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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 rodPiston 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)

612

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)

0200400600

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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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