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Journal of the Chinese Institute of Engineers, Vol. 30, No. 2, pp. 275-287 (2007)

275

SHEAR-THINNING EFFECTS IN ANNULAR-ORIFICE VISCOUS

FLUID DAMPERS

Chien-Yuan Hou*, Deh-Shiu Hsu, Yung-Feng Lee, Hsing-Yuan Chen, and Junn-Deh Lee

ABSTRACT

The number of construction projects using viscous fluid dampers for the purpose

of seismic energy dissipation has been increasing in recent years. Usually, resisting

forces provided by a viscous fluid damper are nonlinearly related to the damper op-

eration velocity. In the current study, the mechanism of the nonlinear behavior is

studied. It is found that the fluid shear rate in the orifices of a damper is high enough

to cause shear thinning of the fluid, that is, the non-Newtonian behavior of the fluid

must be considered to capture the viscous damper’s non-linearity. Carreau’s equation

giving the shear-thinning relationship between fluid viscosity and shear rate is em-

ployed in a finite element model. The model is used to calculate the fluid dynamics in

viscous dampers and the calculated results successfully explain the nonlinear behavior.

Effects of the damper geometry and the fluid viscosity on the damper non-linearity

are also tested and discussed. Again, the trend shown in experimental results can be

fully explained by the shear-thinning concept. In addition, the behavior of a damper

operated at ultra high velocity is addressed.

Key Words: viscous fluid damper, viscosity, shear thinning, non-Newtonian fluid,

polydimethylsiloxan, Carreau’s model, structural control, earthquake.

*Corresponding author. (Tel: 886-6-5718888 ext. 234; Email:

cyhou@ms15.hinet.net)

C. Y. Hou is with the Department of Construction Management,

Diwan College, Tainan, Taiwan 721, R.O.C.

D. S. Hsu, Y. F. Lee, H. Y. Chen, and J. D. Lee are with the

Department of Civil Engineering, National Cheng Kung University,

Tainan, Taiwan 701, R.O.C.

I. INTRODUCTION

As energy-dissipation devices, fluid dampers

have been used to reduce earthquake damage in many

construction projects since the 1994 Northridge

earthquake. A simple damper contains a closed cyl-

inder fully filled with a proper fluid, such as silicone

oil, and a piston with rods connecting to the structure

floors. Different types of orifices are used to allow

the fluid to flow between the two chambers on both

sides of the piston head. Dampers with annular and

circular orifices in the simplest form are shown in

Fig. 1.

Compared to the shock absorbers used in the

automobile industry, the architecture of the fluid

dampers used for seismic energy dissipation is much

less complicated. For example, a shock absorber in a

vehicle suspension system not only includes all the

parts described above, in addition, complex valve

systems are also contained in shock absorbers. It is

obvious that shock absorbers are more sophisticated

and many delicate mathematical models (Segal &

Lang, 1981, Mollica & Yocef-Toumi, 1997, Duym et

al., 1997) have been proposed in the last two decades

to predict their behavior. It seems that the knowl-

edge obtained from the automobile industry can be

fully implanted into a viscous damper for seismic

purposes. However, there is still a major difference

between the two types of energy dissipaters: the load

capacity. A shock absorber for a vehicle suspension

system may have a load capacity of several hundred

kilograms, yet a fluid damper for buildings and

bridges may have a load capacity as large as several

hundred tonnes. To achieve such a capacity, increas-

ing the piston head diameter or reducing the orifice

size is probably the best choice.

Scaled seismic dampers tested in many studies

(Constantinou & Symans, 1992, Tsopelas, et al.,

1994, Reinhorn et al., 1995, Seleemah and

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Journal of the Chinese Institute of Engineers, Vol. 30, No. 2 (2007)

Constantinou, 1997, Constantinou and Symans, 1996)

have shown that dampers with nonlinear behavior are

common and the FD – V relationship should be writ-

ten as:

FD = C|V|n . sgn(V).(1)

The value of n is important both in damper manufac-

turing and structure analysis. In the present paper,

cyclic tests of scaled dampers of annular orifice were

performed to obtain the values of n and subsequently

the mechanism of n was studied on the basis of fluid

mechanics principles.

II. CYCLIC TESTS OF THE SCALED FLUID

DAMPERS WITH ANNULAR ORIFICE

The tests were performed on a heavy and rigid

steel frame (see Fig. 2). A 1-ton capacity load cell

was set on a steel wall of the steel frame. A rod firmly

mounted with a piston was connected to the load cell.

An open cylinder and two cover plates were used to

form a closed cylinder in which the piston was placed.

A hydraulic actuator was tightly attached to the rigid

steel frame and was connected to the closed cylinder.

Because the piston was fixed without any motion, the

cylinder motion state, that is, the actuator motion

state, could be used to describe the motion state of

the tested damper.

The closed cylinder was fully filled with sili-

cone oil, which has a kinematic viscosity of 1000 cSt.

A cylinder with an inner diameter of 50 mm and a

piston head with a diameter of 48.9 mm were used to

form a reduced-scale damper. No circular orifice was

drilled in the piston head and the only orifice for fluid

flowing was the annular gap left between the piston

head and the cylinder wall. The width of the annular

orifice is (50 –48.9)/2 = 0.55 mm. Sinusoidal dis-

placements with various frequencies ranging from 1

to 10 Hz were tested. The amplitude was 10 mm in

each test. Fig. 3(a) shows the measured resisting

force-displacement loops of 1, 2, 4, 8, and 10 Hz. Fig.

3(b) shows the relationship of the FD versus opera-

tion velocity (V), which indicates that the relation-

ship is nonlinear. A regression analysis gives the

following equation to describe the behavior of the

tested damper:

FD = 4.57 × V0.71,(2)

where units of FD and V are kgf and mm/sec,

respectively.

III. PRINCIPLES OF FLUID MECHANICS IN

VISCOUS FLUID DAMPERS

One may recall the fundamental principles of a

viscous flow in fluid mechanics. The internal friction,

i.e., the shear stress, is assumed to be proportional to

the shear rate:

τ = ηγ = ηdu(y)

dy

,

(3)

where τ is the shear stress (N/m2); η is the dynamic

viscosity (kg . sec/m); .γ is the shear rate (sec–1); the

y-direction is perpendicular to the flow direction;

u(y) is the flow velocity at various y locations. An

alternative to express the fluid viscosity is the kine-

matic viscosity (ν), which equals η divided by fluid

density. Kinematic viscosity has a unit of m2/sec in

MKS unit. However, a more commonly employed

unit for kinematic viscosity is centistokes (cSt, mm2/

sec). If a fluid has a constant viscosity, that is, if the

τ and du/dy relationship is linear, the fluid is a

Newtonian type: see Fig. 4(a).

Piston rod Piston rodPiston head

Annular orifice

(a)

(b)

Silicone oil

Piston rod

Piston head

Circular orifice

Silicone oil

Piston rod

Fig. 1 Fluid damper in two of the simplest forms

Load cellViscous fluid

damper

Actuator

Actuator control

panel

Heavy and rigid steel

frame

Data acquisition

system

Fig. 2 The experiment setup

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C. Y. Hou et al.: Shear-Thinning Effects in Annular-Orifice Viscous Fluid Dampers

277

In the case of annular orifice, the flow is simi-

lar to a Poiseuille flow, which can be easily obtained

by solving the simplified Navier-Stokes equations:

∂u

∂t= –1ρ

∂p

∂x+ ν∂2u

∂y2,

(4)

where ρ is the fluid density; p is the hydraulic

pressure; and x is the coordinate along the flow

direction. In a steady flow, that is, neglecting the

term related to time in the equation, the solution

shows that p(x) linearly decreases with increasing x

and velocity profile u(y) is parabolic: see Fig. 4(b).

Subsequently, one can also easily calculate the re-

sisting force provided by the damper:

FD = 6πηL(R/h)3 . V, (5)

where L and R are the width and radius of the piston

head, respectively; h is the width of the orifice. This

linear relationship prevails even when the transient

term is included in Eq. (4). Apparently, regarding

the nonlinear issue, the damper behavior predicted

by the solution is contradictory to the test results.

IV. POSSIBLE CAUSES OF FLUID

VISCOSITY VARIATION

The discrepancy should not be attributed to the

orifice configuration since this factor has already

been considered when the Navier-Stokes equations

are solved. A possible cause of the discrepancy is

fluid viscosity variation. This clue is based on the

following observations: (1) the resisting force pro-

vided by a nonlinear damper is smaller than that of a

linear one at higher operation velocity, (2) a com-

mon sense observation that the lower fluid viscosity

causes smaller resisting forces, and (3) the analytical

solution shown in Eq. (5) was obtained by assuming

constant fluid viscosity. There are three possible reasons

which may cause the viscosity variation: (1) fluid

temperature, (2) hydraulic pressure, and (3) shear rate.

It is well known that higher temperature decreases

the viscosity of mineral oil, i.e., the oil becomes thinner

as fluid temperature increases. Temperature has the

same effect on silicone oil as well. If temperature is

the major cause of non-linearity, then one should ex-

pect in the cyclic tests that the resisting forces be-

come smaller from cycle to cycle since the energy

dissipated by the damper increases the temperature

of silicone oil. In the previously mentioned tests, it

was found that the resisting-force decrease did take

place in the first two or three cycles. After these cycles,

the recorded resisting force-displacement loops were

found to become stable, that is, the resisting forces

were no longer decreased. Temperature measurement

showed that the oil temperature increase was not

-12 -60

Displacement (mm)

(a)

6 12

600

300

0

-300

-600

Resistance force (kgf)

10 Hz

1 Hz

2 Hz

4 Hz

8 Hz

-600 -400-200

Velocity (mm/sec)

(b)

0 200 400600

600

300

0

-300

-600

Resistance force (kgf)

Linear damper

Resisting force difference

between a linear and a

nonlinear damper

Nonliner

damper

Fig. 3The results obtained from sinusoidal displacement tests,

(a) the measured resisting force-displacement loops, (b)

the resisting force versus velocity.

Newtonian fluid

piston wall

(b)

Cylinder wall

Piston head

p(x)

p1

(a)

Silicone oil:

shear-thinning and

non-newtonian fluid

1

1

x

y

x

y

u(y)

p2

Fig. 4(a) The relationship between shear stress and shear rate of

a Newtonian fluid and a shear-thinning fluid, (b) flow ve-

locity profile across the annular orifice of a Newtonian

fluid.

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Journal of the Chinese Institute of Engineers, Vol. 30, No. 2 (2007)

significant. The recorded actuator velocity data showed

that the actuator velocity was not stable in the first

several cycles because the machine was making self-

adjustments to reach the prescribed sinusoidal-displace-

ment condition. In other words, the actuator always

went too fast in the first cycle and had to slow down

in the following cycles, which produced the unstable

resisting forces in the first several cycles. As the

actuator finished the adjustment, the measured force-

displacement loops became stable. Hence, the force

decrease in the first several cycles was not caused by

oil temperature increase, which should not be con-

sidered as the major cause of the damper non-linearity.

In many textbooks regarding fluid machines, one

can easily find the relationship between hydraulic

pressure and oil viscosity. Higher hydraulic pressure

usually increases the oil viscosity. However, the

damper non-linearity cannot be attributed to hydrau-

lic pressure, either. If hydraulic pressure causes the

non-linearity, the force difference between the non-

linear and linear dampers (see Fig. 3(b)) should be

more significant at higher velocities because the pres-

sure effect is more pronounced at high velocities.

This observation is consistent with the test results

shown in Fig. 3(b) that the FD-V relationship bends

down more as the operation velocity increases.

However, the resisting force provided by the pres-

sure-caused nonlinear damping should be larger than

the forces provided by linear damping because the

hydraulic pressure induces thicker fluid. Thicker fluid

always produces larger resisting forces. Therefore,

one should expect that the measured FD-V relation-

ship bends up and exhibits an n value larger than 1 if

the non-linearity is caused by hydraulic pressure. Of

course, this is totally contradictory to the damper

behavior observed in the test results: n < 1.

V. EFFECTS OF SHEAR RATE ON FLUID

VISCOSITY

The third possible reason for viscosity variation

is the shear rate. In mechanical engineering applica-

tions where mineral oil is commonly used in many

machines, the oil exhibits a Newtonian behavior,

i.e., the fluid viscosity is a constant. However, the

fluids treated by plastic and food industries are thicker

and are far from Newtonian. The viscosity of these

fluids is shear rate dependent. Some of them are shear

thickening and some of them are shear thinning. Be-

cause the silicone oil used in a damper is a liquid type

polymer, the ν-.γ relationship must be known before

one can judge if the shear rate is a cause for damper

non-linearity.

The chemical structure of silicone oil is

polydimethylsiolxane: CH3-[(CH3)2SiO]x-CH3. The

number x shows how many (CH3)2SiO structural units

in the series. Larger x indicates that more (CH3)2SiO

structural units form a chain structure in a more en-

tangled manner and hence the oil has a larger mo-

lecular weight. In the polymer industry, it is well

known that the viscosity of a liquid polymer is

strongly dependent on its molecular weight. Silicone

oil of higher viscosity can be manufactured with

higher polydimethylsiloxan molecular weight. This

is because larger friction is caused between the more

entangled molecules as the fluid flows. As the shear

rate increases, the entangled molecules are stretched

and the molecules move in an aligned manner. This

behavior decreases the friction between molecules and

thus decreases the fluid viscosity.

Figure 4(a) schematically shows the relationship

between shear stress and shear rate of silicone oil.

The slope of the tangent of the curve is the viscosity

of the fluid. It is obvious that the slope decreases as

the shear rate increases, indicating that the silicone

oil is a shear thinning non-Newtonian fluid. Fig. 5

shows some ν-.γ relationships of various silicone oils

reported in the literature (Currie & Smith, 1950, Lee

et al., 1970, http:// www.unitedchem.com/pdf/

silicones_introduction.pdf). All these curves exhibit

a similar trend. At low shear rate, the viscosity is

denoted as ν0, or the zero-shear rate viscosity. In this

shear rate range, the fluid viscosity remains almost

constant and forms a plateau in the relationship,

i.e., the fluid behaves as a Newtonian type because

of the constant viscosity. The viscosity plateau is

followed by a transition region in which the slope of

the curve varies. In this region, the viscosity de-

creases and the fluid becomes non-Newtonian. As

the shear rate continues to increase, the viscosity con-

tinues to decrease and the slope of the curve finally

becomes a constant. It is worth noting that all the

curves merge together even though their ν0 values are

significantly different.

The shear rate in the annular orifice of the tested

50-mm damper can be roughly estimated to determine

whether the silicone oil in the damper tests became

non-Newtonian at high operation velocity. First, the

average flow velocity in the orifice can be calculated

using the fluid continuity condition, i.e., A1V1 = A2V2,

where A1 and A2 are the cross-sectional areas of the

piston head and orifice, respectively; V1 and V2 are

damper operation velocity and the average flow ve-

locity in the orifice, respectively. If the tested damper

moves with a maximum velocity of 300 mm/sec, i.e.,

the damper is operated at approximately 5 Hz with

an amplitude of 10 mm, the calculations show that

the average flow velocity in the orifice is approxi-

mately 6600 mm/sec. At the cylinder wall, the flow

velocity is zero due to the no-slip condition. Assume

that the flow velocity at the mid location of the ori-

fice is 6600 mm/sec and the velocity profile from the

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C. Y. Hou et al.: Shear-Thinning Effects in Annular-Orifice Viscous Fluid Dampers

279

cylinder wall to the mid location varies linearly, the

average shear rate in the orifice is (6600-0)/(0.55/2),

which gives approximately 24000 sec–1. Apparently,

for silicone fluid with a ν0 value of 1000 cSt, this

shear rate is far beyond the region of Newtonian

plateau: see Fig. 5. Therefore, silicone oil in the

tested damper easily became a non-Newtonian fluid.

Because the logν-log.γ relationship is linear be-

yond the transition region, the fluid viscosity is usu-

ally described by a power law:

ν = k.γα – 1, (6)

where α –1 is the slope of the linear relationship in

the logarithmic plot. The shortcoming of the power

law is that the viscosity plateau is not considered.

Carreau (1972) proposed a new equation to overcome

the difficulty:

ν

ν0= (1 + (λγ)2)

α – 1

2

,

(7)

where λ is a time constant to control the location of

the transition region. The value of α –1 has also

been studied experimentally and theoretically by

many researchers, and the values were found to range

from -0.7 to -0.88. Graessley (1965, 1967) devel-

oped a molecular entanglement network model esti-

mating -0.82 for the value of α –1.

VI. FINITE ELEMENT ANALYSIS OF THE

FLUID DYNAMICS IN THE TESTED DAMPER

1. The Finite Element Model

The fluid dynamics in the tested damper were

analyzed using a finite element technique. The ana-

lyzed flow domain and its meshes are shown in Fig.

6. Due to the axis-symmetrical nature of the damper,

a 4-node quadrilateral axis-symmetric element was

used. The density of silicone oil was 0.97 g/cm3 and

the value of ν0 was 1000 cSt. The values of λ and α

– 1 used in Carreau’s equation were 3.8 × 10–5 (sec)

and -0.8, respectively. Module FLOTRAN in the

commercial finite element package ANSYS was used

for the calculations.

In the tests, the damper piston was fixed and the

damper was activated by moving the cylinder. To

simulate the motion of the cylinder, the flow region

shown in Fig. 6(a) must also move back and forth:

see Fig. 6(b), that is, the analyzed domain varies from

time to time. This is a moving boundary problem

which is very complicated in computational fluid

dynamics. To simplify the problem, an alternative

model simulating the nature of the tests was used. The

analyzed domain remained unchanged at all times and

the no-slip condition was employed on the nodes lo-

cated at the cylinder, piston head and piston rod walls

(see heavy shaded lines in Fig. 6(a)). However, the

flow velocity of the nodes located at both ends of the

analyzed domain was specified according to the cyl-

inder motion, that is, ux = V0sinωt and uy = 0. One

can imagine that the alternative model is equivalent

to a fully fixed damper with a pump at each end of

the cylinder. These pumps generate sinusoidal fluid

flows, which is just like the fluid flow produced by

the motion of the cylinder.

2. Verification of the Alternative Model by the

Steady Flow Analysis

Figure 6(b) schematically shows the flow domains

at various times in a real test condition. The distance

between the piston head and the ends of the cylinder

varies from time to time. If the hydraulic-pressure

states on both sides of the piston head are significantly

influenced by the distance, the alternative model would

fail to predict the damper oscillatory behavior because

the alternative one is not capable of capturing the ef-

fects of the distance. To verify if this is true, hydrau-

lic pressure of three flow domains as shown in Fig.

6(b) were analyzed. The distances between the left

wall of the piston head and the left end of the ana-

lyzed domain were 20, 40, and 80 mm. A steady flow

with a flow velocity of 200 mm/sec from the left to

the right side of the damper was used in the calcula-

tions and the nodal pressures on the left side of the

piston head were recorded. Fig. 7 shows the pressure

distribution along the radial direction on the left side

of the piston head. It is evident that the calculated

pressure state has no substantial differences among the

three calculated flow domains. Therefore, it is believed

105

104

103

102

101

102

103

104

105

106

Kinematic viscosity. v (cSt)

v0 = 1000 cSt

v0 = 100 cSt

Viscosity curve used

in the calculations

Viscosity curve used

in the calculations

Currie and Smith (1950)

Lee et al. (1970)

http://www.unitedchem.com/pdf/silicones_introduction.pdf

Fig. 5Viscosity versus shear rate relationships reported in the

literature

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Journal of the Chinese Institute of Engineers, Vol. 30, No. 2 (2007)

that the distance is insignificant to the fluid dynamics

in the damper and the alternative model can be used

in an oscillatory flow analysis.

3. Results of Oscillatory Flow

Finite element fluid transient analyses were per-

formed on the tested damper according to the test

conditions, that is, the damper was tested at 1, 2, 4, 8

and 10 Hz with a peak displacement of 10 mm. Fig.

8(a) shows the calculated viscosity reduction in the

orifice when the frequency is 8 Hz. The upper and

lower thick lines are the cylinder and piston head walls,

respectively. When the cylinder moves at a velocity

of 62 mm/sec, the calculated viscosity varies mildly

across the annular orifice and the variation is more

pronounced when the damper velocity becomes 125.

9 and 501.6 mm/sec. At the mid location of the ori-

fice the viscosity remains almost the same for all op-

eration velocities. This is because the shear rate is

zero at the mid location due to the symmetrical na-

ture of the analyzed problem. At the cylinder wall,

the viscosity becomes as small as 20% of ν0 when V

= 501.6 mm/sec. The calculated normalized flow

velocity profiles (u(y)/V) across the orifice are also

shown in Fig. 8(b). When the velocity is 62 mm/sec,

the profile is similar to a parabolic curve indicating

that the fluid behavior at this operation velocity is

still very close to Newtonian. When the velocity in-

creases to 501.6 mm/sec, the profile deviates from a

ux(t) = V0sin t

ω

ux(t) = V0sin t

ω

y

x

Axis of symmetry

(a)

(b)

Distance between the cylinder left end and left wall of piston head

Cylinder wall, ux = 0, uy = 0

Piston rod, ux = 0, uy = 0Piston rod, ux = 0, uy = 0

Piston head

ux = 0

uy = 0

20 mm 40 mm80 mm

t = t0

t = t1

t = t2

Fig. 6 (a) The finite element model used in the fluid dynamics calculations. (b) The flow domain varies from time to time.

25

20

15

10

5

0

Axis of symmetry

Pressure (kgf/cm2)

Distance from axis of symmetry (mm)

Piston head

Piston radial direction

20 20.2 20.420.6 20.821

Piston rod

Radius =

7.5 mm

80 mm

40 mm

20 mm

Fig. 7The calculated hydraulic pressure distributions along the

piston radial direction for different distances between cyl-

inder end and piston head wall.

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C. Y. Hou et al.: Shear-Thinning Effects in Annular-Orifice Viscous Fluid Dampers

281

parabolic curve more significantly.

Hydraulic pressures of the nodes located at the

left and right piston head walls were recorded. Each

node has a corresponding node on the other side of

the piston head at the same elevation. Pressure dif-

ferences of the paired nodes were calculated. Because

the pressure distribution is roughly uniform along the

radial direction of the piston (see Fig. 7), a net pres-

sure difference was calculated by averaging the pres-

sure differences obtained from the paired nodes. The

average pressure difference was multiplied by the

cross-sectional area of the piston head to obtain the

net force acting on the piston. The net force is the

resisting force FD provided by the damper.

Figure 9 shows the calculated and the experimental

results at 1 Hz. In Fig. 9(a), the calculated resisting

force-displacement loop envelops the measured one.

Fig. 9(b) shows the FD-V relationship and the C and n

values obtained from regression analyses are also given.

The n value of the calculated results is very close to 1.

Corresponding to this result, as shown in Fig. 9(a),

the calculated resisting force-displacement loop is close

Piston wall

600

Viscosity, v (cst)

(a)

Cylinder wall

0 200 400800 10001200

V = 62.0 mm/sec

V = 125.9 mm/sec

V = 501.6 mm/sec

v0 = 1000 cst

Piston wall

20

Cylinder wall

0 10

Normalized velocity, u(y)/V

(b)

30 40

V = 62.0 mm/sec

V = 125.9 mm/sec

V = 501.6 mm/sec

Fig. 8 The calculated (a) viscosity reduction curves, (b) flow

velocity profiles across the orifice width.

200

100

0

-100

-200-12

1260

Displacement (mm)

(a)

-6

Calculated

Measured

Resistaing force, FD (kgf)

200

100

0

-100

-200

-100 100 500

Damper velocity, V (mm)

(b)

-50

Calculated

Fb = 2.26 V0.98

Measured

Fb = 2.75 V0.82

Resistaing force, FD (kgf)

Fig. 9 The measured and calculated (a) resisting force versus

displacement, and (b) force versus velocity relationships

at 1 Hz.

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Journal of the Chinese Institute of Engineers, Vol. 30, No. 2 (2007)

to an ellipse, which is the behavior of a linear damper.

Fig. 10(a) shows the calculated and measured resist-

ing force-displacement loops when the frequency is 8

Hz. It is obvious that both loop shapes are far from

ellipses. Fig. 10(b) shows the FD-V relationships. The

nonlinear behavior of the damper is captured by the

calculated results and the n values obtained from the

calculated and measured results are 0.69 and 0.72,

respectively.

At this stage, it is no longer a mystery why the

resisting force of a nonlinear damper deviates from

that of a linear damper at high velocity. At low oper-

ating velocity (or low frequency), flow shear rate across

the orifice is not large enough to cause significant

viscosity reduction. Because the viscosity variation

is mild across the orifice, the behavior of silicone oil

is close to Newtonian, and hence, the damper is close

to a linear type and the n value is close to 1. At high

operating velocity (or high frequency), a large shear

rate, beyond the viscosity plateau, occurs at locations

across the orifice inducing significant shear thinning

of the silicone oil. Since the fluid becomes thinner,

the resisting forces that the damper produces become

smaller than what is expected if the viscosity remains

unchanged. Therefore, the FD – V relationship bends

down and deviates more from the linear damper at a

larger velocity.

Figure 9(b) shows that the experimental FD – V

relationship is a hysteresis loop. The hysteresis be-

havior is more pronounced at higher frequency: see

Fig. 10(b). Hysteresis behavior is an important fea-

ture of dampers and is obviously not captured by the

current calculated results. Usually, the behavior oc-

curs when the frequency is high and is explained by

a restoring-stiffness concept (Constantinou &

Symans, 1992, Tsopelas, et al., 1994, Reinhorn et al.,

1995, Seleemah and Constantinou, 1997,

Constantinou and Symans, 1996). Using of the re-

storing-stiffness concept implies that the damper be-

haves viscoelastically. Although the silicone oil used

in dampers is in the liquid form, it is actually not

purely viscous but exhibits more or less viscoelastic

behavior. The behavior of this type of material is

usually described by the Maxwell model:

τ + λ1∂τ

∂t= ηγ .

(8)

It is obvious that the only difference between

Eqs. (8) and (3) is the λ1∂τ

∂t term. When a viscoelas-

tic material is subjected to cyclic shear strains, the

induced shear stress-strain relationship shows a hys-

teresis behavior. Thus, it is believed that when Eq.

(8) is used to model silicone oil behavior, the hyster-

esis behavior of dampers can be captured. However,

the current finite element model is entirely a viscous

type, in which the λ1∂τ

∂t term cannot be employed in

the available FLOTRAN module. Therefore, the hys-

teresis behavior of the damper will not be further dis-

cussed in the current study, but, of course, will be a

major issue in the authors’ future work.

VII. DISCUSSION

To understand more about the shear-thinning

effects on damper behavior, additional cyclic tests

were performed. Tables 1(a) to (d) show four types

of damper, that is, cylinders with internal diameters

of 40, 50, 70, and 100 mm filled with three types of

silicone oil, 100, 550, and 1000 cSt, were tested.

Different pistons were used in each type of damper

to form various dampers with annular orifice width

ranging from 0.25 to 1.2 mm. Regression analyses

500

250

0

-250

-500

-12 1260

Displacement (mm)

(a)

-6

Calculated

Measured

Resistaing force, FD (kgf)

500

250

0

-250

-500

-600600 3000

Damper velocity, V (mm/sec)

(b)

-300

Calculated FD = 6.53 V0.69

Measured

FD = 4.44 V0.72

Resistaing force, FD (kgf)

Fig. 10 The measured and calculated (a) force versus displacement,

(b) force versus velocity relationships at 10 Hz.

Page 9

C. Y. Hou et al.: Shear-Thinning Effects in Annular-Orifice Viscous Fluid Dampers

283

were performed on the measured FD-V relationships

to obtain the n value of each damper, which are shown

in the tables. Based on these n values, the effects of

damper geometry and zero-shear-rate viscosity are

discussed.

1. Effects of Damper Geometry on n Value

It is difficult to observe any effects of geometry

from the listed n values. However, with the concept

that more viscosity reduction, i.e., higher shear rate,

in an orifice causes less n value, a geometrical pa-

rameter may be used to correlate with these tested n

values. For annular orifices, one can estimate the

average flow velocity in the orifice by the continuity

equation (A1V1 = A2V2):

V2=A1

πdhV1,

(9)

where the πdh is the area of the orifice. The average

shear rate can thus be calculated by dividing V2 by h/2:

γave=2A1

πdh2V1= βV1.

(10)

Eq. (10) shows that .γave is proportional to the opera-

tion velocity V1 and a geometrical parameter β.

Dampers of different geometry exhibiting identical β

value indicates identical .γave in the orifice if these

dampers are operated at the same velocity. The n

value of each damper filled with 1000 cSt oil in Table

1 is plotted against its β value and the results are

shown in Fig. 11. Although the data are scattered a

clear trend is shown: n decreases with increasing β.

The n values obtained from finite element results of

the 50-mm dampers are also shown. It is evident that

the calculated results also show a similar trend but

smaller n values.

The tested dampers are reduced-scale dampers

with β values ranging from 30 to 180. These damp-

ers provide resisting forces of several hundred

kilograms. Larger capacity dampers are often re-

quired in civil engineering structures and are manu-

factured by increasing the piston head diameter and

reducing the width of the orifice. In other words,

dampers with larger β values are required for practi-

cal uses. Therefore, a damper with a β value of 1575

was constructed and tested. Due to the limited ca-

pacity of the actuator, the maximum operation veloc-

ity which could be achieved in the test was 30 mm/

sec. The test data show that the n value of the damper

is 0.58, which is smaller than the values listed in Table

1. This result is consistent with the trend found in

Fig. 11 because the β value of the damper is maxi-

mum among all the tested dampers.

Although the tested and the calculated n values

show a consistent trend, it cannot be ignored that their

1.0

0.8

0.6

0.4

0.2

0.0

Regressed values obtained

from measured data

Finite element results

0 50100 150 200

n

Fig. 11 The values of n decrease with increasing geometrical pa-

rameter β

Table 1 The regressed n values of various damp-

ers

(a) 40-mm damper

Piston head diameter (mm)

39.1

n

39.2

100

550

1000

0.86

0.86

0.70

0.86

0.82

0.67

ν0

(cSt)

(b) 50-mm damper

Piston head diameter (mm)

48.848.9

n

48.549.149.3 49.5

100

550

1000

N.A.

0.98

0.80

0.98

0.88

0.70

0.86

0.80

0.74

0.95

0.83

0.73

0.87

0.77

0.62

0.92

0.75

N.A.

ν0

(cSt)

(c) 70-mm damper

Piston head diameter (mm)

68.769.0

n

68.4

100

550

1000

0.97

0.89

0.81

0.92

0.87

0.79

1.02

0.88

0.73

ν0

(cSt)

(d) 100-mm damper

Piston head diameter (mm)

98.1 98.6

n

97.6

100

550

1000

1.07

0.97

0.88

0.99

0.96

0.81

1.06

0.93

0.78

ν0

(cSt)

Page 10

284

Journal of the Chinese Institute of Engineers, Vol. 30, No. 2 (2007)

discrepancies become larger for larger β values. The

large discrepancy is primarily caused by using Eq.

(7) for modeling the shear-thinning effects. As dis-

cussed previously, the shear rates in the orifice range

from 0 to the order of at least 104 sec–1 or larger. It is

hard to find a simple mathematical equation that fully

captures the viscosity-shear rate relationship over

such a broad shear rate range. In Fig. 5, the viscosity

curve (obtained from Eq. (7)) used in the calculations

is indicated by the two arrows. Comparing this curve

to that provided by Lee et al. (1970), it is found that

the curve is good from 0 to 104 sec–1 but the viscosity

difference becomes larger for higher shear rates. For

a shear rate larger than 105 sec–1, the viscosity differ-

ence is at least 20%, and in the tested dampers shear

rates larger than 105 sec–1 are common. Therefore,

the calculated FD is more or less inaccurate due to

the viscosity error, which resulted in the n-value

discrepancy.

In Eq. (7), λ is a parameter used to control the

shear rate commencing the thinning effect and α –1

is used to control the final slope in the viscosity

relationship. However, there is no parameter to con-

trol the shear rate corresponding to the end of the vis-

cosity transition region. In Fig. 5, it is obvious that

Eq. (7) estimates the shear rate when the thinning effect

starts well, however, it is poor for the end of the transition

region. To overcome this difficulty, the Yasuda (1979)-

Carreau equation has been proposed:

ν

ν0= (1 + (λγ)κ)

α – 1

κ

.

(11)

In the equation the parameter κ is used for the men-

tioned purpose. There are other equations such as

the Ellis equation (Bird, et al., 1977) and the Cross

equation (Cross, 1965) that are used by the polymer

society and a comparison of these equations has also

been studied by Raju et al. (1993). In the current

study, the Carreau equation was chosen because the

FLOTRAN module used in the calculations only sup-

ports that equation. The Carreau equation is a spe-

cial case of Eq. (11), i.e., when the value of κ is set

as 2, Eq. (11) becomes the Carreau equation.

Therefore, if Eq. (11) is used, it is expected that the

viscosity reduction phenomenon can be more accu-

rately modeled and the calculated and tested n values

will be more consistent.

It is worth noting that β is not a parameter rep-

resenting the real shear-rate distribution, i.e., the real

viscosity reduction, across the orifices because of its

average significance. In other words, dampers with

identical β value may have slightly different n values.

However, β is still useful because it can be easily

calculated and is probably so far the simplest geo-

metrical parameter to indicate the shear-thinning state

in viscous dampers.

2. Effects of Zero-Shear-Rate Viscosity

In Table 1, the fluid viscosity effect is pretty

obvious: dampers filled with higher ν0 oil exhibit

smaller n values. This phenomenon can also be eas-

ily explained by the shear-thinning concept.

Figure 12 shows the normalized viscosity pro-

file (ν/ν0) across the orifice of the 50-mm damper

with 48.9-mm piston head operated at a velocity of

300 mm/sec. It is evident that the viscosity reduc-

tion is more significant when the damper is filled with

1000 cSt silicone oil, which explains why the damp-

ers filled with thicker oil exhibit lower n values.

However, the cause for more viscosity reduction in

thicker oil remains unknown. Damper geometry

should not be attributed to the different viscosity re-

duction curves shown in Fig. 12 since these two

curves are obtained from the same damper.

The only factor left to explain the various vis-

cosity reduction curves is the fluid property. As

shown in Fig. 5, the viscosity plateau of thicker sili-

cone oil extends across a shorter shear-rate range. At

a shear rate such as 10000 sec–1 in Fig. 5, viscosity

reduction already takes place in the 1000 cSt oil,

however, the viscosity of 100 cSt oil remains in the

viscosity plateau. Because the two oils experience

the same shear rate condition in the damper, more

viscosity reductions should take place across the ori-

fice in thicker oil, and therefore, should induce a

larger deviation from the linear behavior and a smaller

n value.

Piston wall

0.6

Cylinder wall

0 0.20.4

Nomalized viscosity, v/v0

0.8 1.01.2

v0 = 100 cst

v0 = 1000 cst

Fig. 12 More viscosity reductions in thicker silicone oil at the same

operation velocity

Page 11

C. Y. Hou et al.: Shear-Thinning Effects in Annular-Orifice Viscous Fluid Dampers

285

3. Damper Behavior at Ultra High Velocity

As mentioned before, one of the methods to

manufacture large-capacity dampers is reducing the

area of the orifice. For an annular orifice, a width of

0.5 mm can be easily manufactured. However, as the

width is reduced to 0.1 mm or even less, the tech-

niques required are more complicated and expensive.

As shown in Fig. 1, an alternative way to produce

small orifice areas is to drill circular orifices in the

piston head. For example, a 150-mm damper with

0.5 mm annular width has an orifice area of 236 mm2.

If two circular orifices with a diameter of 4 mm are

drilled instead of using the annular orifice, the ori-

fice area is 25 mm2, which is almost one tenth of the

annular orifice. Since the orifice area is effectively

reduced using the circular orifice, the fluid flow ve-

locity in a circular orifice should increase

dramatically. Again, we take the 150-mm damper as

an example. If the damper is operated at 200 mm/

sec, which is not unusual for floor motion during an

earthquake, the average flow velocity in the 0.5-mm

annular orifice is approximately 14 m/sec. Under the

same condition, the average flow velocity in the two

4-mm circular orifices is approximately 140 m/sec.

It is interesting to see how the damper behavior is

influenced at such high flow velocity in the orifice.

Because the ultra high flow velocity can also be

achieved in the annular orifice by simply increasing

the damper operation velocity, the 50-mm damper

with annular orifice is used for the analysis. Fig. 13

shows the calculated resisting force provided by the

damper at operation velocities ranging from 20 to

6000 mm/sec. When the operation velocity is smaller

than 1600 mm/sec, the FD-V relationship shows the

familiar increasing trend with a nonlinear behavior

of n < 1. However, as the operation velocity contin-

ues to increase, the increasing trend becomes more

pronounced and it is evident that the n < 1 condition

is no longer prevailing. At these ultra high velocities,

the value of n is larger than 1. If the operation veloc-

ity continues to increase, the n value at the ultra high

velocity range is estimated to be approximately 2,

which is the maximum value that can be achieved by

a damper under an ideal condition without fluid

viscosity. Under such a circumstance, the resisting

force is caused by fluid inertia and the role of fluid

viscosity diminishes. Despite that, operation veloci-

ties 1600 mm/sec or even higher are unlikely to oc-

cur for fluid dampers in a civil engineering structure.

A damper with extremely small circular orifices could

have an extremely high flow velocity in the orifices

and could behave in a manner controlled by inertia,

even if the operation velocity is low.

4. Limitations and Some Issues of the Shear-Thin-

ning Concept

Damper technology has a history of approxi-

mately 100 years and the dampers used in construc-

tion projects often exhibit more complex orifice

design than those tested in the current study. Fluid

in such sophisticated dampers may flow in a super-

sonic manner and behave with insignificant viscosity

effects and then, the proposed shear-thinning theory

is no longer valid. For example, as discussed

previously, the inertia effects in the dampers with

small circular orifices is much more important than

the viscosity effects. Under such a condition, the fluid

inertia term must be considered in the Navier-Stokes

equations (in Eq. (4) only the viscosity terms are

considered) to model the damper behavior.

In addition, the maximum hydraulic pressure in

the tested dampers was approximately 40 kg/cm2,

which is far below the design pressure commonly used

by damper manufacturers. For dampers subjected to

large hydraulic pressures, the fluid is extremely com-

pressed causing volume change thus the fluid den-

sity is no longer a constant, which should be also

accounted for in the Navier-Stokes equations. An-

other undesired phenomenon is cavitation erosions,

in which the air dissolved in the high pressure fluid

evaporates in the low pressure region. This event

makes modeling damper behavior more complicated.

Many modern dampers involve complicated ori-

fice configurations with fluid flowing at much higher

velocities, thus, the usefulness of the shear-thinning

concept is limited. However, the theory is definitely

important for viscous type dampers. There is another

reason that damper researchers may consider the con-

cept useful. For many years, the authors have tried

Fig. 13 At extremely high operation velocity, the value of n be-

comes larger than 1

2000

1500

1000

500

0

0 80006000 4000

Operation velocity (mm/sec)

2000

n < 1

n > 1

Resistaing force (kgf)

~ 1600 mm/sec

Page 12

286

Journal of the Chinese Institute of Engineers, Vol. 30, No. 2 (2007)

to find clues, through literature surveys, to explain

the nonlinear behavior of dampers but could only find

limited information and the behavior remained unex-

plained until the shear-thinning effect was disclosed.

This is probably because damper manufacturers like

to keep their own damper configurations or even their

analytical techniques as trade secrets, which resulted

in the research community being unfamiliar with the

shear-thinning effect. For example, Markis et al. (1996)

tested an electrorheological damper of annular ori-

fice and found that, at a zero electrical field, the damper

was linear at low and nonlinear with n < 1 at higher

operational velocities. They attributed this phenom-

enon to the damper allowing fluid flow through the

piston-head orifice at higher operation velocity. This

is a good explanation of the phenomenon. However,

it is also found that the phenomenon can be fully cap-

tured by shear-thinning effects. It is not the authors’

purpose to identify which mechanism is correct for

the behavior of the electrorheological damper, but to

reveal a mechanism that could be considered by re-

searchers for the behavior of advanced dampers in

future studies.

VIII. CONCLUSIONS

1. Fluid shear rate in the orifices of a viscous damper

used for seismic energy dissipation is large enough

to cause a shear-thinning effect in silicone oil. The

shear-thinning behavior causes viscosity reduction

in the orifices and is responsible for the non-lin-

earity of the damper.

2. Shear-rate condition in the orifices controls the

value of n. The higher the shear rate, the more

viscosity reduction is caused. The value of n be-

comes smaller when a damper is operated under

high frequencies because of higher operation

velocities, i.e., the higher shear rate.

3. Viscous dampers filled with thicker silicone oil

exhibit smaller n values.

ACKNOWLEDGMENTS

The authors would like to thank the National

Science Council for funding this research under

project number NSC93-2211-E434-008. The National

Center for High-Performance Computation is also

acknowledged for providing the computation

facilities.

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Manuscript Received: May 03, 2005

Revision Received: Apr. 25, 2006

and Accepted: May 24, 2006