A NEW APPROACH FOR IDENTIFICATION OF

VORTEX-SHEDDING PARAMETERS IN TIME DOMAIN

by

HIMANSHU GUPTA, B.S.C.E.

A THESIS

IN

CIVIL ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

CIVIL ENGINEERING

Approved

Accepted

May, 1995

T3 ^^ ; I"' ^o ACKNOWLEDGEMENTS ^/^^ 7 ^ ^ ' NO' i

I would like to express my sincere gratitude and thanks to my committee

chairman, Dr. Partha P. Sarkar for his patience, support and encouragement

throughout this research. His guidance and willingness to help not only in this

research but also throughout my academic career at Texas Tech University are

sincerely appreciated. Special thanks are extended to my thesis committee

members Dr. Kishor C. Mehta and Dr. W. Pennington Vann for their support for

this research.

Financial support provided by Wind Engineering Research Center, Texas Tech

University to conduct this research is acknowledged.

I would also like to express my sincere appreciation to Mr. Pat Nixon,

Research Associate with Mechanical Engineering Department at Texas Tech

University for his help in putting together the experimental setup. I am also

grateful to Mr. Jianming Yin for his help during my experiments. Finally, I thank

Professor R. H. Scanlan of Johns Hopkins University for his guidance and support.

My deepest appreciation is extended to my parents and my family for their

unending support and encouragement throughout my life. It is to my parents that

I wish to dedicate this thesis.

n

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

LIST OF FIGURES v

CHAPTER

I. INTRODUCTION 1 1.1 Background 1 1.2 Review of Past Studies 2 1.3 Features of Vortex-Induced Vibrations 4

1.3.1 Mechanism of Vortex Shedding 4 1.3.2 Strouhal Number 4 1.3.3 Lock-In Range 6 1.3.4 Steady-State Oscillations 7 1.3.5 Response Hysteresis 7

1.4 Scope of Current Work 8

II. ANALYTICAL MODEL FOR VORTEX SHEDDING 9 2.1 Models Applicable to Vortex-Induced Oscillations 9 2.2 Scanlan's Nonlinear Model 11 2.3 Identification of Parameters 13

III. A NEW APPROACH FOR IDENTIFICATION OF PARAMETERS 16 3.1 Van der Pol Oscillator Equation 18 3.2 Theory of Invariant Imbedding 19 3.3 Nonlinear Filtering Theory 20 3.4 Development of Invariant Imbedding Equations 23 3.5 Equations for Van der Pol Oscillator 28 3.6 Numerical Simulation 30 3.7 Discussion 30

IV. DESCRIPTION OF EXPERIMENTAL WORK 32 4.1 Objectives 32 4.2 Experimental Setup 33

4.2.1 Wind Tunnel 33 4.2.2 Test Section 33 4.2.3 Load Cell Arrangement 39 4.2.4 Data-Acquisition System 39 4.2.5 Electromagnetic Damper System 41 4.2.6 Anemometry 43

m

4.2.7 Generation of Roughness for the Cylinder 45 4.2.8 Generation of Turbulence 49 4.2.9 Data Processing 49

4.3 Wind-Tunnel Section Models 52 4.3.1 Cylinder 52 4.3.2 Bridge Deck Model 53

4.4 Experimental Procedure 56

V. EXPERIMENTAL RESULTS AND PARAMETER EXTRACTION 57 5.1 Section-Model Configurations 57 5.2 Experimental Observations 57

5.2.1 Strouhal Numbers 57 5.2.2 Lock-In Ranges for the Test Sections 58 5.2.3 Lock-In Response 59 5.2.4 Decaying Response and Self-Excited Response 62 5.2.5 Beating Response 68 5.2.6 Vortex-Induced Response in Turbulent Flow 68

5.3 Identification of Cylinder Parameters 72 5.4 Identification of Bridge Deck Parameters 76

VI. CONCLUSIONS AND RECOMMENDATIONS 78 6.1 Conclusions 78

6.1.1 Existing Techniques for Parameter Extraction 78 6.1.2 Proposed Identification Technique 79 6.1.3 Identification of Parameters from Experimental Data 80

6.2 Recommendations for Future Work 80

REFERENCES 82

APPENDIX

A. MASS IDENTIFICATION 86

B. STROUHAL NUMBER 89

C. GRID GENERATED TURBULENCE 92

IV

LIST OF FIGURES

1.1 Karman Vortex Street behind a circular cylinder 5

4.1 Texas Tech wind tunnel (Matty, 1979) 34

4.2 Centrifugal blower of wind tunnel 35

4.3 Test section 37

4.4 Suspension system for model 38

4.5 Load cell 40

4.6 Electromagnetic damper setup 42

4.7 Propeller anemometer used for wind speed measurements 44

4.8 Typical position of hot-film probe 46

4.9 A unit of drywall screen paper 47

4.10 Cylinder covered with roughness elements 48

4.11 Grid ,171 for generation of turbulence 50

4.12 Grid ^^2 for generation of turbulence 51

4.13 Cylinder model 54

4.14 Bridge deck model 55

5.1 Lock-in range for the smooth cylinder 60

5.2 Lock-in range for the bridge deck with upstream sign 60

5.3 Response around the lock-in range (bridge deck with upstream sign) 61

5.4 Typical spectra at lock-in flow speed (smooth cylinder) 63

5.5 Typical spectra at lock-in flow speed (bridge deck with upstream sign) 64

5.6 Lock-in oscillations (smooth cylinder) 65

5.7 Vortex-induced response (smooth cylinder at Scruton number of 1.1) 66

5.8 Vortex-induced response (bridge deck with upstream sign) 67

5.9 Beating response (bridge deck with upstream sign) 69

5.10 Response in turbulent flow (cylinder at Scruton number of 1.1) 70

5.11 Spectra in the wake of smooth cylinder in turbulent flow 71

5.12 Parameters for smooth cylinder 73

5.13 Parameters for rough cylinder 74

5.14 Steady-state amplitude (/?) versus Scruton number 75

A.l Mass identification of cylinder 88

A.2 Mass identification of bridge deck 88

B.l Strouhal number for cylinder 90

B.2 Strouhal number for bridge deck 91

C.l Turbulence spectra (Grid #1) 94

C.2 Turbulence spectra (Grid #2) 94

VI

CHAPTER I

INTRODUCTION

1.1 Background

The emphasis of modern day construction is to achieve the highest standards

of performance within the lowest possible cost. This quest for better construction

has enhanced technology to levels that were never dreamed before. The

innovations in material science, construction technology, and design methodology

have made it possible to conceive of structures which are as tall as the Empire

State Building and as complex as the Golden Gate Bridge. The emerging

technology ha.s promised a new era of development which would aiFect the welfare

of people and the economies of nations. Although the new technology has

produced an array of benefits, it has raised several concerns for designers. Some of

the concerns are related to the increased flexibility of modern day structures,

which has introduced a risk of wind-induced structural damage. One of the

problems that has attracted much attention from designers is wind-induced

vibrations of a bridge. Structural collapse of the Tacoma Narrows bridge remained

a major concern of the designers for many years. The controversy that surrounded

the collapse contributed to the development of the science of bridge aerodynamics.

Aerodynamics and aeroelasticity aim at studying the phenomenon of

flow-induced vibrations of flexible or flexibly supported structures. In essence

2

these disciplines are aimed at studying the complex nature of fluid-structure

interaction problems. Aerodynamics and aeroelasticity draw elements from

disciplines as diverse as fluid mechanics, fluid dynamics and structural dynamics.

The studies in the areas of aerodynamics and aeroelasticity include engineering

problems like aircraft wing flutter, galloping of transmission lines and

vortex-induced vibrations of chimneys and bridges. Aerodynamics is primarily

concerned with studying dynamic forces acting on rigidly supported structures,

while aeroelasticity is concerned with forces induced on a structure due to its

motion in a fluid flow. One of the interesting phenomena observed in a fluid flow is

that of vortex shedding. The vortices shed from a bluff body induce a fluctuating

force on the structure which leads to vibration if the structure is flexible. The

intensity of the force acting on the structure or the amplitude of the vibration of

the structure depends primarily on the shape of the structure and the wind

velocity. Research interest in the vortex-shedding phenomenon is centuries old.

1.2 Review of Past Studies

The first major scientific investigation concerning vortex shedding was reported

in 1878 by Czech physicist Vincenz Strouhal (Strouhal 1878). Strouhal studied the

effect of uniform flow on a stretched wire. He observed a linear relation between

the frequency of sound (/,) generated by the wire oscillation and the uniform flow

velocity (U). This relationship gave birth to a popular nondimensional number

3

known as the Strouhal Number (S). The Strouhal number is a function of the

Reynolds number (3f?). The Strouhal number is given by

S = hDIU, (1.1)

D being the characteristic dimension of the section perpendicular to the flow. The

Reynolds number is defined a^

Ik = UD/u, (1.2)

1/ being the kinematic viscosity of the fluid flow. Lord Rayleigh (1879) established

that vortex-induced oscillations of a body in a uniform flow are primarily across

flow. The first analytical study of vortex shedding was presented by von Karman

(1911) ba^ed on an idealized staggered vortex street. Birkoff (1953) developed the

first phenomenological model for the vortex-induced vibrations. Bishop and

Hassan (1964) found that vortex-induced vibrations of a bluff body can be

represented by a nonlinear oscillator. Since then a number of theoretical and

experimental studies have been reported contributing to the understanding of the

phenomenon. Vortex-induced vibrations of bridge structures have also received

much attention. Some of the prominent studies related to bridge structures are

those reported by Scanlan (1981), Vickery and Basu (1983), and Ehsan(1988).

The understanding of vortex shedding effects on structures has come a long way

since the pioneering experiments of Strouhal.

1.3 Features of Vortex-Induced Vibrations

1.3.1 Mechanism of Vortex Shedding

The mechanism of vortex shedding has been a subject of study for many years.

The complicated nature of flow around a body makes it even more challenging.

Many researchers, through flow visualization and numerical studies, have provided

an explanation of this complex phenomenon. The vortices shed from a body

immersed in a fluid flow are often characterized into two different categories,

Karman vortices and Secondary vortices. Secondary vortices are also known as

motion-induced vortices. These vortices create large scale periodic patterns in the

flow which generate a fluctuating force on the body. Gerrard (1966) ha^ given one

of the earliest physical descriptions of the shedding mechanism. He postulated

that vortex formation is a direct consequence of interaction between two shear

layers separated from the edges of the body. Figure 1.1 (Van Dyke, 1982) shows

Karman vortex street behind a circular cylinder (0.4 in. diameter) at 3f? = 140.

The picture is taken in water at a flow velocity of 0.3 ft/min.

1.3.2 Strouhal Number

The frequency with which the vortices are shed from a body can be made

dimensionless with the flow velocity and a characteristic dimension of the body.

This dimensionless parameter known as the Strouhal number, is defined in

Equation 1.1. It depends on the Reynolds number of the flow (given by Equation

0

>>

CO X

-a

6

1.2). The Strouhal number is named after Physicist Strouhal, who first observed

the relation through his experiments. The range of values for the Strouhal number

lies between 0.12 and 0.28 for circular cylinders. The Strouhal number remains

fairly constant at a value of 0.20 for the Reynolds number range from 300 to

2 X 10^ (Subcritical range). The Strouhal number increases to 0.28 for Reynolds

numbers greater than 3.5 x 10^. Various researchers have established values of

Strouhal number for different cross-sections. Typically, Strouhal number ranges

between 0.12 and 0.15 for most bluff cross-sectional configurations.

1.3.3 Lock-In Range

In forced oscillations of a bluff body the frequency of vortex shedding (fs) and

the arrangement of the vortices can change, mainly because the positions of the

separation points on the body surface and the amount of vorticity shed from those

locations vary. If the frequency of vortex shedding (/s), is close to the natural

frequency (/„) of the oscillating body, amplitude of vibration increases. It has

been observed that for a certain range of values of the frequency ratio (fs/fn)

close to 1.0, the oscillations are quite large and dominate the shedding process.

This phenomenon is known as lock-in or synchronization range. In the lock-in

range the Strouhal relationship is violated as the frequency of vortex shedding

remains constant even with an increase in the speed of the fluid flow.

1.3.4 Steady-State Oscillations

One of the most interesting phenomena associated with vortex-induced

oscillations is the steady-state oscillation of a bluff body in the lock-in range. This

oscillation is a consequence of the nonlinearity of the vibrating system. The

ampHtude of the oscillation increases considerably as the flow speed reaches the

lock-in range. The oscillations are self limiting and reach a steady state. The

steady-state amplitude varies within the lock-in range and is determined by a

nondimensional number known as Scruton number. The Scruton number is

defined as

Sc = mC,lpD'' (1.3)

where m is mass per unit length of the body, ^ is percent damping, D is

characteristic dimension of the body, p is density of air.

1.3.5 Response Hysteresis

Another aspect of the nonlinearity of the vortex-induced vibrating system is

response hysteresis. The steady-state amplitude of the vibrations in the lock-in

range depends upon whether the windspeed is approached from the higher side or

the lower side. The hysteresis effect is dependent on the mechanical damping of

the system. At relatively higher damping ratios the hysteresis response is not

observed.

8

1.4 Scope of Current Work

The current study is motivated toward proposing an alternative system

identification technique for identifying vortex-shedding parameters to eliminate

the limitations of existing methods. Chapter II discusses the model applicable to

vortex-shedding vibrations. The importance of various parameters in the model

equation is also discussed. Some of the existing techniques for identification of the

vortex-shedding parameters are also discussed and their limitations are outlined.

Chapter III discusses the proposed identification technique. Chapter IV describes

the experiments conducted on a circular cylinder and on two configurations

representative of bridge decks. The results are discussed in Chapter V in the light

of the proposed identification technique. The outcome of the research is

summarized in Chapter VI.

CHAPTER II

ANALYTICAL MODEL FOR VORTEX SHEDDING

Prediction of the vortex-induced response of a flexible structure is a two step

process. In the first step, a geometrically scaled section model of the structure is

tested in a wind tunnel and parameters associated with the mathematical model

which describes the response of the section model are identified. In the next step,

the parameters identified from section-model tests are applied to a mathematical

model of the prototype to estimate the response of the full-scale structure.

A detailed mathematical model for vortex-induced response involve the

dynamic equations of motion and the Navier-Stokes equation. These equations

would have to be solved with arbitrary moving boundary constraints. The solution

of such a mathematical problem is still intriguing researchers and no solution is

available so far. Hence, to describe vortex-induced oscillations, many researchers

have resorted to empirical models which account only for the essential physical

features of the phenomenon. The empirical models should be able to predict a

response which is in agreement with the experimentally observed facts.

2.1 Models Applicable to Vortex-Induced Oscillations

Over the years many researchers have sought mathematical models to describe

vortex-induced oscillations. The models that are available in the literature can be

grouped into three distinct categories, namely the following:

10

1. Uncoupled single-degree-of-freedom models in which a forcing term is

included to account for the force due to alternate shedding of vortices and

other aerodynamic lift and damping forces.

2. Mechanical oscillator models to simulate the fluid-elastic effects in which

terms are included to describe the wake-structure interaction.

3. Flow-field models which include parameters that describe the flow field

around the oscillating body.

Models developed by Disilvio (1969), Scanlan (1981), and Vickery and Basu

(1983) are some examples of uncoupled single-degree-of-freedom models. Funakawa

(1969), Tamura and Amano (1983), Hartlen and Currie (1970), and Iwan and

Blevins (1974), are examples of mechanical oscillator models. Models developed

by Parkinson (1974), and Sarpkaya and Ihrig (1986) are flow-field models.

A mathematical model which can be used for tests in a wind tunnel should be

relatively simple to implement. The desirable qualities of such a model would be:

small number of parameters that need to be identified, easily measurable

quantities describing the aerodynamic behavior of the model, and which can be

easily extended to predict prototype response. One such model is that given by

Scanlan (1981). It has been tested through wind-tunnel tests (Ehsan, 1988) and

has a potential for describing the overall lock-in phenomenon. The model is

described by a dynamic equation which has a nonlinear damping term.

11

2.2 Scanlan's Nonlinear Model

The equation of motion in the across-flow degree-of-freedom y can be written

as,

m[y + 2Cu;„t/ -h u;„2y] = ^ ( y , y, y, U, t), (2.1)

where m is the mass per unit span, C, is the mechanical damping ratio, a;„ is

natural frequency, and ^ is a forcing term due to vortex shedding. .^ is a function

of flow velocity t/, displacement y, velocity y, acceleration y, and time t of the

model vibration. Scanlan (1981) proposed that ^ can be broken into two lift

forces, instantaneous fluctuating lift caused by shedding of vortices and is

independent of the motion of the body and the other is the aerodynamic lift

caused by the motion of the body.

The instantaneous lift force per unit length, w^ify.^ can be expressed as,

^v{t) = ]^pU^DCL(t), (2.2)

where CL{t) is the instantaneous lift coefficient, p is the density of the air, and D

is the characteristic dimension of the body.

The aerodynamic lift force can be represented by,

«"«(<) = \pU\2D)[Y^(l - e-^) J + Y,^l (2.3)

where Fi, V2, and e are unknown parameters to be determined experimentally.

12

The total lift force, w{t), acting on the body is a summation of the

instantaneous lift force and aerodynamic lift force. It can be assumed that the

instantaneous lift coefficient is sinusoidal and leads the displacement by the phase

angle (f>:

CL(t) = CL{K)sm{u,t-\-<j>), (2.4)

where K, = LVgD/U is the reduced frequency of vortex shedding and Ug is the

circular frequency of vortex shedding. Thus, the total lift force can be written as

w{t) = ^-pU'(2D)[Y^{l - t^)^ + Y2^ + i C i s i n M + «>)I- (2-5)

Scanlan (1981) proposed a semi-empirical model which takes into account

nonlinear behavior and the self-limiting amplitude of the oscillation by

representing the forcing term T in Equation 2.1 by the total lift force w(t) in

Equation 2.5. Hence the complete nonlinear model for vortex shedding can be

written as,

m[y + 2Ca;„y -f Un'^y] =

^-pU'{2D)[Yr{l - e - ^ ) J + Y^^ + -Cis in(a , , t + 4>)]. (2.6)

This model is represented by parameters Fi, 1 2, e, and CL- The parameter Yi is

the linear aeroelastic damping, Y2 is the linear aeroelastic frequency, e is the

nonlinear aeroelastic damping and CL is the lift coefficient. Ki, ¥2^ and e are the

parameters that determine the motion-induced lift force while CL determines the

13

instantaneous lift. Ehsan (1988) found that the magnitude of the instantaneous

lift force is much smaller than motion-induced lift force and hence the term

containing CL can be neglected in the model. The parameter Y2 is associated with

the shift in the frequency and it can be determined directly from the experimental

values. It has also been found that the parameter Y2 has a very small effect on the

frequency and it can be neglected compared to other terms (Ehsan, 1988). The

shifted frequency of oscillation can be represented by.

.2 2 PU' UJ' = ur.' - ^—Y2. (2.7)

m

The oscillation frequency can be observed easily in an experiment. Substitution of

Equation 2.7 into Equation 2.6 further simplifies the model as the unknown

parameters in the equation are reduced to just Yi and e.

2.3 Identification of Parameters

The simplified form of Equation 2.6 can be written as

y + [2C/C - m . r , ( l - e - ^ ) ] ^ 3 / + ^K^y = 0, (2.8)

where Kn = uJnD/U is the reduced natural frequency, m . = pD^/m is the reduced

mass of the model, and K"^ = Kn^ — mrY2 is the reduced oscillation frequency.

The exact solution of this equation is very difficult to achieve. However an

approximate solution of the form,

y{t) = A(t) cos{Kt - <t>(t)), (2.9)

14

(where A{t) and </>(<) are some functions of time) can be sought using a

quasi-linear approach. Van der Pol (1934) proposed a slowly varying parameter

approach which yields a solution of the form,

y{t) = A{t) X c o s ( - ^ / - ^[mrY2 + {K' - Kn')]t - M, (2.10)

where

-4(0= ^ ' l - ( ^ ) e x p ( - ^ i )

and 13 and a are defined as:

^ = ^V(^-S^)' -- (2.11)

a = rUrYit. (2.12)

A closer look at Equation 2.10 reveals that /? ( — oo, y -^ 5) is the steady-state

oscillation amplitude.

Based on this solution a parameter identification approach has been developed

known as the amplitude ratio method (Ehsan, 1988). In this amplitude ratio

method a ratio of amplitudes separated by n cycles can be used to develop an

expression for parameters assuming that j3 is known from experiments. The

expressions are,

Y^ = —[a^^ + %Kr.] and (2.13)

15

e = - ^ . (2.14)

The amplitude ratio method has a capability of identifying the parameters and

is simple to implement, but it has certain limitations:

1. The method fails for cases where steady state amplitude (/S) cannot be

observed in an experiment with sufficient accuracy when /? is very small.

2. The method does not yield accurate results for noisy observations, and is very

sensitive for certain values of parameters.

Due to these limitations it is necessary to seek an identification method which can

eliminate these limitations. The proposed identification approach is presented in

Chapter III.

CHAPTER III

A NEW APPROACH FOR IDENTIFICATION OF

PARAMETERS

The vortex-induced response of a prototype structure can be estimated from

wind-tunnel tests only when the parameters of the model are determined from

experiments. The accuracy of the calculated prototype response inherently

depends upon the accuracy of the model parameters. It was pointed out in

Chapter II that the existing technique for identification has certain limitations. In

this chapter an alternative approach is proposed which is based on concepts from

nonlinear filtering theory.

In the process of developing a new approach several ideas were tested and

possible modifications to the existing technique (Ehsan, 1988) were attempted.

The mathematical model for vortex shedding response (Equation 2.8) is a

formidable nonlinear equation. The approaches to the problem of parameter

estimation, considered before proposing the current technique presented in this

chapter, are briefly discussed:

1. It was observed in the derivation of Equation 2.10 that the amplitude of

oscillation is assumed to be constant over a cycle. This quasi linear

assumption leads to the concept that vortex-induced oscillations can be

assumed to be linear for every cycle of oscillation. Using this concept, the

amplitude of the next cycle can be estimated by any standard linear system

16

17

identification approach (Ibrahim and Mikulcik, 1977; Sarkar, 1994). The

identified amplitudes can then be used in the amplitude ratio method to

extract the parameters.

2. The equations of the amplitude ratio method can be mathematically

manipulated so that the steady-state amplitude {/S) can be eliminated from

the identification equations. This formulation requires knowledge of at least

three amplitudes as opposed to only two amplitudes in the existing method.

3. The expected value of the product of a noisy time series shifted by n cycles

with itself can be related to the expected value of consecutive amplitudes of

the response. The expected value of the amplitudes can be then related to

the parameters.

4. The parameters can be extracted by using optimization techniques. This is

essentially a trial and error process. Certain values of parameters are

assumed and the differential equation is integrated with these values. Based

on the error between the estimated time series and the observed time series

new values of parameters are chosen until convergence occurs.

5. A quasilinearization and system identification approach developed by

Bellman (1965) is also of interest. This approach has been later modified by

Kagiwada et al. (1986) to make it more efficient by using the simultaneous

calculation of approximations method (SCAM) and fast and efficient

evaluation of derivatives (FEED).

18

The methods discussed above proved either to be inefficient for a certain range

of values of the vortex-shedding parameters or did not eliminate all the problems

of the existing method. The proposed technique is based on the concept of

invariant imbedding and was originally developed by Bellman et al. (1966). The

method uses concepts in mathematics and nonlinear filtering theory to estimate

the unknown parameters. This method was modified for extraction of vortex

shedding parameters in the current research.

3.1 Van der Pol Oscillator Equation

The Van der Pol oscillator equation can be written as,

X-\-{pi - p2X^)i-\-fJ'SX = 0. (3.1)

A close look at Equation 2.8 reveals that it is similar to the Van der Pol oscillator

(Equation 3.1) where,

fii = ^{2CKn-mrY^), (3.2)

/ 2 = -^{'T^rYic), and (3.3)

MS = '^K'- (3-4)

The Dutch physicist Balthasar Van der Pol formulated this equation around 1920

to describe oscillations in a triode-circuit. The equation occupies a prominent

place in mathematics and is widely applied in diverse fields.

The phase plane of the Van der Pol equation contains one closed orbit which

corresponds to a periodic solution. The closed orbit is known as a limit cycle. The

19

case of periodic oscillations is of interest in vortex-induced response. A limit cycle

defines the steady-state amplitude in this case. The equation shows a hysteresis

behavior which is in agreement with the vortex shedding phenomenon. This

equation falls under the category of autonomous equations (Verhulst, 1989), in

which the independent variable t does not occur explicitly in the equation. An

exact solution of this equation is not available but an approximate method of

solution was discussed in Section 2.3. Another method of solution is the

perturbation method (Cunningham, 1958), but it is not very conducive from the

system identification point of view.

3.2 Theory of Invariant Imbedding

Integration of ordinary differential equations with given initial conditions is

done very efficiently by computers. This suggests that problems can be formulated

in a similar way with physical situation in mind. The invariant imbedding

provides a flexible manner in which to relate properties of one process to those of

neighboring processes. Invariant imbedding concept helps in expressing complex

mathematical systems by a simpler system of initial value ordinary differential

equations.

Invariant imbedding is a useful formalization of problems, theoretically and

computationally. Principles of invariance were first introduced by Ambarzumian

(1943) and developed by Chandrasekhar(1950) and Sobolev (1960). The

invariance concept was extended and generalized by Bellman and Kalaba (1956).

20

Further advances have been made by Kalaba and Kagiwada (1967) and others

(Scott, 1972). As an example of the concept of invariant imbedding, consider a

flux, r, passing through a rod of length, x. The investigator wishes to determine

the flux for a unit input in the rod for all points along the length at every second.

To study this processes in an experiment, it could be very difficult to measure flux

at every point along the length of the rod. So it would be desirable to vary the

length of the rod and measure the flux at the ends only. The rod length is thus

made a variable of the problem, so that r = r{x). The original situation is

imbedded in the cla^s of similar ca^es, for all lengths of the rod, and one obtains

directly the flux at any point along the length of the rod.

The concept of invariant imbedding has been applied to many problems in

diverse areas. The prominent examples are radiative transfer (Chandrasekhar,

1950), wave propagation (Bellman and Kalaba, 1959), mathematical physics

(Bellman, Kalaba, and Wing, 1960), transport theory (Wing, 1962), nuclear

physics (Shimizu, 1963), lunar photometry (Minnaert, 1941), Cardiology (Bellman

and Collier, 1967), and filtering theory (Bellman and Kagiwada, 1966). In this

chapter the application of invariant imbedding in nonlinear filtering theory will be

considered in detail.

3.3 Nonlinear Filtering Theory

The estimation of states and parameters in noisy nonlinear dynamic systems

ba^ed on observations on noisy nonlinear combinations of all or some of the states

21

is a problem whose solution is frequently sought. Many investigators like Kalman

and Bucy (1961), Weiner (1950), and Pugachev (1960) have dealt with linear

systems and have assumed that a complete statistical knowledge of the state of

the system is available. The concept of invariant imbedding can be used with

optimal filtering theory to attack a wide variety of nonlinear problems. The

concept being discussed in this chapter was developed by Bellman et al. (1966) for

orbit determination and adaptive control problems.

Consider a system which undergoes a process described by the nonlinear

differential equation

X = g{x,t). (3.5)

The observation of the process are not perfect; i.e. if y{t) is the observed time

history of the process on the interval 0 < < < T, then

y{t) — x{t) = observational error. (3-6)

It is desired to make an estimate of the system at time T on the basis of

observations carried out on the interval 0 < ^ < T. An estimate of the system can

be made by selecting

x{T) = c (3.7)

22

to be such that if y{t) is the observed function, and the function x{t) is

determined on the interval 0 < / < T by the differential equation given by

Equation 3.5 and the condition given by Equation 3.7, then the integral

J= f [x{t) - y{t)Ydt (3.8)

is minimized.

The above formulation of filtering theory can be restated for application to the

problem of system identification. The problem takes the form:

i = f(t-> ^t M, a), (3.9)

where x is the state vector with derivative i , u is the force vector, and a is an

unknown parameter vector to be estimated. The parameter vector, a, has to be

determined based on observations

y{t) = g{t.,x)-\- observational error (3.10)

in order to minimize

I \\y{t)-g[t,x)\\ldt, (3.11)

23

where || • \\Q is a suitable semi-norm. The development of invariant imbedding

equations for this problem is discussed in the next section.

3.4 Development of Invariant Imbedding Equations

Consider a system described by the nonlinear differential Equation 3.5 where x

and £ are n-dimensional vectors with components (a:i, 3:2,..., Xn) and

(^15^2,... ,5rn), respectively. The observations are

y(^t) = h(t,x) •}- observational error, (3.12)

where y and h are m-dimensional vectors with components (yi, y 2 , . . . , ym) and

{hi, /i2, • . . ,/im) with m <n and 0 < t < T.

As stated before

x{T) = c (3.13)

has to be estimated where c has components (ci, C2,. . . , c„) such that

J= I \\y{t)-h{t,x)\\l^,^dt (3.14)

is minimized. This is a more general formulation than given in Equation 3.11 as

the vector h does not have to be of the same dimension as vector x (or p), which

implies that observations of all the components of the state vector are not

required. This is a more practical situation as it is often difficult in an experiment

to measure all the quantities that describe the system simultaneously.

24

The value of the integral in Equation 3.14 is a function of T, the length of time

of observation, and c, the value assigned to x{t) at time T, i.e. ,

r \\y{t)-h{t,x)\\l^,^dt = f{c,T). (3.15)

Now the original process, involving fixed values of T and c, is imbedded in a class

of processes for which 0 <T and —oo < c < -|-oo and the costs, f{c,t), are

interconnected for these processes. This leads to

/ ( c , T -f A) = / ( c - g{c, T)A. T)^ || y{T) - h{T., c) ||^(,) + o (A), (3.16)

or in the limit as A —> 0,

h + (V J , g{T. c)) =11 y ( r ) - h(t, c) ||^(^) (3.17)

whereVc/ represents the n-dimensional vector with components df/dc^ and (•)

represents the Euclidean inner product. Equation 3.17 is a linear partial

differential equation for the cost function f{c,T). To obtain an auxiliary condition

on the function f{c,T), consider / (c ,0) , which is the cost associated with

estimating x(0) = c solely on the basis of a priori available information. Select the

condition

/ (c ,0) = A:(c-co)' , (3.18)

where CQ is the best estimate of the quantity x{0), and k is a measure of confidence

in that estimate. The linear partial differential Equation 3.17 together with the

condition in Equation 3.18 determine the function / ( c , T).

25

If one assumes that the process has been observed in 0 < ^ < TQ. The estimate

of the current state is the value of c which minimizes the cost function / ( c . To). In

fact, it is desirable to determine the minimizing value of c for each value of T > 0.

These minimizing values of c are the optimal estimates, e(T). These minimizing

points satisfy the equation

/ c ( e , r ) = 0, (3.19)

or

V,J{e,T)de + {VJ{c,T))TdT = 0, i.e. (3.20)

^ = -lV„f(e,T)]-\VJ{c,T))r (3.21)

where e{T) is an n-dimensional vector with components (ei, 62 , . . . , Cn), Vccf is

the matrix

^J ] (3.22)

and (Ve/ ) r is the vector (d/dT)(Vcf). ["l" denotes the inverse of a matrix.

Taking the partial derivatives with respect to c , i = 1,2,. . . ,n , in Equation

3.17 yields

(V J ) T + [Vec/]^ + b c ] ( V J ) = -2[KQy{T) - KQh{T, c)], (3.23)

where

t=l ,2 , . . . ,n

K = { ^ \ (3.24) \ nc: / . -

m

26 and

i9c] = {-~) t',i = l , 2 , . . . , n . (3.25)

Combining Equation 3.21 and Equation 3.23 yields

•^ = 9{T, e) -h 2[V,J]-i/i,(T, e)0(r)[y(r) - h(T, e)]. (3.26)

Defining

q{T) = 2[V,J]-\ i.e. (3.27)

\q{T)[V,J] = I (3.28)

(/ being an identity matrix of appropriate dimensions) and taking the total

derivative with respect to T in Equation 3.27 yields

^ [^[Vcc / ] + q{T)-^[V,J]] = 0 (3.29)

and hence,

^ = -^9(7^)^[Vcc/]g(r) . (3.30)

Taking the partial derivative with respect to c,, i = 1,2,..., n on both orders of

Equation 3.23 yields

[ V C C / ] T + ^ c [ V e J ] + [ V c c / ] ^ J + K C , T ) =

-2[hccQy{T) - hccQh - KQh^]. (3.31)

27

In Equation 3.31 the matrix p(c, T) has elements which consist only of the terms

which have factors of the form df/dc^ or d^f/dci dcj dck and hccQHT) is an

n X 71 matrix with ith column d/dcihcQy{T). The term hccQh is similarly defined.

The term gj refers to the transpose of the matrix gc-

When c takes on its optimal estimate e, the terms in p(c, T) with factors of the

form df/dci drop out , and terms with factors of d^f/dci dcj dck are negligible in

the neighborhood of the optimal estimate e{T). In Equation 3.30 the term

{dldT)\Vccf\ can be written as

A [ v „ / ] = [V„/]T + r ( c , r ) , (3.32)

where the matrix r[c, T) has elements which consist only of the terms which have

factors of the form d^f/dci dcj dck.

Combining Equations 3.30, 3.31, and 3.32 and dropping the matrices p{c,T)

and r{c, T) yields

177; = [qgc + gjq] + qlKcQiy - h)]q - qhcQhlq. (3.33)

al

The term [qgc + g^q] is the modified term used for the current research to yield

better results over the original equations (Bellman et al. , 1966). Equation 3.26

can be rewritten as

^ = 5 + qhcQ(y - h). (3.34)

Equations 3.33 and 3.34 are the estimator equations. As mentioned in Section 3.3,

if parameter vector a is included with the estimate vector e then these equations

28

can be used for system identification purposes. The development of equations for

Van der Pol oscillator is dealt in next section.

3.5 Equations for Van der Pol Oscillator

Consider again the Equation 3.1 with a forcing function u{t),

x-\-(pi- p2x'^)x -h p^x = u{t) (3.35)

In the state space representation the system of equations can be written as

ii = X2 (3.36)

^2 = —piX2 + P20clx2-p3Xl-\-U (3.37)

/ii = 0 (3.38)

/i2 = 0 (3.39)

where pi and p2 are the parameters to be estimated, and

xi = x{t),

X2 = x(t).

The optimal estimates of xi, X2, //i, and p2 will be denoted by ei, 62, €3, and 64,

respectively.

It is assumed that observations only of x{t) are available, and these

observations are denoted by y{t). Hence vectors y and h are 1-dimensional vectors

where the only component of vector h\s Ci, the optimal estimate of x{t). This

29

implies that matrix Q{t) in Equation 3.14 is a 1 x 1 matrix i.e. , a scalar number,

the value of which can be conveniently chosen to be equal to 1. Substituting

appropriate expressions for g and h^ defined in Equation 3.5 and Equation 3.24,

respectively, into Equation 3.34 the following equation can be written:

d dT

ei

^2

63

64

62

— 6362 + 646^62 — psei + u

0

0

+5 0

0

0

(y-ei). (3.40)

In Equation 3.33 the matrix h^^ is a null matrix. £ is a 4 x 4 matrix, and g . as

defined in Equation 3.25, can be determined as:

9c =

0 —2646162 — ^3 0 0

1 —63 — 646^ 0 0

0

0

- 6 2

-6^62

0 0

0 0

(3.41)

Equation 3.33, with substitutions given above, together with Equation 3.40,

can be integrated to give the optimal estimates. The converged values of 63 and 64

30

are the true values of the parameters. Initial conditions for £(0) and e(0) can be

chosen suitably for a fast convergence.

Once pi and p2 are obtained the vortex shedding parameters can be calculated

using Equations 3.2 through 3.4.

3.6 Numerical Simulation

To test the method numerically, Equation 3.35 was integrated with pi = 0.15,

P2 = 1000.0, p3 = 1280.0 and u{t) = 0.0. The pure time series was added to a

Gaussian white noise to generate a noisy observation of the process. The initial

conditions were:

e,(0)

6^(0)

^3(0)

e.(0)

y(0)

0.0

0.0

0.0

(3.42)

and q{0) was a diagonal matrix with diagonal elements as 10^, 10^, lO'', and 10^,

respectively. The sampling rate was 100 Hz. The fourth-order Runge-Kutta

method was used for integration of the equations. Excellent convergence for the

parameters was obtained even with a noise to signal ratio up to 25%.

3.7 Discussion

1. The convergence of the parameters does not depend upon the sampling rate

of the observations.

31

2. The rate of convergence was found to be sensitive to the q{0) matrix. The

best initial condition has to be selected by trial and error. This is one of the

shortcomings of the proposed method.

3. A priori knowledge of the steady-state amplitude is not required. The

method remains the same for a decaying or self-excited time series. The

proposed approach can also be used for a beating type response by

appropriately selecting the value of u{t) in Equation 3.35.

4. The parameters converge to their true values very rapidly if initial conditions

are well selected.

CHAPTER IV

DESCRIPTION OF EXPERIMENTAL WORK

4.1 Objectives

The wind-tunnel experiments described in this chapter were performed with

certain objectives in mind. The idea of this research is to propose an alternative

system identification approach for extraction of vortex-shedding parameters. The

robustness of any system identification approach can be determined through

experimental data only. Hence, it was imperative to verify the proposed approach

through experiments. Certain shortcomings of existing identification techniques

were discussed in Chapter II, and in Chapter III it was mentioned that these

shortcomings can be overcome using the proposed technique. The experiments

were thus designed to test the effectiveness of the proposed method under different

experimental circumstances.

The best way to verify the method was to design an experiment with known

parameters so that the results of the identification method could be easily verified.

Also, it was necessary to test the method with different cross-sectional

configurations. Hence, it was decided to conduct experiments on a circular

cylinder and on two configurations representative of bridge decks.

32

33

4.2 Experimental Setup

4.2.1 Wind Tunnel

The wind tunnel located in the Technology building at Texas Tech University

was used for the experimentation. The wind-tunnel facility operates in an open

loop with a maximum flow rate of around 65,000 cubic ft/min. A 5 ft wide, 4.5 ft

diameter centrifugal blower (forced-draft system) drives the wind-tunnel. The

blower is powered by a 50 hp, 3 phase, 220 volt motor. In order to control the flow

rate (and consequently the wind speed), the blower unit has a powered actuator

which opens the vanes from fully closed to fully open position. The actuator is

controlled by a electric switch located at the test section. The wind speed can be

changed continuously using this mechanism. The wind tunnel has a nozzle with a

contraction ratio of 10: 1 following the turbulence-reducing mesh screens and the

settling chamber (Figure 4.1). Figure 4.2 shows the centrifugal blower of the wind

tunnel.

4.2.2 Test Section

The experiments were conducted in a test section which is dedicated to

section-model testing. The test section is equipped with all the necessary

instrumentation to record aeroelastic and aerodynamic forces. The test section

can be attached to the nozzle of the wind tunnel and detached when not required.

The section is 11 ft long with a cross-section 4 ft wide and 3 ft high. The bottom

34

U /y-. t-l

o u a;

O

05

«3

Cl

X

bO

•^^^

G

a;

a bO

a;

CM

' ^ 0/' J—I

bO

36

of the section is fixed to a wheeled support. The test model can be mounted at a

distance of 7.75 ft from the wind-tunnel nozzle. The section has a glass wall on

one side to provide visibility. On the other wall, a door is provided for easy access

to the model. A hot-wire mount is fixed to the inner wall of the section which can

be moved along the vertical direction.

The model suspension system and the load cell frame is fixed on the outer side

of the wall. The benefit of such an arrangement is that their is no interference to

the flow and only the model experiences the wind flow which helps in obtaining

higher accuracy of results. Figure 4.3 shows the dynamic test section and Figure

4.4 shows a close-up view of suspension system and load cell arrangement. The

suspension system is designed such that stiffness of the model support can be

changed through addition of springs or rigid bars. Provisions are made in the test

section to allow movement of the model along two degrees of freedom, i.e., vertical

and torsional, and to allow measurement of forces along three degrees of freedom,

i.e. lift (vertical), moment (torsional) and drag (along-wind). It is possible to

adjust the setup to a single degree of freedom or a coupled degrees of freedom in

vertical and torsional as desired. The experiments described herein were

conducted for only one degree of freedom (lift). Another feature of the test section

is the excitation mechanism, which allows, the operator to give a constant initial

amplitude to the model for any experiment.

:v.

O

^

P bO

:'.<s

Figure 4.1. Suspension system for mode Icl.

39

4.2.3 Load Cell Arrangement

A total of four load cells were used in the setup. Two load load cells were used

on each side of the model. A picture of a load cell is shown in Figure 4.5. The

forces transferred from springs to the load cell are detected by strain gages. The

output from each strain gage is recorded after passing it through the signal

conditioner and amplifier.

Instruments division-2310 model strain gage and signal conditioner boards

were used for the operation of the load cells. The boards can provide excitation to

strain gages ranging from 0-15. The signal can be filtered at lOHz, lOOHz, IKHz,

lOKHz. The gain settings can be varied from 1 to 11000. The boards have a

provision of eliminating the D.C. component.

4.2.4 Data-Acquisition System

Labtech Notebook Version 7.1.1 (Laboratory Technology Corporation,

Wilmington, MA) was the software used for data collection. This software allows

an online plot of the data on the computer terminal. The data can be stored

simultaneously in a computer file in ASCII or binary mode. A Metrabyte DAS-16

interface board was used to convert the analog signals to digital signals.

40

a; o

c

41

4.2.5 Electromagnetic Damper System

The additional mechanical damping in the system was obtained with an

arrangement of a copper plate oscillating between the poles of fixed

electromagnets. The copper plate was suspended from the center point along the

length of the model, in a plane perpendicular to the model axis. When the model

oscillates so does the plate cutting the lines of flux of the magnetic field at a

velocity equal to that of the oscillating model. This motion produces eddy

currents on the surface of the copper plate, the integrated effect of which is a force

proportional to and opposing the model velocity. Thus, the arrangement of the

copper plate and electromagnets produces additional viscous damping in the

system. Figure 4.6 illustrates the setup of the copper plates and the

electromagnets.

The iron leg used for electromagnet had a cross-section of 1 in. x 1 in. and the

total length of iron used was 27 in. The opening between the iron pieces, serving

as opposite poles of the magnet, was 0.5 in. The side dimensions of the copper

plate were 2 in. x 2.5 in. and the thickness was 0.1 in. The wiring on the

electromagnet was done to the saturation point of iron.

12

u

c

i^

1, p

43

4.2.6 Anemometry

Two different types of anemometers were used for the experiments. A propeller

anemometer was used to measure the upstream velocity of the flow and a hot-film

anemometer was used to measure the turbulence and the spectrum of the vortices.

The propeller anemometer, shown in Figure 4.7, was fixed at the beginning of

the wind tunnel. The model of propeller anemometer used is HHF710

manufactured by Omega Engineering, Inc. The Omega HHF710 propeller

anemometer combines relative humidity, temperature, and air velocity

measurements. The unit of anemometer consists of a sensor probe, humidity

probe, and a display meter. The unit has a built-in memory to store up to 1000

measurements and the data can be transferred to a printer or PC via RS-232C

interface. The flow measurement reading can be recorded in either FPM or MPS

units. Average of several readings can also be displayed on the unit. These

features are available for temperature and relative humidity measurements also.

The air flow probe can make measurements in the range of 40 to 6800 FPM with

an accuracy of ± 1 % . The resolution of the display is 1 FPM. The operating

temperature of the air flow probe is —22 to +212°F

A TSI 1210-20 hot-film probe was used for measurement of the turbulence and

the vortex shedding frequencies. The standard probe resistance was 5.67 D and

the standard operating resistance was 8.7 ft, at a temperature of 482°F. A TSI

Intelligent Flow Analyzer model (IFA 100) was used to operate the probe. IFA

44

J <>.vvw N \ w V\J.N.XNV\>.V NV\\VNV.SNVN\>>J.NW ^^5>v>*^ \y^oss^N'»^ , sv

^^pr^^Sif^s^

Figure 4.7. Propeller anemometer used for wind speed measurements.

45

100 is a high performance, low noise, constant temperature anemometer system. It

can be used easily in any fluid flow with high accuracy for a high frequency

response. The anemometer features a digital display, 4 channels for probes, and a

signal conditioner. The signal conditioner has a built-in amplifier and filter. The

filter (low pass) can be set at a cut-off frequency from 1 Hz to 400 KHz. The gain

settings can be varied from 1 to 900 (DC or AC). The unit also contains a test

signal option for tuning the frequency response. The anemometer has a low input

voltage noise (w 1.85 nV/VHz) and a low input drift. The output voltage range

is from 0 to 12 Volts.

The hot-film probe was mounted on a stand which allowed vertical and

longitudinal movement of the probe. The probe was placed in the wake of the

model to measure the vortex-shedding frequency. Figure 4.8 shows a typical

placement of the probe for vortex-shedding frequency measurements. The probe

was placed upstream of the model to measure the turbulence.

4.2.7 Generation of Roughness for the Cylinder

One of the objectives of the study was to determine the effectiveness of the

proposed identification scheme through experiments on a rough cylinder. The

roughness was generated using medium 3M-drywall screen paper. The surface of

the cylinder was covered by the drywall screen paper. Figure 4.9 shows a unit of

the drywall screen paper. Figure 4.10 shows the cylinder covered by the

o

o

o

o ex

06

p bO

47

t-i <v a ex p <v <v ;-! O CO

P P

05

0)

p bO

'IS

CL

X X

•oO

1;

-f.

49

roughness elements. The drywall screen paper is manufactured to sand contours

and irregular shapes on drywall and plaster.

4.2.8 Generation of Turbulence

The experiments were carried out in smooth as well as turbulent flow for the

vortex induced oscillations of the cylinder. Two different grids were used to

generate turbulence of different intensities and length scales. Figure 4.11 and

Figure 4.12 show the two different types of grids. The grid # 1 (Figure 4.11) has

bars placed horizontally. The bars are separated by a center to center distance of

3.75 in. The section of the bars is 1.5 in. deep and 0.5 in. wide. The bars have

holes of diameter 0.5 in. with a center to center distance of 4 in.

The grid # 2 (Figure 4.12) has bars with a square section of 0.75 in. x 0.75 in.

with a square spacing of 4 in. x 4 in. Both grids have total dimensions of 4 ft. x

3 ft. and are made of wood.

4.2.9 Data Processing

The Texas Tech Wind Tunnel facility is equipped with modern data processing

equipment which includes a computer and a signal analyzer. The facility allows

on-line screening and analysis of experimental data.

A Packard Bell (model legend 1160 plus) computer is used for data acquisition

and analysis. The computer is equipped with a 486 DX2/66 microprocessor, an

-0

bO

p .2 -1-5

(L'

52

Intel 80486 DX CPU, a 420 MB hard drive, 4 MB of RAM, and a 1280 x 1024

Extended VGA resolution monitor.

A dual channel signal analyzer Type 2032 (Briiel and Kjaer Instruments, Inc.)

is used for on-line processing of signals. The signal analyzer has an array of

built-in frequency response functions and input/output analysis functions. The

frequency range of the signal analyzer is 0 to 25.6 KHz. A maximum sampling

frequency of 66 KHz is available with 12-bit A/D conversion. The analyzer has

several time averaging and weighting functions and has options of instantaneous

time and spectrum displays.

4.3 Wind-Tunnel Section Models

The experiments were conducted on a cylinder and two configurations of a

bridge deck.

4.3.1 Cylinder

A hollow aluminum tube of 4 in. external diameter was used for the cylinder

model. The wall thickness of the cylinder was 0.0625 in. The length of the model

was 40 in. End plates of aluminum were attached to the ends of the model to help

create a two-dimensional flow over the model. The model was supported by rod

connections to the elastic support system of the test section. Net weight of the

model was 8.4 pounds. The model was elastically supported by four springs of

total stiffness of 560 pounds per foot. The natural frequency of the system was

53

7.375 Hz and the critical damping ratio of the system was 0.26%. Figure 4.13

shows the cylinder mounted in the test section.

4.3.2 Bridge Deck Model

Bridge deck model was built of aluminum which was hollowed out as much as

possible to make the section lighter. Two aluminum channels are attached to the

deck to represent the bottom chord of the bridge. The top of the deck of the

model is covered with a rubbery material to represent the pavement. The railing

of the section has vertical columns with seven horizontal rods spaced along the

vertical direction. End plates of aluminum are attached to the model to maintain

two-dimensionality of the flow. The length of the model is 40 in. and it is attached

to the suspension system of the dynamic test section through rods. The weight of

the elastically supported bridge deck is 14.4 pounds. The model is suspended by

four springs with a total stiffness of 409 pounds per foot. The natural frequency of

the system is 4.7 Hz and the critical damping ratio is 0.36%. Figure 4.14 shows

the bridge deck model.

An aluminum plate of 2.25 in. height was used to model the highway sign on

the bridges. Two configuration of the bridge deck were achieved by fixing the sign

on the upstream and on the downstream sides of the bridge model, respectively.

54

o a (-1

a;

O

CO r—I

<D

P bO

55

o

bO

pq

^

a; p bO

56

4.4 Experimental Procedure

The experimental work was carried out during the period between May 1994

and September 1994. The following steps describe the experimental procedure in

general.

1. Calibration of anemometers and load cells (for displacement and force).

2. Determination of system mass, stiffness, damping, and frequency.

3. Determination of the aerodynamic lift coefficient.

4. Determination of the Strouhal numbers for the different cross-sections.

5. Recording of the time histories of the hot wire anemometer for the

downstream spectra.

6. Recording of time histories of displacement for the different models with

appropriate sampling rates and durations.

CHAPTER V

EXPERIMENTAL RESULTS AND PARAMETER

EXTRACTION

5.1 Section-Model Configurations

The experiments described in Chapter IV were conducted on four different

section-model configurations,

1. Smooth cylinder:

i Uniform flow, and

ii Turbulent flow.

2. Rough cylinder.

3. Bridge deck with sign upstream.

4. Bridge deck with sign downstream.

Experiments were conducted on all of these configurations according to the

procedure discussed in Section 4.4. Vortex shedding parameters for these test

sections were identified using the approach discussed in Chapter III. The results of

the experiments and the identification process are presented in this chapter.

5.2 Experimental Observations

5.2.1 Strouhal Numbers

Strouhal numbers were calculated for all of the test sections. The Strouhal

number, defined in Equation 1.1, depends primarily on the cross-section of the

57

58

body. Estimation of the Strouhal number was done by observing the vortex

shedding frequency / , at several different flow speeds U and then determining the

slope of the regression line of / , vs. U/D curve, D being the flow depth of the test

section. The regression curves for estimation of the Strouhal numbers for the

different sections are shown in Appendix B.

The Strouhal number for the smooth cyhnder was found to be 0.19, which is

close to the theoretical value of 0.20. The rough cylinder had a Strouhal number

of 0.16, which is different from the smooth cylinder perhaps because signature

turbulence is generated by the rough cylinder. The smooth cylinder in turbulent

flow had the same Strouhal number as in uniform flow. These facts lead to the

conclusion that only signature turbulence changes the Strouhal number. The

bridge deck with the sign upstream had a Strouhal number of 0.11 and the bridge

deck with the sign downstream had a Strouhal number of 0.17.

5.2.2 Lock-In Ranges for the Test Sections

The lock-in ranges of the test sections were determined by changing the flow

speed around the point where the frequency of vortex shedding (fs) was close to

the natural frequency of the model (/„). The following observations were made

regarding the lock-in ranges of the test sections:

• The onset flow speed of the lock-in range for the smooth cylinder decreased

with an increase in the Scruton number, mC/pV^, where m is the mass per

59

unit length of the model, ( is the system damping, p is the density of fluid

(air), and D is the flow depth of the model. This observation is consistent

with previous work done on a cylindrical section (Goswami, 1991).

• The lock-in range in turbulent flow was same as that in smooth flow.

• The bridge deck with the sign upstream showed a larger lock-in amplitude

and a wider lock-in range (w 8.33 ft/sec to 10.0 ft/sec) compared to the the

bridge deck with the downstream sign.

In the lock-in range the ratio / , / / „ is equal to 1.0. For flow speeds below the

lock-in flow speed this ratio is less than 1.0 and it increases to the value of 1.0 at

the onset of the lock-in range. For flow speeds greater than the lock-in flow speed

the ratio is greater than 1.0 and it increases with an increase in flow speed until

the vortices become turbulent. Figure 5.1 shows the curve of / s / / „ vs. flow speed,

U, for the cylindrical section and Figure 5.2 shows the same curve for the bridge

deck with the sign upstream.

5.2.3 Lock-In Response

One of the most important features of vortex-induced oscillations is the lock-in

response of the body. The body shows a steady-state oscillation amplitude. The

amplitude varies within the lock-in range and reaches a maximum value. For flow

speeds outside the lock-in range, the oscillations of the body decay down to the

stationary condition.

60

30 40 UjD (l/sec)

50 60

Figure 5.1. Lock-in range for the smooth cylinder.

2.0

0 . 8 -

0 . 6 -

30 40 50 60

(^/D (l/sec)

Figure 5.2. Lock-in range for the bridge deck with upstream sign.

61

U 1(1 >,

ll.'ilpi !

: > i ! A I I

0.00

-0.25

-0,50

0,50

0.25

0.00

-0,25

• 0 . 5 0 - f -

Hi l/WWV#A/^A^MAM-

Pre lock-in (U = l.Sft/sec)

l)^iyy^y^VW'\/*'''VVV^'V•^lVJ•^J\A;VyVlA,|^,^.'^.V/''-^-•^^•'AA,V

Post lock-in (U = IS.Oft/sec)

10 15 20

Time {sec)

Figure 5.3. Response around the lock-in range (bridge deck with upstream sign).

62

Figure 5.3 shows typical time series of displacement for pre-lock-in, lock-in,

and post-lock-in response (recorded for the bridge deck with the sign upstream).

Typical spectra of the response of the model and of the vortex shedding at the

lock-in speed for the cylinder are shown in Figure 5.4, corresponding spectra for

the bridge deck (sign upstream) are shown in Figure 5.5.

Another feature of the lock-in response is that the lock-in oscillations are

sinusoidal in nature. This is demonstrated by Figure 5.6.

5.2.4 Decaying Response and Self-Excited Response

The steady-state amplitude of the response of a body due to vortex shedding

can be reached either by exciting the body from a certain initial amplitude greater

than the steady-state amplitude it will achieve or by letting the oscillations grow

themselves (self-excited). Many times it is observed that the steady-state

amplitudes achieved through decay method and through the self-excited method

do not have the same value. This is a consequence of the nonlinearity of the

system.

Figure 5.7 and Figure 5.8 show the decaying and self excited time series for the

cylinder model and bridge deck model (sign upstream), respectively. It can be

seen in these figures that the steady-state oscillation amplitudes for the cylinder

are the same for the decaying and self-excited methods, while the bridge deck

amplitudes show hysteresis.

63

to 1 0 -

;3

Co

0 0

4-

Vortex-Shedding Spectrum

^^.t^— -t.^.--v^>A K..^^Kmr-^i . ^ J ^

4 6

Frequency {Hz)

8

A^^A.

10

Uec

)

Co"

I b O -

100-

50-

0-

h ^ -

Response Spectrum

^ \ y^-

0 -4-6

Frequency {Hz)

Q 10

Figure 5.4. Typical spectra at lock-in flow speed (smooth cylinder).

10,0-1 —+-

7, S -

«5

5 0

3

CO

2 , 5 -

0.0

Vortex-Shedding Spectrum

0 2

' • ^ \

f - -

Frequency (Hz)

64

[AA.A A - A | - A lA,

8 10

CO

Co

0 4 6

Frequency {Hz)

10

Figure 5.5. Typical spectra at lock-in flow speed (bridge deck with upstream sign).

65

t - - + - — f — -t — i -

- - • 4 - - -I-

CD

(J)

GO

r

(JD

in

^

K l

(Aj

o

«5

(U

a

o o CO

CO

P O

u to

O

o

CO

<v (-1

p bO

C D I

(•"?) JUDlU^DDjd^tQ

66

I , (J

Decaying response

1,0

0 , 5 -

a

to

0,0 ]^vmmjmMimN^mWHm

-0

0

!

0 • f

5 10

Self-excited response

Time {sec)

.5 0

Figure 5.7. Vortex-induced response (smooth cylinder at Scruton number of 1.1).

67

0,50

0.25

o 0.00 a, to

-0.25

-0.50

Decaying response

Time {sec)

0.50

• S 0 . 2 5 -

to

0,00

-0 ,25 - -

-0,50 0

Self-excited response

10 15 20

Time {sec)

Figure 5.8. Vortex-induced response (bridge deck with upstream sign).

68

5.2.5 Beating Response

In the pre lock-in range, when the flow speed is close to the onset velocity, the

force exerted by the vortices is significant. The frequency of vortex shedding is

very close to the natural frequency of the body. This force causes a beating

response and the oscillations increase and decrease with time.

Figure 5.9 shows the beating response observed for the bridge deck with the

sign downstream. The natural frequency of the bridge was 4.625 Hz and the

vortex-shedding frequency was 5.25 Hz.

5.2.6 Vortex-Induced Response in Turbulent Flow

Experiments were conducted in two types of turbulence levels with respect to

the longitudinal velocity fluctuations as generated by two types of grids. Grid :^1

generated a turbulence intensity of 9.5% and grid # 2 generated a turbulence

intensity of 8.0% at the test section. The details of grid-generated turbulence are

given in Appendix C. The cylinder model was tested in turbulent flow for the

same Scruton numbers as those used for the smooth flow. The effect of turbulence

on response of the cylinder was negligible as far as the steady-state amplitudes

were concerned.

Figure 5.10 shows a typical time series of cylinder response in turbulent flow.

The time series in these figures appear to be noisy because of the turbulence in the

flow but the overall oscillations are strictly monofrequency. Figure 5.11 the shows

69

=-+.. ( " • • • J

LO

o to

q j

LO

(\J

CD O CD L_J

-\- o

I

P bO CO

(V

CO

ex P

O (U

bO

y. JO

(U CO

P o CO 0) i-i

bO P ..—<

PQ

as

p bO

( l i z ) JU9lU9DVjdsi(J

70

0,50

0,25

a, to

0.00

-0.25-

-0.50

0.50

0.25

to

0,00

-0.25

-0.50

Time {sec)

Time {sec)

Figure 5.10. Response in turbulent flow (cylinder at Scruton number of I . l) .

71

15,0

4 6

Frequency {Hz) a 10

to

15.0

1 2 . 5 -

1 0 , 0 -

,^. 7 , 5 -

Co 5 , 0 -

2 . 5 -

0.0 0 4 6

Frequency {Hz)

10

Figure 5.11. Spectra in the wake of smooth cylinder in turbulent flow.

72

vortex shedding spectra measured in the wake of the cylinder. It can be observed

from the spectra that it has higher frequency content than the spectrum measured

in smooth flow.

5.3 Identification of Cyhnder Parameters

The identification approach proposed in Chapter III was to be tested for

several different experimental possibilities. The identification results for the

cyhnder parameters were obtained from decay type as well as self-excited type

time series. The approach was also tested for the response in turbulent flow.

Figure 5.12 shows a comparison between the cylinder parameters identified for

the smooth flow and for the turbulent flow. The results show excellent agreement

as the turbulent flow had a negligible effect on the behavior of the system. Figure

5.13 shows the parameters identified for the rough cylinder.

The vortex-induced response of cylinder can be predicted using Griffin and

Ramberg's (1982) empirical formula:

D ~ [l-h0.43(87r252 5e)P-35' ^ - ^

where S is the Strouhal number and Sc is the Scruton number. This analytical

expression has been proven experimentally by several researchers (e.g., Goswami,

1991). Figure 5.14 shows curves of steady-state amplitude and Scruton number for

different test configurations of cylinder. The steady state amplitudes were

calculated using the expression given by Equation 2.11 and the identified

73

on -^ 'Q

1 5 -

^ 10

5 -

0 0 50

- h - -

o uniform flow

+ Grid # 1

* Grid # 2

0.75 1 .00

Scruton number

1.25 1,50

10

8 -

o bO

S 4

0,50 0. 75

o uniform flow

+ Grid # 1

* Grid # 2

00

Scruton number

1,50

Figure 5.12. Parameters for smooth cylinder.

74

20 -4-

1 5 -

> : : 1 0 -o o

5 -o

0 0.50 0.75 1.00 1,25

Scruton number

1.50

bO O

0,50 0,75 1.00 1,25

Scruton number

1,50

Figure 5.13. Parameters for rough cylinder.

75

o ND

CD

r-.j

CO

CD

C\J

O

00

O

CD

O

( H

0) J3

P fl P O

P (-1

u CO

<D 42 s p p p O

- t - j

P i-i

u CO

CO

P

vers

^ — V

^ <Li

- U P

-1-3 • 1—1

3. a «3 0)

-1-3 C^

+ J CO

> . -T! ce CD

CO

'^

p bO

6H

a/£/

76

parameters. A Griffin and Ramberg analytical prediction curve is also shown in

this figure. The points calculated using experimental data show a good agreement

with the theoretical curve. The values of the identified parameters are tabulated

in Table 5.1.

5.4 Identification of Bridge Deck Parameters

The parameters for the bridge deck with the sign upstream were calculated

from the maximum response observed in the lock-in range. The parameters Vi and

e were estimated to be 16.6 and 1050.0, respectively. The steady state ampHtudes

calculated using these parameters and Equation 2.11 are in close agreement with

the experimentally observed amplitudes.

The parameters Yi and c for the bridge deck with the sign downstream were

found to be 12.5 and 1600.0, respectively. These results also show a close

agreement with results obtained by using Equation 2.11. The parameters for this

bridge configuration were also identified from the beating response observed close

to the lock-in range. Although the parameters vary within the lock-in range, since

the lock-in band of this configuration was very narrow the parameter estimates

came out very close. The parameters Vi and e for the beating response case were

13.2 and 1605, respectively.

Table 5.1

Vortex-Shedding Parameters of the Cylinder

77

Scruton Number

0.72

1.02

1.10

1.22

Smooth Cyhnder

Yi

11.0

8.0

12.2

12.6

logio(0

2.15

2.58

2.20

3.08

Rough Cylinder

Y,

7.5

6.2

9.1

9.6

logio(e)

2.60

2.88

3.07

3.87

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

The basic objective of this research was to propose a new approach for

identification of vortex shedding parameters. The entire work can be grouped into

three categories:

1. Investigation of existing techniques for identification of the parameters,

2. Detailed discussion of the proposed technique, and

3. Identification of parameters from experimental data using the proposed

technique.

This chapter provides a general overview of the work, leading to general

conclusions and directions for future research.

6.1 Conclusions

6.1.1 Existing Techniques for Parameter Extraction

A review of existing identification techniques was conducted. The technique

based on solution of the autonomous Van der Pol equation (Ehsan, 1988) was not

found to be very reliable for a certain range of parameters. The method was

inaccurate for situations where observations of the experiment were noisy. This

existing method can not be used in experimental situations where the steady-state

amplitude is very small and therefore cannot be observed in an experiment with

sufficient accuracy.

78

79

The steady-state amplitude of vibration decreases with an increase in the

Scruton number of the section. In many cases of experimental modeling the

Scruton number may become high, thus reducing the steady-state amplitude.

Hence a desirable identification technique is one which does not depend upon the

magnitude of steady-state amplitude of vibration.

6.1.2 Proposed Identification Technique

Several modifications to the existing technique and new approaches were

considered in this research. The proposed identification technique is based on

concepts in nonlinear filtering theory and the theory of invariant imbedding. The

proposed technique overcomes the limitations of the previous method. The new

approach was found to be effective even in noisy observations. The approach can

be used in several different situations that may arise in an experiment. The

advantages of the the new approach are:

1. The method can be used for either a decaying response or a self-excited

response. It is also possible to identify the parameters from the lock-in

oscillations only. Thus, the method is independent of the initial conditions of

the experiment.

2. The method is independent of the steady-state oscillation amplitude. Hence,

it can be used effectively in cases where the magnitude of steady-state

oscillation amplitude is very small.

80

3. The method can be used for a beating response with a slight modification to

the equations.

4. The method is effective for noisy observations.

6.1.3 Identification of Parameters from Experimental Data

A number of experiments were conducted on different bridge deck and cylinder

models. The general observations during the experiments were found to be

consistent with earlier works on vortex shedding. The parameters associated with

the sections were identified using the approach proposed in this research. Some

aspects of identification are mentioned below:

1. The identified parameters for the cylinder were consistent with earlier works

done on cylinder sections.

2. The parameters for the cylinder were successfully identified from the

turbulent flow. These values are also consistent with previous researches.

3. The bridge deck with the sign downstream presented a case in which the

magnitude of the steady-state amplitude was very small. The parameters for

this case were also successfully extracted using the new approach.

4. The parameters were also identified from a beating response.

6.2 Recommendations for Future Work

Further investigations in the following directions would help in advancing the

usefulness of the proposed identification technique:

81

1. The proposed identification technique has been developed only for a single

degree of freedom system in this research. It is possible to extend it for two

degrees of freedom. This would help in identifying the parameters associated

with lift and moment simultaneously.

2. Further investigation and refinement of the technique needs to be done to

make it more efficient. A more accurate integration procedure might improve

the efficiency of the method.

3. Additional research needs to be done to select the initial conditions of the

method.

4. Validation of the technique through experiments on other types of

cross-sections not considered in this study would increase the reliability of the

method.

REFERENCES

[1] Ambarzumian, V.A. "Diffuse Reflection of Light by a Foggy Medium." Compt. Rend. Acad. Sci. U.S.S.R., Vol 38, No. 8, pp. 229-232, 1943.

[2] Bellman, R.E., Collier, C , Kagiwada, H.H., Kalaba, R.E., and Selvester, R. "A Digital" Computer Model of the Vectorcardiogram with Distance and Boundary Effects- Simulated Myocardial Infraction." American Heart Journal. Vol. 74, No. 6, pp. 792-808, 1967.

[3] Bellman, R.E., Kagiwada, H.H., Kalaba, R.E., and Sridhar, R. "Invariant Imbedding and Nonlinear Filtering Theory." Journal of Astronautical Sciences. Vol 13, No. 3, pp. 110-115, May-June, 1966.

[4] Bellman, R.E., and Kalaba, R.E. "On the Principle of Invariant Imbedding and Propagation through Inhomogeneous Media." Proc. Nat. Acad. Sci. USA. Vol. 42, pp. 629-632, 1956.

[5] Bellman, R.E., and Kalaba, R.E. "Functional Equations, Wave Propagations, and Invariant Imbedding." J. Math. Mech.. Vol. 8, pp. 683-704, 1959.

[6] Bellman, R.E., and Kalaba, R.E. Quasilinearization and Nonlinear Boundary-Value Problems. Elsevier, New York, 1965.

[7] Bellman, R.E., Kalaba, R.E., and Wing, G.M. "Invariant Imbedding and Mathematical Physics-I: Particle Processes" J. Math. Phys., pp. 280-308, 1960.

[8] Birkoff, G. "Formation of Vortex Streets." Journal of Applied Physics, Vol. 24, No, 1, pp. 98-103, 1958.

[9] Bishop, R.E.D., and Hassan, A.Y. "The Lift and Drag Forces on a Circular Cylinder Oscillating in a Flowing Fluid." Proc. Royal Soc, Ser. A, Vol 277, pp 51-75, 1964.

[10] Chandrasekhar, S. Radiative Transfer. Oxford University Press, London, 1950.

[11] Cunningham, W.J. Introduction to Nonlinear Analysis. McGraw-Hill, New York, 1958.

[12] diSilvio, G. "Self-Controlled Vibration of Cyhnder in Fluid Stream." Proc. ASCE EM2. pp 347-361, April 1969.

82

83

[13] Ehsan, F. "The Vortex-Induced Response of Long, Suspended-Span Bridges." Ph.D. Dissertation, Johns Hopkins University, Baltimore, Maryland, 1988.

[14] Funakawa, M. "The Vibration of a Cylinder Caused by Wake in a Flow." Bull. JSME. Vol. 12, No. 53, pp. 1003-1010, 1969.

[15] Gerrard, J.H. "The Wakes of Cylindrical Bluff Bodies at Low Reynolds Number." Journal of Fluid Mechanics. Vol. 25, pp. 401-403, 1978.

[16] Goswami, I., Scanlan, R.H., and Jones, N.P. "Vortex-Shedding: Experimental Results and a New Model." 8 ^ International Conference on Wind Engineering, London, Canada, 1991.

[17] Griffin, O.M., and Ramberg, S.E. "On Vortex-Street Wakes of Vibrating Cylinder." J. Fluid Mech.. Vol. 66, pp. 553-576, 1974.

[18] Hartlen, R.T., and Currie, I.G. "A Lift-Oscillator Model of Vortex-Induced Vibration." Proc. ASCE, J. Eng. Mech. Div., Vol. 96, pp. 577-591, 1970.

[19] Ibrahim, S.R., and Mikulcik,E.C. "A Method for the Direct Identification of Vibration Parameters from the Free Response." The Shock and Vibration Bulletin. Bulletin 47, Part 4, 1977.

[20] Iwan, W.D., and Blevins, R.D. "A Model for Vortex-Induced Oscillation of Structures." J. Appl. Mech., Trans. ASME, Vol. 41, pp. 581-585, 1974.

[21] Kagiwada, H.H., Kalaba, R.E., Rasakhoo, N., and Spingarn K. Numerical Derivatives and Nonlinear Analysis. Plenum Press, New York,

1986.

[22] Kalaba, R.E., and Kagiwada, H.H. "An Initial Value Method Suitable for the Computation of Certain Fredholm Resolvents." J. Math, and Phys. Sciences, Vol. 1, No. 1, pp. 109-122, 1967.

[23] Kalman, R., and Bucy, R. "New Results in Linear Filtering and Prediction Theory." Journal of Basic Engineering, Trans. ASME, Series D, Vol. 83, No. 1, pp. 95-108, March, 1961.

[24] Minnaert, M. "The Reciprocity Principle in Lunar Photometry." Astrophysics J., Vol. 93, pp. 403-410, 1941.

84

[25] Parkinson, G.V. "Mathematical Models of Flow-Induced Vibrations of Bluff Bodies." Proc. lUTAM-IAHR Symp. on Flow-Induced Structural Vibrations, Karlsruhe, 1974.

[26] Pol, B. van der "Nonlinear Theory of Electric Oscillations." Proc. IRE. Vol. 22, pp. 1051-1086, 1934.

[27] Pugachev, V. Theory of Random Functions and its Applications to Automatic Control. Gostekhizdat^ Mnsrnw IQfin

[28] Rayleigh, J.W.S. Philosophical Magazine 7. 149, 1879.

[29] Sarkar, P.P., Jones, N.P., and Scanlan, R.H. "Identification of Aeroelastic Parameters of Flexible Bridges." Journal of Engineering Mechanics, ASCE, Eng. Mech. Div., Vol. 120, No. 8, pp. 1718-1742, August,1994.

[30] Sarpkaya, T., and Ihrig, C.J. "Impulsively Started Steady Flow about Rectangular Prisms: Experiments and Discrete Vortex Analysis." J. Fluids Eng., Trans ASME, Vol. 108, pp. 47-54, 1986.

[31] Scanlan, R.H. "On the State-of-the-Art Methods for Calculations of Flutter, Vortex-Induced and Buffeting Response of Bridge Structures." FHWA Report, FHWA /RD-80/050, Washington D .C , 1981.

[32] Scott, M.R. A Bibliography on Invariant Imbedding and Related Topics. Sandia Laboratories, 1972.

[33] Shimizu, A. "Application of Invariant Imbedding to Penetration Problems of Neutrons and Gamma Rays." NAIG Nuclear Research Laboratory, Research Memo-1, December, 1963.

[34] Sobolev, V.V. A Treatise on Radiative Transfer. Cambridge University Press, New York, 1960.

[35] Strouhal, V. " Uber eine besondere Art der Tonerregung. " Analytical Physics, Neue Folge, Band V, pp. 216-250, 1878.

85

[36] Tamura, Y., and Amano, A. "Mathematical for Vortex-Induced Oscillations of Continuous Systems with Circular Cross-sections." Journal of Wind Engineering and Industrial Aerodynamics, Vol. 14, pp. 431-448, 1983.

[37] Van Dyke, M. An Album of Fluid Motion. ParaboHc Press, Stanford, CaHfornia, 1982.

[38] Verhulst, F. Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag, New York, 1989.

[39] Vickery, B.J., and Basu, R.I. "Across-Wind Vibrations of Structures of Circular Cross-Section. Part I. Development of a Mathematical Model for Two-Dimensional Conditions." Journal of Wind Engineering and Industrial Aerodynamics, Vol. 12, pp 49-73, 1983.

[40] von Karman, Th. " Uber den Mechanismus des Widerstandes den ein bewegter Korper in einer Fliissigkeit erfahrt." Nachrichten den Koniglichen Gessellschaft der Wissenschaften. Gottingen, pp. 509-517, 1911.

[41] Wiener, N. Extrapolation, Interpolation and Smoothing of Stationary Time Series. John Wiley Inc., New York, 1950.

[42] Wing, G.M. An Introduction to Transport Theory. John Wiley Inc., New York, 1962.

APPENDIX A

MASS IDENTIFICATION

86

87

The equation for calculation of system mass can be written as:

1 _ m M

where

u; is the circular frequency of the system,

M is the system mass to be determined,

m is the additional mass added to the system, and

K is the stiffness of the system.

A set of data for u is observed experimentally by adding extra mass to the

system. Regression between UJ and m yields the system mass and stiffness. The

regression curve for the cylinder is shown in Figure A.l and that for the bridge

deck is shown in Figure A.2. The slope of the curve for the cylinder was

5.55 X 10~^ and the intercept was 4.66 x lO"'*. These numbers yield

M as 8.4 lbs and

K as 560 Iblft for the cyhnder.

The slope of the curve for the bridge deck was 7.60 x 10"^ and the intercept

was 1.10 X 10"^. These numbers yield

M as 14.4 lbs and

K as 409 Iblft for the bridge deck.

88

CJ C5

X

: SO -

1 / ': . •

1 , 00 -

0 , 7 5 -

0 . 5 0 -

0 , 2 5 -

0 . 0 0 -

p"^'

f

" " " ^

\

v -

_.--*- ^^^

\

-- -\ - -

..^-'"^"^""

\ 0

m {lbs) 10 12

Figure A.l. Mass identification of cylinder.

2.5

2 . 0 -

8 1,5

X

'i 1 , 0 -

0 r:; _._

0.0 0

y

--^'

- • t r '

m (lbs)

0

• t

Figure A.2. Mass identification of bridge declc.

APPENDIX B

STROUHAL NUMBER

89

90

7 -

5

4 -

15

Smooth cylinder

JiK'

-+-

H h-:'5 30

U/D {l/sec)

1

S = 0.19

35 40

•5

8-

7 -

6-

5-

4 -

^-

^

Rough

J'

1

\

cylinder

—1

—1

— \

\

\—

K /

5 =

h--

^

Ji

0.16

— - - h •

15 20 ;'5 ^0 35

U/D {l/sec)

40 50

Figure B.l: Strouhal number for cylinder

91

8 -

6 -

Bridge deck (upstream sign)

5 ^ 50

5 = 0.11

55 60 65

UjD {l/sec)

70 75

10-

9 -

8-

7-

6 -

5 -

, \ 1 1

Bridge deck (downstream sign)

t--

^

y

> /

* •

^ H f - -

30

S = 0.17

40 4 , 50

(^/D {l/sec)

55 60

Figure B.2: Strouhal number for bridge deck

APPENDIX C

GRID GENERATED TURBULENCE

92

93

The turbulence intensity is given by,

where

lu is the longitudinal turbulence intensity,

y/u^{t) is the root mean square value of the wind speed fluctuations,

U is the mean wind speed.

The yu^{t) is calculated by a^, of the fluctuations of the wind velocity.

The (7u for grid # 1 was 1.98 and

U for grid # 1 was 20.7 ft/sec

yielding turbulence generated by grid # 1 as 9.5%.

The CTy, for grid # 2 was 1.8 and

U was 22.3 ft/sec yielding turbulence generated by grid # 2 as 8.0%.

The turbulence spectra of grid # 1 and grid # 2 are shown in Figure C.l and

Figure C.2, respectively.

i oo

0 . 7 5 -

to 0,50

3

Co 0 .25 -

0 00

94

,0 15 20 25 ^0 35

Frequency {hz)

40 45 50

Figure C.l. Turbulence spectra (Grid #1) .

1.00

0 . 7 5 -

=0

0,50

3 3

Co

0 25

0 i(V.44y\^L/i.1^lM^^y.•v.4i.. 5 10 15 . 0 25 50 ?5 40 45 50

Frequency {hz)

Figure C.2. Turbulence spectra (Grid #2) .

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