viscous heating of fluid dampers: experimental studies

Viscous Heating of Fluid Dampers: Experimental Studies

Cameron J. Black, Nicos MakrisDepartment of Civil and Environmental Engineering, UC Berkeley, CA

ABSTRACT

A preliminary experimental investigation on the problem of viscous heating of fluid dampers that find applications in thevibration reduction of civil engineering structures is presented in this paper. Time histories of the temperatures are recordednear the piston head and at the outer surface of a 3-kip fluid damper that undergoes harmonic loading. The experimentalresults uncover some of the limitations of simple analytical expressions that have been derived in the past based on one-dimen-sional approximations of the energy balance equation.

Keywords: Viscous Heating, Fluid Dampers, Experimental Studies, Seismic/Wind Protection of Structures

1. INTRODUCTION

Fluid dampers which generate fluid flow through orifices or valves were originally developed for the shock isolation of mili-tary hardware. During the last decade their applications have been extended to civil engineering where they have beenaccepted as a promising alternative to dissipate the energy that earthquakes and wind induce in structures. When fluid damp-ers are used to suppress wind-induced vibrations of buildings, the piston displacements and velocities are relatively small;whereas, in seismic protection applications they can be much larger. In particular, when fluid dampers are incorporated eitherin the seismic isolation system of a structure or in suspended bridges, the design piston displacement can be several times thepiston diameter. Because of this wide range in operational amplitudes, theoretical and experimental studies on the problem of

viscous heating distinguish between small and large piston motions. 1,2

Most hydraulic dampers that find applications in civil structures have specially shaped orifices which yield a nonlinear force-velocity relationship of the form

P(t) = CI(t)IUsgn[ü(t)] , (1)

where P(t) = piston force, piston velocity, a = fractional exponent, 0 <a � 1 , C = adamping constant with units

(force) . (time/length)X and sgn[ ] is the signum function. When a = 1 equation (1) reduces to the viscous case

P(t) = Cü(t) . (2)

This paper presents experimental results on the temperature rise at various locations of a small, double-ended viscous damper(C = 65 lb sec/in, maximum stroke of in) that has been subjected to long stroke motions (up to 94% of its total stroke).The recorded temperature histories are compared with the predictions of an approximate solution that is based on a one-dimen-sional approximation of the energy balance equation.

2. EXPERIMENTAL SETUP

Viscous heating of the above mentioned damper was achieved with the experimental setup shown in Figure 1 . A hydraulicactuator imposes a prescribed displacement history along the axis of the damper. The force developed in the damper is mea-sured through a stationary load cell that is connected between the damper and the reaction frame (right end in Figure 1). Theimposed displacement history is measured with a linear variable differential transducer (LVDT) that is located within the actu-ator shown at the left of Figure 1.

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Temperature histories are recorded at three locations of the damper as indicated in Figure 2. Thermocouple No. I was posi-tioned near the moving surface of the piston head. This was achieved by drilling a 6.5 in (1(1.5 cm) long bore along the pistonrod. Thermocouple No. 2 was positioned on the outer surface of the damper housing at the zero-stroke location. Thermocou-ple No. 3 was placed on the outer surface of the damper at 1/2' from the iero-strokc location.

Harmonic time histories, u(t) = U0sin( t) where imposed. Figure 3 shows plots of the temperature time histories recorded

with the three thermocouples under harmonic motion with amplitude U0 = I in and frequenciesf 0.25, 0.5 and 1 Hz; whileFigure 4 plots the corresponding time histories when U() = 1.4 in. In both figures, the less steep temperature histories, appear-

ing in each graph, have been recorded under forced convection; whereas, the steeper temperature histories (upper lines in eachgraph) have been recorded under free convection. Forced convection was achieved by means of 2 high powered fans placed insuch a manner as to force the surrounding air away from the damper.

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Figure I: Expenmenial setup.

Figure 2: Schematic of the fluid damper that indicates the locations of the thermocouples and the hore through the leO part of the piston rod.


U-

Temp 1 Temp 2 Temp 3

Figure 3: Recorded temperature time histories under harmonic motion with 1" amplitude.

3. SIMPLE ANALYTICAL MODEL

Preliminary theoretical studies on the problem of viscous heating of fluid dampers have been presented for small1 and large2amplitude motions. In these studies analytical expressions for the temperature rise were derived using an approximate one-dimensional version of the energy equation, where it was assumed that the temperature of the fluid within the damper is afunction of position, x and time, t; so that all fluid particles that belong to a given cross section of the damper have the same

temperature, O(x, t) . In particular under large amplitude motions the energy equation was averaged over the entire damperusing the crude approximation that the temperature at every section of the damper is nearly the same at

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a given time, equal to 9(t) . The reasoning that motivated this sweeping assumption is that when U0 is sufficiently large, most

of the fluid enclosed within the damper housing travels through the piston head. However, this approximation neglects that,under harmonic excitations, the piston velocity near the center of the damper is the maximum; whereas at u = U0 is zero.

Nevertheless, under a triangular wave motion this assumption becomes more realistic since the piston maintains a constantvelocity along it's entire stroke. By adopting this assumption macroscopic energy balance on the entire fluid of the dampergives 2

itd k •2UmC_e(t) +

S0(9(t) — 9(t)) = P(t)ü(t) , (3)

in which m =2pAU0 is the mass of the fluid that travels through the piston head as the piston moves from —U0 to U0 ,C,,

is

the heat capacity coefficient at constant pressure, k is the coefficient of thermal conductivity of steel (radial flow only), c is

Temp 1

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Figure 4: Recorded temperature time histories under harmonic motion with I .4" amplitude.

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the thickness of the shock tube and A is the area of the piston head. 0(t) is the average time-dependent temperature of the

fluid and 00(t) is the time-dependent temperature of the outer surface of the steel shock tube. The first term of the left hand

side of equation (3) is the energy absorbed by the fluid per unit time, while the second term of the left hand side of equation (3)is the energy that conducts radially per unit time. The right hand side of equation (3) is the power input by the actuator. For

dampers with thick shock tubes ( e/d > 0.1 ) the radial heat flow is expressed more appropriately3 by

2rck •2U (9(t)—8 (t))Q(t)= ° \

0• (4)

2e1n1 +—p

In such cases the second term of (3) should be replaced with the expression given by (4). Figure 5 shows the differencebetween the thick and the thin tube formulations as a function of c/dr.

18 - - - f --i------Th---h---t--- — dr/c j16 -

1 4———-

12

-—

——----—-— ''h6

,

I I I4 --1-T;uull.I, -

I I I

2 ———-———-———-

1' 1 I I 1 I I I0 0.05 0.1 015 0.2 0.25 0.3 0.35 0.4 0.45 0.5

id

Figure 5: Difference between the thick and thin tube formulation.

3.1 Boundary ConditionsThe boundary conditions between the outer surface of the damper housing and the surrounding environment are very complexsince they combine the results of heat conduction, convection and radiation. In many occasions radiation can be superimposed

on the conductive and convective heat flow through most gases3'4. For the case of forced conduction one can assume that theair in the immediate vicinity of the damper maintains a nearly constant temperature, Tajr and in this case the convection

boundary condition (Newton's law of cooling) is applied.

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k = h(T . -0 (t)) , (5)Si1n air oin which h is a combined heat transfer coefficient , or occasionally termed as the boundary film conductance5, that com-

bines the effects of convection, conduction and radiation. The left hand side of (5), , is the radial heat flux which in our

case can be approximated with

2k (9(t)—O (t)) k (O(t)—O (t))-k= 0 S 0(6)Sn ( 2E"l C

d lnIl+—p d

The second relation given by (6) is appropriate for thin tubes only (see Figure 5); otherwise the first relation is more appropri-ate for the general case. Whatever equation is used to express the radial heat flux the form of the governing equation thatdescribes this one-dimensional model is the same.

Substitution of the thin tube approximation of (6) into (5) gives

k (e(t)—O (t))S 0 = h(9 (t)-T ), (7)C 0 airfrom which

k9 (t) = S

9(t)+Ch T . (8)0 eh+k ch+k airS S

Equation (8) relates the temperature history of the outer surface of the damper to the temperature history of the fluid and thesurrounding constant air temperature. Substitution of (8) into (3) gives

2ickdU 27tkdU--O(t) + p o Le(t) = —1P(t)t(t) + S P 0 —'—T

, (9)dt mC C l+y mC mC C 1+y airp p p

where y = k/(ch). Introducing the parameter

2ickdU itkdX = S D 0• 1 = S 40 (thin tube) (10)mC C 1+y pC A C l+yp pp

which is a quantity with units in rad/sec, equation (9) simplifies to

j9(t)+A9(t) = ;J_P(t)U(t)+ATair . (11)p

In the case of a thick tube, parameter A. is given by

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27tk

x = . _L (thick tube), (12)( 1+ypC A 1nI1+—pp d

where y =2k/(dh1n(1

+))With the introduction of the parameter, y ,Newton's law of cooling can be rearranged in the form

0 (t)—TY =

O(t)—9(:) (13)

Figure 6 plots the time history of y as the damper is subjected to continuous cycling at various frequencies. It is observed that? eventually reaches a nearly steady value. Moreover; the asymptotic value of y is not independent of the rate of loadingsince y(t = large) 2 when the harmonic excitation is at 0.25 Hz; whereas y(t = large) — 3 when the excitation is at 0.5 Hz.

Cyclic testing with frequency 1.0 Hz was limited to 125 cycles so that the fluid temperature would not exceed 200 °F as tem-peratures in excess of this would be harmful to the teflon seals of the damper. Despite the differences in the asymptotic val-ues of y , in this study we adopt an approximate value of y = 3.

Having established the average value of y from experiments one can estimate X by using the geometrical characteristics of thedamper and the physical properties of the steel and silicone oil which are summarized in Table 1.

Table 1: Geometrical characteristics of tested damper and selected physical properties of steel and silicone oil.

Thickness of Damper Housing = 7.75x103m= 0.305 in

Piston Diameter d = 27.63x103m = 1.088 in

Area of Piston Head A = O.5O5x103m2

Thermal Conductivity of Steel k5 15-30 Joules/(sec m ° C)

Mass Density of Silicone Oil p 950 kg/rn3

Heat Capacity of Silicone Oil C 2000 Joule/kg °C

With the above values and an approximate value of y = 3 , equation (12) yields a value of X 0.05.

3.2 Predictions of Approximate Analytical SolutionsEquation (11) is a linear, first-order differential equation. Its solution under a harmonic and triangular wave motion have been

presented by the senior author 2 Under harmonic motion, u(t) = U0sin(on) , the resulting force in a viscous damper is

P(t) = CU0sin(ou) and the solution of(11) is

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Figure 6: Time history of y as the damper is subjected to continuous cycling at various frequencies.

9(t) = T 1PrnaX1[((() + 2sin(wt))) + 2 - + 2)e_Xt] (14)

where Pmax CoU0/A 15 the maximum pressure drop between the two sides ofthe piston head. Rearranging equation (13)

the temperature history at the outer surface of the damper is given by

8 (t) = —--e(t)+——-T . . (15)0 1+y 1+y air

The predictions ofequations (14) and (15) are shown in Figure 7 next to the experimental data obtained with harmonic loadingof amplitude U0 = 1.4 and frequencies o = 0.25, 0.5 and 1 .0 Hz. For the care wheref= 0.25 and 0.5 Hz the simple analytical

solution underestimates both the internal and external long term temperatures; whereas it over predicts the temperature rise atearly times. On the other hand for the case wheref= 1 .0 Hz the analytical solution overestimates the temperature rise allalong. Possible reasons for this discrepancy are that (a) the analytical solution was derived by assuming radial heat flow only;whereas, in reality there is heat conduction along the longitudial direction of the damper; and (b) that the damper was notentirely full with fluid, therefore not exhibiting a purely viscous behavior and dissipating less energy per cycle than it's viscousidealization.

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Future studies will focus on improving and refining the analytical model so that the correct trends are captured for the entirefrequency range of interest. The challenge of this effort is to develop a dependable method that will allow one to calculate arealistic estimate of the internal fluid temperature by measuring the temperature on the damper housing and back figuring.

?500 1000 1500

0.5

Figure 7: Recorded temperature histories (wavy lines) and predictions resulting from a one-dimensional approximation of the energy balanceequation (smooth line).

4. CONCLUSIONS

In this paper an experimental investigation on the problem of viscous heating of fluid dampers under long stroke motion hasbeen presented. Time histories of temperatures recorded at various locations indicate that a one-dimensional radial flowassumption does yield dependable estimates of the temperature rise, particularly at higher frequencies.

ACKNOWLEDGEMENTS

Financial support for this study was provided by the National Science Foundation under grant CMS-9696241.

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REFERENCES

1. .N. Makris, "Viscous Heating ofFluid Dampers I: Small Amplitude Motions, J. ofEng. Mech., ASCE, Vol. 124, No. 11.pp. 1210-1216, 1998.

2. N. Makris, Y. Roussos, A.S. Whittaker and J. Kelly, "Viscous Heating of Fluid Dampers II: Large-Amplitude Motions",J. of Eng. Mech., ASCE, Vol. 124, No. 11. pp. 12 17-1223, 1998.

3. M. Jakob and G.A. Hawkins, Elementsof Heat Transfer and Insulation, John Willey and Sons, London, U.K., 1942.4. 5. Kakac and Y. Yener, Heat Conduction, Third Edition, Taylor and Francis, Washington, DC, 1993.5. B.A. Boley and J.H. Weiner, Theory of Thermal Stresses, Dover Publications, mc, Mineola, NY, 1997.

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viscous heating of fluid dampers: experimental studies

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