps = tOUoix 27trail1 ~
+ 3= H-~zJ . . . . . . . . . . . . . . . . . . . . . . . . .
(39)
Substitution of (38) and (39) into (35) gives the damping constant, which
in the limit of zero frequency takes the form
df(rl)
Co = limC(~0) = Ix 2 * r r l H , ~
urn,0
q-
H1
[ r 1 df(q)
r2 - q
dr
,rrr 1 -
rjr2
+ 3= H2
r2df(r2)]'~ . . . . . . . . . . . . . . . . . . . . . . . . .
dr _~J
(40)
Eq. (40) is a dosed-form expression for Ca. The first term in (40) is the
contribution to the damping constant from stresses developing on the lateral
surface of the piston. The second term in (40) is the contribution to the
damping constant from the stresses that develop on the bottom of the piston
in the absence of radial stresses, and the third term is the contribution to
the damping constant from the additional stresses that develop on the bottom
of the piston due to the presence of radial stresses. Only the zero shear rate
viscosity Ix and the geometric characteristics of the damper appear in (40).
The density of the fluid is not present because the inertia terms vanish at
zero frequency.
The range of validity of (40) is now investigated. The second term in (40)
accounts for the contribution of the stresses at the bottom of the moving
piston when the radial stress, "rrr in (15) and (19), is neglected. This term
should always be larger than or equal to the damping constant obtained
from the solution of the problem of a slowly moving circular disc of radius
ra in an infinite medium of viscosity Ix. The drag force on the circular disc
is (Landau and Lifshitz 1987)
P = 16Ixqv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
where v is the velocity of the disc. The damping constant is
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J. Struct. Eng. 1993.119:3310-3325.
(41)
C0e = 161xrl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(42)
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which represents the lower bound of the contribution to Co from the bottom
of the piston when radial stresses are neglected. Accordingly, (40) is valid
for
r2
16
>
H2 3~r
..................................................
(43)
For values of r2/H2 less than 16/3~r, the cone model is invalid and constant
Co may be determined from the contribution of the outside stresses together
with the Landau and Lifshitz (1987) solution. That is,
df(rl)
Co = 2~p~rlHx T
+ 161~r1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(44)
RIGOROUS ANALYSIS AT ZERO FREQUENCY LIMIT
The equations of motion and continuity for an incompressible fluid are
P
\or
Ovj = 0
axj
vj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
oxj/
oxj ox,
(45)
...................................................
(46)
where vi = velocity vector, To = stress tensor; and p = pressure at a point.
The fractional derivative constitutive model of the fluid is now rewritten in
tensor form
dq
'Ti] -~ ~k ~
(Ovi
Ovit
"rij : ~L ~OXj -~ OXi/
...............................
(47)
The model of (47) is valid only for infinitesimal displacement gradients
(uo/rl < < 1). Under this condition, the nonlinear terms in the substantial
derivative are negligible for all values of Reynolds number (Landau and
Lifshitz 1987) and (45) become linear. In the limit of vanishingly small
frequencies, the entire left side of (45) vanishes resulting to the so-called
creeping motion of Stokes flow. Furthermore, the dynamic viscosity reduces
to the zero-shear-rate viscosity Ix. At this limit, the governing equations of
motion (45) reduce to
OZv'
3xix~
012 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ox~
(48)
The infinite-space fundamental solution for Stokes flow governed by (46)
and (48) exists. In fact, it is equivalent to the Kelvin fundamental solution
for incompressible elastostatics (Banerjee and Butterfield 1981) with the
shear modulus replaced by the viscosity. As a consequence, boundary-element methods are directly applicable for a rigorous numerical solution at
zero frequency.
With a boundary-element approach, the complexity of the damper geometry presents no particular difficulty, because discretization is required
only on the surface of the fluid. Furthermore, since the fundamental solution
automatically enforces incompressibility, there is no need to introduce pen3319
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alty methods, as is done in other numerical techniques. Details on the
implementation of boundary-element methods for steady incompressible
viscous flow can be found in Dargush and Banerjee (1991).
For present purposes, an axisymmetric formulation, available in the computer code GPBEST (Banerjee 1991), was employed to analyze vertical
motion of the damper. As discussed in the next section, the boundaryelement results were used principally to verify the values of the constant
Co. However, the calculated velocity profiles and stress distribution also
provided some physical insight beneficial to the development of the approximate analytical macroscopic model.
VERIFICATION OF MACROSCOPIC MODEL
The accuracy of the approximate analytical solution in describing the
steady-state response of viscous dampers is investigated first. Makris and
Constantinou (1990, 1991) reported on the experimental behavior of the
viscous damper with the geometry of the damper No. 1 in Table 1. The
damper was filled with silicon gel having density p = 930 Kg/m 3, zero-shear
rate viscosity ~ = 1,930 Pa. s, fractional relaxation time X -- 0.26 (s)q and
fractional order q = 0.57. Steady-state force-displacement loops, as predicted by the analytical model of (12), (31), and (32), are compared to
experimental loops in Fig. 3. The two sets of loops, at frequencies of 0.05
and 1 Hz, are in excellent agreement. Further results reported by Makris
(1991) demonstrated the validity of the analytical solution for frequencies
below 4 Hz. This indicates that the predictions of the analytical solution for
the zero-frequency damping constant Co will be accurate.
For the verification of the macroscopic model of (1), three viscous
dampers were tested and their mechanical properties determined. The experimental results for constant Co were then compared to the predictions
of the analytical macroscopic model and to the results of the rigorous boundary-element formulation. For the latter analysis the axisymmetric model
involved a detailed representation of the piston, including the ribs (in all
cases, mesh was refined until converged solutions were obtained). The piston
was subjected to motion of very low velocity equal to 0.5 mm/s. The stresses
were integrated to obtain the force needed to maintain the motion, w h i c h - TABLE 1. Geometric Characteristics of Tested and Analyzed Dampers
Damper
(1)
1
2
3
4
5
6
7
8
9
10
11
12
Construction
(2)
Standard piston
Standard piston
Hollow piston
Standard piston
Standard piston
Standard piston
Standard piston
Standard piston
Standard piston
Standard piston
Standard piston
Standard piston
rl
(m)
(3)
0.058
0.062
0.051
0.047
0.052
0.062
0.080
0.095
0.100
0.110
0.125
O.130
r2
(m)
(4)
0.130
0.116
0.116
0.095
0.105
0.130
0.160
0.190
0.200
0.220
0.250
0.260
r3
(m)
(5)
--0.047
----------
3320
J. Struct. Eng. 1993.119:3310-3325.
H1
(m)
(6)
0.120
0.070
0.098
0.090
0.100
0.120
0.150
0.170
0.180
0.200
0.230
0.240
H2
(m)
(7)
0.060
0.061
0.032
0.050
0.050
0.060
0.065
0.075
0.080
0.085
0.090
0.095
25FREQUENCY = 0.05 Hz
2015-
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10'7
5-
i,I
r,.)
0-
0
-5-
LL
-10,
-15-20.
-25
.-6
:4
-'2
o
~
;,
DISPLACEMENT ( m m )
300"T
I.L
Lt
--6
-4
-2
0
2
4
6
DISPLACEMENT ( m m )
FIG. 3. Comparison of Experimental and Analytical Steady-State Force-Displacement Loops
upon division by the velocity--resulted in the zero-frequency damping constant Co.
Two of the tested dampers had their piston constructed with 16 inner
pipes of 15 mm diameter so that the bottom part of the piston could be
assumed to be in full contact with the fluid below. The two pistons were
identical, however they were placed in different-size containers and had
different embedment in the fluid (height H1). The third damper had a hollow
piston. In the analytical solution for the hollow piston, the contribution from
3321
J. Struct. Eng. 1993.119:3310-3325.
TABLE 2. Comparison of Values Co of Viscous Dampers
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Damper
(1)
Analytical model
(N s/m)
(2)
Boundary-element
solution
(N s/m)
(3)
13,973
15,783
8,331
14,331
16,196
17,225
25,713
29,460
31,104
34,686
40,250
41,881
14,541
15,071
9,870
14,480
16,440
17,560
26,840
31,160
32,760
36,800
43,540
45,060
1
2
3
4
5
6
7
8
9
10
11
12
10
I
10~_-
I
20
I
[
15,000
15,200
9,400
4O
30
t
Experimental
(N s/m)
(4)
I
t
I
~
50
10 ~
E
Z
Co
u3
LLI
Z
LL
LL
iLl)
Z
K1
10 2
10
2
I-Z
LLI
LP
10
10
Lt.J
L_9
<~
Cts
O
F-(/3
*
--
EXPERIMENTAL
ANALYTICAL MACROSCOPIC
FRACTIONAL DERIVATIVE MODEL
,-
-1
LL
LL
LLI
O
r.D
L_9
Z
EL
s
1 0 -1
0
'
1{0
2~0
,3'0
4to
50
10 -~
FREQUENCY Hz
FIG. 4. Mechanical Properties of Viscous Damper No. 1 and Properties Predicted
by Macroscopic Fractional Derivative Maxwell Model
below is replaced with the contribution from inside. The interested reader
is referred to Makris (1991). The geometric properties (with reference to
Fig. 1) of these dampers are listed in Table 1. All dampers were filled with
the silicon gel described earlier.
Table 2 presents a comparison of the analytical, numerical (boundaryelement solution) and experimental values of constant Co. It is seen that
the agreement between the three sets of values is very good. It should be
noted that the boundary-element solution predicts the damping constant
with excellent accuracy (difference of less than 5% of experimental value).
3322
J. Struct. Eng. 1993.119:3310-3325.
3I
10
1O
I
i
:
20
I
i
3O
t
10 3 s
E
r.n
z
Z
w"
10 2
Downloaded from ascelibrary.org by New York University on 05/14/15. Copyright ASCE. For personal use only; all rights reserved.
0o 10 ~
oo
LLI
Z
Z
LL
LL
Lf)
W
(..)
/
10
* ~ -*. L . . . _ . L . *
,
*
,
Ld
,
,
,
L
L_
10
fC
lml
0
,
0
<
*
EXPERIMENTAL
ANALYTICAL MACROSCOPIC
- - F R A C T I O N A L DERIVATIVE MODEL
n"
0
F-O9
(_9
1
z
D_
<
10
-1 0
1'0
2'0
FREQUENCY
30 1 0
-1
Hz
FIG. 5. Mechanical Properties of Viscous Damper No. 2 and Properties Predicted
by Macroscopic Fractional Derivative Maxwell Model
10
10 3
,
22
,
50
10
3
E
E
O3
z
10
6O 1 0 2,
Cf)
LI
Z
L
L
10
I-CO
LLI
9
.<
Od
0
2
10
z
.2s
p-z
Ld
O
L
LL
LO
O
L_)
*
--
EXPERIMENTAL
ANALYTICAL MACROSCOPIC
FRACTIONAL DERIVATIVE MODEL
1
(/3
(D
Z
<
1 0 -1
1 '0
FREQUENCY
2'0
Hz
Jo t 0 -4
FIG. 6. Mechanical Properties of Viscous Damper No. 3 and Properties Predicted
by Macroscopic Fractional Derivative Maxwell Model
In an extension of this comparison study, nine more dampers of substantially different dimensions were studied. All were of the standard type
with closed bottom. Their geometrical properties are also listed in Table 1.
The damping constant Co of these dampers was determined by the numerical
boundary-element procedure and values are listed in Table 2. Assuming
that the boundary-element solution is accurate, it is apparent in the results
of Table 2 that the analytical solution predicts the value of constant Co with
error of no more than 7.5% of the exact.
Having verified the accuracy of the analytical model in predicting the
3323
J. Struct. Eng. 1993.119:3310-3325.
0.6~
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EXPERIMENTAL
MACROSCOPIC
MODEL
0.34
Z
//
I_.LI 0.00
C,_)
n/
9
LL
-0.34
-0.68
i
-8
I
i
-4
i
0
I
4
8
D SPLACEMENT (ram)
FIG. 7. Recorded Force-Displacement Loop of Damper No. 1 for Four-Cycle-Beat
Displacement Input and Comparison to Loop Predicted by Macroscopic Fractional
Derivative Maxwell Model
parameter Co, the verification proceeds with an examination of the macroscopic model over the entire range of frequencies. The macroscopic model
is given by (1) with parameters Co, X, and q. Parameter Co is evaluated by
the approximate analytical model, whereas parameters X and q are assumed
equal to those of the damper fluid (for silicon gel X = 0.26 (s)q, q = 0.57).
Figs. 4-6 compare the experimentally and analytically determined mechanical properties of storage modulus K1 and damping constant C of the three
tested dampers (Table 1). Evidently, the analytical macroscopic model predicts well the mechanical properties over a wide range of frequencies.
Finally, Fig. 7 compares experimental and analytical loops of force versus
displacement of damper 1 for a beat displacement input--see Makris and
Constantinou (1990) for test description. The analytical and experimental
loops are in excellent agreement.
CONCLUSIONS
In this paper an approximate analytical solution is presented to calculate
the force that develops on the piston of viscous dampers when subjected to
low frequency motions. The tractions of the viscoelastic fluid on the damper
piston are computed using physically motivated approximations.
3324
J. Struct. Eng. 1993.119:3310-3325.
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At the zero-frequency limit, the presented solution provides a closedform expression for the zero-frequency damping constant. With the determination of this constant, a macroscopic fractional derivative model can be
constructed. This model describes accurately the behavior of viscous
dampers over a wide frequency range, and consequently could prove to be
a useful tool within the design process.
ACKNOWLEDGMENTS
Financial support for this work has been provided by the National Science
Foundation (Grant No. BCS-8857080). G E R B Vibration Control, Inc. donated the viscous dampers used in the experiments.
APPENDIX.
REFERENCES
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Banerjee, P. K. (1991). GPBEST users manual. BEST Software Corp., Getzville,
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Dargush, G. F., and Banerjee, P. K. (1991). "Steady thermoviscous flow by the
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Makris, N., and Constantinou, M. C. (1990). "Viscous dampers: testing modeling
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"Pipework dampers." (1986). Tech. Rep., GERB Vibration Control, Westmont, Ill.
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Wolf, J. P. (1988). Soil-structure-interaction analysis in time domain. Prentice-Hall,
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3325
J. Struct. Eng. 1993.119:3310-3325.